Khadidja Benzemrane
email: khadidja.benzemrane@ibisc.fr

Gilney Damm
email: gilney.damm@ibisc.fr

G L Santosuosso
email: santosuosso@ing.uniroma2.it

Nonlinear Adaptive Observer for Unmanned Aerial Vehicle without GPS measurements

The present work deals with the classical problem of speed estimation for a small 4 rotors helicopter drone based on acceleration, rotation angles and angular speed measurements only. This problem concerns indoors applications as well as outdoors applications close-by large structures. Small Unmanned Air Vehicles (UAV) are commonly used for inspection applications such as beneath bridges, water dams, industrial facilities and interior of large structures and pipes. An observer is designed based on nonlinear adaptive techniques providing a global exponential solution to this problem when exact measurements are available. Simulation results illustrate the observer performance and suggest that the estimator presents good robustness in the more realistic case of noisy acceleration measurements.

I. INTRODUCTION

This work adresses problem of speed estimation of a six degrees of freedom vehicle when only accelerations, angular speeds and angles are available for measurement. This problem appears in several applications of robotics and vehicle control. In most vehicle control applications, this problem may be solved by coupling GPS measurements to a Kalman Filter integration for short time periods. The resulting estimation works well as far as GPS precision and sample rate are acceptable for the proposed application. This is usually true when large size vehicles are considered, but not for small unmanned air vehicles (UAV). In the same way, this technique can't be used in indoors applications or even just close-by large structures. In both cases GPS signal is not available and the system must rely only on its local measurements. Even in the cases when GPS is available, the produced error may reach many meters and the sample rate may easily be around 1-2 Hz. While not critical when dealing with large airships, using GPS measurements is not acceptable in the stabilization of small drones less than 1 meter wide. In some cases, a solution may be provided using D-GPS measurements. This has some drawbacks since such devices are in general very expensive; large and heavy; only available around a small region about 500 meters radius, and for these reasons this technique is not considered in this work.

An other well known approach is the use of vision based controllers [START_REF] Guenard | A practical Visual Servo Control for a Unmanned Aerial Vehicle[END_REF]. The main drawback of this method is the large amount of computer resources needed. It is not possible to run such vision algorithms in the very small micro-controllers used in small UAVs. One solution could be to use remote computers at a land base to perform part of the algorithms. This solution is usually not applied first by the time delay associated to this procedure, remote computing and communication, and second by the danger of losing the contact and as a consequence losing stability. Vision can be used for path planning , in a hierarchical display where a slow controller gives path planning references to a fast stabilizing one.

The recent results in non-linear observers have made possible to apply those techniques to small UAVs. The increasing interest in such techniques is well illustrated by the large number of recent works in this field as (see [START_REF] Bonnabel | A non-linear symmetry-preserving observer for velocity-aided inertial navigation[END_REF], [START_REF] Vik | A nonlinear observer for GPS and INS integration[END_REF] and [START_REF] Zhao | Discrete nonlinear observers for inertial navigation[END_REF]). The present work follows this way mainly based in the recent theoretical results presented in [START_REF] Marino | Nonlinear Control Design -Geometric, Adaptive and Robust[END_REF]. These observer's design strategies based on nonlinear adaptive control were used in some previous results where different observers were proposed to deal with the speed estimation problem [START_REF] Benzemrane | Unmanned aerial vehicle speed estimation via nonlinear adaptive observers[END_REF]. Those algorithms are very interesting but may be complicated to implement in small microcontroller. This was the main motivation to develop more simple observer. Furthermore, the present observer is also faster and simpler to tune.

The proposed scheme is applied to a 4 rotors helicopter robot shown in Fig. 1 -produced to operate in an urban environment at the Laboratoire IBISC -CNRS, Universite d'Evry, which is not equipped with GPS related devices and relies on the Inertial Measurement Unit only that provides accelerations, angles and angular speeds of the drone.

The simulation results obtained on the model of the drone -even in the presence of noise-are very promising, and will be soon implemented on the physical prototype. It is important to notice that the proposed scheme is very simple to implement and represents light computation burden. This will be very important for the following implementation steps. The proposed algorithm must be able to run in rather small microcontrollers such as allow a stabilizing controller based only on local measurements.

Section II states the problem and describes the drone model. Section III introduces the observer scheme as well as a sketch of estimation convergence proof. In section IV some simulations are presented to illustrate the observer performances, indicating good robustness toward measurement noise, and to compare it with an Extended Kalman Filter. Finally in section V conclusion remarks and future steps of this work will be presented. The motion of a four rotor aerial robot can be described by the following equations (see [START_REF] Azouz | Modeling and development of a 4rotors helicopter UAV[END_REF]).

ẋ = cosθ cosψ u + (sinφ sinθ cosψ -cosφ sinψ) v + (cosφ sinθ cosψ + sinφ sinψ) w ẏ = cosθ sinψ u (1) + (sinφ sinθ sinψ + cosφ cosψ) v + (cosφ sinθ sinψ -sinφ cosψ) w ż = -sinθ u + sinφ cosθ v + cosφ cosθ w φ = p + (sin(φ) q + cos(φ) r) tan(θ) θ = cos(φ) q -sin(φ) r (2) ψ = (sin(φ) q + cos(φ) r) cos(θ) -1 Ixx ṗ = l b k T (ω 2 1 cosβ 1 -ω 2 3 cosβ 3 ) -qIr(ω 1 cosβ 1 + ω 2 + ω 3 cosβ 3 + ω 4 ) -k M (ω 2 1 sinβ 1 + ω 2 3 sinβ 3 ) -(Izz -Iyy)rq Iyy q = -(Ixx -Izz)pr -rIr(ω 1 sinβ 1 + ω 3 sinβ 3 ) +pIr(ω 1 cosβ 1 + ω 2 + ω 3 cosβ 3 + ω 4 ) (3) +l b k T (ω 2 2 -ω 2 4 ) + u G k T (-ω 2 1 sinβ 1 + ω 2 3 sinβ 3 ) Izz ṙ = -(Iyy -Ixx)pq -l b k T (ω 2 1 sinβ 1 -ω 2 3 sinβ 3 ) +k M (ω 2 1 cosβ 1 -ω 2 2 + ω 2 3 cosβ 3 -ω 2 4 ) +qIr(ω 1 sinβ 1 + ω 3 sinβ 3 ) u = -qw + rv -g sinθ - k T m (ω 2 1 sinβ 1 + ω 2 3 sinβ 3 ) v = -ru + pw + g sinφ cosθ (4) ẇ = -pv + qu + g cosφ cosθ - k T m (ω 2 1 cosβ 1 + ω 2 2 + ω 2 3 cosβ 3 + ω 2 4 )
where η 1 [x y z] T is the position vector represented in the global reference frame, η 2 [φ θ ψ] T is the Euler angles vector represented in the global reference frame (roll pitch and yaw respectively), ν 1 [u v w] T is the speed vector represented in the local reference frame (surge, sway and heave respectively) and ν 2 [p q r] T is the angular speed vector represented in the local reference frame. In this model, the control input vector is

Ω = [ω 1 ω 2 ω 3 ω 4 β 1 β 3 ] T
where ω i , i = 1 . . . 4, are the angular speed of the 4 rotors, and β 1 and β 3 represent the orientation of the rotors 1 and 3. The constants k T and k M respectively are the constants relating rotor speeds to resulting thrust and torque, l B is the lenght of each drone's arm, and I r is the rotor's inertia moment constant. This vehicule is only equipped with an Inertial Measurement Unit (IMU) which provides the orientation, angular velocities and acceleration of the drone. In this context, defining the system state vector X = [φ, θ, ψ, p, q, r, u, v, w] T and the measurement output y = [φ, θ, ψ, p, q, r, u, v, ẇ] T , equations ( 3)-( 5) can be rewritten as :

Ẋ = f (X, Ω), X ∈ R 9 y = h(X, Ω), y ∈ R 6 (5)
where f : R 9 × R 6 → R 9 and h : R 9 × R 6 → R 9 are smooth functions depending on the states and the control input Ω = [ω 1 , ω 2 , ω 3 , ω 4 , β 1 , β 3 ] T . The measured accelerations u, v, ẇ are expressed as functions of X and Ω via (5).

Definition 1: Let G denote the set of all finite linear combinations of the Lie derivatives of h 1 , . . . , h p with respect to f for various values of u = constant. Let dG denote the set of all their gradients. If we can find n linearly independent vectors within dG, then the system is locally observable.

If one can find n linearly independent vectors within dG, then the nonlinear system is said to be locally observable ; the observability matrix is given by :

ϑ =   L 0 f h ... L p-1 f h  
The system is locally observable if ϑ has full rank.

In the case of system we consider, the observability matrix has full rank if and only if (p(t), q(t), r(t)) T are not identically equal to (0, 0, 0) T . In the case of p(t),q(t) and r(t) equal to zero, rank(ϑ) is 6 and the system is not observable anymore.

As a consequence, the control algorithm that the angular speeds [p, q, r] T has to track bounded and non vanishing reference trajectories to garantee the observability of system 5

The paper objective may be stated as : to present an observer designed to provide an estimation of the unmeasurable state variable ν 1 , based on the measurable variables η 2 , ν1 and given by the standard sensors embeded in the drone. The proposed algorithm will be compared to an extended Kalman filter. In section IV, simulation results illustrate illustrate the performances of the proposed observer.

III. OBSERVER DESIGN

In this section, we study the observability of the system described by equations ( 3) -( 5). We also describe the estimation strategy following the techniques presented in [START_REF] Marino | Robust Adaptive Observers for Nonlinear Systems with Bounded Disturbances[END_REF] and [START_REF] Marino | Nonlinear Control Design -Geometric, Adaptive and Robust[END_REF] as well as an outline of its stability proof.

We now consider the system (5). It can be rewritten as :

ν1 (t) = A(t)ν 1 (t) + b(t) (6)
where A, b and y are defined by

A(t) =   0 r -q -r 0 p q -p 0   (7) b(t) =   -g sinθ - k T m (ω 2 1 sinβ1 + ω 2 3 sinβ3) g sinφcosθ g cosφ cosθ - k T m (ω 2 1 cosβ1 + ω 2 2 + ω 2 3 cosβ3 + ω 2 4 )  
We introduce an estimate of the linear velocity

ν1 =   û v ŵ  
along with a filter vector state ζ ∈ R 3 that satisfy the differential equations :

ζ = -cζ + [ ν1 -b(t)] Aν1 -Aν 1 (8)
and

ν1 = ν1 + ΛA T ζ (9)
where c > 0 and Λ is the (diagonal) adaptive gain.

The estimation error ν1 = ν 1 -ν1 then satisfies the following equation :

ν1 = -ΛA T ζ (10)
The observer described by the following equations :

ζ = -cζ + Aν 1 ν1 = -ΛA T ζ
garantees exponential convergence to zero of estimation error under some conditios on matrix A that will be discussed.

Proof:

The stability proof is standard and follows Lyapunov techniques. Consider the Lyapunov function :

V = 1 2 ζ T ζ + 1 2 νT 1 Λ -1 ν1 (11) (12)
By computing the time derivative of V , we have :

V = -c ζ 2 + ζ T Aν 1 + νT 1 Λ -1 (-ΛA T ζ) = -c ζ 2 + νT 1 A T ζ + νT 1 A T ζ V = -c ζ 2 (13)
The previous equations imply that ζ and ν1 are bounded. As a consequence, ζ is also bounded, and then

V = -cζ ζ is bounded.
As a consequence V is uniformly continuous, and according to Barbalat's Lemma, since V ≥ 0, V tends to zero. This in turns implies, from (13), that ζ also converges to zero. Then, according to [START_REF] Pounds | Design of a Four-Rotor Aerial Robot[END_REF] and the convergence of ζ to zero, results that ν1 also converges to zero. Now, stating the assuption that Ȧ is also bounded, it is shown using the equation:

ν1 = Λ(A T ζ + ȦT ζ)
that ν1 is also bounded and then ν1 is uniformly continuous. As a consequence it can now be stated that ν1 converges to a constant, since ν1 converges to zero.

From [START_REF] Marino | Robust Adaptive Observers for Nonlinear Systems with Bounded Disturbances[END_REF], since ζ converges to zero, and its forcing term Aν 1 is uniformly continuous:

Aν 1 = 0 (14)
It was shown that ν1 converges to a constant, and as a consequence this equation has a unique zero solution if and only if the matrix A(t) satisfies the persistency of excitation condition, which means that there exist strictly positive constants α and T such that such that for any t > t 0 :

t+T t A(τ )A T (τ )dτ ≥ kI (15)
where k > 0 Condition (15) implies that the only solution of equation ( 14) is ν1 = 0 Previous arguments can ne summarized in the following Proposition 1: The UAV described by ( 3)-( 5) with output measurement satisfies the following conditions :

1) The matrix A given by ( 7) is bounded.

2) The matrix Ȧ given by

A(t) =   0 ṙ - q -ṙ 0 ṗ q -ṗ 0   ,
where [ ṗ, q, ṙ] T are defined in (4), is also bounded.

3) The persistency of excitation condition given by ( 15) is satisfied.

Then the observer ( 8), ( 9), ( 10) assures global assymptotical estimation of the unmeasurable states representing the linear velocities of the considered UAV. Remark 1: Conditions 1, 2, 3 are easily satisfied since the controller assures the boundness of the angular speeds [p, q, r] T with bounded control magnitude.

Remark 2: Conditions 3 reflects the observability study presented in section II. As presented before, it can be easily verified since [p, q, r] T are not identically equal to zero in the considered time period. For instance, a small periodic orbit around the desired trajectory assures the persistency of excitation condition.

IV. SIMULATION RESULTS

In this section we illustrate the performances of the observer designed above to estimate the linear velocity of an UAV based on the measurable states (angles and angular velocities) and on the linear accelerations.

In all simulations, the estimations have initial conditions set to zero, while the desired states are time varying (and different from zero at t = 0). We have used the following parameters values: In Fig. 2 the orientations, angular speeds, and accelerations of the UAV are plotted. Small periodic orbits were considered in order to satisfy the persistency of excitation condition. To garantee the stability of the physical system, some constraints must be satisfied: the angles have to be less than 0,1 rad (about 6 degrees) and the frequency of the needed oscilations must be less than 1Hz. The time history of the three speeds to be observed is shown in Fig. 3a, and observer estimates in Fig. 3b, while estimation errors going exponentially to zero are shown in Fig. 3c. We have considered the observer in the case of additive measurement noise. The measured acceleration is presented in Fig. 4 where the noise µ is 10 percent of the measured acceleration. One may see in Fig. 5a the velocity estimated by the observer, while Fig. 5b describes the estimation time history, and Fig. 5c illustrates the observation error going exponentially to a residual set given by the noise amplitude. In order to evaluate the performances of the proposed observer, it has been compared to an EKF, an commonly used technque in the field of estimators. On Fig. 6, the estimation given by the adaptive observer is equivalent to the one given by Kalman filter estimation for the noisy acceleration considered in this section.

V. CONCLUSIONS AND FUTURE WORKS

A. Conclusions

In this paper the proposed observer scheme, based only on linear acceleration and angular measurements, assures global exponential estimation of the linear velocity of the UAV. This theoretical result is illustrated by computer simulations and presents good behaviour even in the presence of measurement noise. This good performance as well as the small size of the algorithm reinforces the idea of implementing it in the considered drone. In the following it will be studied the coupling of this observer with a control system in order to stabilize the drone and to perform tracking. Finally, the considered algorithms will be implemented in the 4-rotors drone available at IBISC laboratory.

In this paper the proposed observer scheme, based only on linear acceleration and angular measurements, assures global exponential estimation of the linear velocity of the UAV. This theoretical result is illustrated by computer simulations and presents good behaviour even in the presence of measurement noise. This good performance as well as the small size of the algorithm reinforces the idea of implementing it in the considered drone. Future work will be carried out to compare the proposed observer with other observer's schemes. In the following it will be studied the coupling of this observer with a control system in order to stabilize the drone and to perform tracking. Finally, the considered algorithms will be implemented in the 4-rotors drone available at IBISC laboratory.

Fig. 1 .

 1 Fig. 1. Representation of the UAV

m = 2

 2 .500Kg l b = 23cm I R = 10010 -7 Kgm 2 Ixx = 22493110 -7 Kgm 2 Iyy = 22261110 -7 Kgm 2 Izz = 32513010 -7 Kgm 2 k T = 10 -5 N.s 2 k M = 910 -5 N s 2 mI R is the inertia moment of the rotating elements in the rotors, while I xx , I yy and I zz are the diagonal terms of the body's inertia tensor. Finally k T represents a coefficient related to the lift torque (F k T ω 2 ) and k M is the coefficient related to the drag torque (M D k M ω 2 ).

Fig. 2 .

 2 Fig. 2. (a) Angles (b) Angular Speed" (c) Acceleration

Fig. 3 .

 3 Fig. 3. (a) Linear Speed, (b) Speed Estimation, (c) Estimation Error

Fig. 4 .

 4 Fig. 4. Noisy acceleration

Fig. 6 .

 6 Fig. 6. Comparison of both observer

TABLE I SIMULATION PARAMETERS

 IPARAMETERS


G L Santosuosso
email: santosuosso@ing.uniroma2.it

Khadidja Benzemrane
email: khadidja.benzemrane@ibisc.fr

Gilney Damm
email: gilney.damm@ibisc.fr

† K Benzemrane

‡ G Damm

Nonlinear speed estimation of a GPS-free UAV

In this paper the problem of robust state observer design for a class of Unmanned Aerial Vehicles (UAV) is addressed. A prototype 4 rotors helicopter robot for indoors and outdoors applications is considered: the drone is not equipped with GPS related devices, so that we describe a strategy to estimate its translational velocity vector based on acceleration, angles and angular speeds measurements only. Since the linearized system is non observable at the equilibrium point, a nonlinear observability verication is performed for persistently exciting trajectories. A global exponential solution to this open problem is provided in the framework of adaptive observation theory when exact measurements are available. A modied observer is presented to enhance velocity estimation robustness in the realistic case of noisy measurements. The results are compared with a classical estimation strategy based on the Extended Kalman Filter to test the algorithm's performance.

Introduction

The speed estimation problem has an important role in the context of vehicle control. For land moving robots, the odometry time derivative has a satisfactory performance while for large aerial vehicles (manned or unmanned) velocity estimations can be obtained via approximate derivation of the successive measurements from GPS sensors, motivated by the small resulting errors compared to the measured variables. For fast aircrafts the standard procedure is integrating the acceleration and coupling this result to the derivative of GPS measurements. In fact, signals (translational acceleration, angles and angular velocity) delivered by the embedded Inertial Measurement Unit (IMU) can be used to obtain position information through a double integration process. However, because of sensor drift, such procedure requires high precision IMU. This sensor can't be used for small size UAVs where cost, weight and volume are the most important constraints.

For such applications, two critical issues arise in an open loop strategy like direct acceleration integration: an unknown constant estimation error is produced even when exact acceleration measurements are available while a random drift is induced by noisy acceleration estimations. In practice, numerical integration along with measurement noise induces a very fast growing velocity measurement error.

Reasonable approaches to overcome these problems are represented by sensor fusion techniques, in order to compensate the lack of precision of low cost IMUs. In that context the information obtained from the GPS is used to bound the integration error and initial conditions for acceleration integration are provided by GPS devices [START_REF] Vik | A nonlinear observer for GPS and INS integration[END_REF]. This technique provides bounded errors that are related to the GPS order of magnitude precision and sample rate. Currently used GPS may assure a precision of 5-10 meters with a sample rate of 1-2 Hz. The resulting errors are usually small compared to the size of aerial vehicles and the distance to obstacles. In the same way, the control systems usually applied to these vehicles are robust enough to accept the residual disturbance on the speed estimation provided by these methods.

Unfortunately, the estimation approach above cannot be implemented on small drones less than 1 meter wide and ying at low speed that we consider in this note. This issue is even more relevant in indoors or simply urban applications as inspecting bridges and industrial facilities which cannot rely on GPS, since satellite's signals are shaded by the inspected structure.

A technical solution when no obstacles are present would involve the use of a D-GPS system. These systems are known as centimetric GPS and have a precision of some centimeters, but they are very expensive also, heavyweight and low-range operating equipment. In the same way, Doppler measurements coupled with GPS can actually increase velocity estimation accuracy, but they can only be used in special conditions of smoothness and continuity of obstacles, as well as a good knowledge of their position and shape.

In [START_REF] Metni | A UAV for bridge inspection: Visual servoing control law with orientation limits[END_REF], a control law based on computer vision is proposed for quasi-stationary ights above a planar structure like bridges to perform inspection tasks. Likewise, estimates of the pose and velocity of a rigid body can be provided by an algorithm based on IMU data and images provided by a camera (see [START_REF] Baldwin | Complementary lter design on the special Euclidian group SE(3)[END_REF]). The main drawback of this method is the large amount of computer resources needed. It is still not possible to apply vision algorithms in the small micro-controllers used in the considered UAV's. Displacing the image processing algorithms to computers at a land base is hardly implemented for two reasons: rst, the time delay induced by remote computing and communication, along with the danger of losing the contact with the drone could inevitably imply disruptive device instability; a second important drawback is the sensibility of cameras and vision algorithms to light variations, associated to outdoor drone displacement.

Vision can be used in an hierarchical display where a slow controller gives path planning references to a fast stabilizing one, as in [START_REF] Langelaan | Passive GPS-free navigation for small UAVs[END_REF], where an Unscented Kalman Filter is applied to the navigation problem and a solution for simultaneous localization and mapping when no GPS signal is available is proposed.

The result described in [START_REF] Langelaan | Passive GPS-free navigation for small UAVs[END_REF] relies on an Inertial Measurement Unit along with a monocular camera only and possible losses of vision or radio contact do not aect the UAV's stability. In the eld of fusion techniques, an interesting solution to the problem of estimation of the position and velocity of UAVs is proposed in [START_REF] Vissiere | Using magnetic disturbances to improve IMU-based position estimation[END_REF] and [START_REF] Vissiere | Using distributed magnetometers to increase IMU-based velocity estimation into perturbed area[END_REF]. The algorithm based on Kalman ltering techniques uses the magnetic eld disturbances to improve the estimation error due to the use of low cost IMU; it may be useful for indoor applications.

In the present paper we consider the problem of translational velocity estimation from measurements of translational accelerations in the local reference frame, Euler angles in the global reference frame and angular speeds in the local reference frame. Since translational velocity knowledge is necessary for an ecient drone control design, most literature about small ying robots control assumes that at least the translational speeds are available (see [START_REF] Castillo | Real-time stabilization and tracking of a four-rotor mini rotorcraft[END_REF], [START_REF] Guenardt | Dynamic modeling and intuitive control strategy for an "X4yer[END_REF], [START_REF] Lozano | Global stabilization of the PVTOL : Real-time application to a mini-aircraft[END_REF], [START_REF] Pounds | Design of a Four-Rotor Aerial Robot[END_REF]). This is motivated by the fact that it is hardly possible for a human pilot (inboard or in tele-operation) to regulate a drone by giving references and trajectories as inputs based on the sole acceleration measurement. Notice that in the general case, the reliable estimation of the speed vector is still an open problem. In fact, the acceleration in the local (body xed) frame is not a standard measurement from IMU, but even if this information is available the classical open loop integration using this information would roughly lead to the same results obtained with the standard acceleration measurement in the global reference frame.

Recently, [START_REF] Bonnabel | A non-linear symetry-preserving observer for velocityaided inertial navigation[END_REF], [START_REF] Vik | A nonlinear observer for GPS and INS integration[END_REF], [START_REF] Zhao | Discrete nonlinear observers for inertial navigation[END_REF] have proposed observers to be implemented on UAV's. The critical issue in this domain comes from the fact that the translation velocity is classically known as non observable. A key contribution of the present work is to show that although the system linearized approximation is non-observable in the origin, the full nonlinear system may become observable under suitable sucient conditions, thus allowing its estimation. Motivated by the above issues, in this note we describe a novel observation strategy that solves the problem of the speed estimation of an Unmanned Aerial Vehicle, when the translational acceleration (in the local reference frame), the angles and the angular speeds (in the global reference frame) are available for measurement. We focus our analysis on a prototype drone -a 4 rotors helicopter robot shown in Fig. 1 -produced to operate in an urban environment at the Laboratoire IBISC -CNRS, Universite d'Evry, which is not equipped with GPS related devices (see [START_REF] Pradel | Modeling and development of a quadrotor UAV[END_REF]). We provide a global exponential solution to this open problem in the light of classical adaptive estimation theory when exact measurements are available. A modied estimator is presented to enhance robustness in velocity estimation in the case of noisy acceleration measurements. The simulation results obtained on the model of the real drone -even in the presence of noise-provide an additional algorithm validation test. The proposed scheme is relatively simple and this is a key issue since the algorithm has to be implemented in small embedded micro-controllers without heavy computation burden. The results in this paper are inspired by two design principles. The rst one is that by increasing the use of model based informations, it is possible to better exploit sensors estimations, which is obviously obtained via a much larger modeling eort and theoretical complexity. Second, by exploiting the reference frame changes of coordinates, there are new informations that may arise when some rather mild conditions of persistence of excitation (PE) are fullled. This property in turn is satised when the considered aerial vehicle has decoupled rotational and translational dynamics, since we aim at constructing an observer which estimates translational states based on rotational information. These design principles can be implemented on any vehicle such that rotational dynamics can be stabilized independently on the estimation of translational velocities, like helicopters and Vertical Take O and Landing (VTOL) airplanes.

The nonlinear observability study introduced in this paper nds a natural complement in the design of an Extended Kalman Filter (EKF) for the considered UAV, which is used in the simulation section to illustrate and compare the behavior of the proposed adaptive observer. The tests performed show that the adaptive observer has a better performance in the case of internal and external disturbances like wind, unmodelled dynamics and measurement noise. Furthermore the proposed observers were designed based on adaptive theory and as consequence, present a self-tuning behavior much easier to set than the EKF.

The paper is organized as follows: in Section 2 we describe the drone model and formulate the problem addressed in this paper, analyzing the observability property of the system. This is guaranteed introducing a suitable PE assumption which is used in Section 3 to construct the observation algorithms and to prove that the estimation error converges exponentially to zero. In Section 4 we report some simulations that illustrate the observer performance and stress the inuence of some parameters on the observer behavior.

Finally in Section 5 we conclude the note with few remarks along with an outline of our future research on this topic.

Problem formulation

In this section we describe (see [START_REF] Azouz | Modeling and development of a 4rotors helicopter UAV[END_REF]) the UAV dynamic model equations and state the problem that we solve in the next sections. It is important to remark that the considered drone is a modied quadrotor that presents the interesting feature of swinging two of its motors, similar to a Vertical Take O and Landing (VTOL) airplane, that is represented as two additional inputs than the standard quadrotors. This feature is not used in the present work, since it does not address the UAV's control, but it could be an interesting feature in a dierent setting.

Let η 1 [x, y, z] T be the UAV position vector represented in the global reference frame, let η 2 [φ, θ, ψ] T be the Euler angles vector represented in the global reference frame (roll pitch and yaw respectively); we also assume that ν 1 [u, v, w] T is the speed vector represented in the local reference frame (surge, sway and heave respectively) and ν 2 [p, q, r] T is the angular speed vector represented in the local reference frame. The UAV model in state space form collects the set of the rst order dierential equations expressing η1 , η2 , ν1 , ν2 , as a function of η 1 , η 2 , ν 1 , ν 2 . To be more specic, the linear velocity in the xed reference frame η1 = ( ẋ, ẏ, ż) T can be expressed as a function of η 2 , ν 1 , via the set of dierential equalities ẋ = cos(θ) cos(ψ) u

+ (sin(φ) sin(θ) cos(ψ) -cos(φ) sin(ψ)) v + (cos(φ) sin(θ) cos(ψ) + sin(φ) sin(ψ)) w ẏ = cos(θ) sin(ψ) u + (sin(φ) sin(θ) sin(ψ) + cos(φ) cos(ψ)) v + (cos(φ) sin(θ) sin(ψ) -sin(φ) cos(ψ)) w ż = -sin(θ) u + sin(φ) cos(θ) v + cos(φ) cos(θ) w (1)
The time derivatives of the Euler angles η2 =( φ, θ, ψ) satisfy the following equations :

φ = p + (sin(φ) q + cos(φ) r) tan(θ) θ = cos(φ) q -sin(φ) r ψ = (sin(φ) q + cos(φ) r) cos(θ) -1 (2)
We assume that the UAV four rotors turn respectively at speed ω 1 , ω 2 , ω 3 and ω 4 . Support of rotors 1 and 3 can also swivel around their pitch axes (angles ς 1 and ς 3 represent the orientation of each support).

Theses quantities represent the system's control input functions and are related to the time derivatives of the speed vectors, respectively ν1 = [ u v ẇ] T and ν2 = [ ṗ q ṙ] T which are accelerations in the local reference frame and comply with the mechanical equations of the UAV. In particular, we have

u = (-q w + r v -g sin(θ)) -k T m (ω 2 1 sin(ς 1 ) + ω 2 3 sin(ς 3 )) v = (-r u + p w + g sin(φ) cos(θ)) ẇ = (-p v + q u + g cos(φ) cos(θ)) -k T m (ω 2 1 cos(ς 1 ) + ω 2 2 + ω 2 3 cos(ς 3 ) + ω 2 4 ) (3)
where g is the gravity acceleration constant, m is the drone mass, k T is a constant relating the UAV rotor speeds and resulting thrust. In the angular setting we obtain

I xx ṗ = -(I zz -I yy ) r q + l b k T (ω 2 1 cos(ς 1 ) -ω 2 3 cos(ς 3 )) -q I r (ω 1 cos(ς 1 ) + ω 2 + ω 3 cos(ς 3 ) + ω 4 ) -k M (ω 2 1 sin(ς 1 ) + ω 2 3 sin(ς 3 )) I yy q = -(I xx -I zz ) p r -r I r (ω 1 sin(ς 1 ) + ω 3 sin(ς 3 )) + p I r (ω 1 cos(ς 1 ) + ω 2 + ω 3 cos(ς 3 ) + ω 4 ) + l b k T (ω 2 2 -ω 2 4 ) + u G k T (-ω 2 1 sin(ς 1 ) + ω 2 3 sin(ς 3 )) I zz ṙ = -(I yy -I xx ) p q -l b k T (ω 2 1 sin(ς 1 ) -ω 2 3 sin(ς 3 )) + k M (ω 2 1 cos(ς 1 ) -ω 2 2 + ω 2 3 cos(ς 3 ) -ω 2 4 ) + q I r (ω 1 sin(ς 1 ) + ω 3 sin(ς 3 )) (4)
where I xx , I yy , I zz are the inertia moments, k M is the constant relating rotor speeds and torque, l b is the length of each drone's arm, and I r is the rotor's inertia moment constant.

In this paper we assume that the following quantities are available for measurement: orientation (Euler Angles in the global reference frame), angular velocities (in the local reference frame) and translational acceleration (in the local reference frame) of the vehicle. The coecients k T and k M as well as dierent parameters of the model are experimentally identied. In this context 1 , dening the vectors

   X = [φ, θ, ψ, p, q, r, u, v, w] T y m = [φ, θ, ψ, p, q, r, u, v, ẇ] T Ω = [ω 1 , ω 2 , ω 3 , ω 4 , ς 1 , ς 3 ] T (5)
1 We will not consider position estimation. For this reason, the state vector does not include it.

equations ( 2)-( 4) can be written in a compact notation as

Ẋ = f (X, Ω), X ∈ R 9 y m = h(X, Ω), y ∈ R 6 (6)
where f : R 9 × R 6 → R 9

is a smooth function depending on the state and the control input Ω, and h(X, Ω) is obtained by combining the second equality in (5) along with (3), expressing u, v, ẇ as functions of X and Ω.

The paper objective may then be stated as: given system (6), design an observation strategy yielding exponentially converging estimates of the state X from the measurable output y m , to be eventually included in any control strategy of system [START_REF] Benzemrane | Unmanned aerial vehicle speed estimation via nonlinear adaptive observers[END_REF].

In practical terms, the observation algorithm focuses on the estimation of the unmeasurable state variables ν 1 , based on the measurable variables η 2 , ν1 and ν 2 given by the standard sensors embedded in the drone. We remark that a necessary condition for the synthesis of any observer is the local observability property to be veried. To this purpose, we recall the following denition.

Denition 1 (see [START_REF] Isidori | Nonlinear Control Design -Geometric, Adaptive and Robust[END_REF], [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF]). Let G denote the set of all nite linear combinations of the Lie derivatives of h 1 , . . . , h p with respect to f for various values of u = constant. Let dG denote the set of all their gradients. If we can nd n linearly independent vectors within dG, then the system is locally observable.

The observability matrix O = dG is given by :

O =   L 0 f h ... L p-1 f h  
The system is locally observable if O has full rank. Thus in the case we consider in this paper, the estimation strategy has to be coupled with a tracking control algorithm driving the angular speeds [p, q, r] T to bounded and non vanishing reference trajectories [p r (t), q r (t), r r (t)] T that guarantee the rank of matrix O to be dierent from zero. To be more specic, let Φ(t) be the matrix dened as

Φ(t) =   0 r -q -r 0 p q -p 0   (7)
In this paper we assume Φ(t) is persistently exciting (PE), as formally stated below.

Hypothesis 1 The matrix Φ(t) is bounded and persistently exciting (PE), i.e. there are positive integers

T * , K * 1 and K * 2 such that K * 1 I ≥ t+T * t Φ(τ ) T Φ(τ )dτ ≥ K * 2 I for all t ∈ R (8)
Claim 1 Hypothesis (H1) is satised for any speed vector represented in the local reference frame ν 2 = [p, q, r] T , which is bounded and PE, i.e. such that

c * 1 I ≥ t+T * t ν 2 (τ ) T ν 2 (τ )dτ ≥ c * 2 I for all t ∈ R (9)
for suitable positive integers

Property ( 10) can be expressed concisely as ν 2 (t) = T r(t)V + (t) where V ∈ R 3 , V = 0 is any non zero vector, T r(t) is any scalar time function and (t) is any vector function such that lim t→∞ (t) = 0. It is straightforward to show that any ν 2 (t) in the form above does not satisfy [START_REF] Castillo | Real-time stabilization and tracking of a four-rotor mini rotorcraft[END_REF], and this proves the Claim.

Remark 1 By virtue of Claim 1, Hypothesis 1 is guaranteed for any angular speed vector ν 2 (t) that spans periodically a 3-dimensional surface, for instance a non-vanishing periodic orbit. Besides, the nature of the vehicle provides vibrations in real applications, that may already give enough persistence of excitation which relyes on angular speeds (oscillation frequency) rather than angles (oscillation amplitudes).

Remark 2 The necessity of Hypothesis 1 claries that our estimation strategy can be implemented on that set of aerial vehicles satisfying PE condition. All helicopters and quadrotors fulll this requirement because their rotational and translational dynamics are decoupled and the rotational states are available for measurement with standard sensors (IMU, infrared sensors, magnetometers). For this reason, it is possible to track a persistently exciting reference independently of estimation results. These could then be used in an outer loop that may be used to stabilize the translational dynamics.

Observer Design

In this section we describe the estimation strategy following the techniques presented in [START_REF] Marino | Robust Adaptive Observers for Nonlinear Systems with Bounded Disturbances[END_REF] and [START_REF] Marino | Nonlinear Control Design -Geometric, Adaptive and Robust[END_REF], as well as an outline of its stability proof. Two dierent algorithms that estimate the linear velocity of the system described by equations ( 2) -( 4) are presented. The rst algorithm considers the ideal case with no noise on the measurement outputs while the second takes into account additive measurement noise on the translational accelerations.

The measured output can be expressed as

y m = [y 1 , y 2 ] T = [φ, θ, ψ, p, q, r y1 , u, v, ẇ y2 ] T
where the vector y 2 = ν1 = [ u, v, ẇ] T is available for measurement, so that equation ( 3) is re-written as

ν1 (t) = Φ(t)ν 1 (t) + Λ(t) (11)
y 2 = ν1

with the matrix Φ as in [START_REF] Besnard | Control of a quadrotor vehicle using sliding mode disturbance observer[END_REF] and the vector Λ dened as

Λ(t) =   -g sin(θ) -k T m (ω 2 1 sin(ς 1 ) + ω 2 3 sin(ς 3 )) g sin(φ) cos(θ) g cos(φ) cos(θ) -k T m (ω 2 1 cos(ς 1 ) + ω 2 2 + ω 2 3 cos(ς 3 ) + ω 2 4 )  
In the following we design a reduced-order observer which estimates the translational velocity of the UAV recalling that, in order to guarantee that p(t), q(t) and r(t) are dierent from zero, the control inputs have to be chosen such that the system oscillates around the equilibrium point.

Basic algorithm

In this section we construct an algorithm to estimate the time function ν 1 (t) assuming, by virtue of [START_REF] Hamel | Image based visual servo control for a class of aerial robotic systems[END_REF], that the quantity Y 1 [ ν1 (t) -Λ(t)] = Φ(t)ν 1 (t) is measurable, recalling that both the time derivative of the function to be estimated y 2 = ν1 (t) and the regression matrix Φ(t) are available for measurement. We remark that many adaptive estimation techniques (see [START_REF] Marino | Nonlinear Control Design -Geometric, Adaptive and Robust[END_REF]) are shown to guarantee exponential function estimation when the time derivative of the regression matrix is also available for measurement, thus the rst step in the estimation strategy is to replace the regression identity Y 1 = Φ(t)ν 1 (t) with another identity Y 2 M (t)ν 1 (t), where M (t) is a matrix (to be dened) whose time derivative Ṁ is available for measurement. To this purpose, let M (t) ∈ R 3 × R 3 be the state of the lter

Ṁ = -αM + Φ(t) (12)
with arbitrary initial condition M (0) ∈ R 3 × R 3 , where α ∈ R + is a positive tuning parameter chosen by the designer. The matrix M (t) is also PE, as stated below.

Claim 2 The matrix M is persistently exciting, i.e. there exist positive integers

K * M 1 and K * M 2 such that K * M 1 I ≥ t+T * t M (τ ) T M (τ )dτ ≥ K * M 2 I for all t ∈ R (13)
where T * is the positive real dened in Hypothesis H1.

Proof. By contradiction, assume that there exists a vector s ∈ R 3 with s = 1 such that lim t→∞ M (t)s = 0. Then by setting ∆M (t) = M (t + T * )s -M (t)s we have that lim t→∞ ∆M (t) = 0. Recalling that by virtue of ( 12) ∆M (t) = -α t+T * t M (τ )sdτ + t+T * t Φ(τ )sdτ , we deduce that lim t→∞ t+T * t Φ(τ )sdτ = 0, contradicting hypothesis (H1).

Dene the system ρ = -αρ + ν1 -Λ(t) + M ν1 = -αρ + y 2 -Λ(t) + M y 2 [START_REF] Khalil | Nonlinear Systems[END_REF] with state ρ ∈ R 3 , arbitrary initial condition ρ(0) ∈ R 3 . The task of ( 12) and ( 14) is to obtain a measurable estimate ρ of the vector Y 2 = M ν 1 with M measurable along with its time derivative. In fact, by setting σ = ρ -M ν 1 we have σ = -α (ρ -M ν 1 ) = -ασ [START_REF] Lozano | Global stabilization of the PVTOL : Real-time application to a mini-aircraft[END_REF] which implies that σ exponentially converges to zero.

Consider the lter

Q = -βQ + M T M (16) with state Q ∈ R 3 × R 3
and any initial condition such that Q(0) > 0, where β ∈ R + is a positive tuning parameter chosen by the designer. From ( 16) by using the arguments in [START_REF] Marino | Nonlinear Control Design -Geometric, Adaptive and Robust[END_REF] we demonstrate the following result.

Claim 3 Assume that Q(0) > 0; then the matrix Q(t) is bounded and positive denite, i.e. there exists

a positive constant c 2 such that Q(t) > c 2 I for all t ≥ 0.
Proof. Each entry of the matrix Q(t) is the state of a linear asymptotically stable lter driven by a bounded input, so that Q(t) is bounded. In order to prove that Q(t) is positive denite, from [START_REF] Lebras | Nonlinear observer-based visual control of a VTOL UAV[END_REF] we have that

Q(t) = e -βt Q(0) + t 0 e -β(t-τ ) M (τ ) T M (τ )dτ
Recall that e -βt Q(0) > 0 and t 0 e -β(t-τ ) M (τ ) T M (τ )dτ ≥ 0 for all t > 0, thus Q(t) ≥ e -βT * Q(0) for all 0 ≤ t ≤ T * ; however from [START_REF] Lebras | Nonlinear observer-based visual control of a VTOL UAV[END_REF] we also infer that Q(T * ) ≥ T * 0 e -β(T * -τ ) M (τ ) T M (τ )dτ ≥ e -βT * T * 0 M (τ ) T * M (τ )dτ, that by virtue of (13) in Claim 2 yields Q(T * ) ≥ e -βT * K * M 2 I. By repeating iteratively these arguments we have that Q

(t) ≥ e -2βT * K * M 2 I in each interval iT * ≤ t ≤ (i + 1) T * with i = 1, 2...∞, concluding that Q(t) ≥ c 2 I where c 2 = min{e -βT * Q(0), e -2βT * K * M 2 }.
Thus the idea behind the observation strategy in this note is to seek an estimate

ν1 =   û v ŵ  
of ν 1 such that by setting the estimation error variable ν1 = ν 1 -ν1 , its time derivative can be written in the form

ν1 = γ(-Qν 1 + χ) (17)
where γ ∈ R + is a positive tuning parameter to be chosen by the designer and χ is a quantity to be dened within the synthesis process. Notice that χ must not be chosen necessarily as a measurable quantity, rather it has to converge exponentially to zero. If this is the case, then the positive deniteness of Q(t) implies that lim t→∞ ν1 (t) = 0 exponentially. Since ν1 = ν 1 -ν1 , the computation of ν1 along with [START_REF] Langelaan | Passive GPS-free navigation for small UAVs[END_REF]

yields ν1 = γ(-Qν 1 + Qν 1 -χ) + y 2 . ( 18
)
>From the last expression we infer that χ has to include the non measurable term Qν 1 for the dynamics ν1 to be implementable. Notice that d (Qν 1 ) /dt = -β(Qν 1 ) + M T M ν 1 + Qy 2 ; this circumstance suggests that χ has to be set equal to Qν 1 plus the states of some stable linear lters with eigenvalue -β fed by terms compensating M T M ν 1 and Qy 2 . In particular, dening these stable lters as

δ = -βδ + ρ -βδ + σ + M ν 1 (19) ξ = -βξ + Qy 2 + [αM -Φ(t)] T δ -βξ + Qy 2 -Ṁ T δ (20)
with vector states δ ∈ R 3 , ξ ∈ R 3 , arbitrary initial conditions δ(0) ∈ R 3 ξ(0) ∈ R 3 and setting

χ = Qν 1 -M T δ -ξ, (21)
by taking into account [START_REF] Marino | Robust Adaptive Observers for Nonlinear Systems with Bounded Disturbances[END_REF], [START_REF] Marino | Nonlinear Control Design -Geometric, Adaptive and Robust[END_REF] we obtain χ = -βχ -M T σ.

(

) 22
>From [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF] we infer that χ is the state of a stable lter driven by the exponentially vanishing input σ, so that lim t→∞ χ(t) = 0 exponentially, which implies that lim t→∞ ν1 (t) = 0 exponentially. Equality [START_REF] Metni | A UAV for bridge inspection: Visual servoing control law with orientation limits[END_REF] along with (18) yields the dynamics for ν1 , the estimate of ν 1 , given by

ν1 = γ(-Qν 1 + M T δ + ξ) + y 2 . (23)
Previous arguments can be summarized as follows.

Proposition 1 Consider the UAV model described by ( 2)-( 4). Assume that the variables η 2 , ν 2 , ν1 are available for measurement and that Hypothesis 1 holds. Then for all α ∈ R + , β ∈ R + , γ ∈ R + , the dynamic observer described by ( 12), ( 14), ( 16), ( 19), ( 20) and ( 23) with state

M (t) ∈ R 3 × R 3 , Q(t) ∈ R 3 × R 3 , ρ(t) ∈ R 3 , δ(t) ∈ R 3 , ξ(t) ∈ R 3 and ν1 (t) ∈ R 3 is such that the vector ν 1 -ν1 converges to zero globally exponentially for any initial conditions M (0) ∈ R 3 × R 3 , Q(0) ∈ R 3 × R 3 , ρ(0) ∈ R 3 , δ(0) ∈ R 3 , ξ(0) ∈ R 3 ν1 (0) ∈ R 3 with Q(0) > 0.
Proof. The result has been already outlined in the previous arguments; a rigorous demonstration follows from standard Lyapunov theory. In particular, consider the class of Lyapunov functions

V = 1 2 κ 1 ν1 2 + 1 2 κ 2 χ 2 + 1 2 κ 3 σ 2 (24)
where κ 1 , κ 2 , κ 3 , are positive real numbers to be dened later. By computing the time derivative V and recalling (15), ( 22), ( 17) we obtain

V = -ακ 3 σ 2 -βκ 2 χ 2 -κ 2 χ T M T z -γκ 1 νT 1 Qν 1 + γκ 1 νT 1 χ. (25)
Notice that Hypothesis 1 implies that matrix M (t) is also bounded and persistently exciting. From ( 16) by using the arguments in [START_REF] Marino | Nonlinear Control Design -Geometric, Adaptive and Robust[END_REF] it is shown that if Q(0) > 0, then matrix Q(t) is positive denite for all t ≥ 0. Setting the positive constant c

1 = sup t∈[0,∞) M (t) 2
, we deduce

κ 2 χ T M T σ ≤ κ 2 β 4 χ 2 + κ 2 β c 2 1 σ 2 γχ T ν1 ≤ γc 2 2 ν1 2 + γ 2c 2 χ 2
where c 2 is the bound dened in Claim 3. The substitution of the previous inequalities in (25) yields

V ≤ -(κ 3 α - κ 2 β c 2 1 ) σ 2 -(κ 2 β - κ 2 β 4 + κ 1 γ 2c 2 ) χ 2 -κ 1 γ(c 2 - c 2 2 ) ν1 By setting κ 1 = 1, κ 2 = 2γ βc 2 , κ 3 = 2κ 2 c 2 1 βα = 4γc 2 1 β 2 αc 2 , we nally obtain V ≤ -( γc 2 2 ) ν1 2 -( γ 2c 2 ) χ 2 -( 2γc 2 1 β 2 c 2 ) σ 2 (26)
Let λ 1 = min γc 2 , β 2 , α . Previous inequality implies that V ≤ -λ 1 V, which guarantees the convergence of all errors exponentially to zero.

Extension of the basic algorithm

In this section, the former algorithm is modied in order to quantify the observer performance in case of measurement noise applied on the velocity and acceleration signals. We assume now that the noisy acceleration a µ and the noisy velocity matrix Φ µ (t) to be available for measurement, given by

a µ =   u +
with µ M ∈ R + a suitable positive bound. In this framework, the measurement noises play the role of small non vanishing exogenous inputs aecting the estimation error dynamics. The observation strategy in this setting focuses on the synthesis of a robust observer, i.e. such that noise inuence on estimation error is small. To comply with this requirement, extra tuning parameters will be introduced in the observer design, which also aims at Input-to-State Stability property (see [START_REF] Khalil | Nonlinear Systems[END_REF] for ISS denition):

in practice, the observer estimation error is attracted inside an open ball whose radius depends on the measurement noise. To adapt ISS property to the design we consider in this note, we recall the following result, that is demonstrated via standard Lyapunov theory.

Lemma 1 Consider the system ẋ = A(t)x + B(t)u with state x(t) ∈ R n , arbitrary initial condition x(0) ∈ R n and input u(t) ∈ R m . Assume that:

(i) there is a positive scalar parameter λ m such that v T A(t)v ≤ -λ m v 2 for all t ≥ 0 and for any v ∈ R n ;

(ii) there is a positive scalar parameter β M such that B(t) i.e. x(t) is also exponentially attracted to a ball whose radius is proportional by a suitable factor k 2 to the bound u M of the input u(t).

Proof. Consider the Lyapunov function V (t) = 1 2 x(t) 2 . By computing its time derivative, we have

V (t) = x T A(t)x + x T B(t)u. (29)
By completing the squares, recalling (i), (ii), we obtain x T A(t)x ≤ -λ m x

Figure 1 :

 1 Figure 1: Prototype Drone

  T * , c * 1 and c * 2 .Proof. To prove the Claim we show that any ν 2 (t) which does not comply with (8) does not comply with (9) also. In fact, if Φ(t) is not PE, then there exists a nonzero vector s = [s 1 , s 2 , s 3 ] T such that lim

t→∞ Φ(t)s 2 = 0. By expanding the term Φ(t)s 2 , recalling (7), we deduce that the limit identity above implies

s 3 lim t→∞ q(t) = s 2 lim t→∞ r(t); s 3 lim t→∞ p(t) = s 1 lim t→∞ r(t); s 2 lim t→∞ p(t) = s 1 lim t→∞ q(t)

  21 (t) -q -µ 22 (t) -r -µ 21 (t) 0 p + µ 23 (t) q + µ 22 (t) -p -µ 23 (t) 0  where µ ij (t), i = 1, 2, j = 1, 2, 3 are bounded measurement noises, i.e. are such that µ ij (t) ≤ µ M , for all t > 0, i = 1, 2, j = 1, 2, 3,

	µ 11 (t)				0	r + µ
	v + µ 12 (t)	 ;	Φ µ (t) =	
	ẇ + µ 13 (t)

  2 < β M for all t ≥ 0. If there are suitable positive constants k u , u M , λ u such that u(t) 2 ≤ k u exp(-λ u t) + u 2 M i.e. u(t) is exponentially attracted to a ball with radius u M , then there are suitable positive constantsk 1 , k 2 , λ x such that x(t) 2 ≤ k 1 exp(-λ x t) + k 2 u 2

	M	(28)

with initial condition W (0) = V (0) = 1 2 x(0)

2

. The solution of [START_REF] Zhao | Discrete nonlinear observers for inertial navigation[END_REF] is

1 -e -λmt .

Neglecting the negative terms in previous equality, we have

recalling that 1 2 x(t) 2 = V (t) ≤ W (t) and W (0) exp(-λ m t) ≤ 1 2 x(0) 2 exp(-λ * t), we deduce that [START_REF] Vissiere | Using distributed magnetometers to increase IMU-based velocity estimation into perturbed area[END_REF] is veried by setting λ x = λ * , k

The rst step in the estimation strategy is the denition of the lters with states M µ ∈ R 3 × R 3

,

, whose role is the same of lters [START_REF] Herisse | Hovering ight and vertical landing control of a VTOL Unmanned Aerial Vehicle using Optical Flow[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF] in previous section:

with arbitrary initial conditions, where k ∈ R + , α ∈ R +

, are tuning parameters chosen by the designer.

By setting σ µ = ρ µ -M µ ν 1 and µ 1 (t) = [µ 11 (t), µ 12 (t), µ 13 (t)] ∈ R 3 µ 2 = [µ 21 (t), µ 22 (t), µ 23 (t)] T ∈ R 3

and

, w] T is the speed vector represented in the local reference frame, it is straightforward to verify that

It is shown in the following that the dynamics of σ µ (t) in ( 33) comply with the hypotheses of Lemma 1, i.e. σ µ (t) is exponentially attracted inside a ball whose radius depends on the bounds of µ 1 (t) and µ 2 (t). Consider the lter with matrix state Q µ ∈ R 3 × R 3 , with initial condition such that |Q µ (0)| > 0, dened as:

where β ∈ R + is a tuning parameter chosen by the designer. The lter (34) is a generalization of system [START_REF] Lebras | Nonlinear observer-based visual control of a VTOL UAV[END_REF]; by following arguments similar to the ones of previous section we show a result that generalizes Claim 3.

Claim 4 Assume that Q µ (0) > 0; then the matrix Q µ (t) is bounded and positive denite, i.e. Q µ (t) > 0 for all t ≥ 0.

Proof. First, recall that Q µ is bounded. In fact, set x µ = Q µ s where s ∈ R 3

is any vector such that s = 1. The computation of the time derivative of x µ = Q µ s, by virtue of (34) yields

>From (35) we infer that x µ complies with Lemma 1 with βI

in place of u(t). Thus x µ = Q µ s is exponentially attracted inside a ball whose radius depends on sup t∈[0,∞) M T µ (t)M µ (t) , so that (being s ∈ R 3 arbitrary), we conclude that Q µ is bounded; in particular, there exists c

In order to prove that Q µ is positive denite, set η µ = s T Q µ s, where s ∈ R 3

is any vector such that s = 1. The computation of the time derivative of η µ , by virtue of (34) along with (36) yields

where β * = β + k 4 c 2 Q . From the inequality above we infer that

Recall that e -β * t η µ (0) > 0 and t 0 e -β * (t-τ ) M µ (τ )s 2 dτ ≥ 0 for all t > 0, thus η µ (t) ≥ e -β * t η µ (0) for all 0 ≤ t ≤ T * , where T * is the positive real dened in Hypothesis H1. From [START_REF] Lebras | Nonlinear observer-based visual control of a VTOL UAV[END_REF] we have that η µ (T * ) ≥

Consider the two lters with states δ µ ∈ R 3 and ξ µ ∈ R 3

, whose task is analogous to the one of ( 19), [START_REF] Marino | Nonlinear Control Design -Geometric, Adaptive and Robust[END_REF] respectively in previous section,

with arbitrary initial conditions.

We introduce an estimate of the linear velocity

that satisfy the dierential equations :

with γ ∈ R + tuning parameter to be chosen by the designer.

Under Hypothesis 1, if the variables η 2 , ν 2 , a µ are available for measurement, the dynamic observer described by (31), (32), (34), ( 38), (39), and (40

is a generalization of the observer presented in Section 3.1. In fact, by setting the estimation error variables ν1 = ν 1 -ν1 and χ µ = Qν 1 -M T µ δ µ -ξ µ , we obtain:

The iterative application of Lemma 1 to the systems (33), ( 41) and (42) leads to the demonstration of the convergence properties of the estimation scheme introduced above, where the parameter k adds a degree of freedom to the system and the observer guarantees the exponential convergence of the estimation error to a region around zero as well as the robustness with respect to the noises µ 1 and µ 2 .

Proposition 2 Consider the UAV model described by ( 2)-( 4). Assume that the variables η 2 , ν 2 , a µ are available for measurement and that Hypothesis 1 holds. Then for all k ∈ R + , α ∈ R + , β ∈ R + , γ ∈ R + , the dynamic observer described by (31), (32), (34), (38), (39), and (40

for suitable positive reals λ 2 , g 1 , g 2 , where µ M is given in [START_REF] Vissiere | Using magnetic disturbances to improve IMU-based position estimation[END_REF], for any initial conditions M µ (0

Proof. System (33) complies with the hypotheses of Lemma 1 with σ µ in place of x and [µ 1 (t), µ 2 (t)]

in place of u(t), noting that hypothesis (i) is trivially veried and point (ii) is a consequence of the boundedness of the matrices M µ (t) and N 2 (t). Thus, by [START_REF] Vissiere | Using magnetic disturbances to improve IMU-based position estimation[END_REF], σ µ is exponentially attracted in a ball whose radius depends on µ M . This property implies that system (41) also complies with Lemma 1 with χ µ in place of x(t) and [σ µ (t), µ 1 (t)] in place of u(t). In fact, (41) complies with hypothesis (i) with

in place of A(t) and (ii) follows noting that M T µ (t) and Q µ (t) are bounded, so that the vector χ µ (t) is exponentially attracted in a ball with radius depending on µ M . Notice that from Claim 4 we have that if Q µ (t) > 0, then matrix Q µ (t) is bounded and positive denite for all t ≥ 0. Hence system (42) veries the hypotheses of Lemma 1 with ν1 (t) in place of x(t) and [χ µ (t), µ 1 (t)] in place of u(t), leading nally to inequality (43).

Simulation Results

Illustration of the adaptive observer performances

In this section we illustrate the observer designed above to estimate the linear velocity of an UAV based on the measurable angles, angular velocities and linear accelerations. These simulations also illustrate the eect of the tuning parameters α, β and γ on the observer performances. In all simulations the observer estimations have initial conditions set to zero, while the desired states are time varying (and dierent from zero at t = 0). In all simulations both observers track the desired time varying states, with dierent performances and noise rejections. We have used the following parameters values : Consider the rst version of the observer, that does not take into account the measurement noise. In Fig.

2 the angular speeds, the angles, and the accelerations are plotted.

We choose in our simulation an angular speed vector ν 2 (t) that spans periodically a 3-dimensional surface, in particular a non-vanishing periodic orbit. By virtue of Claim 1, this choice guarantees that PE Hypothesis 1 is satised. The time history of the three speeds to be observed is shown in Fig. 3a, and observer estimates in Fig. 3b, while estimation errors going exponentially to zero are shown in Fig. 3c. We have then considered the rst observer in the case of additive measurement noise. The measured acceleration is presented in Fig. 4 where the noise µ is 10 percent of the measured acceleration.

Comparison with an Extended Kalman Filter

The aim now is to compare the observer proposed in Section 3.2 with an Extended Kalman Filter (EKF), which is an established benchmark for estimation techniques. The EKF described here is based on the dynamic model of the UAV given by the continuous-time equations ( 2)-( 4). We considered the corresponding discrete time model that can be written as

The discrete-time state vector is

W k and Z k respectively represent process noise and measurement noise at time step k ; they are zero mean white Gaussian sequences of random variables.

The behavior of the EKF designed to estimate the linear velocity of an UAV based on the measurable angles, angular velocities and linear accelerations is illustrated by numerical simulation and compared to the estimate provided by the observer described in Section 3.2. In all simulations, the observer estimations have initial conditions set to zero, while the desired states are time varying (and dierent from zero at t = 0). Model parameters are the same as those presented in Section 4.1 and we have considered additive measurement noise on the acceleration (see Fig. 5). First, we compare the performance of our estimation strategy with EKF by setting k = 50, reporting the results in Figure 6. They show that the algorithm in Section 3.2 signicantly outperforms the EKF, since it guarantees in steady state condition a much smaller estimation error. Notice in this simulation (with k = 50), the estimation error convergence of the algorithm in this note is slower than EKF; this is reasonable, since our strategy involves a cascade of lters, the transient performance of the entire algorithm being the sum of transient performances of the lters in the cascade. However, the total number of tuning parameters in our estimator is smaller than the number of parameters to be set in the EKF. As a consequence, the tunning procedure of our algorithms, veried in our simulations, is much more simple than that of the EKF. This is easily explained rst by the small number of parameters, btheirut as weel as their clear eect on the convergence (through the analysis of the Lyapunov fonction). Secondly, the several dynamical variables behave as an automatic gain, this characteristic comming from adaptive theory make our observers rather self tunning compared to the EKF. By decreasing the value of k, in particular by setting k = 10 and performing the simulation for this setting, we have a faster convergence rate, as displayed in Figure 7. Finally, Figure 8 shows that by choosing k = 0, we obtain a steady state estimation error increase. This phenomenon can be explained as follows: >From expression (34) we infer that the bigger is k, the smaller is the norm of Q µ ; the latter quantity in turn is related, by virtue of (42), to the inverse of the estimation error convergence rate. Thus by decreasing k we may obtain a faster convergence rate but may have a greater sensibility to disturbances also; this property is inherently associated with a high gain estimation strategy.

In conclusion, the simulation tests show that the best performance on the algorithm in Section 3.2 are obtained via a ne tuning of the parameter k in order to guarantee a small steady state estimation error outperforming EKF and simultaneously a convergence rate typical of a classic EKF approach.

Conclusion

In this work the UAV drone speed estimation problem has been addressed when no GPS related device is available. This is a realistic feature in many practical applications and has motivated recent works on the eld. The results established for the specic prototype in this study hold also for the entire class of UAV such that rotational and translational dynamics may be decoupled, as helicopters and VTOL aircrafts.

In this context, the rst theoretical issue arises by the fact that the linearized system is non-observable at the equilibrium point. Thus a nonlinear observability verication is performed for persistently exciting trajectories so that under some mild sucient conditions an estimation strategy is shown to guarantee exponential estimation of the translational speed, by using the tools of adaptive observation theory.

A modied scheme is also proposed in order to obtain enhanced robustness property with respect to measurement noise. The theoretical results are illustrated via computer simulations, stressing the noise attenuation features. They also provide useful insight about the parameter tuning eect to achieve a satisfactory observer performance. Furthermore, computer simulations are used to compare the proposed estimation strategy with the classical Extended Kalman Filter. They show that the adaptive observer guarantees good performances even in presence of measurement noise, with greater disturbances attenuation with respect to the one provided by the Extended Kalman Filter.

Future research will be dedicated to the design of a simplied algorithm to possibly trade o noise immunity against faster convergence along with coupling the proposed observer to a control system designed to stabilize the 4-rotors helicopter. Additional work will be to implement the considered scheme in the UAV available at IBISC laboratory and to extend current results to the standard acceleration measurements provided by low cost IMUs.


Van Cuong Nguyen

Gilney Damm

Conditional Servo-compensator of an Airlaunch System*

A Multiple Input Multiple Output (MIMO) controller based on the conditional servo-compensator technique is designed for the robust stabilization of a new satellite launching strategy called (unmanned) airlaunch. This strategy consists in using a two-stages launching system. The first stage is composed of an airplane (manned or unmanned) that carries a rocket launcher which constitute the subsequent stages. The control objective is to stabilize the aircraft in the launch phase. It is developed separately for two nonlinear motion modes of the model, the longitudinal and lateral modes, and is applied to the full multi-input multi-output model of the aircraft. The controller is indeed able to assure system stability for rather large disturbances. Performance of the proposed control algorithm is illustrated through simulations.

I. INTRODUCTION

This work presents the design of a controller for the robust stabilization of a new satellite launching strategy called (unmanned) airlaunch. This strategy consists in using a two-stages launching system. The first stage is composed of an airplane that carries a rocket launcher which constitutes the subsequent stage. The aircraft brings the rocket to a desired drop area, consequently avoiding many costs and risks related to land rocket launching. On the other hand, this procedure brings up many other difficulties connected to the instant of releasing the rocket.

Currently, several airlaunch systems are under development (see [START_REF] Sarigul-Klijn | Selection of a carrier aircraft and a launch method for air launching space vehicles[END_REF], [START_REF] Gary | Quickreach responsive launch system[END_REF]). Most current airlaunch projects use standard or lightly modified airplanes as first stage. For example, there has been tests using F15, C17, B52, L-1011 in Rascal, QuickReach, Proteus and Pegasus projects. Unlike those, other projects aim to develop an airlaunch system that uses an unmanned aerial vehicle (UAV) instead of a standard aircraft with a human pilot inboard. The objective is then to use an UAV to fly the launcher to the desired drop point. There are many advantages in doing so, in first place safety since no human lives are involved during the delicate launching phase. In addition, since there is no need for life supporting devices, weight is restricted to the strict minimum. Finally, mission may take as long as necessary without human restrictions as tiredness.

The present paper addresses the stabilization of the drop phase. It intends to introduce a robust control scheme for this complex procedure. In fact, airlaunch may be very delicate for many reasons. For example, since the rocket may be almost as heavy as the UAV, this means that the aircraft will instantaneously lose almost half of its mass. Current airlaunch systems present a much smaller ratio launcher/aircraft and rely on human pilot to stabilize the aircraft during and immediately after the launching instant. The proposed system must replace the human pilot in this stabilization task, with a much more adverse mass ratio. In the same way, the two-stages system is strongly nonlinear and can even for small perturbations be brought quite far from the initial equilibrium point. Furthermore, available models are based on experiments for different flight conditions, with lookup tables and polynomial interpolation between these points. For this reason, parameters and even models are not very well known, and need very robust control schemes like those found in [START_REF] Hassan | Nonlinear Systems[END_REF].

Our previous works proposed a new strategy based on MIMO conditional integrator (see [START_REF] Damm | Mimo conditional integrator control for a class of nonlinear systems[END_REF]) for the nonlinear model to stabilize the airlaunch system after the launching phase, which is modeled by the Initial conditions aproach [START_REF] Van | Mimo conditional integrator control for unmanned airlaunch[END_REF] and by Perturbation on aerodynamic force and moment in [START_REF] Van | Modeling and conditional integrator control of an unmanned aerial vehicle for airlaunch[END_REF]. The designed controller resulted in a better behavior in these extreme situation that are nevertheless expected in the unmanned airlaunch. This paper is an extension of the previous works using the MIMO Conditional Servocompensator based controller developed in a series of papers ( [START_REF] Mahmoud | Robust control for a nonlinear servomechanism problem[END_REF], [START_REF] Hassan | On the design of robust servomechanisms for minimum phase nonlinear systems[END_REF] and [START_REF] Damm | Mimo conditional servocompensator control for a class of nonlinear systems (accepted)[END_REF]) to stabilize the airlaunch system in the case where the launching phase is modeled by perturbation on aerodynamic force and moment (see [START_REF] Van | Modeling and conditional integrator control of an unmanned aerial vehicle for airlaunch[END_REF]).

The paper is organized as follows: in section II, we describe the nonlinear mathematical system model based on [START_REF] Roskam | Airplane Flight Dynamic and Automatic Flight Controls, Part 1. Design[END_REF], [START_REF] Sonneveldt | Nonlinear F-16 Model Description[END_REF] and [START_REF] Nguyen | Simulator study of fighter airplane with relaxed longitudinal static stability[END_REF]. The control design literature is discussed in section III, and its application to the full nonlinear system model in section IV. The paper is completed by some computer simulations and conclusions.

II. MODELING

Drop phase is delicate to model, and requires a large amount of data and previous knowledge about the real system. It can also be represented as a hybrid system composed by two (or three) continuous models that are switched. These models represents the system before, (possible during) and after the separation phase. In the present work we have adopted this strategy, we have considered three phases, using two aircraft models.

1) before the separation ⇒ The first aircraft model (representing the UAV and the rocket) is in an stable operating condition 2) during the separation ⇒ a second aircraft model representing only the UAV, starting on the previous operating condition is disturbed by impulses on forces and moments. These disturbances are inside a time interval T int and represent a not perfect separation. Furthermore the initial conditions, inherited from the first phase, are not an equilibrium point for the second aircraft model. 3) after the separation ⇒ the disturbances stop (UAV and rocket are not in physical contact anymore). It can be shown that the effect of launching the rocket from the UAV impacts most the lift force, and the roll and pitch moments. We suppose that these perturbing force and moments are constant during interval T int , and we represent then F zp , L p and M p for the perturbations on the lift force, on the roll moment and pitch moment respectively. In the present work we will study a worst case of disturbance. We consider that the separation phase is not simultaneous in all links that attach the rocket and the UAV. For this reason, the rocket remains attached to one end of the UAV during T int . We have then studied how long the disturbance could last and that the control algorithm could still stabilize the aircraft back. We have then assumed that:

• the perturbation on lift force during T int is equal the rocket's mass times gravity, that means F zp = mg. • the perturbation on pitch moment during T int is M p = mgl r /2 where l r is the rocket length.

• the perturbation on roll moment during T int is small, because of the geometry of the rocket (thin and long).

• the model following the launch phase is that of an F-16. Its initial condition is the equilibrium point of the model previous the launch phase. This is taken as the F-16 model with twice the F-16's mass. Following this procedure, the F-16 aircraft in the instant following the dynamic airlaunch is described in the aerodynamic axes (the aerodynamic axes OX W Y W Z W in the Fig. 1) i.e. the reference frame attached to the airspeed vector (V ). The system dynamics can be expressed as (see [START_REF] Sonneveldt | Nonlinear F-16 Model Description[END_REF] and [START_REF] Peter | Modeling and Simulation of Aerospace Vehicle Dynamics[END_REF]):

In (1), I xx , I yy , I zz , I xz are the moments of inertia, m is the mass of the system (kg) and g the gravity constant.

α, β, V, p, q, r, φ, θ, ψ are the state variables of the airlaunch aircraft model, they are the angle of attack, sideslip, airspeed, roll rate, pitch rate, yaw rate, roll angle, pitch angle and yaw angle respectively. α, β, φ, θ, ψ are expressed in rad, p, q, r in rad/s and V in m/s. T is the thrust force, F x , F y , F z and L, M, N are aerodynamic forces and moments respectively. All forces and moments are expressed in N and Nm.

                                                             α = -cos α tan βp + q -sin α tan βr -sin α mV cos β (T + Fx) + cos α mV cos β Fz + g V cos β [sin α cos θ + cos α cos φ cos θ] β = sin αp -cos αr -cos α sin β mV [T + Fx] + cos β mV Fy -sin α sin β mV Fz + g V [cos α sin β sin θ + cos β cos θ sin φ -sin α sin β cos φ cos θ] V = cos α cos β m [T + Fx] + sin β m Fy + sin α cos β m Fz + g[cos α cos β sin θ + sin β sin φ cos θ + sin α cos β cos φ cos θ] ṗ = 1 IxxIzz -I 2 xz [(IyyIzz -I 2 zz -I 2 xz )rq -Ixz(Ixx +Izz -Iyy)pq + IzzL -IxzN ] q = 1 Iyy [(Izz -Ixx)pr + Ixz(p 2 -r 2 ) + M ] ṙ = 1 IxxIzz -I 2 xz [(-IxxIyy + I 2 zz + I 2 xz )pq +Ixz(Ixx + Izz -Iyy)rq + IxxN -IxzL] φ = p + tan θ(q sin φ + r cos φ) θ = q cos φ -r sin φ ψ = q sin φ+r cos φ cos θ (1)
These aerodynamic forces and moments are function of all the considered states. In this model, these aerodynamic forces and moments are under look-up table from wind tunnel data measurements as may be found in [START_REF] Nguyen | Simulator study of fighter airplane with relaxed longitudinal static stability[END_REF]. Finally, the control inputs are respectively the aileron (δ a ), rudder (δ r ) and elevator (δ e ) angles.

This model is based on wind tunnel data from NASA, considering the following conditions:

• angle of attack is in the range of [-10 • all actuators are modeled as a first order model (τ = 1/0.0495s) with limit rates 60 • /s for aileron and elevator, and 120 • /s for rudder. In particular, we use the low quality mode of the F-16 model, and the aerodynamic data is interpolated and extrapolated linearly in simulation from tables found in [START_REF] Nguyen | Simulator study of fighter airplane with relaxed longitudinal static stability[END_REF].

III. CONTROL DESIGN

A. Conditional servo-compensator control design

The MIMO conditional servo-compensator controller design for the output regulation of a class of minimumphase nonlinear systems in case of asymptotically constant references is studied in [START_REF] Mahmoud | Robust control for a nonlinear servomechanism problem[END_REF], [START_REF] Hassan | On the design of robust servomechanisms for minimum phase nonlinear systems[END_REF] and [START_REF] Damm | Mimo conditional integrator control for a class of nonlinear systems[END_REF]. These papers concern a servo-compensator performing as a sliding mode controller outside the boundary layer, and performing as a conditional one that provides servo-compensation only inside the boundary layer. First results have studied the SISO case with a scalar sliding surface and the asymptotic stability of the system inside a boundary layer. results were extended in [START_REF] Seshagiri | Robust output feedback regulation of minimum-phase nonlinear systems using conditional integrators[END_REF] and [START_REF] Attaullah | Output regulation of nonlinear systems using conditional servocompensators[END_REF] for linearized MIMO systems under some additional assumptions. Our present work is dedicated to use a nonlinear extension of these results developed in [START_REF] Damm | Mimo conditional servocompensator control for a class of nonlinear systems (accepted)[END_REF], for stabilizing an unmanned aircraft after the airlaunch phase.

Consider the system:

ė1 = e 2 ė2 = f (e 1 , e 2 ) + g(e 1 , e 2 )u (2)
where e 1 (t) ∈ R n is an output error vector, e 2 = ė1 , u ∈ R n control input and f (e 1 , e 2 ) ∈ R n , g(e 1 , e 2 ) ∈ R n×n are continuous functions.

Let us define the sliding surface as:

s = K 0 σ + K 1 e 1 + e 2 (3)
where σ ∈ R n is the output of the conditional servo-

compensator σ = -K 0 σ + µsat(s/µ) (4)
in which µ is the boundary layer, K 0 is a positive definite matrix, K 1 ∈ R n×n is chosen such a way that K 1 + sI n is Hurwitz.

The saturation function is determined as:

sat(s/µ) = s/ s if s ≥ µ s/µ if s < µ (5)
The previous work [START_REF] Damm | Mimo conditional integrator control for a class of nonlinear systems[END_REF] has shown that system (2) is exponentially stabilized by the controller called Conditional Integrator in the case where K 0 is a scalar. The paper [START_REF] Damm | Mimo conditional servocompensator control for a class of nonlinear systems (accepted)[END_REF] extends the result for the case of K 0 being a matrix. The controller is called Conditional Servo-compensator controller that we remind as below.

We denote O µ as the region neighborhood of (e 1 , e 2 ) = (0, 0) with a radius R µ for s < µ (6) We state the following assumptions on the forcing terms f (e 1 , e 2 ) and g(e 1 , e 2 ) to design the control algorithm. Following these conditions, the controller u defined below can be applied to (2) to stabilize the system:

O µ = {e = (e 1 , e 2 ) ∈ R n × R n | e ≤ R µ }
u = -Π(e 1 , e 2 )sat(s/µ) Π(•) = g -1 (•)(Π 0 + µK 0 + (γ(•) + ∆ 0 )I n ) (7)
Π 0 is a positive definite matrix, µ is the boundary layer and K 0 is a positive definite matrix as defined above.

The stability of the control law [START_REF] Mahmoud | Robust control for a nonlinear servomechanism problem[END_REF] for system (2) can be demonstrated following the results of [START_REF] Damm | Mimo conditional servocompensator control for a class of nonlinear systems (accepted)[END_REF].

B. Longitudinal control design

In the longitudinal control design, we assume that all lateral state variables are null or constant, only longitudinal states are time varying. Moreover it is assumed that the airspeed's response is much slower than other states, and that the control surface deflection has no effects on the aerodynamic force components (lift and drag) but only on moments.

Aerodynamic forces F x , F z and moment M can be calculated by its aerodynamic coefficients (see more in [START_REF] Lee | Nonlinear adaptive flight control using backstepping and neural networks controller[END_REF]). [START_REF] Damm | Mimo conditional servocompensator control for a class of nonlinear systems (accepted)[END_REF]. The model for longitudinal dynamic can be written as: [START_REF] Attaullah | Output regulation of nonlinear systems using conditional servocompensators[END_REF] in which S is wing area, q air pressure, c equivalent width,

F x = (C x (α) + cC xq (α)q/(2V ))qS, F z = (C z (α, β) + cC zq (α)q/(2V ))qS, M = (C m (α) + C mq (α)qc/(2V ) + C m δe (α)δ e )qSc . By replacing F x , F z , moment M and β = 0, φ = 0, p = 0, r = 0 in
           α= 1 mV [-sin α(T + C x (α)qS) + cos αC z (α)qS] +q + ρS 4m (-sin αC xq (α)c + cos αC zq (α)c)q + g V cos (θ 0 -α) q =I 7 qS(C m (α)c + C mq (α)cq + C m δe (α)cδ e ) θ =q
I 7 = 1/I yy , C x (α), C xq (α), C z (α), C zq (α), C m (α), C mq (α) C m δe (α)
are aerodynamic coefficients taken from [START_REF] Morelli | Global nonlinear parametric modeling with application to f-16 aerodynamics[END_REF]. Equation ( 8) can be rearranged as:

In section III, the design methodology of the conditional servo-compensator controller to stabilize the angle of attack, sideslip and roll angle is proposed when full knowledge of the aerodynamic characteristics is available. This section presents numerical simulation results for the controller to demonstrate the performance of the proposed conditional servo-compensator control laws in the drop phase.

As mentioned in section II, we have considered the launch phase as disturbances on aerodynamic force and moments during a time interval T int , and that the model following the launch phase is that of an F-16. This model is used since it has already been applied for (manned) airlaunch, and because its nonlinear model, wind tunnel informations and data are widely known and used for control design. It is important to remark that the model used in the following simulations is even more complete than that used in the control design, for example it includes actuator dynamics and their limitations. As a consequence, simulations also illustrate some properties of robustness to unmodeled dynamics.

In the following simulations, we have simultaneously applied the SISO longitudinal controller for angle of attack, and the MIMO lateral one for the sideslip and roll angles in the full nonlinear F-16 aircraft model. We may note that the control inputs are limited by their physical characteristics introduced in section II.

The control law in [START_REF] Mahmoud | Robust control for a nonlinear servomechanism problem[END_REF] whose Π(•) can be written a simplier way as:

Π(•) = (G i ) -1 (Π 0 + γ(•)I n ) (22)
in which, γ(•) = γ 1 e 1 2 + γ 2 e 2 2 , γ 1 and γ 2 are positive constant. n = 1 for the longitudinal case and n = 2 for the lateral case.

Application of this control law to two motion modes presented in subsections (III-B) and (III-C) is done by determining the set of parameters Π i 0 , γ i 1 , γ i 2 , µ i , K i 1 and K i 0 with i = α, β corresponding to longitudinal mode and lateral mode respectively. We will stabilize the second model following the launch phase to its equilibrium point (V, h) = (154m/s, 5000m) corresponding to (angle of attack α r to 4.6 • , sideslip β r to 0 • , and roll angle φ r to 0 • ).

Its initial condition is the final state of the first model (α = 10.6 • , β = 0 • and φ = 0 • ) as in Section II. Moreover, we add on its initial condition a small disturbance on system output. That means the initial condition of second model is (α 0 = 17.5 • , β 0 = 4 • and φ 0 = 15 • ) for all numerical simulations. The second model is disturbed on its aerodynamic force and moments during an interval T int as in Section II. We simulate three sets of time lengths:

1) T int = 0.2s (corresponding to solid lines in Fig. 2 to Fig. 5) produces damped oscillations (see in [START_REF] Van | Modeling and conditional integrator control of an unmanned aerial vehicle for airlaunch[END_REF]). 2) T int = 0.3s, (the dashed lines in Fig. 2 to Fig. 4), it was shown in [START_REF] Van | Modeling and conditional integrator control of an unmanned aerial vehicle for airlaunch[END_REF] that the system controlled by a LQR controller becomes completely unstable for this length. 3) T int = 0.4s (corresponding to dash dotted lines in Fig. 2 to Fig. 5) one can see that the controller is still able to stabilize the system.

Fig. 2 illustrates the convergence of the system output to the operating point of the aircraft at the end of 5s without static steady error for the three cases of T int = (0.2s, 0.3s, 0.4s). All system outputs are still under their physical limitation. In Fig. 3, it can be seen that the control variables are saturated by their physical limitations due to a high perturbation on aerodynamic forces and moments.

For T int = (0.2s, 0.3s) the system is well stabilized, while for T int assuming larger values the system becomes more oscillatory and attains its limits of stability, the case of T int = 0.4s is an example. We show in Fig. 6 that the system will be unstable even with the robust conditional servo-compensator controller for an interval T int = 0.49s.

V. CONCLUSION

This work has studied the airlaunch system at the stage separation from the reusable airlaunch vehicle from the down stage and the effects of the stage separation phase in the stability of the airlaunch system. The separation phase may produces large impulses on forces and moments, making the system unstable. These impulses are considered to last a time interval, that is then evaluated in simulations.

To stabilize the airlaunch system after this stage separation phase, a conditional servo-compensator control is considered. This controller is designed using an F-16 model representing the aircraft just after dropping the second stage, but disturbed by large impulses. For a perturbation on aerodynamic forces and moments during an interval T int , the stability of the system after the drop stage may be assured for small time intervals. When T int becomes large, the system becomes unstable even with the proposed controller.

In future works other disturbances can be considered, as well as other control strategies for this particularly interesting and difficult problem.

Fig. 1 .

 1 Fig. 1. Frames: Body fixed axes OX B Y B Z B , Stability axes OX S Y S Z S , Aerodynamic axes OX W Y W Z W

2 Assumption 3 . 2 :

 232 Assumption 3.1: f (e 1 , e 2 ) is bounded by a function of γ( e 1 + e 2 ) (where γ(•) is a class K function) and a positive constant ∆ 0 : f (e 1 , e 2 ) ≤ γ( e 1 + e 2 ) + ∆ 0 and as a consequence, f (e 1 = 0, e 2 = 0) = f (0, 0) ≤ ∆ 0 for (e 1 , e 2 ) ∈ R n × R n . Inside the boundary layer, the function f (e 1 , e 2 ) is required to be Lipschitz for (e 1 , e 2 ) ∈ O µ , as a consequence f (e 1 , e 2 ) -f (0, 0) ≤ L 1 e 1 + L 2 e 2 γ( e 1 + e 2 ) is also required to be Lipschitz for (e 1 , e 2 ) ∈ O µ : γ( e 1 + e 2 ) ≤ γ 1 e 1 + γ 2 e Function g(e 1 , e 2 ) is continuous and invertible for all (e 1 , e 2 ) ∈ R n × R n .

Fig. 2 .Fig. 4 .

 24 Fig. 2. Angle of attack, Sideslip Angle and Roll angle stabilized

Fig. 5 .

 5 Fig. 5. State variables: Euler's angles of system

  • , 45 • ] and sideslip of [-30 • , 30 • ] • flag deflection is ignored • physical constraints for aileron (|δ a | ≤ 21.5 • ), rudders (|δ e | ≤ 25 • ) and elevator (|δ r | ≤ 30 • )

where f α 11 (α), f α 12 (α), f α 13 (α, θ), f α 21 (α), f α 22 (α) and g α 2 (α) represent the terms of (8) respectively.

Let us define x α 1 = α, x α 2 = ẋα 1 = α and u α = δ e , which allow us to rewrite [START_REF] Morelli | Global nonlinear parametric modeling with application to f-16 aerodynamics[END_REF] into:

where

We define now the reference for the angle of attack α ref considered as constant in this study, and the error vector of angle of attack e α 1 = x α 1 -x α 1ref = α -α ref and the variable e α 2 = ėα 1 . Equation (10b) can be transformed into [START_REF] Van | Mimo conditional integrator control for unmanned airlaunch[END_REF]:

Here we remark that G α (x α 1 , x α 2 ) is invertible, and that F α (x α 1 , x α 2 , θ) and G α (x α 1 , x α 2 ) are Lipschitz for the entire domain of actuation of the system.

Applying the control algorithm presented in [START_REF] Mahmoud | Robust control for a nonlinear servomechanism problem[END_REF] for system [START_REF] Van | Mimo conditional integrator control for unmanned airlaunch[END_REF] (in this case a nonlinear single input single output system) gives the controller:

with

where µ α , K α 1 are positive constants. Π α 0 and K α 0 are positive constant.

The controller can be shown to assure the stability of angle of attack and its derivative. For the sake of brevity we skip the proof, that is straightforward and based on a Lyapunov function. It can be shown to go to a residual set that can be attenuated by higher gain. The conclusions we can have are that all errors will be ultimately bounded, where the remaining signals stands for the disturbance on the aircraft speed. It is interesting to remark that variable θ was left free, in order to allow situations as a looping, where θ is continuously varying. Its derivative on the other hand is bounded, and also goes to a residual set (by equation 12).

Since the airspeed control is only a secondary objective, we design a simple PI controller for the thrust to regulate airspeed. Its form is:

where V r is the airspeed reference, k P = 711 and k I = 6.3.

C. Lateral control design

As in the case of longitudinal control design, in the lateral case it is considered that only lateral state variables are time varying.

in which b is equivalent length,

are lateral aerodynamic coefficients taken from [START_REF] Morelli | Global nonlinear parametric modeling with application to f-16 aerodynamics[END_REF].

Aerodynamic force F y and moments L, N are calculated by their aerodynamic coefficients (see more in [START_REF] Lee | Nonlinear adaptive flight control using backstepping and neural networks controller[END_REF]).

By replacing F y , moments L, N and α = α 0 , θ = θ 0 in [START_REF] Damm | Mimo conditional servocompensator control for a class of nonlinear systems (accepted)[END_REF]. The lateral nonlinear dynamic model used for the control design procedure is consequently reduced as [START_REF] Seshagiri | Robust output feedback regulation of minimum-phase nonlinear systems using conditional integrators[END_REF]. Equation ( 15) can be rearranged as:

where

, and g β 2 (•) represent the terms of ( 15) respectively. Equation ( 16) is mainly used for controller design and stability analysis.

Let us define

T and u β = (δ a , δ r ) T , that allow us to rewrite equation ( 16) to:

where

We define an output error vector e β 1 = x β 1 -x β 1ref and e β 2 = ėβ 1 where x β 1ref = (β ref , φ ref ) T is the output reference considered as constant. Equation (18) can be transformed into [START_REF] Van | Mimo conditional integrator control for unmanned airlaunch[END_REF] with two new state variables e β 1 and e β 2 .

2 with β ∈ (-30 • , 30 • ) and φ ∈ (-90 • , 90 • ). Application of control law [START_REF] Mahmoud | Robust control for a nonlinear servomechanism problem[END_REF] for system (19) leads to the controller:

where λ β = min( G β (•) ) for β ∈ (-30 • , 30 • ), φ ∈ (-180 • , 180 • ). Π β 0 is a constant large enough, µ β is the boundary layer, K β 0 and K β 1 are positive definite matrices chosen such a way that K + sI 2 is Hurwitz.