On local-global hysteresis-based hovering stabilization of the DarkO convertible UAV

Florian Sansou 1 and Luca Zaccarian 2

Abstract-We characterize an input-affine model of the UAV DarkO: a convertible drone designed and developed at the Ecole Nationale de L'Aviation Civile (ENAC) in Toulouse (France). Starting from a nonlinear model available in the literature, we present an approximate input-affine nonlinear model, whose dynamics simplifies the control design task. For this simplified model, we characterize the hovering equilibria in the absence of wind, and we derive the corresponding linearized dynamics. Then present a hysteresis-based switching mechanism combining a nonlinear feedback (providing a large basin of attraction) with a linearized feedback (providing improved performance but a smaller basin of attraction). Simulation results, using the original nonlinear model, confirm the effectiveness of the proposed feedback design.

I. INTRODUCTION

Convertible drones, within the class of hybrid Unmanned Air Vehicles (UAVs) are a compromise between two architectures : fixed wing and rotorcraft. Being equipped with a fixed wing, they have the ability to fly like an aircraft, with energy efficiency, but also to hover to maintain a position and perform vertical take-off and landing (see the comprehensive survey [START_REF] Saeed | A survey of hybrid unmanned aerial vehicles[END_REF]). In recent years, due to their relevance to applications, different convertible UAV architectures have been proposed, which can arguably be classified in tiltrotors, tilt-wings, quadruplanes and tilt-body (or tailsitter) configurations [START_REF] Olszanecki Barth | Model-free control algorithms for micro air vehicles with transitioning flight capabilities[END_REF], [START_REF] Hegde | Application of robust hinfinity controller in transition flight modeling of autonomous VTOL convertible quad tiltrotor UAV[END_REF]. These drones, like any other, share the important challenges of a cascaded attitude/displacement architecture, as well discussed, e.g., in [START_REF] Hua | Introduction to feedback control of underactuated vtol vehicles: A review of basic control design ideas and principles[END_REF]. In particular, the DarkO UAV recently developed at the Ecole Nationale de L'Aviation Civile (ENAC) in Toulouse (France) is an example of tailsitter comprising two propellers and two elevons with a bat-like architecture [START_REF] Lustosa | Global Singularity-Free Aerodynamic Model for Algorithmic Flight Control of Tail Sitters[END_REF], [START_REF] Olszanecki Barth | Model-free control algorithms for micro air vehicles with transitioning flight capabilities[END_REF], [START_REF] Lustosa | Longitudinal study of a tilt-body vehicle: modeling, control and stability analysis[END_REF].

While convertible drones present several important advantages in terms of energy and maneuvrability, their control remains an active research field, due to the different flight conditions that should be considered in the control design. In particular, a large amount of works (see, e.g., [START_REF] Escareno | Modeling and control strategy for the transition of a convertible tail-sitter UAV[END_REF], [START_REF] Hegde | Application of robust hinfinity controller in transition flight modeling of autonomous VTOL convertible quad tiltrotor UAV[END_REF], [START_REF] Forshaw | Transitional control architecture and methodology for a twin rotor tailsitter[END_REF], [START_REF] Liu | Robust attitude control for tail-sitter unmanned aerial vehicles in flight mode transitions[END_REF] and references therein) address the so-called transition control, wherein different flight modes are stabilized by known approaches (typically linearization-based solutions) and the transition among the different flight modes is the design goal. This control strategy is especially suited for tiltwing configurations.

CNRS, Toulouse, France. zaccarian@laas.fr With tailsitter UAVs, there is the possibility of developing a single feedback controller covering the whole operating range of the drone. This could be done using linear PIDbased strategies [START_REF] Chu | Simulator development for transition flight dynamics of a vtol mav[END_REF], [START_REF] Chen | Control and flight test of a tilt-rotor unmanned aerial vehicle[END_REF], even though limited effectiveness in terms of accuracy and disturbance rejection stems for the limited range of validity of the linearized models. For the DarkO drone, a model-free control technique has been used in [START_REF] Olszanecki Barth | Model-free control algorithms for micro air vehicles with transitioning flight capabilities[END_REF], which allows avoiding the cumbersome modeling effort, but is clearly prone to the well-known improvements stemming from a careful model-aware solution. When the model is taken into account, one may rely on nonlinear dynamic inversion and incremental nonlinear dynamic inversion techniques (see, e.g., [START_REF] Smeur | Incremental control and guidance of hybrid aircraft applied to the Cyclone tailsitter UAV[END_REF], [START_REF] Francesco | Modeling and incremental nonlinear dynamic inversion control of a novel unmanned tiltrotor[END_REF], [START_REF] Silva | Dynamic inversion and gain-scheduling control for an autonomous aerial vehicle with multiple flight stages[END_REF]). A further computationally expensive solution is that of nonlinear model predictive control (see, e.g., [START_REF] Slegers | Nonlinear model predictive control technique for unmanned air vehicles[END_REF]) with its well-known computational related issues. Additional approaches include different linearization-based techniques, including LPV and gain scheduling control, as well as adaptive control strategies, to be designed by addressing the dynamics of the specific UAV model under consideration [START_REF] Silva | Dynamic inversion and gain-scheduling control for an autonomous aerial vehicle with multiple flight stages[END_REF], [START_REF] Swarnkar | Biplanequadrotor tail-sitter UAV: Flight dynamics and control[END_REF], [START_REF] Hegde | Application of robust hinfinity controller in transition flight modeling of autonomous VTOL convertible quad tiltrotor UAV[END_REF], [START_REF] Marcos | Development of linear-parameter-varying models for aircraft[END_REF], [START_REF] Swarnkar | Biplanequadrotor tail-sitter UAV: Flight dynamics and control[END_REF].

The work presented here is part of the approach initiated in [START_REF] Lustosa | Global Singularity-Free Aerodynamic Model for Algorithmic Flight Control of Tail Sitters[END_REF], [START_REF] Olszanecki Barth | Model-free control algorithms for micro air vehicles with transitioning flight capabilities[END_REF], [START_REF] Lustosa | Longitudinal study of a tilt-body vehicle: modeling, control and stability analysis[END_REF] and related to the modeling of the DarkO drone and proposing a unified nonlinear, possibly hybrid, control scheme. The contribution of this paper is multifold. First, in Section II, we present an input-affine DarkO model, stemming from the results in the references above. Our model is suitable for control design. To this end, we characterize the equilibria and the linearized dynamics for hovering flight. Then in Section III, we present two feedback control laws, each one with advantages and drawbacks, and illustrate their blending through a hybrid hysteresis switching scheme. Our results are validated by high fidelity simulations on the identified model given in [START_REF] Lustosa | Global Singularity-Free Aerodynamic Model for Algorithmic Flight Control of Tail Sitters[END_REF], [START_REF] Olszanecki Barth | Model-free control algorithms for micro air vehicles with transitioning flight capabilities[END_REF]. Notation. Given two vectors x 1 and x 2 , we often denote their juxtaposition as

(x 1 , x 2 ) := [x 1 ⊤ x 2 ⊤ ] ⊤ . Given any vectors u, v ∈ R 3 , the skew-symmetric matrix [u] × satisfies [u] × v = u × v.
Symbol I denotes the identity matrix of appropriate dimensions.

II. MODEL OF THE DARKO UAV

The DarkO UAV, designed and developed at the Ecole Nationale de l'Aviation Civile (ENAC) in Toulouse (France), is a clear example of convertible UAV with a so-called tail-sitter architecture. DarkO is assembled from multiple 3D printed Onyx parts (a highly robust material comprising omnidirectional carbon fibers). Its actuators consist in two propellers symmetrically placed at the front of the wing and two elevons, placed at the back of the wing and acting as redundant control surfaces.

We may model the position and attitude of the DarkO by using an inertial reference frame "[I]" linked to the earth's surface, and a body reference frame "[B]" attached to the drone, with x [B] corresponding to the roll axis (the propellers axes), y [B] being the pitch axis (the direction of the wings) and z [B] being the yaw axis. According to the notation in [START_REF] Lustosa | Global Singularity-Free Aerodynamic Model for Algorithmic Flight Control of Tail Sitters[END_REF] the left and right propeller/elevon are denoted by using subscripts i = 1 (left) and i = 2 (right), respectively.

Remark 1: (Plant input saturation) The DarkO actuators have dynamics that limit their actions both in terms of amplitude and rate. For the electric motors generating traction through the propellers, the maximum voltage that can be delivered to the motor and a reasonable lowspeed saturation, which is necessary for the plant input transformation introduced in the following (in addition to ensuring a realistic finite energy model), corresponds to constraining ω i ∈ [200 ; 2000] rad s -1 , i = 1, 2. In terms of maximal rate, a reasonable rate saturation level representing the motor actuation chain (consisting of the ESC, the motor and the propeller) corresponds to imposing ωi ∈ [-3000 ; 3000] rad s -2 , i = 1, 2, which provides a fairly aggressive actuation system. The saturation levels characterizing the elevons stem from the servo motors type of actuation. The elevons input is limited in displacement by the shape of the UAV and the physical limits of the servo motors, which translates into a saturation of the control surface deflection δ i ∈ [-30 ; 30]°, i = 1, 2. Perhaps the most relevant saturation here is the rate phenomenon (due to the servo-motor actuation), which corresponds to δ ∈ {-5.24, 5.24} rad s -2 , i = 1, 2 (stemming from a movement of 60°in 0.2 seconds). These saturation levels are only indirectly taken into account in the control design of this paper, but are explicitly considered in our simulation results.

•

A. Input-affine nonlinear model of DarkO

Since we focus in this paper on the hovering phase, where the speed is small, we may simplify the full nonlinear model (available and described in [18, eqn (2.10), p. 25]) by neglecting the aerodynamic effects, namely all the terms that are quadratic in the speed v b and ω b and the wind effects w. Accounting for the wind effects would allow for useful derivations of the linearized dynamics under the wind action, which we regard as future work. Neglecting these terms and performing the following nonlinear input transformation:

u := τ 1 τ 2 δ 1 τ 1 δ 2 τ 2 ⊤ = k f ω 2 1 ω 2 2 δ 1 ω 2 1 δ 2 ω 2 2 ⊤ , (1)
allows us to derive the following input-affine simplified approximate low-speed model

ṗ = v, m v = -mg + R(q)F u, (2a)
q = 1 2 q ⊗ ω b , J ωb = -[ω b ] × Jω b + M u, (2b)
where p ∈ R 3 is the position of the center of mass (CoM) expressed in the inertial frame and v ∈ R 3 denotes its velocity. g = [0 0 g] ⊤ is the (constant) gravity vector, q is a unit quaternion charactering the attitude and ω b is the rotational speed expressed in the body frame. Finally m is the mass and J is the inertia. The constant matrices F and M can be computed by performing some manipulations of the dynamics, details are available in [18, eqn (2.21), p. 28].

F :=   a f a f 0 0 0 0 0 0 0 0 b f b f   , M :=   a m -a m b m -b m 0 0 c m c m d m -d m 0 0   , (3)
with the following coefficients

  a f b f a m b m c m d m   =    1 -S 4Sp C d0 -S 4Sp (2π + C d0 )ξ f km kf S 4Sp a y (2π + C d0 )ξ f S 4Sp ∆ r (2π + C d0 )ξ m p y + S 4Sp a y C d0    ,
where the numerical values can be found in [18, 2), all the possible input-state equilibrium pairs (u, x) = (u eq , x eq ) = (u eq , p eq , v eq , q eq , ω b,eq ) are parametrized by an arbitrary position p e ∈ R 3 , and an arbitrary roll orientation β ∈ - 1 2 , 1 2 as follows

u eq = mg (1 -S 4Sp C d0 ) [1 1 0 0] ⊤ (4a)
p eq = p e , v eq = 0, ω b,eq = 0 (4b)

q eq = [η eq ϵ ⊤ eq ] ⊤ =   1 2 -β β 2β 2 -1 2 1 2 -β β   ⊤ (4c)
Proof: To define the equilibrium point, we first set all derivatives to zero. v = ṗ = 0 and q = 1 2 q ⊗ ω b = 0. Since the norm of the quaternion is unitary, it never vanishes, therefore ω b = 0, which gives

J ωb = M u -[ω b ] × Jω b = 0 =⇒ M u = 0. Thus, we know that equilibrium is reached when u ∈ ker M . The basis generating the kernel of M is of the form [1 1 0 0] ⊤ , therefore u = λ[1 1 0 0] ⊤ , with λ ∈ R. m v = R(q)F u -mg = 0 =⇒ R(q)F u = mg.
Thus the rotation matrix R(q), must orient the longitudinal axis of the drone towards the vertical axis to align the forces. The set of quaternions inducing this orientation

is q = 1 2 -β β 2β 2 -1 2 √ 1 2 -β β ⊤ , ∀β ∈ -1 2 , 1 2 .
Finally, the value of λ that compensates for the gravity is λ = mg (1-S 4Sp Cd0) , as to be proven. We observe that all of the equilibrium pairs characterized in Proposition 1 are associated with a constant control action ū producing a positive thrust compensating for the gravity effects. In particular, according to the expression of u eq in (4a), the necessary thrust corresponds to mg divided by the correction term (1 -S 4Sp C d0 ) -1 , compensating for the aerodynamic losses.

It is interesting to note that, in addition to an arbitrary position p e , we may define an equilibrium exploiting the additional degree of freedom β, characterizing a rotation about the vertical axis of the inertial frame, which is collinear with the longitudinal axis x [B] of the UAV when it is hovering. This degree of freedom is useful, e.g., when wanting to prepare the UAV for a straight flight in a specific direction. Nevertheless, a specific heading must be imposed by the feedback law whenever wanting to reject a constant and nonzero wind disturbance w affecting the dynamics. This type of study is regarded as future work.

B. Linearized dynamics

We derive here the linearized dynamics starting from the simplified nonlinear model [START_REF] Olszanecki Barth | Model-free control algorithms for micro air vehicles with transitioning flight capabilities[END_REF]. Inspired by [19, Proof of Lemma 1], to deal with the dynamics of the quaternion q = η ϵ ⊤ ⊤ evolving in S 3 , we replace η by its nonnegative value induced by the unit norm of the quaternion. Thus, η = (1 -ϵ ⊤ ϵ)

1 2 = 1 -ϵ 2 1 -ϵ 2 2 -ϵ 2 3
. The linearized equations of motion are then characterized by the incremental state vector:

x := p ṽ ε ωb ⊤ ũ := uu eq (5)

:= p -p eq v ϵ -ϵ eq ω b ⊤ ,
where the constants p eq ∈ R 3 , u eq ∈ R 4 and ϵ eq ∈ R 3 come from any of the equilibrium pairs characterized in Proposition 1.

We may then characterize the linearized dynamics about any such equilibrium pair as follows

ẋ = Ax + Gũ (6)
where the expression of A, can be determined, after some manipulations, as

A =     0 3 I 3 0 3 0 3 0 3 0 3 A v,ϵ 0 3 0 3 0 3 0 3 A q,ωb 0 3 0 3 0 3 0 3     A q,ωb = √ 2 4   1 0 -1 0 1 0 1 0 1   A v,ϵ = √ 2   0 2g 0 -g 0 g 0 -g 0   (7)
In matrix A, we thus obtain a pure integrator on the first three rows of the matrix of the equation [START_REF] Escareno | Modeling and control strategy for the transition of a convertible tail-sitter UAV[END_REF], which represents the link between the velocity and the linear acceleration. We observe that the linear acceleration depends only on the orientation of the UAV. This orientation is only dependent on the angular velocity, while the angular velocities are not affected by the state vector, as one may expect.

The expression of G can be determined following similar derivations, and corresponds to

G =             0 3×1 0 3×1 0 3×1 0 3×1 0 0 a g a g 0 0 0 0 b g b g 0 0 0 3×1 0 3×1 0 3×1 0 3×1 c g -c g d g -d g 0 0 e g e g f g -f g 0 0             , with   a g b g c g d g e g f g   =    S 4mSp (2π + C d0 )ξ f 1 m (1 -S 4Sp C d0 ) J1km kf ay J1 (2π + C d0 )ξ m S∆r 4J2Sp (2π + C d0 )ξ m 1 J3 (p y + S 4Sp a y C d0 )    .
From an actuation point of view, we observe in the linearized model that the input ũ only acts on the acceleration of the UAV. Moreover, the input matrix G shows that it is not possible to generate any acceleration along the y b axis, which is reasonable, given the architecture of the DarkO UAV.

III. FEEDBACK CONTROL DESIGN

We derived in Section II the simplified nonlinear model [START_REF] Olszanecki Barth | Model-free control algorithms for micro air vehicles with transitioning flight capabilities[END_REF], which presents a convenient input-affine relation with respect to u, and the linearized dynamics about the equilibria, represented by [START_REF] Lustosa | Longitudinal study of a tilt-body vehicle: modeling, control and stability analysis[END_REF].

We propose in this section two control design strategies for stabilizing a hovering position. The first one is inspired by the nonlinear stabilizer presented in [START_REF] Michieletto | Hierarchical nonlinear control for multi-rotor asymptotic stabilization based on zero-moment direction[END_REF] and provides a large region of attraction, and the second one is based on the linearized dynamics and allow for more effective gain tuning in the final approaching phase. The two controllers are united via a hybrid mechanism that allows retaining the steady-state performance of the linearized design with the large region of attraction guaranteed by the nonlinear design. Our solution is tested by simulating the full nonlinear model.

Remark 2: We emphasize that vector u in (1) corresponds to a non-invertible transformation of the actual DarkO actuators corresponding to u act := [ω 1 , ω 2 , δ 1 , δ 2 ] ⊤ . Nevertheless, when imposing the saturation constraints discussed in Remark 1, it is possible to uniquely determine u act from a desired value of u in (1), because nonzero positive values of ω 1 and ω 2 can be determined from the first two components of u, and then δ 1 and δ 2 are easily constructed from the last two components of u.

•

A. Nonlinear dynamic feedback controller

We illustrate in this section a nonlinear dynamic control law inspired by the result of [START_REF] Michieletto | Hierarchical nonlinear control for multi-rotor asymptotic stabilization based on zero-moment direction[END_REF]. For the nonlinear control law of [START_REF] Michieletto | Hierarchical nonlinear control for multi-rotor asymptotic stabilization based on zero-moment direction[END_REF] to be applicable, matrices F and M reported in (3) must allow defining a so-called zero moment direction ū ∈ R 4 ensuring |F ū| = 1 and M ū = 0, and a right inverse M r of M satisfying M M r = I and F M r = 0. In our case, it is immediate to see that the zero-moment direction ū = √ 2 2af [ 1 1 0 0 ] ⊤ satisfies the required conditions, whereas the fact that rank(F ) = 2 (so that ker F has dimension 2) makes it impossible to find a right inverse M r of M completely contained in ker F . Due to this fact, we determine M r by (conservatively) parametrizing the right pseudoinverses of M as M r := KM ⊤ (M KM ⊤ ) -1 where parameter K ∈ R 4×4 is symmetric and satisfies M KM ⊤ ≥ I (to ensure invertibility). Under this parametrization, the goal is to minimize the norm of F M r = F KM ⊤ (M KM ⊤ ) -1 , which is well achieved by minimizing the norm of F KM ⊤ , due to the fact that the constraint on M KM ⊤ ≥ I ensures that the factor (M KM ⊤ ) -1 has norm smaller than 1. Performing a Schur complement, this minimization is well obtained by solving the following semi-definite program: leading to κ = 39.7. With this optimality-based selection, the nonlinear dynamic design of [START_REF] Michieletto | Hierarchical nonlinear control for multi-rotor asymptotic stabilization based on zero-moment direction[END_REF] can be effectively applied by obtaining responses that are almost indistinguishable from the fully decoupled case F M r = 0. Note that a similar approach, essentially neglecting the extra terms acting on the translational dynamics is also suggested in the survey paper [START_REF] Hua | Introduction to feedback control of underactuated vtol vehicles: A review of basic control design ideas and principles[END_REF].

min K,κ κ, subject to: M KM ⊤ ≥ I, κI F KM ⊤ M K ⊤ F ⊤ κI ≥ 0, which minimizes κ while ensuring F KM ⊤ M K ⊤ F ⊤ ≤ κ 2 I.
Based on the above-described choice of M r and ū, applying the feedback law in [20, eqn (19)], the input u becomes :

u = u nl := M r τ r + ūf , (8)
where τ r and f are provided by the dynamic feedback controller proposed in [START_REF] Michieletto | Hierarchical nonlinear control for multi-rotor asymptotic stabilization based on zero-moment direction[END_REF].

The optimality-based selection of M r is prone to a few interesting interpretations when observing the product M r τ r = M r [ τr,x τr,y τr,z ] ⊤ . First, to produce a moment τ r,z about the z-axis we mainly use the thrust differential action; secondly, a moment τ r,y about the y-axis is generated by an equal (additive) use of the two flaps, with great efficiency; finally a moment τ r,x about the x-axis comes from a differential use of the flaps.

As a final remark, as compared to the solution proposed in [START_REF] Michieletto | Hierarchical nonlinear control for multi-rotor asymptotic stabilization based on zero-moment direction[END_REF], to partially take into account the saturation effects highlighted in Remark 1, the error feedback interconnection of the outer loop in [START_REF] Michieletto | Hierarchical nonlinear control for multi-rotor asymptotic stabilization based on zero-moment direction[END_REF] has been augmented with a simple error governor strategy never allowing the translational position error e p entering [20, eqn. (22)] to exceed the maximum value of 3 meters. The remaining tuning gains required in the solution of [START_REF] Michieletto | Hierarchical nonlinear control for multi-rotor asymptotic stabilization based on zero-moment direction[END_REF] have been selected following an intuitive PD tuning procedure as k pp = 0.5, k pd = 1.2, k ap = 0.08, k ad = 0.1 and k ∆ = 1.

Figure 1 shows the response of the system in terms of linear and angular positions (top two rows) and actuators efforts (bottom two rows) when the system starts from the initial condition x(0) = [p(0) v(0) q(0) ω b (0)] ⊤ = [0 0 0 0 0 0 0.9140 0.1134 -0.3728 0.1134 0 0 0] ⊤ with a target equilibrium position of p eq = [4 5 6] ⊤ and q eq = [

√ 2 2 0 - √ 2
2 0] ⊤ . A graceful response can be seen, which remains quite far from the actuator saturations (see Remark 1). Increasing the tuning gains can speed up the response but provides undesired attitude oscillations. Therefore it is interesting to combine this nonlinear controller (providing a large region of attraction) with a more aggressive controller, designed based on the linearized dynamics [START_REF] Lustosa | Longitudinal study of a tilt-body vehicle: modeling, control and stability analysis[END_REF] and to be used to improve the fail of the response.

B. Linear control design

Based on the performance-oriented observations of the previous section, given a target position corresponding to an equilibrium p eq , q eq as characterized in Proposition 1, we design here a local linear feedback controller capable of inducing a more aggressive response. To this end, we focus on the linearized dynamics ( 6) and recognize that we can design a state feedback controller

u lin := u eq -K x, (9)
where x has been introduced in (5) and K ∈ R 4×12 is a state feedback gain that can be designed, based on the matrices A and G appearing in [START_REF] Lustosa | Longitudinal study of a tilt-body vehicle: modeling, control and stability analysis[END_REF], in such a way that the closed-loop linear feedback A cl := A -GK be exponentially stable.

For our design, we have used an LQR selection, associated with the simplest possible weight matrices selection Q = I 12 and R = I 4 , which gives desirable closed-loop responses. The LQR design also provides a positive definite Lyapunov certificate matrix S ∈ R 12×12 (solution of the algebraic Riccati equation) ensuring that A ⊤ cl S + SA cl < 0. In particular, it is well known from the linear approximation theorem that function V (x) = x⊤ S x is also a Lyapunov function certifying local exponential stability of x eq for the nonlinear dynamics. More specifically, there exits a positive scalar v ∈ R such that, along dynamics (2), we have :

V (x) ≤ v ⇒ V (x) := ⟨∇V (x), ẋ⟩ < 0, (10)
for all x ̸ = 0; in other words, the sublevel set V (x) ≤ v is contained in the basin of attraction of the equilibrium x eq . Determining the largest possible scalar v ensuring (10) is a challenging problem and conservative lower bounds of this quantity can be determined by quantifying the effect of the nonlinearities on the dynamics. Since ẋ is a function of x, then it is fairly easy to algebraically evaluate V (x) for a large amount of random extractions of the variable x, so as to get a probabilistic estimate of the largest v. Rigorous guarantees about these selections can be obtained by applying the results in [START_REF] Tempo | Randomized algorithms for analysis and control of uncertain systems: with applications[END_REF], which is out of the scope of this paper, but an evaluation of 10000 samples confirmed that the value v = 400 is a good candidate selection satisfying [START_REF] Chu | Simulator development for transition flight dynamics of a vtol mav[END_REF]. Figure 2 shows a simulation starting at the origin with zero orientation on the three axes (horizontal UAV) and zero initial velocities, with a target position p eq = [4, [START_REF] Lustosa | Global Singularity-Free Aerodynamic Model for Algorithmic Flight Control of Tail Sitters[END_REF][START_REF] Lustosa | Longitudinal study of a tilt-body vehicle: modeling, control and stability analysis[END_REF] with a hovering stabilization (vertical UAV) with β = 0. The dotted line represents the target position on each axis. Note that the initial linear and angular velocities are zero. The last graph shows the desirable exponential decay of V . Figure 2 shows both the simulation of the full model (solid) of [START_REF] Sansou | Commande hybride d'un drone convertible pour des déplacements sous optimaux[END_REF] and of the simplified nonlinear model (2) (dashed) showing some significant differences in the initial response. When providing a larger target position p eq = [8, 9, 10] (with the same orientation), the initial condition is outside the basin of attraction and diverging solutions are experienced as shown in Figure 3.

C. Hysteresis-based local-global control design

Inspired by the local/global strategies presented in [22, Ex. 1.7], similar to the solution presented in [START_REF] Andreetto | Quasi time-optimal hybrid trajectory tracking of an n-dimensional saturated double integrator[END_REF], we use a hybrid mechanism to switch between the high performance local feedback (9) (as long as the state is in the basin of attraction of the equilibrium) and the less aggressive nonlinear controller [START_REF] Forshaw | Transitional control architecture and methodology for a twin rotor tailsitter[END_REF], which provides a larger region of attraction (and can be called with an abuse of notation the "global controller"). To this end, we augment the controller state with a logical state variable ℓ ∈ {0, 1}, governing the choice of the control input between ( 8) and ( 9) as

u = u hyb := ℓu nl + (1 -ℓ)u lin , (11)
We ensure, through the hybrid dynamics, that ℓ can only take values in {0, 1}. Its dynamics is defined by:

l = 0, χ ∈ C ℓ + = 1 -ℓ, χ ∈ D
where χ = [p, v, q, ω, l] is the complete closed loop state and C and D are, respectively, the flow and the jump sets, defined as

C := C 0 ∪ C 1 , D := D 0 ∪ D 1 , C 0 := {χ ∈ R 14 : V (x) ≤ v and ℓ = 0} C 1 := χ ∈ R 14 : V (x) ≥ v and ℓ = 1 D 0 := χ ∈ R 14 : V (x) ≥ v and ℓ = 0 D 1 := χ ∈ R 14 : V (x) ≤ v and ℓ = 1
where V (x) := x⊤ S x has been defined in the previous section, v = 400 has been determined in the previous section to satisfy [START_REF] Chu | Simulator development for transition flight dynamics of a vtol mav[END_REF], and v is any positive constant satisfying v < v (a smaller choice of v increases the hysteresis margin but postpones the desirable high performance tail of the feedback response). In our case we choose v = 350. The following result is an immediate consequence of the results in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]Ex. 1.7] and the properties of our linear and nonlinear designs.

Proposition 2: Under the action of the hybrid feedback [START_REF] Chen | Control and flight test of a tilt-rotor unmanned aerial vehicle[END_REF], the closed loop exhibits the same basin of attraction as the one associated with the nonlinear controller (8), while always using the high-performance linear feedback [START_REF] Liu | Robust attitude control for tail-sitter unmanned aerial vehicles in flight mode transitions[END_REF] in the tail of the response.

We performed several simulations of the closed loop using the Matlab toolbox [START_REF] Sanfelice | A toolbox for simulation of hybrid systems in matlab/simulink: Hybrid equations (HyEQ) toolbox[END_REF]. The simulations are carried out with the complete model of the UAV [START_REF] Sansou | Commande hybride d'un drone convertible pour des déplacements sous optimaux[END_REF], including all the nonlinear aerodynamic effects. A sample simulation is reported in Figure 4, where we initialize the UAV at the origin with zero roll and yaw orientation, and with a pitch angle of 45 degrees. The target orientation is in the vertical hovering configuration and the target position is assigned to p eq = [50, 25, 12.5]. We observe that in the time phase t ∈ [0, 38], the UAV exhibits a graceful but slow convergence to the desired target position using the global controller (ℓ = 1). At that time, the state enters set D 1 and the more aggressive local controller is activated up to the convergence to the desired equilibrium.

To perform realistic simulations, the measurements are affected by sensors noise. The intrinsic robustness of the hybrid feedback, established in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]Chapter 7] is confirmed by the graceful performance degradation as a function of the amplitude of the measurement noise.

IV. CONCLUSIONS AND FUTURE WORK

We presented two simplified models of the DarkO UAV that are useful for hovering control design. First, an inputaffine nonlinear model allows adopting a nonlinear hierarchical stabilizing feedback. Then, a linearized model allows designing a simple state-feedback solution inducing desirable performance but a small basin of attraction. A hybrid control solution based on a hysteresis mechanism combining these two approaches. work will include for the effect of the wind. In particular, extending our simplified input-affine model to consider the wind effect is straightforward, while studying perturbed linearized motion is currently under investigation. Future work also includes experimental validation of our control laws in the ENAC flight arena, in addition to generalizing the hovering stabilization solutions to the more challenging trajectory tracking problem.

Fig. 1 .

 1 Fig. 1. Simulation with the nonlinear dynamic feedback controller.

Fig. 2 .

 2 Fig.2. Simulation of the full model (solid) and (2) (dashed) with u = u lin as in (9) from an initial condition x0 within the basin of attraction.

Fig. 3 .

 3 Fig. 3. Diverging simulation of the full model with u = u lin as in (9) from an initial condition x0 outside the basin of attraction.

ACKNOWLEDGMENTFig. 4 .

 4 Fig.4. Closed-loop simulation with the hybrid controller[START_REF] Chen | Control and flight test of a tilt-rotor unmanned aerial vehicle[END_REF].

table 1

 1

, p. 61]. Model (2) provides a desirable input-affine description of the DarkO dynamics. Among other things, the simplified nonlinear dynamics (2) allow us to characterize all the possible equilibria induced by input u. The corresponding result is formalized next.

Proposition 1: For model (

  Solving this optimization, we obtain, for the specific matrices under consideration,

		0 -737 171	-171		0	0	-3.19
	K =	-737 0 171 -171 1583.5 -43.73 -171 171	, M r =	0 -4.51 -27.75 -1.48 0 3.19
		-171 171 -43.73 1583.5		4.51 -27.75 1.48

Research supported in part by ANR via grant HANDY, number ANR-18-CE40-0010.

1 Florian Sansou is with the Ecole Nationale de L'Aviation Civile (ENAC),