H De Plinval
email: henry.deplinval@onera.fr

P Morin

P Mouyon

Stabilization of a Class of Underactuated Vehicles with Uncertain Position Measurements and Application to Visual Servoing $

Keywords: Underactuated vehicle, aerial vehicle, stability, robustness, measurement uncertainties

Stabilization of a class of underactuated vehicles with uncertain measurements of the position tracking error is addressed. Nonlinear feedback laws ensuring semi-global stability for a large class of uncertainties on these measurements are derived based on properties of saturated controls. Practical relevance of the proposed results is illustrated by two application examples for Vertical Take-Off and Landing aerial vehicles equipped with a mono-camera sensor: point stabilization in front of a planar target and visual way-points navigation based on interpolation of homography measures.

Introduction

Underactuated vehicles have long been a source of inspiration for nonlinear control theory. Recent applications with aerial or underwater vehicles have renewed the interest on this topic. This study is motivated by applications with VTOL UAVs (i.e. Vertical Take-Off and Landing Unmanned Aerial Vehicles) but it is relevant to any underactuated vehicle that can be modeled as a rigid body with a body fixed thrust control force and full torque control (so-called "thrust propelled vehicles" [START_REF] Hua | A control approach for thrustpropelled underactuated vehicles and its application to vtol drones[END_REF]). Stabilization of the vehicle's pose (i.e. position and orientation) is an important issue in this context. Large stability domains are needed for small vehicles due to their sensitivity to perturbations (wind, sea currents, etc), and several nonlinear control designs have been proposed to address this issue (see, e.g., [START_REF] Hauser | Nonlinear control design for slightly non-minimum phase systems: Application to v/stol[END_REF]; [START_REF] Isidori | Robust autonomous guidance : an internal model approach[END_REF]; [START_REF] Pflimlin | Position control of a ducted fan vtol uav in crosswind[END_REF]; [START_REF] Hua | A control approach for thrustpropelled underactuated vehicles and its application to vtol drones[END_REF]). Good robustness properties of the closed-loop system is at least as important in practice. Robustness to external perturbations (e.g. wind effects for aerial vehicles or currents for underwater vehicles) has been addressed in [START_REF] Pflimlin | Position control of a ducted fan vtol uav in crosswind[END_REF]; [START_REF] Marconi | Robust full degree-of-freedom tracking control of a helicopter[END_REF]; Aguiar and Pascoal (2007); [START_REF] Hua | A control approach for thrustpropelled underactuated vehicles and its application to vtol drones[END_REF]. Robustness to parameter uncertainties (mass, inertia, etc) has been considered e.g. in Aguiar and Pascoal (2007). Robustness to input disturbances has been considered in Aguiar et al. (2007). This paper concerns robustness w.r.t. (with respect to) uncertainties on the measurement model.

Motion capture systems provide high-quality pose measurements that can yield impressive performance for small aerial vehicles [START_REF] Lupashin | A simple learning strategy for high-speed quadrocopter multi-flips[END_REF]; [START_REF] Mellinger | Trajectory generation and control for precise aggressive maneuvers with quadrotors[END_REF]. Alternatively, ground based passive visual markers have been used to estimate onboard the pose as in [START_REF] Masselli | A novel marker based tracking method for position and attitude control of mavs[END_REF]. For most applications, however, motion capture systems or ground based localization systems cannot be used and localization must rely exclusively on embarked sensors. Good orientation estimates can be obtained with embarked IMUs (Inertial Measurement Units). Estimation of the vehicle's position/velocity is more challenging. GPS may be used to this purpose but it is not always available. Furthermore, in many applications (e.g., inspection) a measurement of the relative position of the vehicle w.r.t. its environment is needed, rather than an absolute position measurement (GPS-like). The former is best obtained from embarked exteroceptive sensors (cameras, lasers, etc). With such sensors, however, the relation between the output function (i.e. measurement) and the relative position error is seldom known precisely. Uncertainties may come from calibration errors, lack of depth information with mono-camera sensors, uncertainties on the environment structure, etc... This leads to the control problem addressed in this paper: Given a class of uncertainties on the position measurements, can one design feedback control laws that guarantee stability of the system for any position measurement in this class?

To our knowledge this problem has only been addressed in very specific cases, like when uncertainties reduce to a positive scale factor on the position vector [START_REF] Metni | A uav for bridges inspection: Visual servoing control law with orientation limits[END_REF]; [START_REF] Bras | Output feedback observation and control for visual servoing of vtol uavs[END_REF]. The results here proposed address a much larger class of uncertainties. They make use of properties of saturated controls. There is a large litterature on this topic, especially for linear systems (see, e.g., [START_REF] Teel | Global stabilization and restricted tracking for multiple integrators with bounded controls[END_REF]; [START_REF] He | Composite nonlinear control with state and measurement feedback for general multivariable systems with input saturation[END_REF]). In those works, it is assumed that the system's dynamics and state are perfectly known. Saturated controls have also been used for UAVs in order to ensure some type of robustness property, e.g., [START_REF] Teel | A nonlinear small gain theorem for the analysis of control systems with saturation[END_REF] and [START_REF] Lòpez-Araujo | Global stabilisation of the pvtol aircraft with lateral force coupling and bounded inputs[END_REF] for robustness w.r.t. input saturation, [START_REF] Marconi | Robust full degree-of-freedom tracking control of a helicopter[END_REF], [START_REF] Hua | A control approach for thrustpropelled underactuated vehicles and its application to vtol drones[END_REF][START_REF] Lòpez-Araujo | Global stabilisation of the pvtol aircraft with lateral force coupling and bounded inputs[END_REF] for robustness w.r.t. unmodelled dynamics. We show that saturated controls can be instrumental in ensuring robustness properties w.r.t. measurement errors. More precisely, we propose nonlinear feedback laws that ensure semi-global stability of the pose tracking error for a large class of uncertainties on the position measurement. These results are reminiscent of Robust Stability results for linear systems (Doyle et al., 1992, Ch. 4), where the objective is to guarantee stability for a family of plants that satisfy some uncertainty bound w.r.t. a nominal system. Two application scenarios are addressed. In the first one we assume that velocity measurements are available, e.g., a GPS provides these measurements and an exteroceptive sensor (e.g., camera) provides position measurements w.r.t. the environment. In the second scenario no velocity measurement is available, i.e., "GPS-denied" environment. Note that position control of UAVs without velocity measurements has already been addressed [START_REF] Abdessameud | Global trajectory tracking control of vtoluavs without linear velocity measurements[END_REF], but position was assumed to be perfectly known. Finally, let us remark that a preliminary version of this paper was presented in de [START_REF] De Plinval | Nonlinear control of underactuated vehicles with uncertain position measurements and application to visual servoing[END_REF].

The results here proposed are illustrated by visual servoing applications with mono-camera measurements. This type of application has been considered in several works (see, e.g., [START_REF] Pebrianti | Autonomous hovering and landing of a quad-rotor micro aerial vehicle by means of on ground stereo vision system[END_REF]; [START_REF] Saripalli | Visually guided landing of an unmanned aerial vehicle[END_REF] and additional references in Section 6) usually with a camera pointing downward and observing a flat and horizontal ground, and under the assumption that altitude is measured independently. Our results provide stability guarantees for much more general application scenarios.

The paper is organized as follows. Background and problem statement are presented in Section 2. A preliminary result is provided in Section 3 for the fully actuated case. The main results on the underactuated case are provided in Sections 4 and 5: in Section 4 we assume that linear velocity measurements are available; in Section 5 such measurements are not available. Application to visual servoing is considered in Section 6, with simulation results given in Section 7. Proofs are given in the Appendix.

Background and problem statement

The n × n identity matrix is denoted as I n . The transpose of a matrix M is denoted as M T . For any square matrix M, M s := M+M T 2 and M a := M-M T 2 respectively denote the symmetric and antisymmetric part of M. The maximum singular value of a matrix M is denoted as |M| and when M is a matrix-valued time-function,

|M| s := sup t |M(t)|. Given a matrix-valued time- function M : t -→ M(t) ∈ R n×n with M(t) ≥ 0 ∀t, we define |M| i := sup αI≤M(t) ∀t α. Note that |M| i = |M s | i . Given a smooth function f defined
on an open set of R, its derivative is denoted as f . Throughout the paper, AS, GAS, and LES stand for Asymptotically Stable, Globally Asymptotically Stable, and Locally Exponentially Stable respectively. CD stands for Convergence Domain.

Definition 1 Given

δ := [δ m ; δ M ] with 0 < δ m < δ M , sat δ : R n -→ R n is called a saturation function if: i) There exists a class C 1 function s δ : R + -→ R + such that sat δ (x) = s δ (|x| 2 )x for all x ∈ R n ;
ii) The function defined on R + by τ -→ s δ (τ 2 )τ is nondecreasing, upper-bounded by δ M , and is equal to the identity function on [0, δ m ];

iii) s δ (τ) ≤ 0 for all τ.

From i) sat δ is fully defined from the associated function s δ . From i)-ii) saturation functions in the sense of Def. 1 inherit the classical properties of a saturation function: sat δ is upperbounded in norm by δ M and sat δ

(x) = x for |x| ≤ δ m (because s δ (τ 2 )τ = τ for τ ∈ [0, δ m ]). Also, s δ (τ) ≤ 1 , ∀τ ∈ R + τ s δ (τ) -→ +∞ when τ -→ +∞ (1)
where the first relation follows from ii) and iii) and the second relation from ii). Then, ii) implies that the derivative of the function τ -→ s δ (τ 2 )τ is non-negative. This property and iii) imply that 2τ|s δ (τ)| ≤ s δ (τ) ∀τ ∈ R + . Thus, from (1),

C δ := sup τ∈R + s δ (τ) + 2τ|s δ (τ)| ≤ 2 < +∞ (2)
An example of a function s δ is given by

s δ (τ) :=        1 i f τ ≤ δ 2 m δ M √ τ - (δ M -δ m ) 2 √ τ( √ τ+δ M -2δ m ) i f τ > δ 2 m (3)

Dynamics of thrust-propelled underactuated vehicles

We focus on the class of so-called "thrust-propelled underactuated vehicles" [START_REF] Hua | A control approach for thrustpropelled underactuated vehicles and its application to vtol drones[END_REF], i.e., rigid bodies moving in 3D-space under the action of one body-fixed force control and full torque control. This class contains most VTOL UAVs (quadrotors, ducted-fans, etc). The dynamics of these systems is described by the following equations, expressed in a "North-East-Down" (NED) frame:

         p = -uRe 3 + ge 3 Ṙ = RS (ω) J ω = Jω × ω + Γ (4)
with p the position vector of the vehicle's center of mass, expressed in a reference (inertial) frame, R the rotation matrix from the body frame to the reference frame, ω the angular velocity vector expressed in the body frame, S (.) the matrix-valued function associated with the cross product, i.e. S (x)y = x × y , ∀x, y ∈ R 3 , u the normalized thrust input, i.e. u = T m where m is the mass and T the thrust input, e 3 = (0, 0, 1) T , J the inertia matrix, Γ the torque vector, and g the gravity constant. In this paper we mainly focus on the system p = -uRe 3 + ge 3 Ṙ = RS (ω)

(5) with u and ω as control inputs, i.e., considering ω instead of Γ as orientation control. Extension of the paper's results to System (4) is discussed in Section 4.

Problem statement

The main objective of this paper is to investigate the stabilization of p to a reference trajectory p r from some relative position measurements of the following form:

σ(t) := R(t) T M(t) p(t) (6)
with M(t) an unknown matrix and p := p-p r the position error.

In other words σ provides information on the position error in body frame and M(t) accounts for measurements uncertainties. Measurements σ are typically obtained from embarked exteroceptive sensors (cameras, lasers, etc). Examples are provided in Section 6. Due to the system's underactuation, stabilization of p to p r fixes two degrees of freedom on the vehicle's orientation. There remains a degree of freedom (yaw angle). Its control is not addressed here since it does not affect the position control.

Let us introduce some assumptions. The first one is made throughout the paper. The other ones concern particular cases. A1: A4: p r is a constant vector and M s and M a commute.

M(t) > 0 ∀t, |M| i > 0, |M| s < +∞, | Ṁ| s < +∞,

A preliminary result

Before addressing the control of underactuated systems we consider a fully actuated system modelled by: p = -Ru (8)

where u ∈ R 3 denotes the body-fixed thrust input and R satisfies the third kinematic relation in (5). We assume that the following measurements are available:

σ , v := R T ṗ (9)
with σ defined by ( 6) and v the linear velocity in body frame.

A reference trajectory is defined by:

p r , v r := R T ṗr , a r := R T pr (10)
and the associated tracking error by: p := pp r , ṽ := vv r (11)

Proposition 1 Let sat δ , sat¯δ denote two R 3 -valued saturation functions with associated functions s δ , sδ.

Consider control gains k 1 , k 2 > 0 such that k 2 2 |M| i > k 1 |M a | s max {C δ |M| s , |M a | s } k 2 δm > k 1 δ M (12)
and define the control law

u := k 1 sat δ ( σ) + k 2 sat¯δ(ṽ) -a r (13)
Then, i) if A1 and A2 hold, then (p, ṗ) = (p r , ṗr ) is a (uniformly) GAS and LES equilibrium trajectory for the closed-loop system (8)-( 13).

ii) if A1 holds then, for any ρ > 0, there exists ϑ > 0 such that, for any M with | Ṁ| s < ϑ, (p, ṗ) = (p r , ṗr ) is a (uniformly) AS and LES equilibrium trajectory for the closed-loop system (8)-( 13) with CD containing

A ρ := {(p, ṗ)(0) : |( p(0), ṽ(0))| ≤ ρ}, with |( p, ṽ)| := | p| 2 + |ṽ| 2 .
Proposition 1 provides bounded feedback laws that ensure global (or semi-global) stability in the presence of uncertain measurements for System (8). Condition (12) can be used to specify admissible control gains given upper bounds on the uncertainties (the norm of M and of its skew-symmetric part M a ). This kind of result is very similar to classical Robust Stability theory for linear systems (Doyle et al., 1992, Ch. 4), [START_REF] Doyle | Multivariable feedback design: Concepts for a classical/modern synthesis[END_REF], [START_REF] Chen | Necessary and sufficient condition for robust stability of linear distributed feedback systems[END_REF], where the objective is to guarantee stability for a set of plants that satisfy some uncertainty bound w.r.t. a nominal system. In our case, uncertainty corresponds to the difference between M and the identity matrix. If the uncertainty is small (which implies in particular that M a is close to the zero matrix and |M| i ≈ 1), then Condition (12) puts little constraints on the control gains. If M a is large, however, large values of k 2 are needed. In summary, Proposition 1 can provide stability guarantees given an a priori bound on uncertainties. Note also that when M is constant, global asymptotic stability can be obtained while only semi-global asymptotic stability is obtained when M varies with time (Case ii)).

Underactuated case with velocity measurements

Let us consider the control system (5), and assume that the following measurements are available:

σ , γ := gR T e 3 , v , ω (14)
Compared to the fully actuated case, there are two additional measurements, i.e. γ and ω. The latter is typically obtained from the gyrometers of an IMU, while the former is obtained by fusing accelerometer and gyrometer measurements (see, e.g. [START_REF] Mahony | Multirotor aerial vehicles: Modeling, estimation, and control of quadrotor[END_REF]). Let q := Re 3 denote the thrust direction, so that the first equality in (5) can also be written as

p = -uq + ge 3 = R(-ue 3 + γ) ( 15
)
If A3 is satisfied then, along the reference position trajectory p r the thrust direction is well defined (this is no longer true if pr = ge 3 since any thrust direction q is solution to (15) for u = 0). More precisely, assuming that u is positive, this reference thrust direction is q r := ge 3pr |ge 3pr | (16)

Proposition 2 Let sat δ , sat¯δ denote two saturation functions. Consider control gains k 1 , k 2 > 0 satisfying (12) with C δ defined by (2), and the additional condition

k 1 δ M + k 2 δM + | pr | s < g (17)
Define a dynamic augmentation

η = η × ω -k 3 (η -σ) , k 3 > 0 (18)
together with the control law

             ω 1 = -k 4 |µ|µ 2 (|µ|+µ 3 ) 2 -1 |µ| 2 µ T S (e 1 ) ω 2 = k 4 |µ|µ 1 (|µ|+µ 3 ) 2 -1 |µ| 2 µ T S (e 2 ) u = µ 3 (19)
with k 4 > 0 and µ and defined by:

µ := γ + k 1 sat δ (η) + k 2 sat¯δ(ṽ) -a r := -k 1 k 3 s δ (|η| 2 )I 3 + 2s δ (|η| 2 )ηη T (η -σ) +k 2 sδ(|ṽ| 2 )I 3 + 2s δ(|ṽ| 2 )ṽṽ T (γ -ue 3 ) (20)
with ṽ and a r defined by (10) and (11). Then, i) if A1, A2, and A3 hold then, there exists k 3 > 0 such that, for any k 3 > k 3 , (p, ṗ, q, η) = (p r , ṗr , q r , 0) is a (uniformly) AS and LES equilibrium trajectory for the closed-loop system (5)-( 18)-( 19)-( 20) with CD

A := {(p, ṗ, q, η)(0) : µ(0) -|µ(0)|e 3 } (21)
ii) if A1 and A3 hold then, for any ρ > 0, there exist ϑ, k 3 > 0 such that, for any k 3 > k 3 and any M with | Ṁ| s < ϑ, (p, ṗ, q, η) = (p r , ṗr , q r , 0) is an AS and LES equilibrium trajectory for the closed-loop system (5)-( 18)-( 19)-( 20) with CD containing

A ρ := {(p, ṗ, q, η)(0) : |( p(0), ṽ(0), η(0))| ≤ ρ , µ(0) -|µ(0)|e 3 } (22)
Let us discuss the links between this result and Proposition 1. First, except for the γ term, µ in (20) is reminiscent of the control law (13) with σ replaced by η. In view of (18), this latter variable can be viewed as a "filtered value" of σ. The important point is that η is known, since it is explicitly given by (18), while the time derivative of σ is not, since M is unknown. Then, the control inputs u, ω 1 , ω 2 are defined so that µ converges to |µ|e 3 . This implies, using the second equality in (15), that p converges to R(-k 1 sat δ (η)k 2 sat¯δ(ṽ) + a r ). This expression is the same as (8)-( 13) with σ being replaced by η. This explains the relation between Propositions 1 and 2. Finally,

|k 1 sat δ (η) + k 2 sat¯δ(ṽ) -a r | ≤ k 1 δ M + k 2 δM + | pr | s < g = |γ| (23)
where the second inequality comes from (17). This inequality implies that:

Lemma 1 µ(0) -|µ(0)|e 3 if γ 3 (0) > -g 2 -k 1 δ M + k 2 δM + | pr | s 2 ( 24
)
Since γ 3 = gq 3 , Lemma 1 implies that in both cases i) and ii)

the CD in roll/pitch contains the upper hemisphere. Note that Condition (24) is conservative. Thus, in both cases i) and ii), a large stability domain in orientation is guaranteed. Global stability is ruled out because q belongs to a compact set (i.e., the unit sphere). If (24) is satisfied, there is no constraint on the initial values of position, linear velocity, and dynamic augmentation variables in case i) (see ( 21)). In case ii), initial values can be made arbitrarily large under conditions on k 3 and Ṁ.

Simplified control law

Another control expression with similar robustness properties is proposed next. It involves a simpler control expression and does not require the dynamic extension (18).

Proposition 3 With the notation of Prop. 2, assume that the following extra condition on the gains k 1 , k 2 is satisfied:

k 1 δ M + k 2 δM + | pr | s ≤ g(1 -κ) , 0 < κ < 1 (25)
and define the control law as:

ω 1 = -k 4 µ 2 , ω 2 = k 4 µ 1 u = µ 3 ( 26
)
with µ := γ + k 1 sat δ ( σ) + k 2 sat¯δ(ṽ) -a r (27)
Then, i) If A1, A2, and A3 hold then, for any ρ > 0, there exists k 4,ρ > 0 such that, for any k 4 > k 4,ρ , (p, ṗ, q) = (p r , ṗr , q r ) is a (uniformly) AS and LES equilibrium trajectory for the closedloop system (5)-( 26)-( 27), with CD containing

A ρ := {(p, ṗ, q)(0) : |ṽ(0)| < ρ , |(µ 1 (0), µ 2 (0))| < gκ/ √ 2 , µ 3 (0) > 0} (28)
ii) If A1 and A3 hold then, for any ρ > 0, there exist ϑ, k 4 > 0 such that, for any k 4 > k 4 and any M with | Ṁ| s < ϑ, (p, ṗ, q) = (p r , ṗr , q r ) is an AS and LES equilibrium trajectory for the closed-loop system (5)-( 26)-( 27) with CD containing

A ρ := {(p, ṗ, q)(0) : | p(0)| < ρ , |ṽ(0)| < δm /2 , |(µ 1 (0), µ 2 (0))| < gκ/ √ 2 , µ 3 (0) > 0} (29)
The main assets of Proposition 3 are a large stability domain, robustness to position measurement uncertainties, and the simplicity of the control expression. Concerning the latter aspect, the fact that the control expression is essentially linear (modulo saturation functions) is clearly an asset with respect to the control law of Proposition 2, e.g., when considering effects of measurement noise. Another asset is related to the extension of the present analysis to the full model (4) (i.e., considering Γ as control input instead of ω). A classical solution in this case would be a linear torque feedback with feedforward action.

Computing the feedforward action requires to differentiate angular velocity inputs proposed above. Differentiating ω 1 , ω 2 in ( 26) is much simpler than for ( 19) and requires much less information. In addition, one may want in this case to replace σ in ( 27) by η given by (18) since η is known. Additional work is needed for the stability analysis of such a torque control law and this issue is left for future research.

Another common approach to extend the controller from kinematics to dynamics is to use a high gain controller: Γ = -Jω × ω -kJ ωω d with k chosen large enough and ω d the kinematic controller [START_REF] Hua | A control approach for thrustpropelled underactuated vehicles and its application to vtol drones[END_REF]; [START_REF] Brescianini | Design, modeling and control of an omnidirectional aerial vehicle[END_REF]. This simple solution is motivated by a time separation argument. To the authors' knowledge its stability analysis remains an open issue.

Extensions to GPS-denied environments

This section considers extension of the results of Section 4 to velocity-free scenarios, i.e., when velocity v is not measured. This is a challenging problem and we only consider a special case of the general framework addressed in Section 4. We will show in the application section, however, that this special case covers important scenarios.

Let v M := R T M ṗ. Assume that M and p r are constant values. Then, v M = R T M ṗ and it follows from (6) that

σ = σ × ω + v M vM = v M × ω + R T M p ( 30
)
We want to obtain an estimation of the non-measured variable v M . To this purpose, consider the following observer:

σ = σ × ω + vM -2αk( σ -σ) vM = vM × ω -k 2 ( σ -σ) (31)
Proposition 4 Let e σ := σσ and e v := vMv M denote the estimation errors. Assume that there exists a constant C such that, for any initial condition, | p(t)| ≤ C, ∀t. Then, for any ε > 0 and any α > 0 there exists k m > 0 such that, for any k ≥ k m and any initial condition

( σ(0), vM (0)), |e σ | + |e v | is ultimately bounded by ε.
From Proposition 4 a good estimate of v M can be built from the measurement σ if p is bounded. Note that the control laws derived in Section 4 ensure the boundedness of p. This suggests to use such control laws with vM as velocity input in place of v.

In the rest of this section, we focus on the stability analysis of these control laws with v M as velocity input. By doing so, we neglect the discrepancy between vM and v M knowing that, from Proposition 4, it can be made arbitrarily small ultimately. Stability analysis of the couple controler/observer is left for future studies.

The rest of this section will invoke Assumptions 1, 2, and 4. Note that, when p r is a constant vector, the control law u in (13) can be written as u := k 1 sat δ ( σ) + k 2 sat¯δ(v). The following propositions show that the results of Section 4 can be extended to the case of velocity measurements v M with minor modifications. Due to space limitations, we only address extension of Propositions 1 and 3.

Proposition 5 Let sat δ , sat¯δ denote two R 3 -valued saturation functions with associated functions s δ , sδ.

Consider control gains k 1 , k 2 > 0 such that k 2 2 |M| 2 i > k 1 |M a M -1 | s max C δ |M| s , |M a M -1 | s |M| i k 2 δm |M| i > k 1 δ M |M| s (32)
and define the control law

u := k 1 sat δ ( σ) + k 2 sat¯δ(v M ) (33)
If A1, A2, and A4 hold then (p, ṗ) = (p r , 0) is a (uniformly) GAS and LES equilibrium point for the closed-loop system (8)-( 33).

Proposition 6 Let sat δ , sat¯δ denote two saturation functions. Consider control gains k 1 , k 2 > 0 satisfying (32) with C δ be defined by (2), and the additional condition

k 1 δ M + k 2 δM ≤ g(1 -κ) , 0 < κ < 1 (34)
and consider the control law (26) with

µ := γ + k 1 sat δ ( σ) + k 2 sat¯δ(v M ) (35)
If A1, A2, and A4 hold then, for any ρ > 0, there exists k 4,ρ > 0 such that, for any k 4 > k 4,ρ , (p, ṗ, q) = (p r , 0, q r ) is a (uniformly) AS and LES equilibrium point for the closed-loop system (5)-( 26)-( 35), with CD containing

A ρ := {(p, ṗ, q)(0) : |v M (0)| < ρ , |(µ 1 (0), µ 2 (0))| < gκ/ √ 2 , µ 3 (0) > 0} (36)

Application to visual servoing of UAVs

Micro UAVs (MAVs) are increasingly used for surveillance and inspection applications. Controlling such systems through vision sensors is an important issue. A first solution consists in using a stereo vision system. When MAVs operate far from ground/obstacles, however, the advantage of a stereo system w.r.t. a mono-camera system is questionable since the quality of depth-estimation is then poor due to the short baseline. In fact, a mono-camera system can be preferred for simplicity and because image processing can be performed at a higher framerate. Thus, much attention has been paid to mono-camera visual servoing of UAVs (see, e.g., [START_REF] Saripalli | Visually guided landing of an unmanned aerial vehicle[END_REF]; [START_REF] Conte | An integrated uav navigation system based on aerial image matching[END_REF]; [START_REF] Caballero | Vision-based odometry and slam for medium and high altitude flying uavs[END_REF]; [START_REF] Cunha | Vision-based control for rigid body stabilization[END_REF]; [START_REF] Mondragon | 3d pose estimation based on planar object tracking for uavs control[END_REF]). These approaches often rely on the estimation of the so-called "Homography matrix", which embeds information on the camera's pose. However, the relation between the Homography matrix and the pose involves quantities that may be poorly known. This issue is often resolved in the litterature by considering restrictive scenarios (e.g., perfectly calibrated camera, camera observing an horizontal ground, knowledge of the vehicle's altitude, etc). We show next that our results provide stability guarantees for a much larger range of scenarios.

Preliminary recalls and problem statement

Consider an underactuated UAV with a video camera facing a planar target. Suppose that a "reference" picture of this target taken at a reference pose is known. This reference pose is represented on Fig. 1 by the reference frame R * . We assume that this pose is a possible equilibrium for the dynamics of this underactuated vehicle, meaning that the z axis of R * is vertical. Otherwise, asymptotic stablization of this pose is not possible. We also assume that the optical center of the camera corresponds to the vehicle's center of mass and the optical axis corresponds to the x-axis of R * . At every time, the "current" picture of the target, taken at the current pose represented by the frame R, is compared to the reference one. From there, the homography matrix is computed (see e.g. [START_REF] Ma | An Invitation to 3-D Vision: From Images to Geometric Models[END_REF]). This matrix, which transforms the target's points coordinates from the reference pose to the current pose, is

H := R T - 1 d * R T pn * T (37)
with d * the distance from the UAV reference position to the target plane and n * the normal to this plane expressed in the reference frame. Both d * and n * are unknown and thus unavailable for the control design. We show next that for any orientation of the visual target one can extract from H position measurements of the form (6). The case of a non-vertical target is first briefly described. Then, the case of a vertical target is studied in more details.

Non-vertical target

In this case, n * 3 := n * T e 3 > 0 and it follows from (37) that

He 3 = R T e 3 -(n * 3 /d * )R T p = γ g - n * 3 d * R T p.
As recalled in Section 4, γ is usually estimated from the UAV's IMU. By substracting From (37) and the assumption n * 3 = 0, one can verify that

σ = R T M( n * d * )p , γ = gR T e 3 ( 39
)
with M(τ) := τ 1 I + S (τ 2 e 3 ). Thus, σ and γ satisfy Eq. ( 6), ( 14) with p r = 0. Note that M s (τ) = τ 1 I and M a (τ) = S (τ 2 e 3 ) commute for any τ. Thus, like for a non-vertical target, M satisfies all the conditions in Assumptions 1, 2, and 4. We detail below application of our results to fixed-point stabilization and way-points navigation.

Vision based point stabilization. Let us first address the stabilization of the UAV at the reference pose. From (39), Proposition 2-i) applies directly with M = M( n * d * ) provided the gain conditions ( 12) and ( 17) are satisfied. We deduce that the control law ( 18)-( 19) ensures asymptotic stabilization of the reference pose (with global convergence domain in position/velocity) if:

a) n * 1 , k 1 , k 2 > 0 b) k 2 δm > k 1 δ M c) k 1 δ M + k 2 δM < g d) n * 1 d * k 2 2 > k 1 |n * 2 | |n * 2 | + 2n * 1 3 √ 3
Condition n * 1 > 0, which ensures that M > 0, means that the camera is "facing" the target at the reference pose (obvious condition in practice). Given bounds on the uncertain param-

eters d * , n * , i.e., d * ∈ [d * , d * ], n * 1 ∈ [n * 1 , 1], Condition d) can be replaced by: n * 1 d * k 2 2 > k 1 (1 + 2/(3 √ 3)
). Thus, one obtains stability conditions on the control parameters given bounds on the uncertain parameters d * , n * . Yaw control. The yaw degree of freedom is not involved in the stabilization objective. In practice, it matters to keep the target inside the field of view of the camera. We propose the following yaw control law: ω 3 = k 5 H 21 . Upon convergence of the position, velocity, roll and pitch errors to zero, the yaw dynamics will be close to ψ ≈ -k 5 sin ψ, thus ensuring the convergence of ψ to zero unless ψ(0) = π (a case contradictory with the visibility assumption).

Visual-based way-points navigation. Consider a sequence of reference images of a planar scene taken from different reference frames (hereafter referred to as way-points). The objective is to make the UAV navigate along this sequence of way-points. Without loss of generality, we consider two way-points. From the two reference images and the current image, one can define two homography matrices, from which are computed two uncertain relative position measurements σ1 , σ2 as defined by Eq. ( 38). Let p i denote the position vector of the current frame w.r.t. the i-th reference frame and R i denote the rotation matrix from the current frame to the i-th reference frame. If χ (resp. χ * i ) denotes the coordinate vector of a point of the scene in the current frame (resp. in the i-th reference frame), then

χ * 1 = R 1 χ + p 1 and χ * 2 = R 2 χ + p 2 .
The transformation between the two reference frames is defined as χ * 2 = Rχ * 1 + p where R is a constant matrix and p is a constant vector, and one has R 2 = RR 1 and p 2 = Rp 1 + p. We implicitly define a reference trajectory by considering a time-varying interpolation σ of σ1 and σ2 : σ := (1 -λ(t)) σ1 + λ(t) σ2 with λ an increasing function ranging over [0, 1]. Consider the generic case where both parameters n * , d * change between the two reference images. Then, we deduce from the above relations that

           σ = R T 1 M(p 1 -p r ) M(t) := (1 -λ(t)) M n 1 d 1 + λ(t) RT M n 2 d 2 R p r (t) := -λ(t) M(t) -1 RT M( n 2 d 2 ) p (40)
Thus, σ is of the form of Eq. ( 6) with M(t) replaced by M(t). Proposition 2-ii) can then be used to ensure semi-global stability of this non-stationary reference trajectory. Note that the existence of | M| i > 0, | M| s < +∞ and | Ṁ| s < +∞ in Assumption A1 for the matrix M(.) follows from the fact that M > 0 and λ(t) ∈ [0, 1]. Thus, the linear interpolation of σ1 and σ2 implicitly defines a reference trajectory with compact image set.

Compacity plays a key role in ensuring that Assumption A1 is satisfied.

Simulation results

We illustrate the results of this paper for the visual servoing applications of Section 6, in the case of a vertical visual target. We first consider the fixed-point stabilization problem. The initial conditions and scene parameters at the reference pose are:

                 p 0 = (-5.5m; 1.2m; 1.3m) T v 0 = -2.3m.s -1 ; -1.6m.s -1 ; 0.4m.s -1 T φ 0 = -0.5˚, θ 0 = 11.4˚, ψ 0 = 4.1n * = (0.7; -0.71; 0) T , d * = 3m (41)
For the simulation reported on Fig. 2, the control law of Prop. 2 is used with η(0) = 0. For the simulation reported on Fig. 3, the simplified controller of Prop. 3 is used. For the simulation reported on Fig. 4, the velocity-free controller of Prop. 6 is used with v M replaced by vM and vM the output of the observer (31). The control gains are [k 1 , k 2 , k 3 , k 4 , k 5 ] = [1, 1.5, 1, 1, 1] and the saturation functions sat δ , sat¯δ are defined from the expression (3) of the associated functions s δ , sδ, with δ = [.9; 1], δ = [1; 1.1]. The gains of the observer (31) are k = 7, α = 0.7 and σ(0) = vM (0) = 0. All controllers stabilize the system. Transient behaviors are qualitatively similar but differences can be noticed. With respect to the control law of Prop. 2, the simplified control law of Prop. 3 yields a shorter settling time, less overshoot, and smaller angular velocities. The velocityfree control law of Prop. 6 yields results similar to that of the first simulation but the convergence of the angular velocity to zero is slower. From Fig. 4 one can also conjecture asymptotic stability of the couple observer/controller, i.e., asymptotic convergence of vM to v M and of the tracking error to zero. Fig. 5. The interpolation function λ is defined as: λ(t) = 0 for t ≤ t s , λ(t) = 1 for t ≥ t e and λ(t) = t-t s t e -t s for t s < t < t e with t s = 50s, t e = 80s. Initial conditions and scene parameters for the first image are still given by (41). Scene parameters for the second image (i.e., second way-point) are n * = (0.87, -0.5, 0) T , d * = 1/3m. The position vector between the two way-points is defined by p = (-5, 10, 20) T . Let us notice the smooth transition and small control input values on the transition interval.

Conclusion

We have proposed a feedback control design and robustness analysis for the stabilization of a class of underactuated vehicles with uncertain position measurements. Strong stability results have been obtained for a large class of position measurements uncertainties and sufficient stability conditions on the control gains have been derived in relation with the norm of the uncertainties. We have shown direct applications of these results to UAVs for two visual servoing problems: fixed-point stabilization and visual way-points navigation w.r.t. a planar scene. Simulation results support the proposed analysis.

Proof of Propositions

Proof of Proposition 1: From (6), x := R σ = M p. From ( 9)-( 11) y := Rṽ = ṗ. Thus, ẋ = My + Ṁ p and, from ( 8), ( 11) and ( 13), ẏ = -R(k 1 sat δ ( σ) + k 2 sat¯δ(ṽ)). From Definition 1-i), R sat δ (ξ) = sat δ (Rξ) for any ξ. Therefore, the closed-loop system in (x, y) coordinates is given by:

ẋ = My + Ṁ p ẏ = -k 1 sat δ (x) -k 2 sat¯δ(y) (42)
Lemma 2 Let y denote a solution of the equation

ẏ = -k 1 sat δ (x) -k 2 sat¯δ(y) + (t) (43)
with x any time-function and a bounded and continuous timefunction such that, for some t 1 ≥ 0 and c > 0,

| (t)| ≤ k 2 δm -k 1 δ M -c , ∀t ≥ t 1 (44)
Then, there exists a continuous function T such that, for t ≥ T (y(0)), sat¯δ(y(t)) = y(t), i.e., the function sat¯δ desaturates after time T (y(0)).

Property i): Assumption A2 implies that Ṁ = 0. Applying Lemma 2 to the second equation in (42) with ≡ 0 and using the second condition in (12), one deduces that along any solution of System (42) the function sat¯δ desaturates after some time and the solution then satisfies:

ẋ = My ẏ = -k 1 sat δ (x) -k 2 y = -k 1 s δ (|x| 2 )x -k 2 y (45)
Consider the CLF (Candidate Lyapunov Function) V defined by:

V(x, y) = k 1 |x| 2 0 s δ (τ) dτ + y T My + 2k 1 k 2 s δ (|x| 2 )x T M a y + κ s δ (|x| 2 )x T y = k 1 |x| 2 0 s δ (τ) dτ + y T My + 2k 1 k 2 sat δ (x) T M a y + κ sat δ (x) T y ( 46
)
where κ is a constant positive number. We show that V is a Lyapunov function for κ small enough. Let

                         κ := k 2 2 |M| i -k 1 |M a | s |M| s C δ > 0 κ 1 := 2 √ k 1 |M| i -2 k 1 k 2 |M a | s > 0 κ 2 := 2κ k 2 |M| s C δ > 0 κ 3 := 2κ k 2 |M| s C δ + k 3 2 4k 1 > 0 0 < κ < min {κ 1 , κ 2 , κ 3 } (47)
where positivity of κ, κ 1 follows from ( 12) and positivity of κ 2 , κ 3 is a consequence thereof. We first prove that V is positive definite and proper. Integrating by part and using the fact that s δ (τ) ≤ 0 ∀τ due to Def. 1iii), we get

|x| 2 0 s δ (τ) dτ = - |x| 2 0 τs δ (τ) dτ + |x| 2 s δ (|x| 2 ) ≥ |x| 2 s δ (|x| 2 )
From (1), s δ (|x| 2 ) ≤ s δ (|x| 2 ). Therefore, from (46),

V ≥ k 1 |x| 2 s δ (|x| 2 ) - 2k 1 k 2 |M a | s + κ s δ (|x| 2 )|x||y| + |M| i |y| 2 Therefore, V is positive definite provided that b 2 < 4k 1 |M| i , b := 2k 1 k 2 |M a | s + κ (48)
which is equivalent to κ < κ 1 . Since s δ (τ) > 0 for τ 0 this ensures that V is a positive definite function of x and y, and V is proper due to (1). Let us now prove that V is non-increasing along the solutions of System (45). Differentiating V along these solutions yields

               V = -2k 2 y T M s y -2 k 1 k 2 y T M a F δ (x)My -κk 1 s 2 δ (|x| 2 )|x| 2 + κy T F δ (x)My -κk 2 s δ (|x| 2 )x T y F δ (x) := s δ (|x| 2 )I 3 + 2s δ (|x| 2 )xx T (49) By (2), |F δ (x)| ≤ C δ , ∀x and we deduce that V ≤ -C 1,1 |y| 2 + C 1,2 |y|s δ (|x| 2 )|x| -C 1,3 s 2 δ (|x| 2 )|x| 2 ≤ -C 1,1 |y| 2 + C 1,2 |y||sat δ (x)| -C 1,3 |sat δ (x)| 2 (50) with C 1,1 := 2k 2 |M| i -b|M| s C δ , C 1,2 := κk 2 , C 1,3 := κk 1 (51)
The right-hand side of ( 50) is a quadratic form in |y| and |sat δ (x)|. Therefore, V is negative definite provided that

a) C 1,1 > 0, b) C 1,3 > 0, c) C 2 1,2 < 4C 1,1 C 1,3 (52)
Condition a) follows from the fact that κ < κ 2 . Condition b) holds true for any κ > 0 since k 1 > 0, and Condition c) follows from the fact that κ < κ 3 . Thus, ∃β > 0 such that

V ≤ -β(|y| 2 + |sat δ (x)| 2 ) ( 53
)
This shows global asymptotic stability of (p, ṗ) = (p r , ṗr ). Local exponential stability readily follows by noting that both V and V are locally quadratic in x and y around the origin, i.e., from Def. 1-ii), s δ (τ) = 1 for τ ≤ δ m .

Property ii): Since |y| = |ṽ|, (p(0), ṗ(0)) ∈ A ρ implies that |y(0)| ≤ ρ. Applying Lemma 2 to System (42) with ≡ 0 and using the continuity property of the function T in Lemma 2, one deduces that for any t ≥ T ρ := max |y|≤ρ T (y) and along any solution with initial condition in A ρ , sat¯δ desaturates and the solution satisfies

ẋ = My + Ṁ p ẏ = -k 1 sat δ (x) -k 2 y (54)
From ( 6) and (7),

∀t, |M| i | p(t)| ≤ |x(t)| = |M(t) p(t)| ≤ |M| s | p(t)| (55) Therefore, |(x(0), y(0))| ≤ ρ max(1, |M| s ) for (p(0), v(0)) ∈ A ρ .
Assumption A1, (54), and (55) imply that |( ẋ, ẏ)| ≤ k|(x, y)| for some constant k. As a consequence, there exists a constant ρ such that, along any solution with initial condition in

A ρ , |(x(T ρ ), y(T ρ ))| ≤ ρ.
Consider V defined by ( 46) as a CLF for System (54). Its derivative along the solutions of this system satisfies

V = V| Case i) + 2k 1 s δ (|x| 2 )x T Ṁ p +2 k 1 k 2 pT ṀT F δ (x)M a y + κ pT ṀT F δ (x)y +y T Ṁy + 2 k 1 k 2 s δ (|x| 2 )x T Ṁa y (56)
with F δ given by ( 49) and V| Case i) the expression (49) of V. Using (55) then implies that for any M with | Ṁ| s < ϑ,

V ≤ V| Case i) + ϑ 2k 1 |M| i s δ (|x| 2 )|x| 2 + |y| 2 + 2 k 1 k 2 |M a | s + κ C δ |M| i + 2 k 1 k 2 s δ (|x| 2 ) |x||y| ≤ -C 1,1 -ϑ |y| 2 -C 1,3 |sat δ (x)| 2 + C 1,2 + 2 k 1 k 2 ϑ |sat δ (x)||y| + 2k 1 |M| i ϑ|x||sat δ (x)| + 2 k 1 k 2 |M a | s + κ C δ |M| i ϑ|x||y| (57)
Let:

V M := max |(x,y)|≤ ρ V(x, y) > 0 x M := max (x,y)/V(x,y)≤V M |x| > 0 s m := min |x|≤x M s δ (|x| 2 ) > 0 (58)
V M exists because V is continuous and V M > 0 because V is positive definite. x M exists because V is radially unbounded and x M > 0 because V M > 0. Finally, s m exists because s δ is continuous and s m > 0 because otherwise s δ vanishes at some point, which contradicts ii) of Definition 1. From the definition of s m , note that

|x| ≤ x M =⇒ |x| = s δ (|x| 2 )x s δ (|x| 2 ) ≤ 1 s m |sat δ (x)| (59)
Therefore, as long as |x| ≤ x M , it follows from (57) that

V ≤ -C 2,1 |y| 2 + C 2,2 |y||sat δ (x)| -C 2,3 |sat δ (x)| 2 (60) with            C 2,1 := C 1,1 -ϑ C 2,2 := C 1,2 + 2 k 1 k 2 + 2 k 1 k 2 |M a | s + κ C δ s m |M| i ϑ C 2,3 := C 1,3 -2k 1 s m |M| i ϑ (61)
From the above expression, the C 2, j 's tend to the C 1, j 's as ϑ tends to zero. Thus, it follows from (52) that for ϑ > 0 small enough and κ satisfying (47), the right-hand side of ( 60) is a negative-definite quadratic form in |y| and |sat δ (x)|. We thus have shown that for ϑ > 0 small enough, 0 |x| ≤ x M =⇒ V < 0. From the definition of x M , we deduce that 0

V ≤ V M =⇒ V < 0. From the definition of V M and the fact that |(x(T ρ ), y(T ρ ))| ≤ ρ along any solution with initial condition in A ρ , this implies convergence of (x, y) to zero along these solutions. Local exponential stability is proved as for Property i).

Proof of Proposition 2: It builds on the proof of Prop. 1. Recall from Prop. 1 that x = M p and y = ṗ. In addition, let z := Rη and μ := µµ 3 e 3 . Using the expression (15) of p, the expression of u in (19), and ( 18), one obtains (compare with (42)):

         ẋ = My + Ṁ p ẏ = -k 1 sat δ (z) -k 2 sat¯δ(y) + R μ ż = -k 3 (z -x) (62) Lemma 3 With ω 1 , ω 2 defined by (19), μ = 0 is LES. More precisely, ∃c, α > 0 : | μ(t)| ≤ c| μ(0)|e -αt , ∀t, for any initial condition such that µ(0) -|µ(0)|e 3 .
Property i): Assumption A2 implies that Ṁ = 0. From Lemma 3 μ tends to zero and from ( 12) k 1 δ Mk 2 δm < 0. Then, Lemma 2 applies to the second equation in ( 62) and along any solution, sat¯δ desaturates after some time and the solution then satisfies:

               ẋ = My ẏ = -k 1 sat δ (x) -k 2 y -k 1 [sat δ (z) -sat δ (x)] + R μ ż -ẋ = -k 3 (z -x) -My (63)
Note that System ( 63) is equivalent to System (45) when z = x and μ = 0. Consider the CLF V 1 defined by

V 1 (x, y, z) = V(x, y) + (z -x) 2 (64)
We deduce from the mean-value inequality and (2) that |sat δ (z)sat δ (x)| ≤ C δ |z -x| ∀(z, x). Therefore, the time-derivative V1 of V 1 along the solutions of (63) satisfies:

                     V1 ≤ -C 3,1 |y| 2 -C 3,2 |z -x| 2 -C 3,3 s 2 δ (|x| 2 )|x| 2 +C 3,4 |y||z -x| + C 3,5 |y|s δ (|x| 2 )|x| +C 3,6 |z -x|s δ (|x| 2 )|x| + ∂V ∂y R μ C 3,1 := C 1,1 , C 3,2 := 2k 3 , C 3,3 := C 1,3 C 3,4 := 2 (C δ k 1 + 1) |M s | s , C 3,5 := C 1,2 , C 3,6 := bC δ k 1
(65) with the C 1, j 's given by ( 51).

Let us first assume that μ = 0. We claim that V 1 is a Lyapunov function for a proper choice of κ and k 3 . Let 0 < κ < min{κ 1 , κ 2 } with κ 1 , κ 2 defined by (47). From the proof of Prop. 1, κ < κ 1 implies that V is positive definite and proper, so that V 1 is positive definite and proper too. Then, 0 < κ < κ 2 implies that C 3,1 > 0. Since C 3,2 , C 3,3 > 0 and V1 is a quadratic form in |y|, |z -x|, and s

2 δ (|x| 2 )|x| = |sat δ (x)|, V1 is negative definite provided that 4C 3,1 C 3,2 C 3,3 > C 3,1 C 2 3,6 + C 3,2 C 2 3,5 + C 3,3 C 2 3,4 + 1 2 C 3,4 C 3,5 C 3,6 (66)
This condition is satisfied by a proper choice of κ and k 3 . Indeed, the only term depending on k 3 in ( 66) is C 3,2 . Thus, ( 66) is satisfied for k 3 large enough provided that 4C 3,1 C 3,3 > C 2 3,5 .

From ( 65), this is equivalent to 4C 1,1 C 1,3 > C 2 1,2 . This inequality, which corresponds to Condition c) in ( 52), is satisfied for κ > 0 small enough.

Let us now take into account the additive perturbation μ. It follows from ( 65) and ( 66) that for some β > 0,

V1 ≤ -β(|y| 2 + |z -x| 2 + |sat δ (x)| 2 ) + | ∂V ∂y || μ| (67)
and from ( 46), there exists c > 0 such that

∂V ∂y (x, y) ≤ c|(y, sat δ (x))| ∀(x, y) (68)
Therefore, by the triangular inequality,

V1 ≤ - β 2 |y| 2 + |z -x| 2 + |sat δ (x)| 2 + c 2 | μ| 2 2β
Convergence to zero of V 1 then follows from the convergence of μ to zero. This, together with Lemma 3 implies the convergence of (p, ṗ, q, η) to (p r , ṗr , q r , 0) from any initial condition in A. Finally, local exponential stability follows from local exponential stability of μ = 0 (Lemma 3) and the fact that saturation functions are identity functions around the origin (i.e., System (62) is locally linear).

Property ii): Lemma 3 still implies that μ exponentially converges to zero. Proceeding as in the proof of Prop. 1-ii), one deduces from Lemma 2 that for any t ≥ T ρ := max |y|≤ρ T (y) and along any solution starting from A ρ , sat¯δ desaturates and the solution satisfies

               ẋ = My + Ṁ p ẏ = -k 1 sat δ (x) -k 2 y -k 1 [sat δ (z) -sat δ (x)] + R μ ż -ẋ = -k 3 (z -x) -My -Ṁ p (69)
Note that System (69) is equivalent to System (54) when z = x and μ = 0. Since μ exponentially converges to zero and

| μ(0)| ≤ |µ(0)| ≤ 2g + k 1 δ M + k 2 δM , there exists T 1 > 0 such that, for any initial condition in A ρ , | μ(t)| ≤ βδ m c √ 6 , ∀t ≥ T 1 (70)
with β satisfying (67). Thus, for t ≥ Tρ := max{T ρ , T 1 }, both ( 69) and ( 70) are satisfied. Proceeding as in the proof of Prop. 1-ii), one also deduces from ( 55) that for some ρ and along any solution with initial condition in

A ρ , |(x( Tρ ), y( Tρ ), (z - x)( Tρ ))| ≤ ρ 1 . Let ρ := max{ρ 1 , δ m } so that |(x( Tρ ), y( Tρ ), (z - x)( Tρ ))| ≤ ρ.
We consider again the CLF V 1 defined by ( 64) and define s m , x M , V M as follows (compare with (58)):

V M := max |(x,y,z-x)|≤ ρ V 1 (x, y, z) > 0 x M := max V 1 (x,y,z)≤V M |x| > 0 s m := min |x|≤x M s δ (|x| 2 ) > 0 (71)
By using (59), one deduces from (69) that for |x| ≤ x M and for

M such that | Ṁ| s < ϑ, V1 ≤ -C 4,1 |y| 2 -C 4,2 |z -x| 2 -C 4,3 sat 2 δ (|x| 2 )|x| 2 +C 4,4 |y||z -x| + C 4,5 |y|sat δ (|x| 2 )|x| +C 4,6 |z -x|sat δ (|x| 2 )|x| + ∂V ∂y R μ with:                  C 4,1 := C 3,1 -ϑ, C 4,2 := C 3,2 C 4,3 := C 3,3 -2k 1 s m |M| i ϑ, C 4,4 := C 3,4 C 4,5 := C 3,5 + 2 k 1 k 2 + 2 k 1 k 2 |M a | s + κ C δ s m |M| i ϑ C 4,6 := C 3,6 + ϑ s m |M| i
Let us choose κ and k 3 as in the proof of case i) above, so that (66) is satisfied. Since the C 4, j 's tend to the C 3, j 's as ϑ tends to zero, for ϑ > 0 small enough the following inequality is satisfied (compare with (67)):

V1 ≤ - 2β 3 (|y| 2 + |z -x| 2 + |sat δ (x)| 2 ) + | ∂V ∂y || μ| (72)
Therefore, by ( 68) and the triangular inequality,

V1 ≤ - β 6 |y| 2 + |z -x| 2 + |sat δ (x)| 2 + c 2 | μ| 2 2β (73)
Recall that this relation is true as long as |x(t)| ≤ x M and t ≥ Tρ . In particular, it is true at t = Tρ because |(x( Tρ ), y( Tρ ), (zx)( Tρ ))| ≤ ρ (see above) and from (71),

|(x, y, z -x)| ≤ ρ =⇒ V 1 (x, y, z) ≤ V M =⇒ |x| ≤ x M
From ( 73) and ( 70), |x(t)| ≤ x M and t ≥ Tρ imply that

V1 ≤ - β 6 |y| 2 + |z -x| 2 + |sat δ (x)| 2 - δ 2 m 2 (74)
We claim that

V 1 (x, y, z) = V M =⇒ |(x, y, z -x)| ≥ δ m (75)
Indeed, otherwise, from the definition of V M in (71) and the fact that ρ ≥ δ m , on the set {(x, y, z) : |(x, y, zx)| ≤ ρ} V 1 reaches its maximum in the interior of this set, which implies that V 1 has a critical point. This contradicts (72) that implies V 1 is a Lyapunov function for μ = 0. From (71), V 1 = V M implies that |x| ≤ x M and thus, (74) holds true. We thus deduce from (74), (75), and the properties of the function sat δ that for any t ≥ Tρ ,

V 1 = V M =⇒ V1 ≤ -β 6 δ 2 m 2 < 0. Since V 1 ≤ V M at t = Tρ , V 1 ≤ V M
everafter and thus, |x| ≤ x M everafter. Then, ( 73) is satisfied for any t ≥ Tρ and convergence to zero of x, y, z follows from the convergence of μ to zero. Local exponential stability is deduced as for Property i).

Proof of Proposition 3: Let

x = R σ, y = Rṽ, Y = (µ 1 , µ 2 ) T , and ε = 1 k 4 .
One obtains in closed-loop, after some calculations:

               ẋ = My + Ṁ p ẏ = -k 1 sat δ (x) -k 2 sat¯δ (y) + R(Y 1 , Y 2 , 0) T ε Ẏ = -µ 3 Y + εω 3 (t)Y ⊥ +εR T 1,2 (k 1 F δ (x) ẋ + k 2 Fδ(y) ẏ) (76)
with

Y ⊥ = (Y 2 , -Y 1 ) T , R T
1,2 the first two lines of R T , and

F δ (x) := s δ (|x| 2 )I + 2s δ (|x| 2 )xx T Fδ(y) := sδ(|y| 2 )I + 2s δ(|y| 2 )yy T (77)
Property i): It relies on the following lemma.

Lemma 4 Assume A1, A2, and A3. Then, for any ρ > 0 there exists ε ρ , T > 0 such that, for any ε ∈ (0, ε ρ ), any t ≥ T , and along any solution of the system with initial condition in A ρ : a) |y(t)| ≤ δm (i.e. the function sat¯δ desaturates);

b) µ 3 (t) > gκ/ √ 2.
Lemma 4 implies that by choosing ε small enough, along any trajectory with initial condition in A ρ , the function sat¯δ desaturates after some time, so that the trajectory becomes solution to the system

               ẋ = My ẏ = -k 1 sat δ (x) -k 2 y + R(Y 1 , Y 2 , 0) T ε Ẏ = -µ 3 Y + εω 3 (t)Y ⊥ +εR T 1,2 (k 1 F δ (x) ẋ + k 2 Fδ(y) ẏ) (78)
The first two equations of this system correspond to (45), modulo the additional term R(Y 1 , Y 2 , 0) T . Then, using (53), property b) of Lemma 4, and (68), one deduces that V 2 (x, y, Y) := V(x, y) + |Y| 2 is a Lyapunov function for System (78) for ε small enough, with V upper-bounded by a negative definite quadratic function of sat δ (x), y, and Y.

Property ii): It relies on the following lemma.

Lemma 5 Assume A1 and A2. Then, for any ρ > 0 there exist ε ρ , T 0 , c5 , β, c, ϑ > 0 such that, for any ε ∈ (0, ε ρ ), any t ≥ T 0 , any M such that | Ṁ| s < ϑ, and along any solution of the system with initial condition in

A ρ : a) |x(t)| ≤ c5 ; b) |y(t)| ≤ δm (i.e. the function sat¯δ desaturates); c) µ 3 (t) > gκ/ √ 2; d) V ≤ -β 2 |(y, sat δ (x))| 2 + c|(y, sat δ (x))
||Y|, with V defined by (46). Lemma 5 implies that by choosing ε and ϑ small enough, along any trajectory with initial condition in A ρ , the function sat¯δ desaturates after time T 0 , so that the trajectory becomes solution to the system

               ẋ = My + Ṁ p ẏ = -k 1 sat δ (x) -k 2 y + R(Y 1 , Y 2 , 0) T ε Ẏ = -µ 3 Y + εω 3 (t)Y ⊥ +εR T 1,2 (k 1 F δ (x) ẋ + k 2 Fδ(y) ẏ) (79)
The first two equations of this system correspond to (54), mod- Proof of Proposition 5: Let x := R σ = M p and y := M -1 Rv M = ṗ. Then, x, y satisfy the following equations:

ulo the term R(Y 1 , Y 2 , 0) T . Property a) of Lemma 5 implies that x is bounded, so that |( ẋ, ẏ)| is
ẋ = My ẏ = -k 1 sat δ (x) -k 2 sat¯δ(My) (80)
This is the same as (42), except for the presence of the matrix M in the term sat¯δ(My). This similitude allows one to duplicate the proof of Prop. 1 modulo minor adaptations detailed below. Lemma 2, with sat¯δ(y) in ( 43) replaced by sat¯δ(My) and the desaturation condition (44) replaced by

|M (t)| ≤ k 2 δm |M| i -k 1 δ M |M| s -c , ∀t ≥ t 1 (81)
still holds true to show that sat¯δ(My) desaturates. Indeed, ȳ := My satisfies the equation

ẏ = -k 1 Msat δ (x) -k 2 Msat¯δ(ȳ) + M (t)
and the proof proceeds like for Lemma 2, by considering the Lyapunov function V 0 defined by V 0 (ȳ) = 1 2 |ȳ| 2 . The second inequality in (32) ensures that ( 81) is satisfied with ≡ 0 and some c > 0. After desaturation, solutions to (80) satisfy the following equations (compare with (45)):

ẋ = My ẏ = -k 1 sat δ (x) -k 2 My (82)
The Lyapunov function in ( 46) is modified as follows:

V(x, y) = k 1 |x| 2 0 s δ (τ) dτ + y T My + 2k 1 k 2 sat δ (x) T M a M -1 y + κ sat δ (x) T y (83)
i.e., V so defined only differs from (46) by the matrix term M a M -1 (in place of M a ). We show that V is a Lyapunov function for κ small enough. Let (compare with (47))

                         κ := k 2 2 |M| 2 i -k 1 |M a M -1 | s |M| s C δ > 0 κ 1 := 2 √ k 1 |M| i -2 k 1 k 2 |M a M -1 | s > 0 κ 2 := 2κ k 2 |M| s C δ > 0 κ 3 := 2κ k 2 |M| s C δ + k 3 2 |M| 2 s 4k 1 > 0 0 < κ < min {κ 1 , κ 2 , κ 3 } (84)
where positivity of κ, κ 1 follows from (32) and positivity of κ 2 , κ 3 is a consequence thereof. Positive definiteness of V is still ensured by (48), with b now defined by b := 2k 1 k 2 |M a M -1 | s + κ. This yields the condition κ < κ 1 . Differentiating V along the solutions of System (82) yields

           V = -2k 2 y T M s My + 2 k 1 k 2 y T (M a M -1 ) T F δ (x)My -κk 1 s 2 δ (|x| 2 )|x| 2 + κy T F δ (x)My -κk 2 s δ (|x| 2 )x T My -2 k 2 1 k 2 sat δ (x) T M a M -1 sat δ (x) (85)
We now use the assumption that M a and M s commute. This implies that, for any ξ ∈ R 3 , ξ T M s M a ξ = 0 and

ξ T M a M -1 ξ = ((M a + M s )M -1 ξ) T M a M -1 ξ = |M a M -1 ξ| 2
Therefore, one deduces from (85) that

           V ≤ -2k 2 |M s y| 2 + 2 k 1 k 2 y T (M a M -1 ) T F δ (x)My -κk 1 s 2 δ (|x| 2 )|x| 2 + κy T F δ (x)My -κk 2 s δ (|x| 2 )x T My (86)
This implies that ( 50) is still satisfied with the C 1, j 's defined by (compare with (51))

C 1,1 := 2k 2 |M| 2 i -b|M| s C δ , C 1,2 := κk 2 |M| s , C 1,3 := κk 1
Thus, V is negative definite provided that ( 52) is satisfied with the above-defined C 1, j 's. Condition a) follows from the fact that κ < κ 2 . Condition b) holds true for any κ > 0 since k 1 > 0, and Condition c) follows from the fact that κ < κ 3 . Thus, ( 53) is satisfied for some β > 0 and the end of the proof follows like for Proposition 1.

Proof of Proposition 6: Let x := R σ, y := Rṽ, Y := (µ 1 , µ 2 ) T , and ε := 1 k 4 . One obtains in closed-loop, after some calculations (compare with (76) for Ṁ = 0):

               ẋ = My ẏ = -k 1 sat δ (x) -k 2 sat¯δ (My) + R(Y 1 , Y 2 , 0) T ε Ẏ = -µ 3 Y + εω 3 (t)Y ⊥ +εR T 1,2 (k 1 F δ (x) ẋ + k 2 Fδ(My) M ẏ) (87)
with Y ⊥ = (Y 2 , -Y 1 ) T , R T 1,2 the first two lines of R T , and F δ , Fδ defined by (77). By setting, like in the proof of Prop. 5, ȳ := My = R T v M , the above equations can also be written as

               ẋ = ȳ ẏ = -k 1 Msat δ (x) -k 2 Msat¯δ (ȳ) + MR(Y 1 , Y 2 , 0) T ε Ẏ = -µ 3 Y + εω 3 (t)Y ⊥ +εR T 1,2 (k 1 F δ (x) ẋ + k 2 Fδ(ȳ) ẏ) (88)
We claim that the conclusion of Lemma 4 is still valid with statement a) replaced by: a) |ȳ(t)| ≤ δm (i.e. the function sat¯δ desaturates) Indeed, based on (87) the proof follows exactly that of Lemma 4 with y replaced everywhere by ȳ, and the constant k 2 δmk 1 δ M in (102) and subsequent equations replaced everywhere by k 2 δm |M| ik 1 δ M |M| s (compare with ( 81)). Note also that the value of the constants c 1 , c 2 must be changed as follows: 35)) and therefore, from (88), c 2 := |M| s (2k 1 δ M + 2k 2 δM + g).

c 1 := k 1 δ M + k 2 δM + g (from (
After desaturation of the function sat¯δ, the trajectories of System (87) become solution to the system

               ẋ = My ẏ = -k 1 sat δ (x) -k 2 My + R(Y 1 , Y 2 , 0) T ε Ẏ = -µ 3 Y + εω 3 (t)Y ⊥ +εR T 1,2 (k 1 F δ (x) ẋ + k 2 Fδ(My) M ẏ) (89)
The first two equations in (89) correspond to (82) modulo the additional term R(Y 1 , Y 2 , 0) T . The end of proof follows as for Prop. 3, using the fact that V in ( 83) satisfies (53).

Proof of Lemmas

Proof of Lemma 1: We proceed by contradiction. Assume that µ(0) = -|µ(0)|e 3 . From ( 20) and ( 23) µ(0) 0. From (20), γ(0) + |µ(0)|e 3 = -(k 1 sat δ (η) + k 2 sat¯δ(ṽ)a r ). Thus,

|k 1 sat δ (η) + k 2 sat¯δ(ṽ) -a r | 2 = |γ(0) + |µ(0)|e 3 | 2 = |γ(0)| 2 + |µ(0)| 2 + 2|µ(0)|γ 3 (0) = g 2 + |µ(0)| 2 + 2|µ(0)|γ 3 (0) > g 2 + |µ(0)| 2 -2|µ(0)| g 2 -k 1 δ M + k 2 δM + | pr | s 2 > k 1 δ M + k 2 δM + | pr | s 2 + g 2 -k 1 δ M + k 2 δM + | pr | s 2 -|µ(0)| 2 > k 1 δ M + k 2 δM + | pr | s 2
where the first inequality comes from ( 24) and the fact that µ(0) 0. This contradicts (23).

Proof of Lemma 2: Consider the function V 0 defined by V 0 (y) = 1 2 |y| 2 . Its derivative along the solutions of Eq. ( 43) is given by

V0 = -k 1 y T sat δ (x) -k 2 y T sat¯δ(y) + y T (t) ≤ k 1 |y|δ M -k 2 y T sat¯δ(y) + |y|| (t)| (90)
Case 1: t ∈ [0, t 1 ]. Let | | s denote the max of on this time interval. From the above inequality,

V0 ≤ k 1 |y|δ M + |y|| | s ≤ (k 1 δ M + | | s ) √ 2V 0
The comparison lemma [START_REF] Khalil | Nonlinear systems, 3rd Edition[END_REF]

:= t 1 + max{0, |y(0)|+(k 1 δ M +| | s )t 1 -δm c }.
In other words, sat¯δ(y(t)) = y(t) for t ≥ T (y(0)).

Proof of Lemma 3: It relies on the following result (Hua et al., 2009, Prop.

      ω 1 = -k 4 ζ2 (1+ ζ3 ) 2 -ζ T S (Re 1 ) ζ ω 2 = k 4 ζ1 (1+ ζ3 ) 2 -ζ T S (Re 2 ) ζ , k 4 > 0 (92)
Then, on the unit sphere, Re 3 = ζ is exponentially stable with convergence domain {R(0)e 3 : ζ T (0)R(0)e 3 -1}.

We apply the above proposition with ζ := Rµ |µ| . First, ζ is well defined because, from ( 20) and ( 23), µ never vanishes. Then, ζ is a smooth function since both R and µ are smooth. Let us check that ζ is independent of ω. First, recall that by Def. 1i), Rsat(z) = sat(Rz) for any R, z. Therefore, from (20) and the definitions of γ, ṽ, and a r ,

ζ = ge 3 + k 1 sat δ (Rη) + k 2 sat¯δ( p) -pr |ge 3 + k 1 sat δ (Rη) + k 2 sat¯δ( p) -pr |
The derivatives of ge 3 , k 2 sat δ ( p) and pr do not depend on ω. As for Rη, it follows from ( 6) and ( 18) that d dt (Rη) = -k 3 (Rη-M p). This term is thus also independent of ω.

T (0)R(0)e 3 -1}. Thus, ∃c, α > 0 : |Re 3 -ζ| ≤ c|(Re 3 -ζ)(0)|e -αt , ∀t. This is equiv- alent to |e 3 -µ |µ| | ≤ c|(e 3 -µ |µ| )(0)|e -αt , ∀t. Since, from A1
, (20), and (23), |µ| is lower and upper-bounded by strictly positive constants independent of the initial conditions, the above inequality is equivalent to | μ(t)| ≤ c| μ(0)|e -αt for some constant c.

Proof of Lemma 4: Since A ρ * ⊂ A ρ when ρ * < ρ, it is sufficient to prove the existence of ε ρ > 0 for any ρ > ρ * , where ρ * is any strictly positive value. Thus, we assume from now on that ρ > δm (93)

Let us first establish a few inequalities. From ( 25) and ( 27),

|Y| 2 + µ 2 3 = |µ| 2 > g 2 κ 2 . Thus, ∀t ≥ 0, µ 3 (t) 2 > g 2 κ 2 -|Y(t)| 2 (94)
From ( 27) and Assumption A3,

|µ| < c 1 := k 1 δ M + k 2 δM + 2g (95) Therefore, |Y| = |(µ 1 , µ 2 ) T | < c 1 and from (76), |ẏ| < c 2 := 2(k 1 δ M + k 2 δM + g) (96)
Recalling that y = Rṽ, it follows from ( 28) and ( 96) that

∀t > 0, |y(t)| < ρ + c 2 t ( 97
)
Since M is constant, (76) implies that ẋ = My. Recalling, from (2), that F δ and Fδ are bounded by C δ , it follows from (76) that

ε Ẏ = -µ 3 Y + εω 3 (t)Y ⊥ + ε(ξ 1 + ξ 2 y) (98)
where ξ 1 , ξ 2 are functions bounded by a constant c 3 independent of ε. Thus, using the triangular inequality,

d dt |Y| 2 ≤ |Y| 2 - 2µ 3 ε + 1 + c 2 3 (1 + |y| 2 ) (99)
Let T 0 denote any strictly positive constant. It follows from ( 97) and ( 99) that

∀t ∈ [0, T 0 ], d dt |Y| 2 (t) ≤ |Y| 2 (t) - 2µ 3 (t) ε + 1 + c 4 (100) with c 4 := c 2 3 (1 + (ρ + c 2 T 0 ) 2 ) (101)
We show that there exists ε 0 > 0 such that, for any ε ≤ ε 0 and any initial condition in A ρ ,

|Y(T 0 )| < κ := min 9(k 2 δm -k 1 δ M ) 10 , gκ/ √ 2 (102)
We claim that there exists ε1 > 0 such that, ∀ε ≤ ε1 ,

∀t ∈ [0, T 0 ], |Y(t)| 2 < g 2 κ 2 /2 (103)
Suppose on the contrary that there exists a sequence (ε n ) n>0 converging to zero such that, for any ε = ε n there exists a time tn ∈ [0, T 0 ] such that |Y( tn )| 2 ≥ g 2 κ 2 /2. Since, from (28), |Y(0)| 2 < g 2 κ 2 /2, we can assume without loss of generality that

|Y(t)| 2 ≤ |Y( tn )| 2 = g 2 κ 2 /2 for t ≤ tn . Therefore, d dt |Y| 2 ( tn ) ≥ 0 (104)
On the other hand, from (94), the fact that µ 3 (0) > 0 (cf. ( 28)), and the fact that |Y(t)| 2 ≤ g 2 κ 2 /2 for t ≤ tn , we deduce that µ 3 ( tn ) > gκ/ √ 2. Thus, we deduce from (100) and (104) that 0

≤ g 2 κ 2 2 - 2µ 3 ( t) ε + 1 + c 4 < g 2 κ 2 2       - √ 2gκ ε + 1       + c 4
This is impossible if ε < ε1 := g 3 κ 3 √ 2/(g 2 κ 2 + 2c 4 ), which shows (103) for ε ≤ ε1 . From ( 103), (94), and the fact that µ 3 (0) > 0, it follows that µ 3 ( t) > gκ/ √ 2 for all t ∈ [0, T 0 ]. In other words, for ε ≤ ε1 ,

∀t ∈ [0, T 0 ], |Y(t)| 2 < g 2 κ 2 /2 , µ 3 (t) > gκ/ √ 2 (105) Therefore, from (100), ∀t ∈ [0, T 0 ], d dt |Y| 2 (t) ≤ |Y| 2 (t)       - gκ √ 2 ε + 1       + c 4 Applying the comparison lemma yields ∀t ∈ [0, T 0 ], |Y| 2 (t) ≤ e -at |Y| 2 (0) - c 4 a + c 4 a (106)
with a = gκ √ 2 ε -1. Since a tends to infinity as ε tends to zero and |Y(0)| < gκ/ √ 2, there exists ε2 such that, ∀ε ≤ ε2 ,

e -aT 0 |Y| 2 (0) - c 4 a + c 4 a < 9(k 2 δm -k 1 δ M ) 10
This inequality, together with ( 106) and ( 103) imply (102) for ε ≤ ε 0 := min{ ε1 , ε2 }. We now assume that

ε ∈ (0, ε ρ ) with ε ρ = min        ε 0 , √ 2gκ κ2 κ2 + c 4        (107)
We claim that

∀t ≥ T 0 ,          |y(t)| < ρ + c 2 T 0 |Y(t)| < κ µ 3 (t) > gκ/ √ 2 (108)
Since ε < ε 0 , it follows from ( 97), ( 102), and (105) that the three inequalities in ( 108) are satisfied at t = T 0 . Suppose by contradiction that there exists T > T 0 such that at least one of the three inequalities in ( 108) is not satisfied at t = T and these inequalities are satisfied for any t ∈ [T 0 , T ). We distinguish three cases: where the last inequality follows from (107). This contradicts (111). Case 3: µ 3 (T ) = gκ/ √ 2. Since ( 108) is satisfied for t < T , it follows by continuity that |Y(T )| 2 ≤ g 2 κ 2 /2. Thus, µ 3 (T ) 2 + |Y(T )| 2 ≤ g 2 κ 2 , which contradicts (94).

Case 1: |y(T )| = ρ + c 2 T 0 . We claim that ∀t ∈ [T 0 , T ), |y(t)| ≤ max{ δm , |y(T 0 )|} (109) Indeed, from (108) for t ∈ [T 0 , T ), |Y(t)| ≤ κ < k 2 δm -k 1 δ M for any t ∈ [T 0 , T ]. Therefore, by (76), ∀t ∈ [T 0 , T ], |y(t)| ≥ δm =⇒ d dt |y|(t) < 0 (110)
This concludes the proof of (108). Lemma 4 follows from (108) and Lemma 2 applied to the second equation in ( 76 where the last inequality comes from (114). We deduce from ( 115), (117), and ( 118) that ( 100) is satisfied with c 4 now defined by c 4 := c 2 3 1 + δ2 m + c2 5 ( 119)

From here, one can proceed as in the proof of Lemma 4 to show that for ε smaller than some ε 0 > 0, (102) and (105) hold true with κ defined by (113).

Recall from (113) that T 0 := δm 2c 2 . Thus, from (115), |y(T 0 )| < δm . We impose on ε the condition (107), with c 4 defined by (119). We claim that there exists ϑ > 0 such that, provided that | Ṁ| s ≤ ϑ,

∀t ≥ T 0 , i) |V(x(t), y(t))| < 2 V ii) |y(t)| < δm iii) |Y(t)| < κ iv) µ 3 (t) > gκ √ 2 (120)
The four properties in ( 120) are satisfied at t = T 0 . Indeed, by ( 113), (115), and (118), |V(x(T 0 ), y(T 0 ))| ≤ V. Property ii), at t = T 0 , follows from (115). Properties iii-iv) follow from ( 102) and ( 105), which are satisfied because ε < ε 0 . To prove (120) we proceed by contradiction. Suppose that for any ϑ > 0 there exists some M with | Ṁ| s ≤ ϑ and some T > T 0 such that at least one of the properties in (108) is not satisfied at t = T and these properties are satisfied ∀t ∈ [T 0 , T ). We distinguish four cases. 120) is satisfied for t ∈ [T 0 , T ), it follows that d dt |Y| 2 (T ) ≥ 0. Furthermore, (120) for t < T and (113) imply that x(T ) ≤ c5 . This implies, from (117) and (120) for t < T , that (112) is satisfied with c 4 defined by (119). We thus obtain a contradiction. Case 4: µ 3 (T ) = gκ/ √ 2. One proceeds as in Case 3 of Lemma 4 to show that this case also yields a contradiction.

We thus have proved that (120) is satisfied if ϑ > 0 is chosen small enough. Property a) of Lemma 5 follows from i) in ( 120) and ( 113). Properties b-c) correspond to ii-iii) in (120).

  and there exists a constant scalar | ṗr | s such that: ∀t, | ṗr (t)| ≤ | ṗr | s (7) A2: M is a constant function. A3: ... p r = 0 and | pr | < g.

  Figure 1: Problem scheme

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: Fixed-point stabilization: control law of Proposition 2.

Figure 5 :

 5 Figure 5: Way-points navigation: control law of Proposition 3.

  bounded by a linear function of |y|, |sat δ (x)|, and |Y| (see (59)). From here, Properties c) and d) of Lemma 4 and boundedness of F δ , Fδ imply that V 2 (x, y, Y) := V(x, y) + |Y| 2 is a Lyapunov function for System (79) for ε small enough.Proof of Proposition 4:Let ēσ := Re σ = R σ -M p, ēv := Re v = Rv M -M ṗ.From (30)-(31), ėσ = ēv -2αkē σ and ėv = -k 2 ēσ -M p. Thus, e σ is solution to a Hurwitz-stable second order linear equation with bounded input u b := -M p, i.e., ëσ + 2αk ėσ + k 2 ēσ = u b . Let τ := kt and denote the derivative w.r.t. τ. Then, the previous equation becomes ē σ + 2αē σ + ēσ = u b k 2 . Since u b is bounded, this implies the ultimate boundedness of |ē σ | and |ē σ | by a value proportional to 1/k 2 . Hence, |ē σ | and | ėσ | are ultimately bounded by a value proportional to 1/k 2 and 1/k respectively. The same property holds for |e σ | and |e v |.

1 )Proposition 7

 17 Consider a smooth function ζ with |ζ| = 1 and ζ independent of ω. Let ζ := R T ζ and 

  Thus, if max{ δm , |y(T 0 )|} = |y(T 0 )| then |y(t)| ≤ |y(T 0 )| for all t ∈ [T 0 , T ), and if max{ δm , |y(T 0 )|} = δm then |y(t)| ≤ δm for all t ∈ [T 0 , T ). This proves (109). By continuity, |y(T )| ≤ max{ δm , |y(T 0 )|}. This contradicts the assumption |y(T )| = ρ + c 2 T 0 since |y(T 0 )| < ρ + c 2 T 0 by (97), and δm < ρ by (93). Case 2: |Y(T )| = κ. Since (108) is satisfied for t ∈ [T 0 , T ), (99), (101), and (108) for t < T , that d dt |Y| 2 (T ) ≤ |Y| 2 (T ) -

  ) with (t) = R(t)(Y 1 (t), Y 2 (t), 0) T .Proof of Lemma 5: It is inspired by the proof of Lemma 4. Let us first define some constant numbers. With c 2 defined by (96) and V given by (46) the Lyapunov function of Prop. 1, let us defineT 0 := δm 2c 2 , c 5 := ρ + δm T max{|V(x, y)| : |x| ≤ c 5 , |y| ≤ δm } c5 := max{|x| : inf y V(x, y) ≤ 2 V} c 6 := min{|(y, sat δ (x))| : V(x, y) = 2 V} κ := min 9(k 2 δm -k 1 δ M ) 10, constant satisfying (68). Note that c5 is well defined because V is proper. Note also, from the definition of V and c5 , that c 5 ≤ c5 (114)Relations (94) to (96) are still valid when M is not constant. As for (97), using (29) it becomes∀t > 0, |y(t)| < δm 2 + c 2 t (115)Due to the term Ṁ p = ṀM -1 x in the expression (76) of ẋ, relation (98) becomesε Ẏ = -µ 3 Y + εω 3 (t)Y ⊥ + ε(ξ 1 + ξ 2 y + ξ 3 x) (116)with ξ 1 , ξ 2 , ξ 3 functions bounded by a constant c 3 independent of ε. Thus, by the triangular inequality (99) becomesd dt |Y| 2 ≤ |Y| 2 -2µ 3 ε + 1 + c 2 3 (1 + |y| 2 + |x| 2 ) (117) Since ṗ = Rṽ = y, | p(t)p(0)| ≤ t 0 |y(s)| ds for all t. Thus, from (29) and (115), | p(t)| < ρ + δm t 2 + c 2 t 2 2 for any t > 0, which implies by the definition (113) of c 5 that ∀t ∈ (0, T 0 ], |x(t)| < c 5 ≤ c5 (118)

  Case 1: |V(x(T ), y(T ))| = 2 V. Since (120) is satisfied for t ∈ [T 0 , T ), one has V(x(T ), y(T )) ≥ 0 (121)One also deduces from (120) on the time-interval [T 0 , T ) that on the time-interval [T 0 , T ], the function sat¯δ desaturates. Thus, from (76), the solution satisfies on [T 0 , T ]:ẋ = My + Ṁ p ẏ = -k 1 sat δ (x)k 2 y + R(Y 1 , Y 2 , 0) T (122)This system corresponds to (54) modulo the additive "pertur-bation" R(Y 1 , Y 2 , 0) T . It follows from (120) for t ∈ [T 0 , T ] that |V(x(T ), y(T ))| ≤ 2 V. From (113), this implies that |x(t)| ≤ c5 ∀t ∈ [T 0 , T ].Using the same argument as for (59), one de-duces that ∀t ∈ [T 0 , T ], |x(t)| ≤ |sat δ (x)| s mwith s m := min |x|≤c 5 s δ (|x| 2 ). With this definition of s m , it follows from (60) that the derivative of V along the solutions of (122) satisfies, on the time-interval [T 0 , T ]:V ≤ -C 2,1 |y| 2 + C 2,2 |y||sat δ (x)| -C 2,3 |sat δ (x)| 2 + ∂V ∂y R(Y 1 , Y 2 , 0) T (123)with C 2,1 , C 2,2 , C 2,3 defined by (61). From (61), the coefficients C 2, j tend to C 1, j as ϑ tends to zero. Therefore, by (53) there exists ϑ > 0 such that, for| Ṁ| s ≤ ϑ, + |sat δ (x)| 2 ) + c|(y, sat δ (x))||Y| (125)It thus follows from (120) for t ∈ [T 0 , T ) that, on the timeinterval [T 0 , T ], V ≤ -β 2 (|y| 2 + |sat δ (x)| 2 ) + c κ|(y, sat δ (x))|(126)From (113) and the assumption V(x(T ), y(T )) = 2 V, one deduces that |(y(T ), sat δ (x(T ))| ≥ c 6 . Then, it follows from (126) and (113) that V(x(T ), y(T )) < 0. This contradicts (121). Case 2: |y(T )| = δm . Since (120) is satisfied for t ∈ [T 0 , T ), it follows that d dt |y| 2 (T ) ≥ 0. Since |Y(t)| ≤ κ on [T 0 , T ], relation (110) still holds, which implies that d dt |y| 2 (T ) < 0, thus a contradiction. Case 3: |Y(T )| = κ. Since (

  Finally, since |x(t)| ≤ c5 , (125) is satisfied, which corresponds to Property d) of Lemma 5.

  Thus ζ is independent of ω. By replacing ζ in (92) by the expression ζ := Rµ |µ| , one obtains after a few calculation the expression (19). By application of Prop. 7, we deduce that Re 3 = ζ is exponentially stable with convergence domain {R : ζ

$ This work has been supported by the "Chaire d'excellence en Robotique RTE-UPMC" and the ANR SCAR Project.


Antoine Adenis

Valentin Dufit

Maylis Douine

Florine Corlin

Gulen Ayhan

Fatiha Najioullah

Vincent Molinie

Paul Brousse

Gabriel Carles

Vincent Lacoste

Raymond Cesaire

Mathieu Nacher
email: mathieu.nacher66@gmail.com

High prevalence of HPV infection in the remote villages of French Guiana: an epidemiological study

Keywords: HPV, epidemiology, French Guiana, prevalence

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Cervical cancer is a global public health problem caused by persistent HPV infection leading to malignancy [START_REF] Walboomers | Human papillomavirus is a necessary cause of invasive cervical cancer worldwide[END_REF] . For women, it ranks second in terms of cancer incidence and mortality worldwide, and first in several developing countries. The global burden estimates suggest that annual incidence is 510 000 and mortality is 288 000 each year worldwide.

In France, cervical cancer is ranked number 8 in terms of cancer incidence. [START_REF] Hantz | Anti-papillomavirus vaccines and prevention of cervical cancer: progress and prospects[END_REF] In the French overseas territories in the Americas the incidence rates are situated between what is observed in mainland France and the Caribbean 3 , one of the most affected regions worldwide [START_REF] Eluf-Neto | Cervical cancer in Latin America[END_REF] . In French Guiana, cervical cancer is the second most frequent cancer in females [START_REF] Roue | Cervical cancer incidence in French Guiana: South American[END_REF] still causing significant mortality [START_REF] Douine | Survival of patients with invasive cervical cancer in French Guiana, 2003-2008[END_REF] . In 2002-2005, the standardized incidence of cervical cancer was 30.3 per 100 000 women, which was close to that of neighboring Brazil [START_REF] Arrossi | Incidence and mortality of cervical cancer in Latin America[END_REF] . French Guiana is a sparsely populated region covered by primary forest, with populations living in remote parts of the territory only accessible by boat. Despite a network of health centers connected to the main hospital, there are some difficulties to access care. It was recently shown that despite the absence of significant difference in incidence between rural areas and urban areas, there were some differences in the diagnostic delays with 77% of the women from the rural areas with lesions metastasizing beyond the cervix versus 44% for women living in urban areas [START_REF] Roue | Cervical cancer incidence in French Guiana: South American[END_REF] . Health professionals concur that given the early start of sex life in populations living in these remote parts [START_REF] Van Melle | Knowledge, attitudes, behaviors, and practices differences regarding HIV in populations living along the Maroni river: particularities of operational interest for Amerindian and Maroon populations[END_REF] , the risk of cervical lesions due to HPV should be high. In these remote areas, sexual transmission is also seen for the HIV epidemic, which has soared from zero to over 1% within a decade [START_REF] Nacher | What is driving the HIV epidemic in French Guiana?[END_REF] . Given that HPV 16-18 are involved in 73% of cervical cancers in Europe, vaccination against these genotypes is now recommended in France. However, little is known about the epidemiology of HPV in French Guiana, a French territory with very different populations and a very different pathogen environment from that of mainland France. The relative isolation of Amerindian and maroon populations may thus lead to singular epidemiologic features. The objective of the present study was thus to describe the prevalence of HPV infections in general, the prevalence of the different genotypes, and of cytological anomalies in women aged 20-65 years living in the remote areas of French Guiana.

Methods

Study population : the source population was women living in the remote villages on the Maroni and Oyapock rivers. The target population was women aged 20-65 years having previously had sexual activity. The inclusions took place between Dec 2012 and Sept 2014.

Inclusion criteria:All women from the target population. Non inclusion criteria: women having had hysterectomy, pregnant women (>3 months pregnant).

Data collection : at the time of inclusion a short questionnaire was filled to collect socio economic and demographic data, gynecological and obstetrical history.

Study conduct: Before starting inclusions, communication missions in all the villages sensitized the local populations on this public health problem. Traditional authorities, health centers' workers were also informed. Local radio messages informed the population of the dates of the study team's presence in the village. During the inclusion, women wishing to be screened came to the health center where the project was explained to them, and where the questionnaire was filled and samples were taken and transferred in a cooler until the end of the mission. The samples were then send to the Virology laboratory of Fort de France Hospital, in Martinique where extraction and genotyping were performed using an automated method in order to reduce the risk of error and contamination. DNA extraction was performed using a minimum of 2 ml of sample in liquid phase.

HPV infection was defined by the detection of HPV DNA using the GREINER-BIO-ONE kit, used at the Virology laboratory in Fort de France University Hospital. This kit allows the identification of High risk HPV genotypes : 16 All results were sent to attending physicians for medical care. When HPV was positive and cytology was negative, a gynecological follow up was recommended to verify if HPV positivity disappeared or if cytologic lesions appeared. If both HPV and cytology was positive colposcopy was proposed.

Ethical and regulatory aspects:

Although it has been shown that HPV tests are superior to cytology for the detection of cervical cancer, in France, the HPV test is only reimbursed when the patient has a cytological examination revealing ASCUS. Here, all patients received the HPV test and cytology free of

Data analysis

Prevalence was obtained by dividing the number of women infected by at least one HPV genotype (high grade or low grade) by the total number of women included. For specific genotypes, prevalence was obtained by the number of women infected by that specific genotype by the total number of women included in the study. Prevalence of cytological anomalies was obtained by dividing the number of women with at least one cytological anomaly by the total number of women with an interpretable cytological examination.

The standardized HPV prevalence rate used the direct standardization method with the world population as a reference.

Unconditional multiple logistic regression was used to predict the variables associated with having a positive HPV test. Different age group-specific models were performed. Categorical variables were transformed into indicator variables. The data was analyzed using STATA 13© (College Station, Texas, USA). Mapping was performed using Mapinfo 12.0© (Troy, NY, USA).

Results

Overall the population of women living in the remote villages was estimated at 5712 on the Maroni and 1209 on the Oyapock. A total of 643 women (406 on the Maroni and 237 on the Oyapock) were included and 51 were not included because they had an exclusion criteria. On the Maroni 117 were included in Apatou, 69 in Grand Santi, 86 in Papaïchton, 65 in Maripasoula, 32 in Taluen, and 37 in Antecume Pata. On the Oyapock 98 were included in Saint Georges, 60 in Camopi, and 79 in Trois Sauts. The ratio between the number of women included in each village and the estimated target population number in each village ranged between a minimum of 6% in Grand Santi to 41.7% in Camopi. Overall, 9.3% of women from the target population entered the study. Overall, sexual relations started at a young age with 27% percent of the surveyed women had their first sexual relation before age 15, and 50% before age 16.

Table one shows very high HPV prevalence notably for high risk viruses.The overall agestandardized prevalence rate was 35%. Figure 1 shows there was a U shaped evolution of overall HPV prevalence by age and high risk HPV prevalence by age, with women over 50 years at highest risk for HPV, followed by the 20 to 29 years group. Maroon language was associated with a greater HPV prevalence whereas Portuguese language was associated with a lower prevalence. Other risk factors such as number of sex partners, parity, age at first sex, education, contraception, or smoking were not significantly linked to differences in HPV prevalence. None of the women had declared receiving the HPV vaccine.

Figure 2 shows the geographical differences in HPV prevalence with the lower part of the Maroni river having the highest prevalence. Table 2 shows that HPV 52 was the most frequent high risk virus. HPV16 was the second most frequent genotype in women aged less than forty years. It seemed less frequent in women above 40 years. Overall, the prevalence of HPV 16 or 18 exceeded 5% of tested women.

Overall 139 women had single HPV infections, 53 had double infections, and 22 had 3 or more different HPV genotypes.

A very large proportion (N=147, 27.2%) of women with normal cytology had in fact a positive HPV test. 4 show multiple regression analysis models predicting HPV positivity for different covariates. Few variables predicted HPV positivity. In the global models women aged 20-29 and women aged 50-64 years were at higher risk of having a positive HPV test, and of having a high risk genotype than other age groups as shown in Fig 1 . Table 4 shows that education was independently associated with HPV positivity in women 20-29 years but that among women aged 30-39 years, intermediate or high education were independently associated with a lower risk of having a positive HPV test. A number of reported sex partners of 2 or more in the past year was associated with an increased risk of having a positive HPV test in women aged 30-39 years.

studies showing a high incidence of cervical cancer suggest that HPV testing would be a better method than the default method which did not detect over 27% of women with HPV infection. HPV testing can also be performed using self tests, which could make it more acceptable for some women [START_REF] Toliman | Field evaluation of the Xpert(R) HPV Point of Care Test for the detection of human papillomavirus infection using self-collected vaginal and clinician-collected cervical specimens[END_REF] and less human resources-intensive. Given the high prevalence of HPV infection and the high incidence of cervical cancer in French Guiana, emphasis on vaccination and expanded and innovative screening seems important [START_REF] Lynge | What happens when organization of cervical cancer screening is delayed or stopped?[END_REF][START_REF] Sankaranarayanan | HPV screening for cervical cancer in rural India[END_REF] . The exact HPV vaccine coverage in these remote territories is unknown, but it is presumably very low. None of the surveyed women reported having been vaccinated. Given the early age of sexual life initiation, the optimal age of vaccination may need to be shifted downwards when compared to recommendations in mainland France. When looking at the variety of HPV genotypes present in these remote territories of French Guiana, the nonavalent HPV vaccines would seem more appropriate that the bivalent or tetravalent vaccines, which do not cover other circulating high risk genotypes [START_REF] Londesborough | Human papillomavirus genotype as a predictor of persistence and development of high-grade lesions in women with minor cervical abnormalities[END_REF] . This nonavalent vaccine has been recommended in France [START_REF] Riethmuller | Potential impact of a nonavalent HPV vaccine on the occurrence of HPV-related diseases in France[END_REF] and it would presumably have an even greater added value in French Guiana. Presently, given the large proportion of the population that do not have health insurance in these remote areas, the cost of the vaccine is prohibitive. Given the present results, Health authorities should tackle these economic obstacles to ensure that populations have access to this vaccine.

Overall, the present results show that over a third of women were infected with HPV, and that over a quarter of women with high risk HPV infection had normal cytology. Given the high incidence of cervical cancer in French Guiana and the relative delay in diagnosis [START_REF] Douine | Survival of patients with invasive cervical cancer in French Guiana, 2003-2008[END_REF][START_REF] Roué | Epidemiology of Cancer in the Tropical Areas[END_REF] , these results emphasize the importance of vaccination, preferably with a nonavalent vaccine. However, in this region where a large portion of the population has no health insurance, this still represents a challenge. Finally, the results underscore the importance of expanded screening using HPV tests given their better sensitivity than cytology, and their potential for

  charge. All included subjects gave written informed consent. Regulatory and ethical approval was given by the Comité d'Evaluation Ethique de l'Inserm (CEEI), approval n° 12-064; the Comité Consultatif sur le Traitement de l'Information en matière de Recherche dans le domaine de la Santé (CCTIRS), n° 12.310 ; the Commission Nationale de l'Informatique et des Libertés (CNIL), n° 912459 ; and the Comité de Recherche Clinique (CoRC) Pasteur Institute, n° 2012-15. The study was funded by the European Regional Development Fund (FEDER presage n° 30814).

  , 18, 31, 33, 35, 39, 45, 51, 52, 53, 56, 58, 59, 66, 68, 70, 73, 82 and of low risk HPV genotypes: 6, 11, 40, 42, 43, 44, 55. This kit allowed the identification of multiple infections. Cytologic anomalies were described using the 2001

	Bethesda classification.

Table 3 and table

 3

Discussion

The overall prevalence of HPV infection in the remote areas of this French territory was much higher than in mainland France. Moreover, the standardized HPV prevalence rate was comparable and even higher than published studies around the world [START_REF] Franceschi | Variations in the age-specific curves of human papillomavirus prevalence in women worldwide[END_REF] . The level of prevalence observed was comparable to what is usually observed in developing countries.

High HPV prevalence levels are consistent with the high incidence of cervical cancer observed in French Guiana, which also resembles more what is observed in developing countries than what is observed in France. The age at first sex was lower than in mainland France with differences between the Maroon regions, and the Amerindian regions.

The HPV prevalence varied according to age, and was highest in the Maroon villages. As observed elsewhere, and notably in Latin America and Africa, there was a U-shaped curve of HPV prevalence by age group [START_REF] Forman | Global burden of human papillomavirus and related diseases[END_REF][START_REF] Lazcano-Ponce | Epidemiology of HPV infection among Mexican women with normal cervical cytology[END_REF] . The prevalence rate reflects the combination of incidence, new infections, and the duration of infection. The U shape may have reflected new infections in these age groups and/or a slower clearance of viruses in older parous women [START_REF] Molano | Determinants of clearance of human papillomavirus infections in Colombian women with normal cytology: a population-based, 5year follow-up study[END_REF] .As other studies elsewhere, apart from age few variables were able to predict the risk of having a positive HPV test [START_REF] Molano | Determinants of clearance of human papillomavirus infections in Colombian women with normal cytology: a population-based, 5year follow-up study[END_REF][START_REF] Lazcano-Ponce | High prevalence of human papillomavirus infection in Mexican males: comparative study of penile-urethral swabs and urine samples[END_REF] .

The study limitations were that inclusion was voluntary, and that women living far from the health centers may have been less likely to be included in the study. Finally, cytology and histology from cervical biopsies would have been important to compare the frequency of the genotypes in all women receiving the HPV tests and in women with actual cervical lesions.

Nevertheless, this is the first study conducted in this border region.

Despite studies repeatedly showing the superior sensitivity of HPV tests over cytology, HPV tests are not reimbursed by the French health insurance as screening tools. They are only reimbursed if cytology shows ASCUS lesions or worse. The present results combined to the operational simplification in a region where health resources are limited and overstretched 15, 21, 22 .


Saving final or primary energy? Lessons from a flow approach of the French energy balance. Implications on the value of energy standards for buildings.

Ghislaine DESTAIS

Assistant professor, LEPII -CNRS -UPMF / Grenoble / France

The proposals of the Parliamentary Office (december 2009)  Modulation of the 50 kWh/m 2 /year limit according to location between -10% and +30% (40 to 65 kWh) for 6 areas, corresponding to the BBC-Effinergie label  Modulation on heigh ground + 10 kWh above 400 m, + 20 kWh above 800 m  Modulation according to the size (S) of the building on 40% of the total, for exemple with a coefficient of 0,6 + 80 / (100 + S)  For non residential buildings: simulated limit of 50 kWh applied only on permanently occuped areas, but activ management of enegy to obtain an effective global performance No modulation according to CO2 balance of energy sources, but a maximum CO2 emission of 5kg/m 2 /year (except for renewables), with the previous modulations  No change of the conversion factor from final to primary electricity which represents a physical reality (as calculated below) but its value should be reexamined periodically.

My own proposals: a guideline for accurate conversion factors

A bigger conversion factor from final to primary electricity

Different conversion factors for fuels

No political use of these values; preferably change the consumption limits.

thermal regulation

Primary Energy Consumption Limit for new residential buildings (for heating, cooling and hot water)

Conventional coefficients from Final to Primary Energy

Same implicit threshold for Useful Energy

* with an average yield of 70% for the boilers

2,7

The 2009 Grenelle 1 Law toward the 2012 thermal regulation  A single Primary Energy Consumption Limit for all new buildings (for heating , cooling, hot water, lighting ,ventilation, pomps) and all energy sources except wood :

kWh/m 2 /year

 There is a controversy against this value qualified by some people of « weapon of the crime» against electricity.  The law also says it has to be modulated according to the location and use of the building and for energies which present a good CO2 balance,  and that the conversion factor from final to primary energy has to be examined.

Electricity (21,8% of total final consumption)

yield energetic final consumption (cc) cons. of the energy sector

																final energetic
															consumption
												3,2						1,2
	coal coal coal	12,1 12,1 12,1 4,4%	CoalProd: Stocks: Imports: Exports: CokeStocks: Imports: Exports:	0,11 -1,38 13,2 0,11 -0,04 0,96 0,66	11,93 11,93 11,93 0,9	8,7 8,7 8,7	3,71 adjustments -0,2	coke production coke production yield 83,0% coke production	3,08 3,08 -0,1 3,08 adjustments	3,5	actual (cc) 1,63 6,61 6,5 2,87 6,61 6,5 0,02 4,1% 0,62 1,02 6,61 6,5 3,8% 0,1 non energetic	5,74 5,74 5,74 3,6%	steel industry steel industry steel industry	0,8
									0,16
	oil	88,4 32,6%	CrudeOilProd: Stocks: Import: Export: RefinedOilPr: Stocks: Import:	0,98 0,02 83,24 0 0,07 0,27 33,07	84,24 30,89	84	-0,02 adjustments	84,26	total input 85,08 yield 92,7% 0,54 0,39 0,11 0,28 refining electricity	adjustments 0,14 79		81,5	81,47 47,0%	43% 68 13,6	1,22 0,43 5,12 11,76 10,41	30,5 19%	industry	1,5
			Exports:	26,74
	electricity gas	43% 117 15% 39,7	bunkers: GasProd: Stocks: Imports: Exports: NuclearProd: HwpProd: ElecImport: ElecExport:	2,52 0,81 0,07 39,9 1,09 114,53 115 6,42 6,42 0,92 0,92 5,05 40,78	115 6,4 3,7	0,33 adjustments -0,01 0,42 hydroelectric, wind, photovoltaïc 6,35 yield 71,7% 1,26 0,75 total input 11,45 yield 41,8% 76,70 34,4 3,39 auxiliaries and U enrichment 0,46 adjustments 0,08 2,61 5,18 1,65 nuclear power stations heat loss power stations 4,79 conventional thermal industrial gas	36,0 0,3 therm imports 5,42 5,96 0,4 Hwp 26,4 2,07 nuclear 21,8% 24% 22% 1,6 37,83 38,0 grid loss conventional 36,00 35,0 20,8%	0,31 0,05 13,39 22,7 24,87 3,32 0,28 0,62	4,27 2,7% 43% 69,4	agriculture residential-services sector	8,0 0,1
	Renewables	5,4% 14,6	RenProd: RenImport: RenExport: primary production 137 14,21 0,35 0	15	PES 220 2,8	30,0% 0,23 0,93 consumption of the energy sector internal use adjustments 98	0,56 pumping 122 available energy from the energetic sector	6,8% 7,5% 11,75 12,0	46,57 0,08 1,09	50,2 31%	transports	2,4
		272	primary cons.		=				98	+ final consumption 173			160
		Hwp: hydroelectric, wind,photovoltaic						non energetic consumption 15,25
		Ren: thermic renewables and waste								climatic correction 2,06

by sector with climatic correction (cc) primary energy consumption by the energetic sector PES secondary energy production by the energy sector final energy consumption by source availabilities = primary consumption rea

In the prospect of limiting not only energy consumption but also entropy increase, it is important to understand the way the energy system operates. This work is an attempt to do so by presenting and analyzing an original flow diagram of the French energy balance. It first provides evaluations of the energy efficiency of the national energy system. And then uses them to discuss the conversion factors from final to primary energy that are used in the French heat regulation. It shows that these conversion factors, especially the electricity one, depart from reality. This leads to a misinterpretation of the 2012 energy standard for new buildings.

	From primary sources to electricity final consumption in France 2008 (Mtoe)	2008 energy flows in France (Mtoe)
	1 MWh = 3.6 GJ = 0.086 toe	1 nuclear gross Mwh = 0.26 toe
			Nuclear		71.0%
	efficiency = 28,04 % gross electricity production net electicity heat loss aux. nuclear 114,54 76,74 1,82 35,97 76,5% pumpinggrid loss grid loss 0,56 0,04 0,34 0,40 0,01 0,40 0,8% pumping aux. net production grid loss U enrichment 1,57 1,69 0,29 5,05 0,02 hydroelectric from to exports to refineries grid loss = primary production production	final 0,37 26,44 consumption
	Non thermal renewables	14.8%
	gross electricity production	12,20%		net	final
	hydroelectric conventional 5,53 wind, photovoltaic 0,49 = primary production		aux. 0,07	5,46 0,49 electicity	0,36 grid loss	5,59 consumption
		efficiency = 92,86 %
	Conventional thermal plants	14.2%
	primary sources coal, petroleum products, gas, 11,46 renewables, waste 2,20	heat loss gross electricity 10,5% production 6,38 5,08 0,29 aux. partial efficiency=41.8%	net electicity production 4,79 imports 0,92	grid loss 0,27 grid loss 0,06	final consumption 4,50 0,86	secondary efficiency = 55 %
		efficiency= 39.23%		global efficiency = 64 %
	PEC max (kWh/m 2 /year)	H1 cold area	H2 temperate area		H3 mediterranean area
	Fuels	130			110		80
	Electricity	250			190		130
		Efficiency	Conversion factor
	Fuels		100%				1
	Electricity	38,8%			2,58
	UEC max (KWh/m 2 /year)	H1 area	H2	H3
	Fuels			91		77	56
	Electricity		97		74	50
							Nuclear	Photovoltaic Hydraulic, Wind	From Fuels	Average
							% in electric consumption	71%	14.8%	14.2%	100%
							Efficiency	28.04%	92.86%	39.23%	32.75%
							Conversion factor	3,57	1,08	2,55	3.05
							Fuels	Coal	Oil	Gas	Biomass, waste
							Efficiency	83.0%	92.7%	100%	100%
							Conversion factor	1,20	1,08	1	1
							fr

Contact: ghislaine.destais@upmf-grenoble.