Veljko Ašković
email: veljkoaskovic@hotmail.com

Emmanuel Trélat
email: emmanuel.trelat@sorbonne-universite.fr

Hasnaa Zidani
email: hasnaa.zidani@insa-rouen.fr

Two-term large-time asymptotic expansion of the value function for dissipative nonlinear optimal control problems

Considering a general nonlinear dissipative finite dimensional optimal control problem in fixed time horizon T , we establish a two-term asymptotic expansion of the value function as T Ñ `8. The dominating term is T times the optimal value obtained from the optimal static problem within the classical turnpike theory. The second term, of order unity, is interpreted as the sum of two values associated with optimal stabilization problems related to the turnpike.

Introduction

The long-term asymptotic properties of the value function have been extensively explored from the perspective of partial differential equations (PDE), primarily within the framework of Hamilton-Jacobi-Bellman (HJB) equations ( [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF], [START_REF] Barles | On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations[END_REF], [START_REF] Alvarez | Ergodic Problems in Differential Games[END_REF]) and ergodic theory ( [START_REF] Arisawa | Ergodic problem for the Hamilton Jacobi-Bellman equation i. existence of the ergodic attractor[END_REF]). For instance, in [1, Chapter VII], the authors investigate the optimal control problem with discounted Lagrange cost over an infinite time horizon. To characterize the ergodic behavior of the value function, the authors take the limit as the discount factor tends to zero and identify the limit value function as the viscosity solution of the limiting equation. In [START_REF] Barles | On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations[END_REF], assuming suitable conditions, including periodicity assumptions on the Hamiltonian, the authors characterize the large-time behavior of the solution to the first-order HJB equation as the solution of a stationary equation. The extension of these results to deterministic zero-sum differential games with two conflicting controllers has been studied, as seen in [START_REF] Alvarez | Ergodic Problems in Differential Games[END_REF]. It is worth noting that more general results are available, such as those presented in [START_REF] Buckdahn | On Representation Formulas for Long Run Averaging Optimal Control Problem[END_REF], where the authors demonstrate that, under appropriate assumptions, there exists at most one potential accumulation point (in the uniform convergence topology) of the values. This occurs when the time horizon of Cesaro means converges to infinity or the discount factor of Abel means converges to zero.

From the classical optimal control perspective, the large-time behavior of the value function is typically inferred as a consequence of a property satisfied by a broad class of optimal control problems when the time horizon is sufficiently large. This property is known as the turnpike property, indicating that, for certain optimal control problems with a sufficiently large time horizon, any optimal solution tends to remain close to the optimal solution of an associated static optimization problem for the majority of the time. This optimal static solution is referred to as the turnpike (the term originates from the concept that a turnpike represents the fastest route between two distant points, even if it is not the most direct route; see Figure 1). The turnpike phenomenon was initially observed and investigated by economists for discrete-time optimal control problems (see [START_REF] Samuelson | Linear Programming and Economic Analysis[END_REF], [START_REF] Mckenzie | Turnpike theorems for a generalized leontief model[END_REF]). Various notions of turnpike properties exist, with some being stronger than others (see [START_REF] Zaslavski | Turnpike theory of continuous-time linear optimal control problems[END_REF]). Exponential turnpike properties have been established in [START_REF] Grune | Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations[END_REF], [START_REF] Porretta | Remarks on long time versus steady state optimal control[END_REF], [START_REF] Porretta | Long time versus steady state optimal control[END_REF], [START_REF] Trélat | The turnpike property in finite-dimensional nonlinear optimal control[END_REF] and [START_REF] Trélat | Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces[END_REF] for the optimal triple resulting of the application of Pontryagin's Maximum Principle (PMP), ensuring that the extremal solution (state, adjoint and control) remains exponentially close to an optimal solution of the corresponding static controlled problem, except at the beginning and at the end of the time interval, when the time horizon T is sufficiently large. As unravelled in [START_REF] Trélat | The turnpike property in finite-dimensional nonlinear optimal control[END_REF] this phenomenon is closely related to hyperbolicity properties of the Hamiltonian flow. For discrete-time problems it has been shown for instance in [START_REF] Damm | An exponential turnpike theorem for dissipative discrete time optimal control problems[END_REF], [START_REF] Grüne | Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems[END_REF] that the exponential turnpike property is also closely related to a strict dissipativity property. Measure-turnpike is a weaker notion of turnpike, meaning that any optimal solution, along the time frame, remains close to an optimal static solution except during a subset of times with small Lebesgue measure. It has been proved in [START_REF] Faulwasser | On turnpike and dissipativity properties of continuous time optimal control problems[END_REF], [START_REF] Trélat | Integral and measure-turnpike properties for infinitedimensional optimal control systems[END_REF] that measure turnpike follows from strict dissipativity or from strong duality properties.

Based on the turnpike property, an equivalent as T Ñ `8 of the value 1 T

.vpT, x, zq "

T Ñ`8 v if v ‰ 0 (1)
where vpT, x, zq is the optimal cost to steer the system from x to z in time T and v is the "steady" cost at the turnpike.

Figure 2: Turnpike phenomenon Significant insights have been achieved in [START_REF] Esteve | On the turnpike property and the long time behavior of the Hamilton-Jacobi equation[END_REF] and [START_REF] Askovic | On the asymptotic behavior of the value function in large time optimal control problems[END_REF], where the authors explore variations of the linear quadratic (LQ) problem. Assuming certain controllability conditions, they derive the large-time expansion of the value function at the order of one in 1{T . In [START_REF] Esteve | On the turnpike property and the long time behavior of the Hamilton-Jacobi equation[END_REF], the various aspects are identified, and some are interpreted within the framework of the Hamilton-Jacobi-Bellman (HJB) theory. Meanwhile, in [START_REF] Askovic | On the asymptotic behavior of the value function in large time optimal control problems[END_REF], all the terms are characterized as functions of the solutions of the Riccati algebraic equation, initial/final states, and the Lagrange multiplier of the static optimization problem.

In this manuscript, we extend the results of [START_REF] Esteve | On the turnpike property and the long time behavior of the Hamilton-Jacobi equation[END_REF][START_REF] Askovic | On the asymptotic behavior of the value function in large time optimal control problems[END_REF] to the class of dissipative nonlinear systems. The concept of (strict) dissipativity, as introduced in [START_REF] Willems | Dissipative dynamical systems, Part I: General Theory[END_REF], is defined in a broad context, accompanied by related notions like the available storage function and the supply rate function. In cases where a system exhibits dissipativity with a specified supply rate function, the inquiry into identifying a suitable storage function has been extensively investigated (refer to, e.g., [START_REF] Brogliato | Dissipative systems analysis and control[END_REF]). This inquiry bears resemblance to the task of determining an appropriate Lyapunov function in the Lyapunov second method, which ensures the stability of a system. A precise mathematical definition will be provided later in this paper.

Setting of the problem

We consider the finite-dimensional optimal control problem, consisting of minimizing the cost functional

J T,x,z puq " ż T 0 f 0 pyptq, uptqqdt (2)
over the time interval r0, T s, T ą 0 being fixed under the constraints

9 yptq " f pyptq, uptqq, (3a)
yp0q " x, ypT q " z,

where f : R n ˆRp ÝÑ R n and f 0 : R n ˆRp ÝÑ R are of class C 1 . The associated value function defined by vpT, x, zq :" min

up¨qPU Ω T J T,x,z puq (4)
where u P U Ω T :" L 8 pr0, T s, Ωq, Ω being a fixed compact subset of R p . In the sequel, we assume that Ω Ă Bp0, cq for some c ą 0.

Remark 2.1. Since Ω does not depend on T , the space of controls U Ω is independent of T . This is because, beyond the time horizon T , the controls can be trivially extended within Ω and consequently, the space of controls will be denoted by U Ω .

We assume that for T large enough, (2)-( 3)-( 4) admits an optimal solution, denoted by pp y T p¨q, p u T p¨qq. The conditions ensuring the existence of such a solution are well known (see for instance [START_REF] Trélat | Contrôle optimal: théorie et applications[END_REF], [START_REF] Cesari | Optimization-theory and applications[END_REF]). For instance, if the set of velocities tf py, uq | u P Ωu is a convex subset of R n for any y P R n , with mild growth at infinity and if the epigraph of f 0 is convex, then there exists at least one optimal solution. These conditions are for example satisfied if the dynamics are control-affine and if the cost functional is convex with respect to u By the Pontryagin maximum principle ([20]- [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]), there exist λ 0 ď 0 and an absolutely continuous mapping p λ T : r0, T s ÝÑ R n (called adjoint vector) satisfying p p λ T p¨q, λ 0 q ‰ p0, 0q such that:

9 p y T ptq " BH Bλ ´p y T ptq, p λ T ptq, λ 0 , p u T ptq 9 p λ T ptq " ´BH By ´p y T ptq, p λ T ptq, λ 0 , p u T ptq BH Bu ´p y T ptq, p λ T ptq, λ 0 , p u T ptq ¯" 0 (5)
for almost every t P r0, T s, where the Hamiltonian H is defined by Hpy, λ, λ 0 , uq :" xλ, f py, uqy `λ0 f 0 py, uq

and x , y is the Euclidean scalar product in R n . We assume throughout that the abnormal case does not occur, and we set λ 0 " ´1. Moreover, we assume the existence and uniqueness of the solution (denoted by pȳ, ūq) to the static optimization problem v :" min f py,uq"0

f 0 py, uq. (7)
This is a usual nonlinear constrained optimization problem settled in R n ˆRp . Note that the minimizer exists and is unique in the linear quadratic case.

By the Karush Kuhn Tucker (KKT) optimality conditions, assuming that the abnormal case does not occur, there exists λ P R n such that f pȳ, ūq " 0

´Bf 0 By pȳ, ūq `xλ , Bf By pȳ, ūqy " 0 ´Bf 0 Bu pȳ, ūq `xλ , Bf Bu pȳ, ūqy " 0 (8)
In the Hamiltonian formalism, the first-order optimality system (8

) is equivalent to BH Bλ pȳ, λ, ´1, ūq " 0 ´BH By pȳ, λ, ´1, ūq " 0 BH Bu pȳ, λ, ´1, ūq " 0 (9)
Because of the turnpike property (see further), since it is expected that the dominating term in the asymptotic expansion of vp¨q is T.v, we "absorb" it by subtracting it to the cost J T,x,z and consider the "shifted" optimal control problem defined as

min up¨qPU Ω C T pu, x, zq :" min up¨qPU Ω ż T 0 wpyptq, uptqq dt (10a)
´Px,z r0,T s ¯9 yptq " f pyptq, uptqq, @t P r0, T s (10b)

yp0q " x, ypT q " z (10c)
where wp¨q is the "shifted cost" defined by wpy, uq :" f 0 py, uq ´f 0 pȳ, ūq.

The readers acquainted with the notion of dissipativity in nonlinear optimal control will recognize in (11) a classical storage function. This is where the link with dissipativity appears.

We introduce also the shifted infinite-time optimal control problems

pP x 8f q v f pxq :" min up¨qPU Ω ż `8 0 wpyptq, uptqq dt 9 yptq " f pyptq, uptqq, yp0q " x (12)
and

pP z 8b q v b pzq :" min up¨qPU Ω ż `8 0 wpyptq, uptqq dt 9 yptq " ´f pyptq, uptqq, yp0q " z (13)
In our notations the index "f" stands for "forward" while the index "b" stands for "backward". We assume the existence of optimal solutions to pP x 8f q and pP z 8b q with a finite cost. Remark 2.2. The function v f p¨q (resp. v b p¨q) is the optimal cost of the infinitetime optimal control problem consisting in steering the system from x (resp. z) forward (resp. backward) in time and minimizing the cost functional ş `8 0 wpyptq, uptqq dt.

Assumptions

Assumptions of global nature:

pA 1 q: (Regularity of f and f 0 ): We assume that f and f 0 are of class C 1 .

pA 2 q: (Existence and uniqueness of optimal solutions): There exists T 0 ą 0 such that for any T ě T 0 each of the optimal control problems ´Px,z r0,T s ¯, pP x 8f q and pP z 8b q admits a unique optimal solution denoted respectively by pp y T p¨q, p u T p¨qq, pp y 8f p¨q, p u 8f p¨qq and pp y 8b p¨q, p u 8b p¨qq.

pA 3 q: (Boundedness of the optimal trajectories): The optimal trajectories of ´Px,z r0,T s ¯, pP x 8f q and pP z 8b q are bounded uniformly with respect to T ě T 0 :

Db ą 0, | @t ě 0, }p yptq} ď b. ( 14
)
pA 4 q: (Boundedness of the optimal costs): The optimal costs of ´Px,z r0,T s ¯, pP x 8f q and pP z 8b q are bounded uniformly with respect to T ě T 0 . pA 5 q: The minimizer pȳ, ūq of ( 7) is unique and there exists a unique Lagrange multiplier λ satisfying [START_REF] Zaslavski | Turnpike theory of continuous-time linear optimal control problems[END_REF]. Moreover, we assume that ū P Ω (interior of Ω).

In the sequel, we will also need a concept of strict dissipativity. We recall that (10a)-( 10b) is dissipative at pȳ, ūq with respect the the supply rate function w if there exists a bounded function S : R n ÝÑ R, called storage function such that for any admissible pair pyp¨q, up¨qq and any T ą 0:

Spyp0qq `ż T 0 wpyptq, uptqq dt ě SpypT qq. ( 15
)
The system is strictly dissipative if, in addition, there exists some function αp¨q of class K (i.e., α : r0, `8q ÝÑ r0, `8q continuous, increasing, and such that αp0q " 0) and for any T ą 0 we have:

Spyp0qq `ż T 0 wpyptq, uptqq dt ě SpypT qq `ż T 0 α ˆ› › › › yptq ´ȳ uptq ´ū › › › › ˙dt. (16)
In this work we assume the following.

pA 6 q: (Strict dissipativity property): The family of optimal control problems (10a)-(10b) indexed by T is strictly dissipative at pȳ, ūq with respect to the supply rate function w defined by [START_REF] Porretta | Long time versus steady state optimal control[END_REF] with a storage function S.

The notion of strict dissipativity was introduced in [START_REF] Willems | Dissipative dynamical systems, Part I: General Theory[END_REF] and already used in [START_REF] Damm | An exponential turnpike theorem for dissipative discrete time optimal control problems[END_REF], [START_REF] Grüne | On the relation between strict dissipativity and turnpike properties[END_REF], [START_REF] Faulwasser | On turnpike and dissipativity properties of continuous time optimal control problems[END_REF] and [START_REF] Grüne | Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems[END_REF] to derive turnpike properties.

Assumption of local nature:

pA 7 q: Setting A :" Bf By pȳ, ūq, B :" Bf Bu pȳ, ūq, the pair pA, Bq satisfies the Kalman rank condition, i.e, the linearized control system at pȳ, ūq is controllable.

pA 8 q: (Local boundedness of the minimum time trajectories and controls near the turnpike): There exists r ą 0 such that for any x P Bpȳ, rq the minimum time trajectory to ȳ starting from x, denoted by y τ f

x p¨q and the associated control, denoted by u τ f p¨q remain in the neighbourhood of respectively ȳ and ū uniformly with respect to x, i.e., Dr, K r ą 0 | @x P Bpȳ, rq @t P r0, τ f pxqs }y τ f

x ptq ´ȳ} `}u τ f ptq ´ū} ď K r (17) where τ f p¨q is the minimum time function to reach ȳ with the dynamics f .

Comments

The assumptions pA 1 q and pA 7 q together imply that there exists r ą 0 such that for any x P Bpȳ, rq, there exists an admissible trajectory steering the control system from x to ȳ in finite time. Thus the minimum time function τ f p¨q is well defined on Bpȳ, rq and is continuous at ȳ. This is classical result that can be found, for instance, in [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF]. The result remains true for the minimum time function associated to the backward-in-time dynamics ´f , denoted by τ ´f p¨q.

The assumption pA 8 q requires the local boundedness of the minimum time trajectories and controls for any trajectory starting in the previously defined neighbourhood of the turnpike. This assumption is satisfied if the minimum time function is C 1 in the neighbourhood of ȳ. We highlight here that the regularity of the minimum time function has been widely studied in the literature: it is well known that under appropriate controllability type conditions the minimum time function has an open domain of definition and is locally Lipschitz on it, see for instance [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF]- [START_REF] Frankowska | Value Function in Optimal Control[END_REF]. It is thus differentiable almost everywhere on its domain. The value function fails in general to be differentiable at points that are reached by at least two minimum time trajectories and its differentiability at a point does not guarantee continuous differentiability around this point. In [START_REF] Frankowska | Local Regularity of the Minimum Time Function[END_REF], the authors show that, under some assumptions on the regularity and target smoothness (which excludes the singleton case), the nonemptiness of the proximal subdifferential of the minimum time function at a point implies its continuous differentiability in a neighborhood of this point. An analogous result has been proved for the value function of the Bolza problem in [START_REF] Cannarsa | Local regularity of the value function in optimal control[END_REF] in the case where the initial state is a prescribed point and the final state is let free. In [START_REF] Stefani | Regularity properties of the minimum time map[END_REF], the author gives a survey of results on the regularity of the minimum time map for control-affine systems with prescribed initial and final points. Finally, for results on the set where the value function is differentiable we refer the reader to [START_REF] Aubin | Set-Valued Analysis[END_REF], [START_REF] Clarke | The relationship between the maximum principle and dynamic programming[END_REF], [START_REF] Rifford | Morse-Sard type results in sub-Riemannian geometry[END_REF], [START_REF] Rifford | On the stabilization problem for nonholonomic distributions[END_REF], [START_REF] Clarke | Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control[END_REF] and references therein.

The (strict) dissipativity property pA 6 q is certainly the less intuitive assumption to check in practice. In general, when the system is dissipative, storage functions are closely related to some viscosity sub-solutions of partial differential inequalities called Hamilton-Jacobi inequalities. We refer the reader to [START_REF] Brogliato | Dissipative systems analysis and control[END_REF]Chapter 4] for more details on this subject. One can remark that, under suitable regularity and boundedness assumptions on the dynamics and the cost, the value function (its opposite more precisely) can be taken as a storage function, and the dissipativity inequality is then deduced from the Dynamic Programming Principle (DPP). Note also that the infinitesimal form of the (non-strict) dissipativity inequality [START_REF] Grüne | Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems[END_REF], with α " 0, is the Hamilton-Jacobi inequality: H 1 py, ∇Spyq, uq ď ´f 0 pȳ, ūq where H 1 is the maximized normal Hamiltonian (indeed, divide by t 1 ´t0 ą 0 and take the limit t 1 ´t0 Ñ 0). The existence of C 1 solutions is therefore related to the so-called weak KAM theory (see [START_REF] Fathi | Weak KAM theorem in Lagrangian dynamics[END_REF]). In this context, the singleton tpȳ, ūqu is the Aubry set and f 0 pȳ, ūq is the Mañé critical value.

Remark 4.1. The strict dissipativity inequality [START_REF] Faulwasser | On turnpike and dissipativity properties of continuous time optimal control problems[END_REF] remains true for the dynamics ´f , provided that one switches the initial and final states. The corresponding storage function is ´Sp¨q.

Main result

Theorem 5.1. Under Assumptions pA 1 q´pA 8 q, the value function (4) satisfies vpT, x, zq " T.v `vf pxq `vb pzq `op1q [START_REF] Esteve | On the turnpike property and the long time behavior of the Hamilton-Jacobi equation[END_REF] as T Ñ `8.

In order to prove Theorem 5.1, we need some preliminary lemmas. f ptq " 0.

Some useful lemmas

Proof. A proof can be found for instance in [START_REF] Khalil | Nonlinear Systems[END_REF]. By contradiction, take ą 0 and assume that f ptq does not converge to 0 as t Ñ `8. In this case, there exists an increasing sequence pt n q nPN in R `such that |f pt n q| ą . By the uniform continuity of f there exists δ ą 0 such that, for any n P N, and any t P R |t ´tn | ď δ ùñ |f ptq ´f pt n q| ď 2 .

So for any t P rt n , t n `δs and any n P N, one has |f ptq| " |f pt n q ´pf pt n q ´f ptqq| ě |f pt n q| ´|f pt n q ´f ptq| ě 2 .

Therefore,

ˇˇˇˇż tn`δ 0 f ptqdt ´ż tn 0 f ptqdt ˇˇˇˇ" ˇˇˇˇż tn`δ tn f ptqdt ˇˇˇˇ" ż tn`δ tn |f ptq|dt ě δ. 2 .
The latter inequality contradicts the convergence of ż t 0 f pτ qdτ as t Ñ `8 and the lemma follows.

Lemma 5.3. The optimal trajectory p y T p¨q of pP x,z 0,T q satisfies DtpT q P r0, T s | p y T ptpT qq ÝÑ ȳ as T Ñ `8.

Proof. For T ě T 0 , the strict dissipativity inequality applied to the optimal pair pp y T p¨q, p u T p¨qq implies

f 0 pȳ, ūq ď 1 T ż T 0 f 0 pp y T psq, p u T psqqds `Spxq ´Spzq T ´1 T ż T 0 α ˆ› › › › p y T psq ´ȳ p u T psq ´ū › › › › ˙ds. (20)
Let us prove that

1 T ż T 0 α ˆ› › › › p y T psq ´ȳ p u T psq ´ū › › › › ˙ds ÝÑ 0 as T Ñ `8. (21)
Let us assume by contradiction that this is not true: then there exists η ą 0 and a sequence T k ÝÑ `8 such that

1 T k ż T k 0 α ˆ› › › › p y T k psq ´ȳ p u T k psq ´ū › › › › ˙ds ě η. (22)
By multiplying the inequality (20) by T k one gets

ż T k 0 α ˆ› › › › p y T k psq ´ȳ p u T k psq ´ū › › › › ˙ds ď ż T k 0 wpp y T k psq, p u T k psqqds `Spxq ´Spzq (23)
which implies from ( 22)

T k .η ď ż T k 0 wpp y T k psq, p u T k psqqds `Spxq ´Spzq. (24)
Assumption pA 4 q leads to a contradiction in the above inequality when k Ñ `8.

Then we have 1 T

ż T 0 α ˆ› › › › p y T ptq ´ȳ p u T ptq ´ū › › › › ˙ds ě 1 T ż T 0
α p}p y T ptq ´ȳ}q ds ÝÑ 0 as T Ñ `8 which implies, using the mean value theorem, that

DtpT q P r0, T s | α p}p y T ptpT qq ´ȳ}q ÝÑ 0 as T Ñ `8. (25)
From the properties of αp¨q, this leads to

p y T ptpT qq ÝÑ ȳ as T Ñ `8. ( 26
)
The lemma is proved.

Remark 5.4. As noted in [START_REF] Trélat | Linear turnpike theorem[END_REF], if one makes the change of variable s " t T , in

T 0 α ˆ› › › › p y T ptq ´ȳ p u T ptq ´ū › › › › ˙dt " ż 1 0 α ˆ› › › › p y T pT.sq ´ȳ p u T pT.sq ´ū › › › › ˙ds ÝÑ 0 as T Ñ `8 (21) then 1 T ż
which implies, by the converse Lebesgue theorem (see [START_REF] Brezis | Functional analysis, Theory and applications[END_REF]Theorem IV.9]) that there exists an increasing sequence of time horizons pT k q kPN such that p y T k pT k .sq ÝÑ ȳ and p u T k pT k .sq ÝÑ ū as k Ñ `8 for almost every s P r0, 1s. The latter looks like a measure turnpike result. However we did not exploit this result in our paper.

Lemma 5.5. The optimal trajectory p y 8f p¨q of pP x 8f q satisfies p y 8f ptq ÝÑ ȳ as t Ñ `8

Proof. From ( 16) we have Op1q.

First, we remark that [START_REF] Rifford | On the stabilization problem for nonholonomic distributions[END_REF] and the positivity of αp¨q imply the convergence of ΦpT q as T Ñ `8.

On the other hand, from pA 1 q and pA 3 q we know that f is of class C 1 on the compact set Bp0, bq ˆBp0, cq thus bounded by a global constant, denoted by k ą 0. Therefore p y 8f p¨q is globally Lipschitz continuous in time t, and consequently t Þ Ñ }p y 8f ptq ´ȳ} as well.

From the boundedness of p y 8f p¨q (see pA 3 q) and the continuity of αp¨q, we deduce the uniform continuity of t Þ Ñ αp}p y 8f ptq ´ȳ}q over r0 `8q. Indeed, αp¨q being continuous on a compact set, it is uniformly continuous (Heine theorem). Applying Barbalat's lemma (see 5.2), one has αp}p y 8f ptq ´ȳ}q ÝÑ 0 as t Ñ `8, which implies p y 8f ptq Ñ ȳ as t Ñ `8 and the proof is over.

Corollary 5.6. Lemma 5.5 remains true for the optimal trajectory of pP z 8b q problem (with the backward-in-time dynamics ´f ).

Proof of the main result

Proof. For T ě T 0 , let us consider the respective optimal solutions pp y T p¨q, p u T p¨qq, pp y 8f p¨q, p u 8f p¨qq and pp y 8b p¨q, p u 8b p¨qq of ´Px,z r0,T s ¯, pP x 8f q and pP z 8b q.

We split the optimal cost C T pp u T , x, zq as

C T pp u T , x, zq " ż T 0 wpp y T ptq, p u T ptqq dt (31)
"

ż tpT q 0 wpp y T ptq, p u T ptqq dt loooooooooooooomoooooooooooooon C f T `ż T tpT q wpp y T ptq, p u T ptqq dt looooooooooooomooooooooooooon C b T ( 32
)
where tpT q is defined by [START_REF] Askovic | On the asymptotic behavior of the value function in large time optimal control problems[END_REF].

We perform the proof in two steps. We first prove that

Step 1: v f pxq `vb pzq ď lim inf

T Ñ`8 C T pp u T , x, zq (33)
Then we prove that:

Step 2: lim sup

T Ñ`8 C T pp u T , x, zq ď v f pxq `vb pzq (34)
which will prove the required result.

The real number r being defined in Assumption pA 8 q, we first remark that py, uq Þ Ñ wpy, uq is continuous on Ω r :" Bpȳ, K r q ˆBpū, K r q which is a compact set of R n ˆRp . Consequently, DM r ą 0 | @py, uq P Ω r , |wpy, uq| ď M r .

where τ 1 :" τ ´f pr y T pT ´tpT qqq is the minimum time to reach ȳ from r y T pT ´tpT qq and p u 1 p¨q the associated optimal control.

We infer from ( 37) and ( 38) that

v b pzq ď ż `8 0 wpyptq, ȗptqq dt ď ż T ´tpT q 0 wpr y T ptq, r u T ptqq dt `ż T ´tpT q`τ1
T ´tpT q wpp y 1 ptq, p u 1 ptq looooomooooon PΩr q dt `ż `8

T ´tpT q`τ1 wpȳ, ūq loomoon 39) and (42), we obtain

0 dt ď C b T `τ1 .M r ď C b T ` 2 (42) Combining (
v f pxq `vb pzq ď C f T `Cb T ` " C T pp u T , x, zq ` (43)
and thus v f pxq `vb pzq ď lim inf

T Ñ`8 C T pp u T , x, zq (44)
Ź Step 2: From Lemma 5.5 and Corollary 5.6, we have p y 8f ptq ÝÑ ȳ and p y 8b ptq ÝÑ ȳ as t Ñ `8. Consequently:

DT 2 ą 0 s.t @T ě T 2 , › › › › p y 8f ˆT 2 ´1˙´ȳ› › › › ď γ (45)
and

DT 3 ą 0 s.t @T ě T 3 , › › › › p y 8b ˆT 2 ´1˙´ȳ› › › › ď γ (46)
Take T ě max pT 2 , T 3 q and denote:

• τ 3 :" τ f ˆp y 8f ˆT 2 ´1˙˙a nd p u 3 p¨q the associated optimal control;

• τ 4 :" τ ´f ˆp y 8b ˆT 2 ´1˙˙a nd p u 4 p¨q the associated optimal control.

We construct the admissible control u T p¨q for pP 0,T q x,z as follows (see Figure (48) Noting that the integrals A and E converge, we take the limit superior as T Ñ `8 of the above inequality and we obtain: Remark 5.7. In the LQ case, a two-term large-time expansion of the value function has been derived in [START_REF] Askovic | On the asymptotic behavior of the value function in large time optimal control problems[END_REF] solely under the Kalman condition. Assumptions such as the strict dissipativity property and the existence/uniqueness of the solution of the static optimization problem are automaticaly satisfied in this case. For the proof, see [START_REF] Askovic | On the asymptotic behavior of the value function in large time optimal control problems[END_REF].

Conclusions

Considering a general nonlinear dissipative optimal control problem in finite dimension and fixed time T , we have established a two-term asymptotic expansion of the value function as T Ñ `8. Essentially based on the strict dissipativity property, the result is a generalization of the expansion established in the variants of the linear quadratic case treated in [START_REF] Esteve | On the turnpike property and the long time behavior of the Hamilton-Jacobi equation[END_REF] and [START_REF] Askovic | On the asymptotic behavior of the value function in large time optimal control problems[END_REF]. We highlight here that a generalization of [START_REF] Askovic | On the asymptotic behavior of the value function in large time optimal control problems[END_REF] to the infinite dimensional case is currently being finalized. The infinite dimensional nonlinear case remains open.

Figure 1 :

 1 Figure 1: Turnpike illustration: (a) small time case (b) large time case

Lemma 5 . 2 ( 8 ż t 0 f

 5280 Barbalat's lemma). Assume that f : r0, `8q ÝÑ R is uniformly continuous and that lim tÑ`pτ qdτ exists and is finite, then lim tÑ`8

0 α p}p y 8f ptq ´ȳ}q dt ďα

 0 ptq, p u 8f ptqqdt `Spxq ´Spp y 8f pT qq[START_REF] Clarke | The relationship between the maximum principle and dynamic programming[END_REF] the right-hand side of the inequality being bounded uniformly with respect to T one gets ż T p}p y 8f ptq ´ȳ}q dt " T Ñ`8

2 ´1wpp y 3 ptq, p u 3 ptq looooomooooon PΩr q dt ď τ 3 .M r ď 2 Cw pp y 4 2 ´1wpp y 4 ptq, p u 4 2 ´1 0 wT 2 ´10

 232424202 pT ´tq, p u 4 pT ´tqq dt " ptq looooomooooon PΩr q dt ď τ 4 .M r ď 2 Finally, we obtain C T pp u T , x, zq ď ż T pp y 8f ptq, p u 8f ptqq dt `ż wpp y 8b ptq, p u 8b ptqq dt `

8 CT Ñ` 8 C

 88 T pp u T , x, zq ď v f pxq `vb pzq. (49) Ź Conclusion: By combining (44) and (49), we obtain the required result: lim T pp u T , x, zq " v f pxq `vb pzq.(50)

Let ą 0. The continuity of τ f p¨q at ȳ gives:

The continuity of τ ´f p¨q at ȳ gives:

We set γ :" minpη, ν, rq ą 0, and we denote by B :" Bpȳ, γq.

Ź Step 1: From Lemma 5.3, we know that:

We select a time horizon T such that T ě maxpT 0 , T 1 q and construct q up¨q admissible control for the pP x 8f q problem as follows:

where τ 0 :" τ f pp y T ptpT qqq is the minimum time to reach ȳ from p y T ptpT qq and p u 0 p¨q the associated optimal control.

We infer from ( 36) and ( 38) that

w pq yptq, q uptqq dt ď ż tpT q 0 w pp y T ptq, p u T ptqq dt `ż tpT q`τ0 tpT q wpp y 0 ptq, p u 0 ptq looooomooooon PΩr qdt `ż `8

tpT q`τ0 wpȳ, ūq loomoon

where pr y T ptq, r u T ptqq :" pp y T pT ´tq, p u T pT ´tqq is such that 9 r y T ptq " ´f pr y T ptq, r u T ptqq with r y T p0q " z (41)

Noting that the r y T pT ´tpT qq " p y T ptpT qq, we construct ȗp¨q the admissible control for the pP z 8b q problem as follows ȗptq :" 3):


Keywords: aliment fermenté, intestin, nutrition, prébiotique, probiotique, santé Keywords -fermented food, gut, health, nutrition, prebiotic, probiotic

La fermentation permet de conserver de manière durable les aliments à moindre coût et participe à la sécurité alimentaire mondiale. Ce mode de conservation est actuellement très prisé par une population en recherche de naturalité. Des vertus santé sont également attribuées aux aliments fermentés parce qu'ils contiennent des micro-organismes vivants ou des macro-et micronutriments spécifiques. Ces produits, ou le processus de fermentation en tant que tel, sont-ils vraiment bénéfiques pour la santé ?

Health benefits of fermented foods . Fermentation allows to preserve food in a sustainable way at a lower cost and contributes to the world food security. This method of preservation is currently very popular with a population in search of naturalness. Health virtues are also attributed to fermented foods because they contain living micro-organisms or specific macroand micronutrients. Are these products, or the fermentation process, as such really beneficial to health?

Intérêt des aliments fermentés pour la santé

Isabelle SAVARY-AUZELOUX a , * Directrice de recherches à l'Institut national de recherche pour l'agriculture, l'alimentation et l'environnement (Inrae)

Françoise RUL b

Chargée de recherches Inrae L a fermentation est, avec le salage et le séchage, l'une des méthodes les plus anciennes de conservation des aliments puisqu'elle remonte à un peu moins de 10 000 ans avant notre ère [START_REF] Marshall | Traditional fermented food and beverages for improved livelihoods[END_REF][START_REF] Marco | The International Scientifi c Association for Probiotics and Prebiotics (ISAPP) consensus statement on fermented foods[END_REF] . Il s'agit d'un procédé simple, durable, qui ne fait pas intervenir la chaîne du froid et qui est réalisable à petite comme à grande échelle, ce qui explique que les aliments fermentés peuvent être fabriqués par l'industrie agroalimentaire et de manière artisanale. Le principe en est le suivant : à partir de matières premières extrêmement variées, des communautés microbiennes endogènes et/ou ajoutées se développent et génèrent un produit très éloigné de ces matières premières. Le procédé technologique de fermentation, largement répandu dans le monde, permet d'élaborer des aliments très différents selon les conditions environnementales (température, humidité, etc.), les micro-organismes présents dans l'environnement ou ajoutés, et les matières premières à disposition sur le site de production. À titre d'exemple, si les Européens sont très friands de yaourts, de fromages ou autres produits laitiers fermentés, les Asiatiques apprécient de nombreux produits fermentés à base de végétaux, comme le tofu, le kombucha et le kimchi. Puisque ces aliments couvrent une gamme extrêmement large, ils diffèrent également fortement en termes de textures (de la sauce soja au fromage, en passant par la choucroute), de goûts (produits doux, acides, amers, etc.) et de compositions en flore (bactéries, levures et champignons).

Depuis quelques années, de nombreux blogs et ouvrages se sont approprié ce mode de conservation millénaire pour en faire un sujet tendance, très prisé par une population soucieuse de durabilité, de maîtrise des procédés et de produits plus naturels, moins industriels, sans additifs et faits maison [START_REF] Renaud | Fermented products: a strong trend in French food markets[END_REF] . Ce marché est porteur (plus de 600 milliards de dollars en 2016-2018) et croît de plus de 5 % par an [START_REF]Research BIS . Global fermented food & ingredients market-analysis and forecast[END_REF] . Il est par conséquent scruté par l'industrie agro alimentaire et les start-up, à la recherche d'idées nouvelles pour la grande distribution ou de marchés de niche. Or, malgré cet engouement indéniable, une étude menée en 2018 a montré que seuls 44 % des sondés connaissaient le procédé de fermentation et que 60 % avaient une perception négative du terme "fermenté", alors que tous mangeaient du fromage [START_REF] Valorial | Le focus thématique "Aliments fermentés : entre verrous réglementaires & attentes consommateurs" est sorti ! 27 février[END_REF] . La fermentation et les aliments qui en sont issus sont ainsi largement méconnu s de la population générale. En parallèle de ces considérations éthiques, culturelles et économiques, il apparaît que sont attribuées aux aliments fermentés des actions bénéfiques sur la santé intestinale ainsi que sur la santé en général. Quand on connaît la difficulté d'objectiver l'effet santé d'un aliment seul ou de diètes (en raison d'études cliniques ou observationnelles nombreuses et parfois contradictoires), le fait de prêter de tels bienfaits aux produits fermentés dans leur ensemble peut sembler illusoire. Néanmoins, il est intéressant de s'interroger sur les vertus santé qui seraient les leurs. Ne concernent-elles que Les aliments fermentés dossier Notes 1 D'après les 22 es Entretiens de nutrition de l'Institut Pasteur de Lille, organisés les 3 et 4 juin 2021. 2 Dans ce cas, la composition du consortium bactérien impliqué dans la fermentation n'est pas contrôlée, il s'agit donc souvent de procédés utilisés artisanalement ou à domicile. 3 Produits laitiers mis à part ou situations dans lesquelles la fermentation permet l'élimination ou l'apport de composés spécifi ques, comme la dégradation du lactose ou des facteurs antinutritionnels.

certains aliments ou le processus de fermentation en luimême ? Si ces effets sont avérés, sur quelles cibles ces produits agissent-ils ? 1

Défi nitions et caractéristiques des aliments fermentés

Pour évaluer l'intérêt potentiel de la fermentation pour la santé, il convient d'expliciter ce processus, qui concerne une large gamme de matières premières, et d'établir un état des lieux de la diversité des aliments fermentés produits.

Défi nitions

F Une revue de l'Association scientifique internationale pour les probiotiques et les prébiotiques [START_REF] Marco | The International Scientifi c Association for Probiotics and Prebiotics (ISAPP) consensus statement on fermented foods[END_REF] a très récemment proposé une définition des aliments fermentés et fait le point sur leur rôle dans l'alimentation humaine. D'un point de vue biochimique, la fermentation se définit comme « un processus générant de l'adénosine tri phosphate [ATP], dans lequel les composés organiques agissent à la fois en tant que donneurs et accepteurs d'électrons » . Cette définition s'applique bien aux fermentations anaérobies lactiques ou alcoolique s (yaourt, vin, etc.), mais pas aux produits faisant appel à des fermentations aérobies, telles celles mises en oeuvre dans la production de la sauce soja (champignons) ou du vinaigre (bactéries acétiques). Par conséquent, les experts ont retenu une définition plus large permettant de décrire les aliments et les boissons fermentés : « Aliments obtenus à la suite de la croissance désirée de microbes et de conversions enzymatiques de matières premières alimentaires » [START_REF] Marco | The International Scientifi c Association for Probiotics and Prebiotics (ISAPP) consensus statement on fermented foods[END_REF] . Ainsi, la présence d'enzymes endogènes ou exogènes des matières premières ne peut à elle seule suffire pour qu'un aliment soit dit fermenté. F À titre d'exemple, on peut considérer comme fermentés les produits dans lesquels les microorganismes utilisés pour la fermentation sont présents vivants (la plupart des fromages et des yaourts, le miso, certaines bières, le kéfir, etc.) ou morts (pain au levain, produits fermentés puis pasteurisés, la plupart des vins et des spiritueux, etc.). En revanche, ce n'est pas le cas de tous ceux issus de la salaison seule, les végétaux en saumure ou le pain obtenu à l'aide de levure chimique.

Une grande diversité

F Plus de 5 000 aliments fermentés sont référencés ( tableau 1 ) [START_REF] Marshall | Traditional fermented food and beverages for improved livelihoods[END_REF][START_REF] Tamang | Fermented foods and beverages of the world[END_REF][START_REF] Wolfe | Fermented foods as experimentally tractable microbial ecosystems[END_REF][START_REF] Marco | Health benefi ts of fermented foods: microbiota and beyond[END_REF][START_REF] Tamang | Fermented foods in a global age: east meets west[END_REF] . Les matières premières animales (lait, viandes, poissons) et végétales (céréales, légumineuses, tubercules, racines, fruits, légumes, graines et fèves) sont susceptibles de subir un processus de fermentation pour se transformer en toute une gamme de produits. F Les micro-organismes intervenant dans ce procédé ont des origines variées : ils sont ajoutés extempora nément (amorçage, fréquemment utilisé dans les industries agroalimentaires car l'ensemencement peut être strictement contrôlé), repiqués à partir de produits précédemment fermentés (repiquage) ou sont naturellement présents dans la matière première (croissance spontanée 2 ). Ce sont essentiellement des bactéries lactiques et acétiques, mais d'autres types bactériens sont parfois présents -les levures et les champignons filamenteux -, surtout lorsqu'il s'agit d'aliments à ensemencement spontané. F Puisque l'aliment fermenté est le résultat de l'association d'une matière première et de microorganismes variés, le nombre de combinaisons est quasiment infini, ce qui explique la très grande diversité des produits disponibles.

Des particularités uniques

F En dehors des considérations microbiologiques, les produits fermentés présentent un profil bio chimique unique lié à la fermentation. En effet, ce procédé se traduit par la synthèse de molécules à partir de celles présentes dans la matière première : peptides antihypertenseurs, acides gras à chaîne courte, vitamines du groupe B et acides gras polyinsaturés de type oméga-3 notamment [START_REF] Lortal | Aliments fermentés & bénéfi ces santé : un défi pour la recherche[END_REF][START_REF] Melini | Healthpromoting components in fermented foods: an up-to-date[END_REF] . Il engendre également la dégradation de composés présents dans la matière première : lactose, protéines allergènes, acide phytique, fermentable oligo-, di-, monosaccharides and polyols (Fodmap) [START_REF] Lortal | Aliments fermentés & bénéfi ces santé : un défi pour la recherche[END_REF][START_REF] Melini | Healthpromoting components in fermented foods: an up-to-date[END_REF] , etc. F Les caractéristiques physico-chimiques des aliments fermentés sont souvent très différentes de celles de la matière première initiale : pH, acidité, potentiel redox, ou activité antioxydante, microet macrostructure. Par conséquent, la texture, le goût et la sensation en bouche de ces denrées sont très éloignés de ceux de la matière première dont elles sont issues.

Les matières premières animales comme végétales sont susceptibles de subir un processus de fermentation. Plus de 5 000 aliments fermentés sont actuellement référencés.

Les aliments fermentés dossier

Fermentation, sécurité alimentaire et risques potentiels

La fermentation, procédé unique utilisé depuis des millénaires, a pour objectif principal de maintenir comestibles des matières premières sur une longue durée. Néanmoins, ce processus de conservation peut présenter certains risques qu'il convient de ne pas négliger.

Implication dans la sécurité alimentaire

F Puisque la fermentation permet la préservation des matières premières, il est évident que les aliments fermentés jouent un rôle majeur en santé publique, en particulier dans les pays chauds et dans ceux où les autres modes de préservation (comme le froid) sont peu répandus. Ces produits contribuent donc parfois à couvrir les besoins nutritionnels de base des populations. F Néanmoins, lorsque l'on s'intéresse, sur des espèces modèles, à l'impact de molécules spécifiques (acides gras à chaîne courte, métabolites de dégradation des polyphénols, etc.) issues de la fermentation ou à l'effet de micro-organismes parti culiers (probiotiques), un impact bénéfique est retrouvé de manière plus systématique sur les pathologies nutriinduites, la santé intestinale et les cancers [START_REF] Marco | Health benefi ts of fermented foods: microbiota and beyond[END_REF][START_REF] Lortal | Aliments fermentés & bénéfi ces santé : un défi pour la recherche[END_REF] . En conséquence, il ne peut être exclu que le fait qu'aucun effet santé -sauf cas particuliers -des produits fermentés ne soit avéré chez l'homme puisse être lié à notre difficulté à le mesurer par des marqueurs pertinents ou des schémas expérimentaux adaptés.

Conclusion

Les aliments fermentés font partie intégrante de notre diète depuis des millénaires. Ils présentent de nombreux avantages en termes de sécurité nutritionnelle, en permettant la conservation sur le long cours de matières premières périssables de manière très peu énergivore. Ce mode de conservation des aliments est aussi très prometteur en termes d'innovation, en substitution à d'autres méthodes moins durables. Néanmoins, du fait de la grande diversité des aliments fermentés, leur effet bénéfique sur la santé demeure peu étayé 3

Tableau 1 .

 1 Principales classes de produits fermentés, matières premières utilisées, micro-organismes réalisant la fermentation, ainsi que lieu majeur de production et de consommation[START_REF] Marshall | Traditional fermented food and beverages for improved livelihoods[END_REF][START_REF] Tamang | Fermented foods and beverages of the world[END_REF][START_REF] Wolfe | Fermented foods as experimentally tractable microbial ecosystems[END_REF][START_REF] Marco | Health benefi ts of fermented foods: microbiota and beyond[END_REF][START_REF] Tamang | Fermented foods in a global age: east meets west[END_REF] .

Références La fermentation permet de dégrader des composés pouvant avoir des effets néfastes sur la santé, comme le lactose du lait.

					dossier
					Les aliments fermentés
	Type de produit F Un aliment fermenté peut être considéré comme Matières premières sûr s'il contient des concentrations en acides orga-niques supérieures à 100 mM combinées avec une faible et l'état de santé des populations considérées. Des Nom du produit Références études interventionnelles et observationnelles ont donc été réalisées chez l'homme afin d'estimer, dans quantité d'eau, du sel, des nitrites et d'autres composés des conditions habituelles de vie, si l'ingestion de ces	Micro-organismes impliqués de soja peuvent être fermentés (miso) ou non (jus de Source des micro-organismes soja ou tofu) et rien ne prouve à ce jour que la fermen-systematic review . Nutrients 2019 ; 11 ( 5 ) : 1189 . tation a un impact positif les concernant. De manière Zone géographique [12] Shiby VK , Mishra HN . générale, et pour de nombreux végétaux fermentés
	Produits laitiers antimicrobiens [2] . C'est aussi le cas des boissons ren-Lait produits (ensemble ou individuellement) était capable Yaourt Lait, sel Fromage fermant plus de 4 % d'alcool et dont le pH est inférieur à 4,5. En outre, les bactéries lactiques présentent la par-d'influer sur la santé de l'hôte. F L'effet santé des aliments fermentés a été étu-	BL testés, les effets santé observés sont plutôt attribués Amorçage Fermented milks and milk Europe, Amérique AB, B Ac, CF, BL, Lev selon les fromages Amorçage, repiquage Europe, Amérique, Océanie products as functional aux macronutriments (protéines, fibres) et aux micro-foods--a review . Crit Rev Food Sci Nutr 2013 ; 53 ( 5 ) : 482 -96 . nutriments, naturellement présents dans la matière
	Lait ticularité de synthétiser des bactériocines qui inhibent dié, chez l'homme, dans les situations patho-Lait fermenté (kéfi r,	Lev, BL, AB, B Ac première ou ajoutés lors de la préparation de l'ali-Repiquage Monde (sous [13] European Food Safety
	lait ribot, lassi, etc.) logiques ou les perturbations métaboliques la croissance d'autres bactéries.	différents noms) Authority. Scientifi c opinion ment (composés antioxydants, tel l'ail pour la prépa-
	Fruits F La fermentation permet de dégrader certains Fève de cacao suivantes : obésité, syndrome métabolique, diabète, Chocolat composés pouvant avoir des effets néfastes sur la maladies cardio-vasculaires et hypertension [10,23] . santé. L'un des avantages les plus connus de ce pro-Leur rôle en cas de dérèglements ou de pathologies	Lev, BL, AB, B Ac ration du kimchi). À l'inverse, la version non fermentée Spontané Afrique, Amérique on the substantiation of health claims related to live yoghurt latine, exportation cultures and improved lactose de certains produits, comme le chocolat et le café, mondiale digestion (ID 1143, 2976) n'existe pas. Leurs effets santé peuvent donc être
	Cerise de café cédé est la dégradation du lactose, naturellement intestinales (allergies, pathologies inflammatoires, Café	Lev, CC, AB partiellement liés au processus de fermentation ; là Spontané Afrique, Amérique latine, Asie, pursuant to article 13(1) of Regulation (EC) N o 1924/2006.
	présent dans le lait, par les bêta-galactosidases bac-intolérances alimentaires, transit) est également	exportation mondiale 19 octobre 2010. www.efsa. encore, la composition de la matière première (fibres,
	Grains de raisin tériennes [12] . D'ailleurs, la seule allégation santé documenté [15,24] . Une littérature moins abondante Vin	Lev essentiellement poly phénols, vitamines) semble être l'un des éléments Spontané, amorçage Monde europa.eu/fr/efsajournal/
	pressés actuellement autorisée par l'European Food Safety s'est aussi intéressée à la santé cognitive [25] et aux	explicatifs majeurs.	pub/1763.
	Jus de pomme ou Authority concernant les aliments fermentés est l'in-cancers [23] .	Cidre, kéfi r de fruits	BL, AB, Lev	Amorçage,	Monde
	d'autres fruits, eau (sucre pour kéfi r) gestion de yaourt pour les populations intolérantes F Concernant les effets globaux, il est bien connu		repiquage
	Graines, racines, tubercules au lactose [13] . La dégradation, par la fermenta-Orge, houblon, eau que la diète méditerranéenne est associée à une Bière tion, des composés indésirables ou antinutritionnels, amélioration de l'insulinosensibilité et à une moindre	Lev essentiellement	Amorçage, repiquage	Monde
	Divers (riz, vin, pommes, pommes de terre, orge, maïs, seigle, blé, orge, etc.) comme l'acide phytique, les alcaloïdes, les inhibiteurs occurrence de pathologies cardio-vasculaires [26] . Alcools forts : saké, cognac, calvados, trypsiques et les composés cyanogènes, est égale-Or, ce régime est connu pour être particulièrement vodka, single malt ment connue [14] ; elle peut limiter la toxicité directe riche en aliments fermentés, en particulier en produits whisky, bourbon, etc. de certaines de ces molécules (glycosides cyano-laitiers fermentés, qui contribueraient à son impact	Lev essentiellement	Spontané, amorçage [15] El Mecherfi KE , Monde Todorov SD , Cavalcanti de Albuquerque MA , et al. Allergenicity of fermented foods: emphasis on seeds
	Farine de blé, eau gènes), favoriser l'absorption des autres nutriments bénéfique [27] .	Pain au levain	Lev, BL	Repiquage, spontané protein-based products . Amérique, Europe,
	alimentaires (absorption du calcium et du magnésium F Les produits laitiers peuvent être considérés			Australie Foods 2020 ; 9 ( 6 ) : 792 .
	Graine de soja, eau, sel pour l'acide phytique) ou minorer le caractère allergène à part, car leurs effets sur la santé ont été largement Sauce soja, miso Graines de céréales, pseudo-céréales, légumineuses Jus fermentés, tempe, natto, idli, ben-saalga, ogi de certaines protéines [15] . étudiés à travers le monde. Ils ont fait l'objet de nom-F La fermentation permet aussi de valoriser cer-breuses méta-analyses et revues systématiques de Autres végétaux Chou, condiments, ail, sel Kimchi taines matières premières qui resteraient, sinon, sous-utilisées. Cela concerne notamment le café et le la littérature, qui constituent les analyses de don-Objectivisation de l'eff et santé CF CF, Lev, BL Lev, BL chez l'homme nées les plus robustes. Sur toutes les pathologies chocolat, mais aussi les végétaux dont la concentration F La fermentation peut permettre de limiter l'ap-Amorçage, spontané [16] Capozzi V , Fragasso M , Asie, exportation mondiale Amorçage, spontané Russo P . Microbiological safety and the management Asie, Afrique, of microbial resources in exportation mondiale Spontané, amorçage Asie, exportation artisanal foods and beverages: the need for a transdisciplinary assessment to conciliate mondiale actual trends and risks métaboliques considérées, seule la consommation
	Chou, sel en composés antinutritionnels est trop élevée pour que de yaourt -et, dans une moindre mesure, celle de Choucroute parition de molécules néfastes pour la santé et, BL Spontané, amorçage avoidance . Microorganisms Europe, Amérique,
	la digestion de l'ensemble des constituants de la diète fromage -est négativement associée à l'apparition à l'inverse, favoriser celle de molécules potentielle-	Chine 2020 ; 8 ( 2 ) : 306 .
	Thé, sucre se fasse correctement ou que la santé de l'hôte ne soit d'un diabète de type 2 [23] , alors que l'ingestion du Kombucha ment bénéfiques. Dans certains cas spécifiques (telle Lev, BL, B Ac, AB Repiquage pas dégradée (acide phytique ou composés cyano-la dégradation du lactose), elle améliore la santé de lait seul n'est pas corrélée positivement ou négative-	Asie, reste [17] Alvarez MA , Moreno-Arribas MV . The du monde problem of biogenic amines
	Viandes gènes dans certaines céréales et légumineuses). ment avec cette pathologie. Cela suggère que la fer-Viande, sel Saucisson, salamis, jambons certaines populations cibles. De nombreuses revues Lev, CF, BL, AB, Amorçage, spontané in fermented foods and the use Europe, Amérique du Nord, Asie Poissons, coquillages Poisson, sel, condiments Sauce nuoc-mâm, poissons fermentés Lev, BL, AB Spontané Inde, Asie, Europe du Nord, Grèce détaillent les impacts connus des molécules issues of potential biogenic amine-mentation du lait per se est responsable du résultat degrading microorganisms Risques potentiels de la fermentation et les mécanismes d'interac-as a solution . Trends Food Sci observé, sans que l'on sache avec certitude si son
	(fréquents), F Les risques associés à l'ingestion des aliments tion potentiels avec l'hôte pouvant engendrer des rôle bénéfique provient des bactéries présentes dans voire sucre fermentés sont de deux ordres : une possible contami-effets bénéfiques sur la santé au sens large [19,20] . l'aliment, de sa composition en lipides ou d'effets liés	Technol 2014 ; 39 ( 2 ) : 146 -55 . [18] Agence française de
	nation par des bactéries non désirées pathogènes et la à la matrice alimentaire, tous très différents de ceux Ces résultats ont généralement été obtenus sur des	sécurité sanitaire des aliments. Évaluation des risques liés
	synthèse, via la fermentation, de molécules connues pour associés au lait natif. Concernant les autres patholo-modèles animaux, plus rarement chez l'homme. Il en	à la présence de mycotoxines
	être néfastes pour la santé. Les risques micro biologiques gies, les études manquent et/ou sont contradictoires, va de même pour l'étude du rôle des bactéries pré-	dans les chaînes alimentaires
	existent pour tous les produits fermentés, mais sont ce qui ne permet pas de valider de manière robuste un sentes dans ces produits présentant des propriétés	humaine et animale. Rapport fi nal. Mars 2009. www.anses.
	accrus lorsque le processus n'est pas complètement effet santé des produits laitiers fermentés. probiotiques [21,22] .			fr/fr/system/fi les/RCCP-Ra-
	maîtrisé -mettant en jeu la fermentation spontanée -, F Force est de constater que les données F Via la connaissance de la composition en	Mycotoxines2009.pdf.
	généralement lorsqu'il est artisanal [16] . Parmi les micro-manquent pour objectiver le bénéfice sur la santé micro-organismes et celle des métabolites et	[19] Van Hylckama Vlieg JET ,
	organismes pathogènes les plus connus, on peut citer des autres produits fermentés, notamment végé-composés bénéfiques ou néfastes de ces aliments,	Veiga P , Zhang C , et al. Impact of microbial transformation
	il devrait être possible d'évaluer leur effet potentiel, taux. À titre d'exemple, il est conféré au soja dans	of food on health-from
	positif ou négatif, sur la santé du consommateur. son ensemble, ou aux protéines et aux isoflavones	fermented foods to
	Mais il faudrait pouvoir prendre en compte les quanti-qu'il contient, des effets bénéfiques dans de nom-	fermentation in the gastro-intestinal tract . Curr Opin
	tés ingérées, leur association avec les autres compo-breuses patho logies (maladies cardio-vasculaires,	Biotechnol 2011 ; 22 ( 2 ) : 211 -9 .
	sants du repas, ainsi que les habitudes nutritionnelles cancers, etc.) [28] . Néanmoins, les produits à base

AB : autres bactéries ; B Ac : bactéries acétiques ; BL : bactéries lactiques ; CF : champignons filamenteux ; Lev : levures. Amorçage : ajout de micro-organismes pour démarrer le processus de fermentation. Repiquage : utilisation des micro-organismes issus de fermentations précédentes de la même matière première. Spontané : fermentation à partir des micro-organismes présents dans la matière première. Escherichia coli , Bacillus cereus , Listeria monocytogenes , Salmonella sp. et Clostridium botulinum . L'hypothèse que certains micro-organismes présents dans les aliments fermentés puissent transmettre des gènes d'antibiorésistance fait l'objet d'un intérêt croissant.

F Certaines molécules posent potentiellement un problème pour la santé, telles que, bien évidemment, l'alcool et le sel (ajouté dans certains aliments fermentés). Dans les deux cas, les conséquences se manifestent sur le long terme et se traduisent par des pathologies ou des dérèglements métaboliques majeurs (obésité, diabète, maladies cardio-vasculaires, stéatose hépatique non alcoolique). D'autres molécules peuvent avoir des effets plus aigus, comme les amines biogènes produites par les bactéries lactiques (migraines) [17] et les mycotoxines (aflatoxines, etc.) produites par les champignons filamenteux (intoxications aiguës avec lésions hépatiques, par exemple) [18] .

[14] Gupta RK , Gangoliya SS , Singh NK . Reduction of phytic acid and enhancement of bioavailable micronutrients in food grains . J Food Sci Technol 2015 ; 52 ( 2 ) : 676 -84 .

  . Ainsi, il est nécessaire de mettre en place des études cliniques observationnelles ou interventionnelles au sein desquelles l'ingestion des aliments fermentés est parfaitement connue ou maîtrisée, combinées d'une part à des outils haut débit de phénotypage de la santé des individus et, d'autre part, à une connaissance approfondie des facteurs confondants (reste de la diète, quantités ingérées, état de santé des sujets, etc.). Seul ce type d'approche permettra d'objectiver avec certitude les impacts réels des aliments fermentés sur notre santé et, à plus long terme, de formaliser des recomman dations nutritionnelles spécifiques.

w Déclaration de liens d'intérêts Les auteurs déclarent ne pas avoir de liens d'intérêts.

© stock.adobe.com/sveta_zarzamora

© stock.adobe.com/muro


C Gabellieri

M Tognon

D Sanalitro

A Franchi

Force-based Pose Regulation of a Cable-Suspended Load Using UAVs with Force Bias

This work studies how force measurement/estimation biases affect the force-based cooperative manipulation of a beam-like load suspended with cables by two aerial robots. Indeed, force biases are especially relevant in a force-based manipulation scenario in which direct communication is not relied upon. First, we compute the equilibrium configurations of the system. Then, we show that inducing an internal force in the load augments the robustness of the load attitude error and its sensitivity to force-bias variations. Eventually, we propose a method for zeroing the load position error. The results are validated through numerical simulations and experiments.

I. INTRODUCTION

Physical interaction using aerial robots is widely studied for its relevant applications, such as contact-based inspection, construction, and delivery [START_REF] Ruggiero | Aerial manipulation: A literature review[END_REF]- [START_REF] Ollero | Past, present, and future of aerial robotic manipulators[END_REF].

Different tools have been proposed, but cables are particularly advantageous in terms of weight and cost. The manipulation of a cable-suspended load by a team of aerial vehicles has been largely addressed [START_REF] Sreenath | Dynamics, control and planning for cooperative manipulation of payloads suspended by cables from multiple quadrotor robots[END_REF]- [START_REF] Sanalitro | Indirect force control of a cable-suspended aerial multi-robot manipulator[END_REF]. Those works study the differential flatness [START_REF] Sreenath | Dynamics, control and planning for cooperative manipulation of payloads suspended by cables from multiple quadrotor robots[END_REF] and the control of payloads suspended with cables below aerial robots; the controllers are either centralized, based on common knowledge of the object's state, or relying on a communication network among the robots. Moreover, the manipulation of beam-like loads has received attention in the literature [START_REF] Goodman | Geometric control of two quadrotors carrying a rigid rod with elastic cables[END_REF]- [START_REF] Spurny | Cooperative transport of large objects by a pair of unmanned aerial systems using sampling-based motion planning[END_REF] for its relevance to several real-world applications, especially in the construction field, where elements like pipes, iron beams for cement walls, and scaffolds are manipulated.

Indeed, multi-robot cooperation is often used to enhance the overall payload [START_REF] Mohiuddin | A survey of single and multi-uav aerial manipulation[END_REF]. While decentralized algorithms, as [START_REF] Mellinger | Cooperative grasping and transport using multiple quadrotors[END_REF], are robust to faults and scalable with the number of robots, explicit communication represents a bottleneck [START_REF] Wang | Force-amplifying n-robot transport system (force-ants) for cooperative planar manipulation without communication[END_REF]. Communication delays and packet loss undermine the performance and stability of multi-robot systems. Furthermore, the hardware and software complexity can be reduced by confining explicit communication. For all these reasons, a manipulation method not relying on explicit communication has been proposed in [START_REF] Wang | Force-amplifying n-robot transport system (force-ants) for cooperative planar manipulation without communication[END_REF] for mobile ground robots.

This work has been partially funded by the Horizon Europe MSCA Flyflic [grant agreement No. ID: 101059875] and Horizon 2020 research and innovation programme [grant agreement No. 871479] AERIAL-CORE. 1 Robotics and Mechatronics (RaM) group, EEMCS faculty, University of Twente, Enschede, The Netherlands; 2 Univ Rennes, CNRS, Inria, IRISA, F-35000 Rennes, France; 3 LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, France; 4 Electrical electronic and computer engineering department, University of Catania, 95125, Catania, Italy; 5 Department of Computer, Control and Management Engineering, Sapienza University of Rome, 00185 Rome, Italy. Email: c.gabellieri@utwente.nl, marco.tognon@inria.fr dario.sanalitro@unict.it,a.franchi@utwente.nl Fig. 1: Two quadrotors connected through cables to a beam load. A motion capture system is used for the robots' localization.

On the other hand, aerial cooperative transportation not relying on explicit communication has been performed in [START_REF] Gassner | Dynamic collaboration without communication: Vision-based cable-suspended load transport with two quadrotors[END_REF] using vision. Alternatively, force feedback has been proposed: [START_REF] Tagliabue | Collaborative transportation using MAVs via passive force control[END_REF] considers a cable-suspended beam-like load and an admittance-based leader-follower scheme, and [START_REF] Tagliabue | Robust collaborative object transportation using multiple mavs[END_REF] extends the results to more robots and provides a method for gain-tuning which increases robustness against unmodeled dynamics and parameter uncertainties. However, [START_REF] Tagliabue | Collaborative transportation using MAVs via passive force control[END_REF] and [START_REF] Tagliabue | Robust collaborative object transportation using multiple mavs[END_REF] achieve transportation of the load disregarding its attitude. In addition, the leader-follower strategy is not used to regulate the robots' height, which is instead predetermined in a centralized way. Another example of communication-less force-based transportation where the velocity of a flexible object is controlled by a group of aerial robots is [START_REF] Thapa | Cooperative aerial load transport with force control[END_REF].

The study of full-pose regulation, with analytical proof of the system equilibrium configurations and their stability, has been carried out for the first time in [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF]. In [START_REF] Gabellieri | A study on force-based collaboration in swarms[END_REF], that work has been extended to a generic load transported by more than two robots. Recently, parameter uncertainties have been addressed in [START_REF] Gabellieri | Equilibria, stability, and sensitivity for the aerial suspended beam robotic system subject to parameter uncertainty[END_REF]. [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF]- [START_REF] Gabellieri | Equilibria, stability, and sensitivity for the aerial suspended beam robotic system subject to parameter uncertainty[END_REF] demonstrate how the regulation of the load's full pose is made possible by non-vertical cables that induce an internal force in the load.

However, the literature has not yet tackled the biases in the forces estimation/measurement that the robots use to cooperate and, thus, on which the task execution largely relies. That is especially interesting in practice, e.g., due to the typical drift that affects the output of force sensors, which are sensitive to temperature increase in repeated usage [START_REF] Abhiram | Thermal drift compensation of load cell reading using linear regression in weighing lysimeters[END_REF].

In this work, we address the aforementioned issue by including in the analysis the realistic case in which the external forces known by the robots and used for control are affected by errors.

The main contributions of this work are: (i) derivation of the new equilibrium configurations of the uncertain system in the presence of force biases; (ii) formal study of the effect of the load internal force in the robustness of the attitude control when forces are affected by biases. It emerges that the internal force is beneficial for the load control robustness against force biases; (iii) study of the sensitivity of the load attitude error to force bias variations. The load internal force is also shown to be beneficial for the error sensitivity; (iv) regulation of the load position in the presence of force biases; (v) validation through numerical simulations and experimental results.

II. MODEL

The considered system, reported in Fig. 1, consists of a beam-like load attached to two aerial robots through cables anchored to two non-coincident points on the load. We summarize here the dynamic model of the closed-loop system. Please note that more details about the modeling can be found in [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF] and [START_REF] Tagliabue | Robust collaborative object transportation using multiple mavs[END_REF], where similar assumptions are made on the description of the system and the control law, excluding the presence of force biases.

We define an inertial frame F W = {O W , x W , y W , z W } with z W oriented in the direction opposite to the gravity. The beam-like load has mass m L ∈ R >0 and a positive definite inertia J L ∈ R 3×3 . Rigidly attached to the load we define a frame

F L = {O L , x L , y L , z L }, where O L is centered in the load CoM.
The position and orientation of F L w.r.t. F W , defined by the vector 1 W p L ∈ R 3 and the rotation matrix R L ∈ SO(3), respectively, describe the full configuration of the load. Because the load is a beam, the yaw and pitch angles of the beam, denoted by ψ and θ , respectively, are enough to describe its entire attitude. We define q L = (p L , R L ) and the

vector v L = [ ṗ⊤ L L ω L ⊤ ] ⊤ , where L ω L ∈ R 3 is the angular velocity of F L w.r.t. F W expressed in F L .
Two cables are attached to the load through passive spherical joints at points

B i with i = 1, 2 with position L b i ∈ R 3 . Using basic kinematics, b i = p L + R L L b i . Moreover, we assume without loss of generality that L b 1 = [b 1 0 0] ⊤ and L b 2 = [-b 2 0 0] ⊤ , where b i ∈ R >0 , for i = 1, 2. We also define the length L = b 1 + b 2 .
The i-th cable is attached through a passive spherical joint to the i-th aerial vehicle at the point O i , coincident with the robot's CoM. Centered in this point, we define a frame

F Ri = {O i , x Ri , y Ri , z Ri } rigidly attached to each vehicle. The position of O i in F W is p Ri ∈ R 3 . We define q R = [p ⊤ R1 p ⊤ R2
] ⊤ and v R its time derivative. The i-th cable is modeled as a spring as done in [START_REF] Goodman | Geometric control for load transportation with quadrotor uavs by elastic cables[END_REF], [START_REF] Yigit | Novel omnidirectional aerial manipulator with elastic suspension: Dynamic control and experimental performance assessment[END_REF]. Its parameters are the constant elastic coefficient k i ∈ R >0 and the constant rest length denoted by l 0i ∈ R >0 . Defining the vector l i = p Ri -b i , we write the force acting on the load at B i using the simplified Hooke's law: 1 The left superscript is the reference frame, equal to F W if omitted.

f i = ∥f i ∥ l i ∥l i ∥ , with ∥f i ∥ = k i (∥l i ∥ -l 0i ) if
∥l i ∥l 0i > 0 and zero otherwise. The force produced on the i-th robot at O i , is equal to -f i . We assume, as commonly done in the literature, that the cables are always taut during the task execution [START_REF] Masone | Cooperative transportation of a payload using quadrotors: A reconfigurable cable-driven parallel robot[END_REF], [START_REF] Li | Cooperative transportation of cable suspended payloads with mavs using monocular vision and inertial sensing[END_REF], [START_REF] Sanalitro | Fullpose manipulation control of a cable-suspended load with multiple uavs under uncertainties[END_REF].

Assuming that the cables are attached to each robot's CoM, and in virtue of the time-scale separation principle, the rotational dynamics of the robots is decoupled from the remaining dynamics and will be neglected in the theory. The interested reader can find more details in [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF], [START_REF] Ruggiero | Aerial manipulation: A literature review[END_REF], [START_REF] Nonami | Autonomous flying robots: unmanned aerial vehicles and micro aerial vehicles -Chapter 12[END_REF].

As done in [START_REF] Tagliabue | Collaborative transportation using MAVs via passive force control[END_REF], [START_REF] Tagliabue | Robust collaborative object transportation using multiple mavs[END_REF], [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF], we use an admittance filter on the robots, which allows reshaping their translational dynamics, at will, to mimic that of a desired second-order system. Eventually, the closed-loop system dynamics can be written as in [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF]

: v = m(q, v, π A ) where q = (q R , q ⊤ L ), v = [v ⊤ R v ⊤ L ] ⊤ and m(q, v, π A ) = M -1 A -B A v R -K A q R -f + π A M -1 L (-c L (v L ) -g L + Gf) , (1)
the first six equations being the robots' translational dynamics under the assumption of perfect tracking of the admittance filter trajectory, and the latter six the load dynamics. We call

M A = diag(M A1 , M A2 ), B A = diag(B A1 , B A2 ) and K A = diag(K A1 , K A2 ), where M Ai , B Ai , K Ai ∈ R 3×3
are positive definite symmetric matrices that are the gains of the admittance controller. They correspond to, respectively, the virtual inertia of the robot i-th, and the damping and stiffness coefficients of a virtual spring-damper system that links the robot to a desired reference [START_REF] Villani | Handbook of robotics, chapter force control[END_REF]. 3 an additional forcing input of robot i-th that must be properly set to steer the load to the desired configuration, as explained in the following. Differently from what is done in previous works,

π A = [π ⊤ A1 π ⊤ A2 ] ⊤ with π Ai ∈ R
f = [f ⊤ 1 f ⊤ 2 ] ⊤ is not known to the robots. Instead, f = [ f ⊤ 1 f ⊤ 2 ]
⊤ contains the measured or observed cable force affected by some errors and is used in the robots' admittance controllers. Especially, we write fi := -f i + δ i (we recall that f i acts on the load by definition, so -f i acts on the robot), with δ i ∈ R 3 denoting an unknown bias. We have that δ i = δ i d i , where δ i ∈ R >0 is the bias intensity and d i := δ i /δ i the unit vector denoting the bias direction. M L = diag (m L I 3 , J L ) with I 3 ∈ R 3×3 the identity matrix, m L the mass of the load, and

J L ∈ R 3×3 its rotational inertia; g L = [m L ge ⊤ 3 0] ⊤ ,
where g is the gravitational acceleration and e i is the canonical unit vector with a 1 in the i-th entry. Coriolis and centrifugal terms and the grasp matrix are, respectively,

c L = 0 S(ω L )J L ω L , G = I 3 I 3 S( L b 1 )R ⊤ L S( L b 2 )R ⊤ L .
where S(⋆) is the skew operator. Remark 1. The admittance controller only requires local information: the robot's own state (p Ri , ṗRi ) and the force applied by its own cable, which can be measured by a force sensor or estimated by model-based observers as in [START_REF] Tagliabue | Collaborative transportation using MAVs via passive force control[END_REF], [START_REF] Ryll | 6D physical interaction with a fully actuated aerial robot[END_REF]. Therefore, the proposed control method is decentralized and does not require explicit communication between the robots. Note that the state of the load is generally not needed by the proposed controller.

A leader-follower approach is used, choosing, without loss of generality, robot 1 as leader and robot 2 as follower. In mathematical terms, this is expressed by setting K A1 ̸ = 0 and K A2 = 0 [14], [START_REF] Tagliabue | Robust collaborative object transportation using multiple mavs[END_REF], [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF]. In this way, only the designated leader drags the load while the follower moves by following the force, compensating for part of the load weight, dampening oscillations, and contributing to the load attitude control. The load internal force t L is defined as

t L := 1 2 f ⊤ I 3 -I 3 ⊤ R L e 1 ,
where

I 3 -I 3 ⊤ R L e 1 ∈ null(G) 2 .
We have that if t L > 0, the internal force causes a tension in the load; if t L < 0, the internal force causes a compression.

III. EQUILIBRIUM STUDY

In this Section, we present the study of the equilibrium configurations when system parameters are uncertainly known and force biases affect the information available to the controller. A desired load configuration qL is given, and ⋆ indicates the desired value of ⋆ in general.

Theorem 1 (Equilibrium Inverse Problem). Consider the closed-loop system (1) and assume that the load is at a given desired configuration q L = qL = ( pL , RL ). For each internal force t L ∈ R, there exists a unique constant value of the forcing input π A = πA (and a unique position of the robots q R = qR ) such that q = ( qL , qR ) is an equilibrium of the system:

for i = 1, 2, πA ( qL ,t L ) = K A qR + f ( qL ,t L ) -δ (2) pRi ( qL ,t L ) = pL + RL L b i + ∥ fi ∥ k i + l 0i fi ∥ fi ∥ , with (3) f ( qL ,t L ) = f1 f2 = b 2 m L g L b 1 m L g L I 3 I 3 e 3 + t L I 3 -I 3 RL e 1 . ( 4
)
The proof is here omitted for the sake of space but the interested reader can find a similar proof in [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF]. However, differently from what emerged in previous works and as suggested by intuition, here the forcing input that brings the system to the desired configuration, πA , must compensate for the force biases δ

= [δ ⊤ 1 , δ ⊤ 2 ] ⊤ , see (2)
. The user chooses the parameter t L in (4) to make qL an equilibrium. Nevertheless, consider the case when the controller has only access to the uncertain values, defined as mL , b1 , L, l0i , ki , ki , instead of the corresponding real ones, m L , b 1 , L, l 0i , and k i , respectively. We shall indicate as ∆ ⋆ the difference between the uncertain and real value, namely, ⋆ -⋆. In such realistic conditions, πA cannot be exactly applied because (3) and ( 4) are unknown due to parametric uncertainties, and (2) is unknown because the force bias δ.

Hence, we shall study the equilibrium configurations of the system when πA is applied, where πA indicates πA computed with the uncertain values of the system parameters and without the use of the unknown force bias.

Theorem 2 (Equilibrium Direct Problem). Given qL = ( pL , RL ) and the internal force t L ∈ R, compute the forcing input πA from (2) using the uncertain parameters ⋆ and without using δ. Applying then πA ( qL ,t L ) to the closed loop system (1), the following conditions describe the sole equilibrium configurations:

p R1 = pR1 -K -1 A1 (∆ m ge 3 + (δ 1 + δ 2 )) := p eq R1 (5) R eq L s.t.S(e 1 )R eq L ⊤ ξ ge 3 + t L RL e 1 -δ 2 = 0 (6)
f 1 = m L ge 3 - mL b1 g L e 3 + t L RL e 1 -δ 2 := f eq 1 ( 7
)

f 2 = b1 mL g L e 3 -t L RL e 1 + δ 2 = f2 + δ 2 := f eq 2 ( 8
)

p L = p eq R1 -R eq L L b 1 - ∥f eq 1 ∥ k 1 + l 01 f eq 1 ∥f eq 1 ∥ := p eq L , (9)
where ⋆ indicates the reference value of ⋆ but computed using the uncertain parameters and ξ = b 1 m L L -b1 mL L is a term that depends on the uncertain load parameters.

The proof is similar to that in [START_REF] Gabellieri | Equilibria, stability, and sensitivity for the aerial suspended beam robotic system subject to parameter uncertainty[END_REF], but Theorem 2 leads to new interesting results when force biases are taken into account. Note that, formally, the system reaches an equilibrium only if δ i is constant at the equilibrium. In the following, we thus assume that the force bias variation is negligible at steady state.

The first fact that clearly emerges from Theorem 2, and especially from [START_REF] Lee | Geometric control of quadrotor uavs transporting a cablesuspended rigid body[END_REF], is that the load equilibrium attitude is only affected by the bias in the follower robot's force, δ 2 . This is also reflected in [START_REF] Li | Cooperative transportation of cable suspended payloads with mavs using monocular vision and inertial sensing[END_REF] and ( 8) since the equilibrium forces, which determine the load equilibrium attitude, only depend on δ 2 . On the other hand, from (5), δ 1 affects the leader robot equilibrium position, and so the load position-see [START_REF] Rossomando | Aerial load transportation with multiple quadrotors based on a kinematic controller and a neural smc dynamic compensation[END_REF]. This result suggests to the user that, e.g., to maximize the accuracy of the load attitude control when using force sensors, a better sensor on the follower robot may be preferred.

Let us now choose a parametrization of δ 2 that highlights the vector components along a basis described by RL e 1 , e 3 , and RL e 2 . This is a valid representation as long as RL e 1 ̸ = e 3 , always true in reality as a vertical load would not be practically realizable. Hence, we write δ 2 = δ L RL e 1 + δ y RL e 2 + δ v e 3 , where δ L , δ y , and δ v are the components of δ 2 along the respective directions, and (6) becomes

S(e 1 )R ⊤ L (ξ g -δ v )e 3 + (t L -δ L ) RL e 1 -δ y RL e 2 = 0. (10
) Consequently, we have that δ v has the same effect on the load attitude equilibrium as the uncertainty on the load parameters. Moreover, δ L has an interesting effect on the system behavior as it adds to the internal force t L . So, an equivalent internal force can be defined as t Lδ L . We have shown in [START_REF] Gabellieri | Equilibria, stability, and sensitivity for the aerial suspended beam robotic system subject to parameter uncertainty[END_REF] that the best choice to guarantee stability and robustness of the sought equilibrium state is t L > 0. As a result, selecting an appropriate value of t L is essential to ensure that t Lδ L > 0 Unlike parameter uncertainties, the force bias has the unique effect of not only affecting the pitch of the load at the equilibrium but also the yaw. This is because δ y has an effect on the system equilibrium that none of the parameter uncertainties has, as shown in [START_REF] Thapa | Cooperative aerial manipulation with decentralized adaptive force-consensus control[END_REF]. Specifically, by solving (6) one has

ψ eq = atan δ y c ψ + (t L -δ L )c θ s ψ -δ y s ψ + tc θ c ψ , (11)
where t := (t Lδ L ) and c ⋆ and s ⋆ are the cosine and sine functions, respectively. Note that [START_REF] Sanalitro | Indirect force control of a cable-suspended aerial multi-robot manipulator[END_REF] confirms that ψ eq = ψ + kπ if δ y = 0. Note also that δ v does not appear in [START_REF] Sanalitro | Indirect force control of a cable-suspended aerial multi-robot manipulator[END_REF] as it does not influence the yaw angle at the equilibrium but only the pitch angle. For completeness, we report the expression for θ eq , provided that θ eq ̸ = π/2 + kπ:

θ eq = atan ξ g -δ v -ts θ -c ψ eq (δ y s ψ + tc θ c ψ ) + s ψ eq (δ y c ψ -tc θ s ψ ) (12)
This analysis has shown that the force bias contributes to the load pose error at the equilibrium. In the following, we shall propose methods to mitigate such an effect.

A. Uncertainty solely on the estimated or measured forces

This section studies the case where the only uncertainty in the system is on the measured or estimated cable force. This can be of interest in all cases in which the system parameters are accurately known and, thus, the bias on the force represents the main source of uncertainty. In this case, at the equilibrium

p R1 = pR1 -K -1 A1 (δ 1 + δ 2 ) (13)
S(e 1 )R eq L

⊤ (t L RL e 1 -δ 2 ) = 0 (14)
f 1 = L -b 1 L mge 3 + t L RL e 1 -δ 2 = f1 -δ 2 ( 15
)
f 2 = b 1 L mge 3 -t L RL e 1 + δ 2 = f2 + δ 2 . (16)
Specifically, at steady state, the leader robot could estimate δ 2 from (15), being f1 known to the robot. Once it retrieves δ 2 , it can also compute δ 1 using (13), being pR1 known. Analogously, the follower robot is able to estimate δ 2 from [START_REF] Pereira | Pose stabilization of a bar tethered to two aerial vehicles[END_REF]. Hence, the robots could independently correct their references and thus bring the system to the desired configuration. Nevertheless, this does not hold true in the more general and realistic cases when parameter uncertainties and force biases are simultaneously present.

IV. THE ROLE OF THE INTERNAL FORCES

In this section, we provide a method to mitigate the effect of force biases on the load error. Especially, we propose a formal analysis of the role that the internal force plays in the load pose at the equilibrium in the presence of force biases and parameter uncertainties.

A. Load attitude error Theorem 3. The maximum value of the load attitude error e R L caused by given parameter uncertainties and force biases is inversely proportional to t L . Furthermore, the same holds for the error sensitivity w.r.t the intensity of the force bias δ 2 , defined as

∂ e R L ∂ δ 2 , while ∂ e R L ∂ δ 1 = 0
Proof: Consider the following definition of the attitude error:

e R L = ∥R eq L e 1 × RL e 1 ∥ 2 . ( 17
)
Note that R L e 1 is enough to describe the entire attitude of the beam-like load. Moreover, the definition in ( 17) is a suitable metric for the attitude error as it is ∥R eq L e 1 × RL e 1 ∥ 2 = 0 for R eq L = RL , and it increases with the displacement between the last two vectors, at least locally (for displacements smaller than π/2). Rewrite now [START_REF] Lee | Geometric control of quadrotor uavs transporting a cablesuspended rigid body[END_REF] in F W as:

R eq

L e 1 × b 1 m L -b1 mL ge 3 + Lt L RL e 1 -Lδ 2 = 0. (18) Define also: 18), we have that R eq L e 1 × RL e 1 = x and, from [START_REF] Spurny | Cooperative transport of large objects by a pair of unmanned aerial systems using sampling-based motion planning[END_REF], that e R L = x ⊤ x = ||x|| 2 . Hence, we analize the upper bound of ||x|| to find that of e R L . From [START_REF] Mellinger | Cooperative grasping and transport using multiple quadrotors[END_REF],

(b 1 -∆b)(m L -∆m) -b 1 m L t L L (R eq L e 1 × ge 3 ) + 1 t L (R eq L e 1 × δ 2 ) = α t L L (R eq L e 1 × ge 3 ) + 1 t L (R eq L e 1 × δ 2 ) := x ( 19
)
with α := (b 1 -∆b)(m L -∆m)(ℓ -∆ ℓ ) -m L b 1 . From (
||x|| = || α t L L (R eq L e 1 × ge 3 ) + 1 t L (R eq L e 1 × δ 2 )|| ≤ (20) || α t L L (R eq L e 1 × ge 3 )|| + || 1 t L (R eq L e 1 × δ 2 )|| ≤ ||α||g t L L + δ 2
t L where all terms in the last line are inversely proportional to t L . This completes the first part of the proof. Regarding the sensitivity, we can write it as:

∂ e R L ∂ δ 2 = 2x ⊤ ∂ x ∂ δ 2 = = 2 αR eq L e 1 Lt L × ge 3 ⊤ R eq L e 1 t L L × d 2 + 2 RL e 1 t L × δ 2 ⊤ R eq L e 1 t L L × d 2
Eventually, recalling that, given three vectors a, b, and c:

(a × b) ⊤ (a × c) = |a| 2 (b ⊤ c) -(a ⊤ b)(a ⊤ c),
we can write

∂ e R L ∂ δ 2 = 2 gα(-sin θ cos β 1 + cos β 2 ) Lt 2 L + δ 2 cos β 1 2 t 2 L ,
where β 1 is the angle between R eq L e 1 and δ 2 , and β 2 that between e 3 and δ 2 . The application of analogous steps as in [START_REF] Wang | Force-amplifying n-robot transport system (force-ants) for cooperative planar manipulation without communication[END_REF] finalizes the proof. □ Theorem 3 shows that a positive t L increases the attitude error robustness and is beneficial for the error sensitivity to variations of the force bias intensity. The larger the internal force the smaller the maximum error and sensitivity, given the uncertainties and the force biases.

In other words, thanks to Theorem 3, one knows that a large value of t L > 0 is beneficial to maintain the expected load attitude error low and constant over different task executions in which the value of the force biases may have changed.

B. Load position error

Differently from the load attitude error, the load position error at the equilibrium does not necessarily decrease when increasing t L [START_REF] Gabellieri | Equilibria, stability, and sensitivity for the aerial suspended beam robotic system subject to parameter uncertainty[END_REF]. However, changing solely the reference position of the leader robot is enough to bring to zero the load position error even in the presence of force biases. To see that, substitute (5) into (9) explicitly writing pR :

p eq L = pL + RL L b1 + ∥ f1 ∥ k1 + l01 f1 ∥ f1 ∥ -K -1 A (∆ m ge 3 + δ 1 + δ 2 ) - ∥f eq 1 ∥ k 1 + l 01 f eq 1 ∥f eq 1 ∥ -R eq L b 1 . (21)
From [START_REF] Tagliabue | Robust collaborative object transportation using multiple mavs[END_REF] we have an expression of p eq L -pL := e p L . Now, if the sole leader robot knows the load position, it can adjust its reference, such that the new reference, is: 2 pR1 = pR1 -e p L . This implies p eq R1 = 2 pR1 -K -1 A (∆ m + δ 1 + δ 2 ) and, thus, (21) gives p eq L = pL . Moreover, the leader robot position does not influence the attitude of the load at the equilibrium, which only depends on the cable forces-see the last three lines of (1). In other words, the leader robot corrects the position error of the load, while the cooperatively generated internal force acts on the attitude error.

The previous result can be exploited also in the case that a user wishes to manually command the position of the load. The reference forces necessarily generated by the combined action of the robots take care of correcting the load final attitude, while the user can have intuitive control over the load position through the leader robot position alone.

V. NUMERICAL RESULTS Extensive numerical tests have been carried in Gazebo ODE physics engine using a URDF description of the system. A realistic numerical simulation model has been preferred to one complying with all simplifying assumptions used to derive the theoretical results. In particular: (i) underactuated quadrotors are deliberately chosen because they represent the worst-case scenario in terms of the validity of the theoretical assumptions; (ii) cables are composed of multiple links, thus they are subject to sagging; (iii) there is no guarantee of perfect trajectory tracking as assumed in the theory but a standard position controller [START_REF] Lee | Geometric tracking control of a quadrotor UAV on SE(3)[END_REF] is implemented for each robot; (iv) the wrench observer proposed in [START_REF] Ryll | 6D interaction control with aerial robots: The flying end-effector paradigm[END_REF] is used to produce an estimate of the external force applied on each robot.

The control software has been implemented in Matlab-Simulink using the Generator of Modules GenoM3 . Matlab-Gazebo is also managed by a Gazebo-genom3 plugin 4 . A state machine is used to simulate all phases of a physical experiment, beginning with takeoff, to ensure that the results mimic a real full-length operation. The robots lift the load after takeoff, and the admittance controller is activated.

The robots have a mass of 1.03 Kg each and every propeller can exert up to 6 N thrust. The bar has a mass of

∆m m = -6.9% ∆ b 1 b 1 = ∆ L L = 9.3% ∆ l 0i l 0i = 9.2% Scenario2: ∆m m = 0.94% ∆ b 1 b 1 = ∆ L L = -4.4% ∆ l 0i l 0i = -8.0% Scenario3: ∆m m = -8.1% ∆ b 1 b 1 = ∆ L L = -9.1% ∆ l 0i l 0i
= -4.5% 0.5 kg and a length of 1 m. Without loss of generality, the robots are asked to bring the load to pL = [1 1 1] ⊤ m with θ = ψ = 0 deg.

First, Fig. 2 contains the results from 27 runs obtained as follows. Three different scenarios have been defined, each of which is characterized by different parametric uncertainties. Specifically, for each scenario, each uncertain parameter has been randomly generated between ±10% of the corresponding real parameter value. Then, for each scenario, the same manipulation task as described above has been carried out 9 times: 3 times for each different value of t L ∈ {0.5, 1.5, 2}N. Indeed, for each value of the internal reference force, 3 different biases on the robots' estimated forces have been tested. Specifically, for i = {1, 2}, δ i = a[1 1 1] ⊤ with a = {0.1, 0.2, 0.3} N, which means δ i = 0.173 N, 0.346 N, and 0.520 N, respectively. The results clearly show the beneficial effect of higher values of t L in terms of the accuracy of load attitude control. Please note that in the chosen setup, δ L is such that the equivalent internal force at the equilibrium is t Lδ L > 0. However, large values of δ 2 and small values (0.5 N) of t L result in a small equivalent internal force and, thus, in a large load attitude error at the equilibrium (e.g., 100 deg in Scenario2). Furthermore, the data shows that the variation of the load attitude error against the force bias, e.g., the error sensitivity to parameter variation, is lower for larger values of internal force, given the same uncertainties. This emerges from the slope of the lines fitting the evolution of e R L in Fig. 2. For the sake of repeatability, we report in Table I the randomly generated values of the uncertainties. Without loss of generality, no uncertainty on k i has been set as it has an equivalent effect to the one on l 0i [START_REF] Gabellieri | Equilibria, stability, and sensitivity for the aerial suspended beam robotic system subject to parameter uncertainty[END_REF].

In Fig. 3, the screenshots from a simulation showing the unstable nature of the desired equilibrium when t L = 0 N but δ L = 0.1 N so that the equivalent internal force is negative, equal to -0.1 N. Hence, the scenario is equivalent to that described in [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF] for t L < 0. This shows the important role of the internal force especially when there are biases in the estimated forces.

Eventually, Fig. 4 shows the results of a simulation in which the leader robot changes its reference position based on the load position equilibrium error, as in Sec. IV-B, and corrects the load position error. The last two plots of Fig. 4 are the force biased estimations and the dotted lines the robots' references, -fi . Note that, from (8) the follower's estimated force at the equilibrium is -f eq 2 + δ 2 = -f2 . Instead, according to [START_REF] Li | Cooperative transportation of cable suspended payloads with mavs using monocular vision and inertial sensing[END_REF], the leader robot estimates -f eq 1 + δ 1 = -f1 + ∆ m ge 3 + δ 1 + δ 2 ̸ = -f1 , in accordance with the plots. Fig. 3: t L = 0 but the equivalent internal force is negative due to δ L . The system does not converge to the desired equilibrium (red bar) but goes to the other one with flipped load attitude (see [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF]), and tension in the load. The leader has a red arm, the follower a blue one.

Fig. 4:

t L = 1.5 N, δ i = [0.2 0.2 0.2] ⊤ N, ∆ m m = -3.6%, ∆ b 1 b 1 = ∆ L L = 9.

4%, and

∆ l 0i l 0i = 6.4%. After 42 s, the leader changes its position reference and p L goes to pL . Other quantities are unaffected.

VI. EXPERIMENTAL RESULTS

This section describes the experiments, and the setup is in Figure 1. Two quadrotors, with a mass of 1.03 Kg and a maximum thrust for each propeller of 6 N, have been equipped with two lightweight cables of length equal to 1 m attached at the endings of a 2-meter-long bar with a mass of 0.2 Kg. These components are shown in Fig. 1. Each robot has an onboard PC and 4 electronic speed controllers that control the propeller speed in closed loop [START_REF] Franchi | Adaptive closed-loop speed control of BLDC motors with applications to multi-rotor aerial vehicles[END_REF]. State-of-theart UKF-based state estimation fusing the motion capture system measurements at 120 Hz with the IMU measurements at 1 kHz and a geometric control [START_REF] Lee | Geometric tracking control of a quadrotor UAV on SE(3)[END_REF] run onboard. Three different task executions have been carried out with pL = [0.5 0 1.5] ⊤ m, ψ = 0.2 rad, and θ = -0.12 rad and three different values of the internal force t L ∈ {1, 2, 3} N. In each run, a force bias was present and the controller knew the load parameters with a 10% error. The force bias has been estimated by bringing the position-controlled robots vertically on top of their cables during an initialization phase at the beginning of each task execution. Because of the short task duration in the laboratory conditions, we assumed the force bias constant throughout the execution. The yaw angles of the robots are controlled to be constantly equal to zero.

We had each component of δ 2 equal to ∼ 0.2 ± 0.1 N, ∼ 0.4 ± 0.1 N, and ∼ 0.1 ± 0.1 N in the execution with t L equal to 1, 2, and 3 N, respectively. Fig. 5 shows the load attitude resulting from the three task executions. Please note that, although the force biases are not exactly the same in the three tests, they are such that the equivalent internal force is higher in the tests with a higher value of t L . The main result to be noted is how the final value of the pitch and yaw errors as well as the overall load attitude errors are smaller for higher values of the internal force.

VII. CONCLUSIONS

This work studied the force-based manipulation of cablesuspended loads using non-communicating UAVs in the presence of biased force estimates/measurements. The equilibrium points of the closed-loop system have been studied. It has been shown that a load internal force induced by nonvertical cables at the equilibrium is beneficial for the robustness of pose control and for the error sensitivity to force bias variations. Numerical and experimental results have been presented. Interesting future directions are the extension of the robustness analysis to general rigid bodies manipulated by more than two robots and the outdoor experiments.

Fig. 2 :

 2 Fig. 2: Each plot contains 9 different simulations with random parameter values, 3 simulations for each value of t L . e R L has been computed as ||ψ eq -ψ|| + ||θ eqθ ||. The lines are the linear fitting that approximates the error values (indicated by an asterisk) obtained for each tested t L ; the goodness of each fit is shown in the legend by the R-squared value, indicated as R 2 .

Fig. 5 :

 5 Fig. 5: On the right, pitch (top) and yaw (bottom) errors in 3 tests for different values of t L . On the left, box plots of the mean value of e R L = ||θ -θ || + ||ψ -ψ|| over the last 20 seconds of the 3 tests. With t L = 1, θ is still increasing towards the equilibrium.

TABLE I Scenario1 :

 I:

Preprint version, final version at http://ieeexplore.ieee.org/ 2 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2023)

null(⋆) indicates the nullspace of ⋆.

Preprint version, final version at http://ieeexplore.ieee.org/ 3 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2023)

Preprint version, final version at http://ieeexplore.ieee.org/ 4 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2023)

https://git.openrobots.org/projects/genom3

https://git.openrobots.org/projects/mrsim-gazebo

Preprint version, final version at http://ieeexplore.ieee.org/ 5 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2023)

Preprint version, final version at http://ieeexplore.ieee.org/ 6 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2023)

Preprint version, final version at http://ieeexplore.ieee.org/ 7 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2023)