Sette Diop
email: diop@l2s.centralesupelec.fr

On sensor selection for differential algebraic systems observability

Keywords: Sensor selection, observability theory, observation problems, differential algebraic systems theory

We address the problem of selecting sensors, that is, output equations, in order to endow dynamic equations of a system with some properties. Among all such desirable properties is the basic one of observability and/or identifiability. Once such a problem is solved one may ask how to choose sensors in order to improve estimation algorithms in terms of reliability, robustness, or simply, low complexity. First, what is the minimal number of sensors that make the dynamics observable? Second, when the sensors are bound to measure state components, what is their minimum number? Third, how may the observability margin be improved by selecting the sensors? In this communication we provide an overview of these questions in the differential algebraic approach of observation problems.

INTRODUCTION

Given state dynamics

ẋ = f (x, u) (1)
with input u and state x there may be opportunity in practice for the designer to choose sensors, that is output equation, in order to endow the system with desirable properties in terms of observation problems. Selecting an output equation amounts to finding function

y = h(x, u) (2)
such that the state becomes observable with respect to (u, y). This is part of various problems that are addressed in the litterature. In the parameter estimation context for linear systems, see for instance pioneering works [START_REF] Mehra | Optimization of measurement schedules and sensor designs for linear dynamic systems[END_REF]; [START_REF] Friedland | On the calibration problem[END_REF].

Notions of differential algebraic geometry used in this introduction are briefly recalled in a later section.

Let y be an arbitrary output of the system. By definition y is componentwise algebraic over k u, x . An output makes the dynamics observable if d • k u,y k u, x, y = 0 , that is, if each component of x is algebraic over k u, y .

Clearly, y = x is an output which makes the system observable.

Let n denote the number of components of x, and

I = p ∈ N : ∃y 1 , • • • , y p ∈ k (u, x) , d • k u,y k u, x, y = 0 .
The set I is the one of integers p such that there exists an output y with p components which makes the system observable.

It is a nonempty (since n ∈ I) subset of N. Therefore, I contains a smallest element which is precisely the minimal number of sensors which make the system observable.

To the author's opinion characterization of sensors minimal number is the first question which needs to be answered in sensor selection problem. And it seems to be an open problem. A very partial answer is given in a later section.

The next question is, when the output y is chosen as a subset of the components of x, instead of a vector rational function of u and x, what are those subsets of components of x which make the system observable.

In the differential algebraic approach of observation problems we may consider generalized implicit state dynamics

P ( ẋ, x, θ, u) = 0 (3)
where u designates a finite collection of derivatives of the input u, and P is a vector function with components P i (non differential) polynomials in x, ẋ, θ and u, and take output equation as polynomial equations

Q(y, x, θ, u) = 0 (4)
and ask for an additional question: assume that we are given the freedom to select sensors among state components, how to choose them such that the system is identifiable, that is, each component of θ is algebraic over k u, y ?

The differential algebraic approach of observation problems that is used here dates back to late eighties and early nineties with works of [START_REF] Pommaret | Géométrie différentielle algébrique et théorie du contrôle[END_REF]; [START_REF] Fliess | Quelques remarques sur les observateurs non linéaires[END_REF]; [START_REF] Glad | Model structure identifiability and persistence of excitation[END_REF]; Diop and Fliess (1991b,a). See [START_REF] References Diop | From the geometry to the algebra of nonlinear observability[END_REF] for a survey.

The main point of this approach, as first clarified in Diop and Fliess (1991b), is that a quantity, say z, of a system is observable with respect to some other one, say w (which is supposed to be available in some time interval), if each component of z is a solution of a (non differential) algebraic equation with coefficients eventually depending on w and finitely many of w's time derivatives.

The theory applies to models of systems in terms of differential algebraic equations only but which may be implicit in the variables to be observed.

In this approach the identifiability of constant parameters is simply viewed as the observability of these parameters in the system equations supplemented by differential equations expressing the fact that the time derivatives of the parameters are zero.

The remaining of this communication is organized as follows. Main lines of the differential algebraic approach of observability are recalled in the the next section. Then the single sensor selection theorem is recalled in section 3.

Next we overview the sensor minimal number problem in section 4.

ON THE DIFFERENTIAL ALGEBRAIC APPROACH OF OBSERVATION PROBLEMS

A thorough introduction to the differential algebraic approach is available in [START_REF] References Diop | From the geometry to the algebra of nonlinear observability[END_REF]. For the sake of completeness the following definition is recalled from there.

By a (differential algebraic) system with coefficients in k it is meant here a quasi-affine variety, X defined over k.

In other words, a system is defined by a set of differential equations together with a differential inequation.

A system is often specified along with its input, u = (u 1 , u 2 , . . . , u m ), output y = (y 1 , y 2 , . . . , y p ), and latent variable z = (z 1 , z 2 , . . . , z ν ). Because the same differential equations usually define distinct systems when the input, output, and latent variable are specified in different ways.

But in this observability study, the crucial distinction between the system variables will be between the supposedly measured or known variables or quantities, and the those whose estimation, or observation is of interest.

The variables of the system will, therefore be denoted by τ = (τ 1 , τ 2 , . . . , τ µ ).

The differential Zariski closure of X is denoted by X , and the differential k-algebra associated to the latter differential affine variety is denoted by k X = k {τ } and with coefficients in the ordinary differential field, k, with characteristic zero. X is defined by giving differential quasi-affine variety in U m × U ν × U p where U stands for a differential universal field extension of k.

With variables u, z, y, the differential k-algebra associated to the latter differential affine variety is denoted by k X = k {u, z, y} as explained below.

Let µ ∈ N, µ ≥ 1, and U {T 1 , T 2 , . . . , T µ } be the differential polynomial U-algebra in the differential indeterminates

T 1 , T 2 , . . . , T µ . If Σ is a subset of U {T 1 , T 2 , . . . , T µ }, V (Σ)
denotes the subset of U µ consisting of the zeros of Σ in U µ , i.e., the elements (t 1 , t 2 , . . . , t µ ) ∈ U µ , such that P (t 1 , t 2 , . . . , t µ ) = 0 (P ∈ Σ). Conversely, if X is a subset of U µ then I (X ) denotes the defining differential ideal of X , i.e., the perfect differential ideal of U {T 1 , T 2 , . . . , T µ } consisting of the differential polynomials P such that

P (t 1 , t 2 , . . . , t µ ) = 0 ((t 1 , t 2 , . . . , t µ ) ∈ X ) .
If a differential algebraic set X of U µ is defined over k then I (X ) /k denotes the defining differential ideal of X over k. For a differential algebraic set X which is defined over k U {X } (k {X }, respectively) denotes the differential coordinate ring of X (the differential coordinate ring of X over k, respectively), and U X (k X , respectively) denotes the complete differential ring of quotients of U {X } (the complete differential ring of quotients of k {X } over k, respectively) and call it the differential ring of differential rational functions on X (the differential ring of differential rational functions on X over k, respectively). Recall that the differential algebra U X (k X , respectively) is a field if, and only if, X is irreducible (irreducible over k, respectively). An irreducible system X over a differential field k is equivalently defined by one of the three following data sets:

(i) A positive integer µ and a set of differential polynomials, P 1 , P 2 , . . ., in k{T 1 , . . . , T µ } such that the perfect differential ideal generated by the P i 's is prime.

The P i 's are usually said to be the equations of the system X , but any (finite) set of generators of the perfect differential ideal, I (X )

/k = {P 1 , P 2 , • • • } /k , of k {T 1 , T 2 , .
. . , T µ } which is generated by the P i 's play the same role. The variables of X are then the residues τ 1 , τ 2 , . . . , τ µ of T 1 , T 2 , . . . , T µ (mod I (X ) /k ), respectively. (ii) A differential universal field U which is universal over k, a positive integer µ, and an irreducible differential k-algebraic set, X , of U µ . The equations of the system X are then any set of generators of the defining differential ideal of the differential k-algebraic set X over k. The variables of X are the generic point, τ 1 , τ 2 , . . . , τ µ , consisting of the differential coordinate functions on the differential k-algebraic set X , but any other generic point, τ 1 , τ 2 , . . . , τ µ , of the differential k-algebraic set play the role of a new set of variables of the system X , the corresponding equations being different from those which are relative to τ 1 , τ 2 , . . . , τ µ . (iii) A differential field k-extension, k X , differentially of finite type. Given k X = k τ 1 , τ 2 , . . . , τ µ , τ 1 , τ 2 , . . ., τ µ are the variables of X . The equations of the system X are then any set of generators of the defining differential ideal of k τ 1 , τ 2 , . . . , τ µ , that is, the kernel of the differential morphism of differential

k-algebras of k {T 1 , T 2 , . . . , T µ } → k {τ 1 , τ 2 , . . . , τ µ } which sends T i into τ i .
Variables of a system should be distinguished from values of these variables (called the trajectories) which verify the equations of the system. The above definitions of systems are incomplete in that the variables of a system are often partitioned into external variables (which consists of the input u (which itself consists of the control and/or the disturbance), and the output y), and the latent variables (that are denoted by z and which consists of the remainder of the variables of the system when the external variables are specified). The external variables are often attached to the system and cannot be arbitrarily changed without altering the definition of the system. In particular, the input variable should be a differential transcendence basis of k X over k, or at least, as containing such a differential transcendence basis.

In observation problems, the system variable is partitioned into the data, or observations, w = w 1 , . . . , w µ , the variable being observed (or estimated) z = z 1 , . . . , z n and the remaining variables, ζ. In the classical observation problem, the data consist exclusively of (u, y), the control u and the measurements y. When the variable ζ is present, the projection X w,z of X along the variable

ζ is considered. It is the set of elements (w, z) ∈ k µ × k n such that there is at least ζ such that (w, z, ζ) ∈ X .
In terms of equations, previously defined systems are those described by

P i (w, z, ζ) = 0 , i = 1, 2, . . . , Q(w, z, ζ) = 0 , (5)
where the P i 's and Q are finitely many polynomials in w, z, ζ and their derivatives.

For a system X the variable z is said to be (algebraically) observable with respect to w if the projection map π : X w,z → X w (sending every trajectory (w, z) of X w,z onto the corresponding observation w) is generically finite.

If z is observable with respect to then the degree of π is called the observability degree of z with respect to w, and is denoted by d • w z.

The variable z is said to be rationally observable with respect to w if it is observable with respect to w with observability degree one.

State systems of the form (1) are said to be observable if x is observable with respect to (u, y).

It was first proved in Diop and Fliess (1991b) (see [START_REF] References Diop | From the geometry to the algebra of nonlinear observability[END_REF] for more details) that the previous definition has a differential algebraic translation, namely: z is observable with respect to w iff z is algebraic over k w , that is, for each component, z i of z there is a polynomial equation

H i (z i , w, ẇ, . . .) = 0 (6)
in z i , and finitely many time derivatives of the data w, with coefficients in k.

A quite general rank condition which applies to implicit differential algebaric systems has been obtained in Diop and Fliess (1991b). When specialized to rational dynam-ics (1) this rank condition is similar to (but is not formally the same as) the rank condition found in [START_REF] Hermann | Nonlinear controllability and observability[END_REF].

The reader is referred to [START_REF] References Diop | From the geometry to the algebra of nonlinear observability[END_REF] for details on differential algebraic geometry terms or notations used here without explanations.

THE SINGLE SENSOR SELECTION THEOREM

When the output equation ( 2) is scalar and linear in x as follows

y = α 1 x 1 + α 2 x 2 + • • • + α n x n (
7) then this rank condition reduces to the rank of the Wronskian matrix of the coefficients α. The existence of a scalar output (7) which makes the state of (1) observable then results from a classical theorem on the linear dependence of vectors of functions over constants.

In addition, the single sensor selection problem thus obtained implies that if the rational function f is with coefficients in a differential field of constants (say, k = R) then any set of functions of the time, α 1 (t), α 2 (t), . . . , α n (t), which is linearly independent over k, will make the state observable. Theorem 1. Let the state dynamics of a rational state system X be given by ẋ = f (x, u) (8) with a vector rational function f of the input u and the state x, and with coefficients in a differential field k of constants. Let m and n be the respective numbers of components of u and x. Let K be a differential extension field of k containing nonconstants. There always is a scalar output

y = n i=1 α i x i (9)
with α 1 , α 2 , . . . , α n in K, which makes x observable with respect to (u, y). Moreover, for y as in output 9 to make X observable it suffices that the associated α's be linearly independent over the subfield of constants of K. Remark 2. Note that the previous theorem is also an answer to the sensor selection identifiability problem mentionned in the introduction.

ON THE SENSOR MINIMAL NUMBER PROBLEM

Let us consider the sensor selection for the identifiability of θ in system 10.

P i (z, θ, u) = 0 , i = 1, 2, . . . , Q(z, θ, u) = 0 , (10)
In the previous definition of the set I the output was supposed to be in k (u, z), this may be relaxed by allowing components of y to be algebraic over k (u, z) instead.

I = {p ∈ N : ∃y 1 , y 2 , • • • , y p algebraic over k (u, z) , d • k u,y k u, θ, y = 0 .
If the output y is chosen as a subset of the components of z, instead then a complete but combinatorial answer to the question is as follows.

Since the set of subsets of components of z is finite, they may be examined one to see if their measure is sufficient for θ to be observable with respect to (u, y). For each subset, y = (z i1 = z 1 , . . . , z ip = y p ), the test merely consists of computing the characteristic set of P i (z, θ, u) (i finitely many), y j = z ij (1 ≤ j ≤ p) with respect to a ranking such that any derivative of u and y is lower than any component of θ, whose derivatives, in turn, are lower than any state component which is not in y.

There are 2 n -1 observability tests thus to be done when the number of components of z is n!

The question thus reduces to how to avoid most of those characteristic set computations, if possible?

The answer to the question is positive if, and only if, θ 1 , θ 2 , . . . , θ q are algebraic over k u, z .

Let a ranking of k {U, Z, Θ} be fixed such that every derivative of U is lower than Z; every derivative of Z is lower than Θ. Such a ranking is denoted as following.

{u}, {z}, {θ}.

(11) Let A be a characteristic set of Z with respect to ranking 11. Lemma 3. A consists of two groups of differential polynomials:

A z (U, Z) , A θ (U, Z, Θ) . A z (U, Z) represents the elements of A whose leaders are derivatives of components of Z. There are as many elements in this group as there are components of z. (Each component of Z has a derivative, necessarily, proper which is the leader of an element in A z (U, Z).) A θ (U, Z, Θ) consists of elements of A whose leaders are components of Θ. Each component of the parameter is the leader of a differential polynomial in A θ (U, Z, Θ).

From this lemma, it comes Proposition 4. If p designates the number of components of Z which are present in one of the elements of A θ (U, Z, Θ) (regardless the order of derivation) then p ≤ p ; and it is sufficient to measure these p components of z to guarantee the observability of θ with respect to (u, y).

To the question

p ≤ p? we make the following remarks. For the following system ẋ1 = x 2 ẋ2 = x 1 + x 2 ẋ3 = x 2 + x 3 It is not true that if the minimal number of sensors of a given dynamics is known to be 1 then the sensor may be placed at one of the state components. A counter-example may be exhibited from the previous remark.

CONCLUSION

Differential algebraic approach of observation problems seems to be well suited for the study of the sensor selection problem for the observability of systems though most questions are still unanswered. The relation to observability margin will be covered in the final version of this communication.

  minimal number of sensors is 1. The minimal polynomials of the latent variable are as follows ẍ1 -ẋ1 -x 1 The minimal polynomials of the latent variable are as follows ẋ1 -a x 1 = 0 , ẋ2 -b x 2 = 0 , ẋ3 -c x 3 = 0 . But y = α x 1 + β x 2 + δ x 3 makes the system observable as long as α β δ(a -b)(b -c)(c -a) = 0 .


Mihaela-Ancuta Mone

Sette Diop
email: sette.diop@l2s.centralesupelec.fr

Dumitru Popescu
email: dumitru.popescu@acse.pub.ro

Optimal multi-model control for nonlinear systems with parametric uncertainties -Diesel Engine case study*

The paper proposes a novel solution to control the combustion regime of Diesel engines in order to obtain NOx reduction and to minimize fuel consumption. The nonlinear dynamic model associated to a Diesel platform equipped with an EGR and VGT is estimated. The linearized and reduced state model structure has been approximated. The optimal control procedure for the air flow in the intake manifold of the cylinders and the pressure inside the combustion chamber of the Diesel engine is proposed. A multi-model control for nonlinear systems with parametric uncertainties solution is developed. Our procedure is implying the minimization of the multi-model structure by reducing the identification effort of the number of models and associated controllers. The simulation results will be implemented and validated on the experimental Diesel engine.

I. INTRODUCTION

The Diesel engine is widely embarked on vans, tractors, tube, motorcycles, underwater and ships. The majority of vehicles are powered by Diesel. A 200 years old invention became the most important engine of the 21st century. Throughout the years, many analyses have been performed as to whether the utilization of Diesel engines is dangerous for the human health or not in terms of pollution. As to reduce the negative impact of the Diesel engines on human's health and environment pollution, we want to obtain the NOx reduction and the reduction of the consumed fuel by these engines [START_REF] Mone | Optimal Control for Diesel Engine Combustion Regime[END_REF].

Section II presents the mathematical models associated to the dynamic the experimental Diesel engine. The data was obtained on the experimental platform located at the University Picardie in Amiens, France [START_REF] Cirstoiu | Multi-model adaptive control for turbocharged Diesel engines[END_REF]. The engine is equipped with an exhaust gas recirculation system and a variable geometry turbocharger. The models presented here are as follows: state-space model with linear parametervarying, invariant model, reduced invariant model. Given the difficulties to access the parameters of the invariant model, the system has been decoupled and we have considered the direct transfer from the most two important inputs of the system to the outputs. Therefore, section II-D presents the input/output engine models, for which state-space minimal realizations are presented in II-E. For the latter, LQR control law is discussed in section III.

Section IV includes a robustness analysis regarding the parametric uncertainties of the models concerning the LQR control. The structural deviations of the command have been obtained so that the closed-loop system remains stable. Three operating points have been proposed on the non-linear characteristic so that the commutation mechanism of the multi-model control is improved.

The results obtained in simulation and the corresponding analysis results are listed in section V.

II. ENGINE'S DYNAMIC MODELS

A. State-space model with linear parameter-varying

Starting from the following linear parameter-varying model:

⇢ ẋ(t) = A(⇢(t))x(t) + B(⇢(t))u(t) y(t) = C(⇢(t))x(t) + D(⇢(t))u(t) (1)
where the non-linear matrices A, B, C and D depend on ⇢(t), an unknown exogenous time-varying parameter, we want to acquire a reduced order model, but whose approximation is to be counteracted by the control laws designed later. The mathematical model of the engine is obtained working on the general gas equation:

pV = ⌫RT ( 2
)
where ⌫ is the amount of substance present in a tube of volume V , pressure p and the temperature T . R is the ideal gas constant (R = 8.314 J molK ). We apply the time derivative on equation (2) and write one differential equation for the intake and one for the exhaust. We obtain the dynamic of the system:

ṗi = RT i V i (W ci + F ie ) + Ṫi T i p i (3) ṗx = RT x V x (F ic + F f F xi F xt ) + Ṫx T x p x (4)
Ṗc = 1
⌧ ( P c + P t ) (5)
where p i is the intake manifold pressure, p x is the exhaust manifold pressure, [START_REF] Dauphin-Tanguy | Modelisation, Identification et Commande des Systemes[END_REF] is the dynamic of the turbocharger'given by the power transfer delayed by ⌧ .

By considering the temperatures in the intake manifold and exhaust manifold at a constant value, the effect of the variables Ṫi , Ṫx , p i and p x is not important, and we obtain the first approximation of the non-linear model (3) - [START_REF] Dauphin-Tanguy | Modelisation, Identification et Commande des Systemes[END_REF].

Using the flow rate formula, we have determined the EGR flow rate(i.e. F xi ) and the VGT flow rate (i.e. F xt ). The variation of the air flow inside the intake manifold to the cylinders (i.e. F ie ) was determined based on the speeddensity equation.

978-1-7281-9809-5/20/$31.00 ©2020 IEEE F xt ) at a constant value, and also the temperatures inside the intake and exhaust manifolds, T i and T x , respectively. Therefore, we can say that (3) -( 5) have constant parameters [START_REF] Cirstoiu | MultiModel Adaptive Control for Turbocharged Diesel Engines[END_REF]. This new model approximation does not affect the engine's dynamics at low and medium engine speed even though, in reality, these parameters vary based on the operating point.

B. Invariant model

The following state-space model, i.e. the invariant model, is obtained based on (3) -( 5):

8 > > > > > > > > > > < > > > > > > > > > > : 0 @ ṗi ṗx Ṗc 1 A = A 0 @ p i p c P c 1 A + B 0 B B @ A r x v N W f 1 C C A ✓ W ci p i ◆ = C 0 @ p i p x P c 1 A + D 0 B B @ A r x v N W f 1 C C A (6)
The inputs are represented by the position of the VGT valve, A r , the position of the EGR valve, x v , the engine speed, N , and the fuel flow, W f . The outputs are the mass air flow, W ci , and the manifold absolute pressure, p i . The states are given by p i , p x and P C . The matrices A, B, C and D have corresponding dimensions, and D is the null matrix. Therefore, we have a three-order model.

C. Reduced invariant model

We considered the following approximation in order to reduce the model given by ( 6), meaning following two inputs were redefined as below:

⇢ A r = A r + 8.9 ⇤ 10 5 [m 2 ] x v = x v + 0.5 (7)
The values of the fuel flow W f and the engine speed N are considered constant.

Reducing the number of inputs from four to two, the following state-space representation has been obtained:

8 > > > > > > < > > > > > > : 0 @ ṗi ṗx Ṗc 1 A = A 0 @ p i p c P c 1 A + B ✓ A r x v ◆ ✓ W ci p i ◆ = C 0 @ p i p x P c 1 A (8)
where each element of the matrices A, B and C are identified using (3) -(5).

D. Input/Output Models

It is difficult to access the parameters of the invariant model, therefore we decouple the system (6) and consider the direct transfer from the EGR to p i and VGT to W ci , respectively. Through model identification, using the recursive least squares method on the data acquired on the experimental Diesel platform, we obtain the input/output models of the two systems.

1) Multi-models associated to the EGR-p i transfer: The discrete transfer function from A r to the pressure p i is:

H d A r p i (z) = 280.8z 3 102.9z 2 273.6z 99.22 z 3 2.271z 2 + 1.617z 0.3458 (9)
The choice of operating points has been made based on the non-linear characteristic of the EGR valve. Therefore, the operating domain has been split into three ranges, corresponding to the movement of the EGR valve: 20-40%, 40-60% and 60-80% opening.

Therefore, we have the following three dynamic models:

H d 1 A r p i (z) =
280.8z 3 102.9z 2 273.6z 99.22

z 3 2.271z 2 + 1.617z 0.3458 (10)
H d 2 A r p i (z) = 210z 3
56.8z 2 198.45z 77.28

z 3 2.271z 2 + 1.617z 0.3458 (11)
H d 3 A r p i (z) =
178.5z 3 12.7z 2 102.33z 24.44

z 3 2.271z 2 + 1.617z 0.3458 (12)
2) Multi-models associated to the VGT-W ci transfer: The discrete transfer function from x v to the air mass flow W ci is:

H dx v W ci (z) = 0.03711 3 0.0785z 2 z 3 2.271z 2 + 1.617z 0.3458 + + 0.09375z + 0.02185 z 3 2.271z 2 + 1.617z 0.3458 (13)
The choice of operating points has been made based on the non-linear characteristic of the VGT valve. Therefore, the operating domain has been split into three ranges, corresponding to the movement of the VGT valve: 0-40%, 40-60% and 60-100% opening.

Therefore, we have the following three liniarized dynamic models:

H dM 1 x v W ci (z) = 0.03711z 3 0.0785z 2 z 3 2.271z 2 + 1.617z 0.3458 + + 0.09375z + 0.02185 z 3 2.271z 2 + 1.617z 0.3458 (14)
H dM 2 x v W ci (z) = 0.02899z 3 0.0622z 2 z 3 2.271z 2 + 1.617z 0.3458 + + 0.08945z + 0.01765 z 3 2.271z 2 + 1.617z 0.3458 (15)
H dM 3 x v W ci (z) = 0.02489z 3 0.0558z 2 z 3 2.271z 2 + 1.617z 0.3458 + + 0.08012z + 0.01247 z 3 2.271z 2 + 1.617z 0.3458 (16)

E. State-space minimal realization

If we apply z-transform to the discrete-time LTI state space model given by (1) and assume that the initial condition of the system x 0 = 0, we obtain the next connection between the input and the output of the system:

Y(z) = H(z)U(z) (17)
The transfer function H of the system is given by:

H(z) = C(zI A) 1 B + D (18)
For a SISO transfer function of a LTI system with b 0 normalized to 1,

H(z) = b 1 z n 1 + • • • + b n z n + a 2 z n 1 + • • • + a n+1 (19)
an equivalent state-space representation is given by the matrices (A, B, C, D) whose elements are:

A = 2
C = ⇥ b 1 b 2 . . . b n 1 b n ⇤ D = ⇥ 0 ⇤ (20)
Therefore, for each I/O model presented in II-D we can obtain the corresponding state-space minimal realization.

III. LINEAR-QUADRATIC REGULATOR

The solution of the LQR control law is the command matrix K LQR so that the command:

u = K LQR x K LQR = R 1 B T P (21)
minimizes the quadratic cost function:

J = Z 1 t=0 (x T Qx + u T Ru)dt (22)
P from (21) is the solution of the continuous time Riccati differential equation:

A T P + PA PBR 1 B T P + Q = 0 (23)
Hence, for the three models corresponding to the three operating points described in II-D, we obtain a corresponding command matrix that ensures the closed-loop stability of the two systems.

IV. MULTI-MODEL CONTROL

The multi-model control structure for the three operating points can be observed in Fig. 1 (adapted from [START_REF] Lupu | Multiple-Model Design and Switching Solution for Nonlinear Processes Control[END_REF]), where r is the set-point, M i represent the models for different operating points r i . P is the process to be controlled, C i represent the controllers, u i represent the commands, y is the process output, y i represent the models output, e i is the model error, SW IT CH is the commutation law, i = 1.. [START_REF] Cirstoiu | Multi-model adaptive control for turbocharged Diesel engines[END_REF].

In order to improve the commutation between the control algorithms, we propose a robustness analysis of the LQR command designed in III. In [START_REF] Dauphin-Tanguy | Modelisation, Identification et Commande des Systemes[END_REF] is presented an approach towards non-structural parametric uncertainties bounded by norm.

For the state-space models presented in ( 6) and ( 8), we consider the following uncertainty model:

ẋ(t) = (A + A(t))x(t) + (B + B(t))u(t) (24)
where A(t) and B(t) represent the parametric uncertainties that affect the matrices A and B, respectively. There is defined the uncertainty domain, D 1 :

D 1 = { A(t), B(t) | A T (t) A(t)  a Q 0 , B(t)  b R 0 } (25)
where Q 0 and R 0 are symmetric positive definite matrices and a and b are positive scalars.

The problem of interest is to find the maximum value of the scalars a and b so that the closed-loop system in (24)-( 25) is stable given the control law (21)-( 23) developed for the nominal system.

Q and R are defined as follows:

Q = Q 0 + Q 1 (26) R = R 0 + R 1 (27)
[5] provides the theorem according to which the closedloop uncertain system (24)-( 25) with the control law ( 21)-( 23) is stable within the domain D 1 which satisfies:

I = ( a + b )I < (28)
where

= P 1 (Q 1 + K T LQR R 1 K LQR )P 1 (29)
is a symmetric positive definite matrix. The control law is developed on the nominal system using the standard LQ criteria. The uncertainties are taken into account through the control matrices (26) and ( 27). The theorem gives the possible upper bound of the uncertainties that are rendering unstable the closed-loop system. Our procedure is implying the minimization of the multimodel structure by reducing the identification effort of the number of models and associated controllers. By enlarging the domain of uncertainties, we obtain the reduction of (model, controller) pairs in the proposed structure [START_REF] Hamidi | Enlarging the robust domain of attraction in nonlinear polynomial uncertain systems[END_REF].

In other words, we have to determine the maximum values of A and B so that the closed-loop system in (24)-( 25) is stable given the LQR control law (21)-( 23) developed for the nominal system.

V. SIMULATION RESULTS

The results presented in this section correspond to the reduced invariant model discussed in II-C.

A. Input/Output models simulation

Using the Matlab environment, the following minimal state-space realizations were obtained.

Therefore, for the transfer function in (9), we obtained:

A p i = 2
D p i = ⇥ 0 ⇤ (30)
For the transfer function in [START_REF] Kalmna | Contributions to the theory of optimal control[END_REF], we obtained:

A W ci = 2
D W ci = ⇥ 0 ⇤ (31)
For each transfer function in ( 10)-( 12) and ( 14)-( 16), respectively, the following minimal realizations were computed:

A 1p i =
D 1p i = ⇥ 0 ⇤ (32)
A 2p i = 2
D 2p i = ⇥ 0 ⇤ (33)
A 3p i = 2
B 2W ci = 2 4 1 0 0 3 5 C 2W ci = ⇥ 0.0036363 0.04257 0.02768 ⇤ D 2W ci = ⇥ 0 ⇤ (36)
A 3W ci = 2 4
2.271 1.617 0.3458 1 0 0 0 1 0

3 5 B 3W ci = 2 4 1 0 0 3 5 C 3W ci = ⇥ 0.0007252 0.039873 0.021077 ⇤ D 3W ci = ⇥ 0 ⇤ (37)
The step response of (30) can be observed in Fig. 2 and the step response of (31) can be observed in Fig. 3.

For the multi-model approach, the state-space minimal realizations step response of ( 10)-( 12) are plotted in Fig. 4.

In the case of the multi-model approach, the state-space minimal realizations step response of ( 14)-( 16) can be observed in Fig. 5.

B. Optimal control

The step response for the closed-loop systems can be observed in Fig. 6.

VI. CONCLUSIONS AND PERSPECTIVES

The paper presents the dynamic of an experimental Diesel engine. Minimal realizations for the linearized input/output transfer functions for both EGR and VGT were proposed.

A LQR control procedure controls the air flow inside the VGT and the pressure inside the EGR, both systems being present on the experimental platform. A robustness analysis for the LQR command has been performed.

A multi-model structure is proposed and for the LQ control law the upper bound of the non-structural uncertainties are computed so that the closed-loop system remains stable. The commutation problem between the models is addressed using the previous mentioned approach regarding the non-structural uncertainties bounded by norm. Therefore, three operating points have been proposed on the non-linear characteristic and the commutation of the command has been improved.

The stability of the closed-loop system with structural uncertainties can be addressed as further research.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Multi-model control structure

Fig. 5 .

 5 Fig. 5. Minimal realization of VGT-W ci transfer step response

ACKNOWLEDGMENT

The work has been funded by the Operational Programme Human Capital of the Ministry of European Funds through the Financial Agreement 51675/09.07.2019, SMIS code 125125

*The work has been funded by the Operational Programme Human Capital of the Ministry of European Funds through the Financial Agreement 51675/09.07.2019, SMIS code 125125


Dr Charchit Kumar

Prof Damien V Favier

Thomas Speck

Vincent Le Houérou
email: v.lehouerou@unistra.fr

Prof T Speck

Prof V Le Houérou

In-situ investigation of adhesion mechanisms on complex micro-structured biological surfaces

Keywords: Adhesion mechanics, JKR theory, PDMS replica, in-situ imaging, bioinspired micro-structures

Recently, plant surfaces have attracted great attention given their fascinating functionalities, particularly unique adhesive properties, which are largely resulting from their diverse surface structuring. This paper contributes to the advanced adhesion mechanics investigation on complex biological surface morphologies. Elastomeric replica of three different plant leaves, comprising morphologies at a broad scale (0.5-100 µm), with distinct shapes and complexity, and of a smooth surface were studied in contact with an adhesive probe. To perform precise adhesion measurements, an ultra-nanoindenter was modified based on the JKR (Johnson-Kendall-Roberts) mechanics and equipped with an in-situ real-contact visualization system.

The adhesion force on all surfaces was systematically investigated regarding the pre-load conditions. The results were analysed in the light of Hertzian and JKR theories, and underlying morphology-specific mechanisms were identified. A close examination of contact image-force data synchronization revealed unique attachment-detachment mechanisms, arising from different pre-loads and topographies. A significant influence of pre-load on adhesion was observed on the surface with fine micro-structuring and complex morphology, no specific influence was recorded for the remaining two. An overall comparison demonstrates a significant reduction in adhesion on coarse cone-shape patterns and complex micro-structures.

The specific adhesion mechanisms arising from biological morphologies may offer assistance to design bio-inspired smart interfaces.

Introduction

Adhesive phenomena are found in a plethora of biological systems as well as in numerous man-made engineering applications, especially when the surface to volume ratio gets extremely increased. [START_REF] Scherge | Biological Micro-and Nanotribology[END_REF][START_REF] Bhushan | Springer Handbook of Nanotechnology[END_REF][START_REF] Kendall | Molecular Adhesion and Its Applications: The Sticky Universe[END_REF][START_REF] Kaneko | Proceedings. IEEE Micro Electro Mechanical Systems[END_REF][START_REF] Myshkin | [END_REF] Undoubtedly, the adhesive characteristics of interacting surfaces are a key feature to control system's performance and durability. [START_REF] Kaneko | Proceedings. IEEE Micro Electro Mechanical Systems[END_REF][START_REF] Fischer-Cripps | Introduction to Contact Mechanics[END_REF] Consequently, it appears of paramount importance to precisely tune adhesive properties, especially in recently emerging micro-or nano-contact applications. [START_REF] Bhushan | Springer Handbook of Nanotechnology[END_REF][START_REF] Jin | [END_REF][START_REF] Tas | Proceedings IEEE Thirteenth Annual International Conference on Micro Electro Mechanical Systems[END_REF] A well-recognised way to adjust the adhesion characteristics relies on modifying the surface chemistry. [START_REF] Maboudian | [END_REF][START_REF] Ashurst | Device and Materials Reliability[END_REF][11][12] Another approach consists in introducing surface texturing on the interacting surfaces. [13][14][15][16][17] The pioneering work by Fuller and Tabor back in 1975, proposed a reduction in adhesion force by incorporating surface roughness. [18] In the last four decades, various approaches have been published utilising notably defined geometrical asperities (square, cylindrical or hexagonal pillars, spherical dimples, etc.) to study the influence of surface morphology on adhesive response. [13,[15][16][17][19][20][21][22][23][24] Few others explored adhesion characteristics and contact formation mechanisms on surfaces decorated with ripple or wrinkle shaped texturing. [START_REF] Jin | [END_REF][25][26][27][28][29] Regardless of all these studies, the adhesion mechanisms are not completely understood yet, since the type and complexity of morphologies can cause unique and distinct adhesion behaviours. [START_REF] Jin | [END_REF]13,17,[30][31][32][33][34][35][36][37][38][39] In nature, adhesion also plays an important role, e.g. for the interaction of animals with plant surfaces or with the inanimate environment. [START_REF] Scherge | Biological Micro-and Nanotribology[END_REF][40][START_REF] Gorb | IUTAM Symposium on Scaling in Solid Mechanics[END_REF][START_REF] Gorb | [END_REF][43] Broadly mentioning, insect attachment pads evolved in a manner to assist in sticking to or climbing on various plant surfaces, [START_REF] Gorb | IUTAM Symposium on Scaling in Solid Mechanics[END_REF][44][45][46][47] while on the counterpart, most of the plant leaves possess surfaces obstructing or reducing insect attachment. [48][49][50][51] Leaf surfaces are often decorated with species-specific surface structuring of different size, ranging from few nanometres to few hundreds of micrometres, having distinct shape and complexity, both at various levels of hierarchy. [52][53][54][55][56] This induces some optimised surface functionalities such as surface wettability, anti-adhesive properties, friction reduction, antifouling, slipperiness against insect attachment and optical properties, just to name a few. [44,50,52,53,[57][58][59][60][START_REF] Bargel | The Evolution of Plant Physiology[END_REF][START_REF] Eder | [END_REF] As a result, these surfaces gained a lot of attention from the contact mechanics community which intended to investigate, get inspired, and thus transfer innovative insights into biomimetic surface applications. [14,59,[63][64][65][66][67][68][START_REF] Barthlott | Springer Handbook of Nanotechnology[END_REF][START_REF] Busch | [END_REF] However, the relationship between structures and functions is not straight forward, keeping in mind that these plant surface phenomena are driven by a complex interplay of material composition, heterogeneous surface chemistry and diverse surface structuring leading to a highly sophisticated system to investigate. [53,[START_REF] Bargel | The Evolution of Plant Physiology[END_REF][START_REF] Eder | [END_REF] A possible simplification, that has been used in the past, consists in the precise replication of the complex structural morphologies of plant surfaces onto polymeric surfaces. [49] This facilitates a methodical investigation of the role of surface morphology, without the influence of physico-chemistry aspects. [49,71,72] Although the advancement in the micro-structured surface fabrication technologies in the past decades made possible to generate a vast kind of micro-or nano-structures, [19,[73][74][75][76][77][78][79] it is still far to realise the diversity and intricacy of biological surface structures. [52][53][54]56] Nevertheless, several papers have been published offering different methodological approaches to replicate the surface structures directly from original plant leaves onto the various polymeric substrates. [80][81][82][83][84][85] Over the recent years, there has been a growing interest towards the in-situ contact visualization when conducting adhesion investigations, and it became possible with the latest innovations in the field of optical imaging. [START_REF] Jin | [END_REF]19,21,22,25,[86][87][88][89] Surely, in-situ imaging offers a detailed insight into the contact junctions over the apparent contact area, as well as attachment/detachment mechanisms. However, much of the research performing in-situ contact visualization has been limited to either smooth surfaces [87,[90][91][92] or regular micro-structured surfaces. [START_REF] Jin | [END_REF]16,19,21,23,25,31,89] Indeed, most of the contact mechanics investigations based on plant leaf surfaces could not achieve real-time visualization of the contact formation, down to cellular or sub-micron-sized cuticular fold level. [49,71] Taken all together, in this work, we use the adhesion mechanics approach to methodically investigate the contact formation (closely inspired from insect-plant interactions) between biological structured replicas and a model adhesive tip. Three model plant leaves were selected and replicated onto polymeric samples, considering a broad morphological range in terms of their structure's size, shape and hierarchy. By considering the advantageous aspects and final positive replica characteristics relevant to our research specifications, we employed the replication technique presented by Kumar et al. 2018, which uses epoxy resin for generating negative moulds and producing the final replica on Polydimethylsiloxane (PDMS)

substrates. [80] The force range that corresponds to plant-insect interactions falls in the few mN range, and therefore calls for a precise experimental setup. [45,93,94] Such a low force range and high sensitivity can be accomplished with a nano-indenter like apparatus. [95][96][97] Thus, we introduce an adhesion force tester (modified nano-indenter) to perform low-range adhesion force measurements under precise load or displacement control. Furthermore, the modified apparatus permits to visualise the in-situ contact on the complex micro-structured surfaces. At first, this paper quantitatively investigates adhesion force characteristics on the PDMS leaf replicas and on a smooth PDMS surface benchmark, and critically examines the influence of pre-load and surface morphology. Furthermore, the dedicated in-situ imaging offers a comprehensive insight into the real and apparent contact areas, along with the understanding of attachment and detachment mechanisms.

Results and Discussion

Surface Morphologies

The results of SEM surface morphological investigation, shown in Figure 1 1.h) were replicated to PDMS with high fidelity. After the n-heptane treatment, all samples were perfectly restored back by de-swelling to their original state without damage, as proven in Figure 1.c, 1.f and 1.i.

Adhesion Force Characteristics

Adhesion force measurements were carried out for each polymeric replica as well as for a smooth PDMS surface. Data sets were recorded by carrying out in total 5 to 7 experiments spotted on the samples for each surface type. Figure 2 shows a plot obtained from a typical pull-off adhesion force measurement on a smooth PDMS sample. Adopting the standard contact mechanics formalism, the absolute maximum negative force value during the retraction phase corresponds to the adhesion pull-off force (Fad), as reported in Figure 2. [START_REF] Jin | [END_REF]98] In order to investigate the effect of pre-load (FL), adhesion experiments were performed at the same spot, for seven different FL values: 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 and 3.5 mN, keeping all other test parameters (loading/unloading rate, retraction speed, and test duration) and conditions constant.

In the following sub-sections, the results on attachment and detachment mechanisms, alongside with the results from the pre-load effects on adhesion force characteristics, are presented and discussed consecutively for each surface type. For reading convenience, hereafter, only genus name is used to address each PDMS replica sample instead of full species name: Hevea replica for H. brasiliensis, Ludisia replica for L. discolor, and Litchi replica for L. chinensis.

Smooth PDMS

A force-displacement-time curve of an adhesion measurement, for a whole test cycle, on smooth PDMS sample at a pre-load (FL) of 1.5 mN is shown in Figure 2.A. The measured adhesion force (Fad) is 0.827 mN. Still image sequences from the in-situ contact video (see the Video 1 in supporting data) of the adhesion test at the various points of interest (a-f) are set out in Figure 2.B. With regard to the kinetics of a smooth PDMS contact, attachment and detachment events appeared very continuous, homogenous and circular in shape over the whole contact cycle. The real contact area increased with increasing normal load and started decreasing once the retraction part began, involving the well-known adhesion hysteresis. [99] The point (a) in the graph indicates the contact point at zero load, right after the snap-in contact has taken place (not shown in the graph) ending the approach step. Therefore a small contact could be seen in the corresponding in-situ contact image (Figure 2.a). At this state, tip and substrate already formed a solid-solid intimate contact. After this, the continuous loading phase began and the tip slowly is pressed onto the substrate surface under a precise load controlled loading, at a rate of 83.3 µN/sec. Point (b) in the graph represents the phase when the system can relax under the constant maximum load: a closer observation of point b (radius ≃ 184 µm) and point c (radius ≃ 188 µm) exhibited a small increase in radius at point c, which witnesses this material relaxation.

The JKR theory is suitable to describe the contact mechanics for soft compliant materials with high adhesion energy. [START_REF] Kroner | Adhesion Measurements on Patterned Elastomeric Surfaces[END_REF][START_REF] Grierson | [END_REF] In order to confirm the applicability of the JKR theory, one could utilize the dimensionless physical parameter, Tabor's parameter (µ $ ), which is defined by:

µ $ = & 𝑅𝑊 ) 𝐸 * ) 𝑍 - . / 0 . ⁄ (1)
Here, R is the relative radius of curvature of the contacting bodies, and can be estimated with

1 𝑅 ⁄ = 1 𝑅 0 ⁄ + 1 𝑅 ) ⁄
(with 𝑅 0 and 𝑅 ) being the radii of the contacting bodies), E * is the effective elastic modulus, which can be calculated using 𝐸 * = 𝐸 2(1 -𝑣 ) ) ⁄ ; E is the PDMS Young's modulus and v is the Poisson's ratio (vPDMS = 0.5). [90] The E value for PDMS was calculated as 1.01 ± 0.01 MPa by using the classical JKR model fitting on smooth PDMS substrate. [102,103] Z0 is the equilibrium separation of the surfaces in the Lennard-Jones potential and usually comes in between 0.3 and 0.5 nm. [90] If µ $ value is greater than 5, then a contact can be well described using the JKR theory. [START_REF] Grierson | [END_REF] By utilising the known values from the given experiment's formation: 𝑅 0 = 1.5 mm (tip), 𝑅 ) = ∞ (flat surface), µ $ value would result in the range of ≈ 3500-5800. This value range clearly suggests the well applicability of JKR model in this experiment. Further investigation of the retraction part of the curve shows, that the measured value of the contact diameter (2ad) at the absolute zero load condition (Figure 2.d) was found to be 348 µm. The contact diameter (2ae) at the very lowest force point, the point of detachment instability, where the contact diameter abruptly vanished to zero, was measured to be 219 µm (Figure 2.e). This finding is in good accordance with the previously established JKR theory, which predicts the following relation ae = 0.63ad. [104,105] Effect of pre-loading: The results reported in Figure 3 show the effect of FL on Fad. Here, Fad values appear to be independent (Fad ≃ 0.809 mN) from FL variations, which is again in good agreement with the Johnson, Kendall, and Roberts (JKR, 1971) model, [102] as reported in previous studies. [106,107] The relaxation time for PDMS is reported to be in the order of 0.05-0.5 s, significantly less as compare to the contact time applied in this investigation. [108] Hence, it could be assumed that this contact system behaves more likely as elastic since loading and unloading phases could be considered as quasi-static. [108] Actually, all the tests in this investigation were performed slowly enough to minimize the viscous effects of material.

Moreover, no effect of loading history on the adhesion force values in the investigated force range was noticed, as supported with the results stating the independence of Fad from FL. Hence, this observation very much supports that our test protocol is adequately complying with the standard JKR adhesive model. Therefore, these considerations on smooth PDMS contacts validate our test protocol, notably concerning the loading/unloading speeds and relaxation time which ensure a quasi-static test.

Hevea Replica

An in-situ video of the adhesion test for an entire contact cycle on Hevea replica at a FL of 1.5 mN is shown in Video 2 (supporting data). To gain a better understanding of the attachment dynamics, we closely analysed the zoomed-in in-situ image captures. The fine structured surface of Hevea replica is easily recognizable (Figure 4) and it is clearly possible to distinguish between the true contact area (in bright contrast) and the area out of contact (in dark contrast).

The contact formation initiated at the second level of micro-structuring (fine cuticular folds), and as the contact border advanced a whole cell was pulled in an intimate contact. [19] At the maximum load, even though each cell appeared to be in full contact, the cell boundaries (outer line surrounding a cell, first level of structuring) were left out of contact (darker line areas).

Afterwards, as the detachment event began, two distinct contributions in contact separation mechanisms could be identified when one closely observes the detachment sequences: (i) cell boundaries initiated and assisted the interfacial crack propagation during the detachment sequence and (ii) each cell, as a whole detached at a time, behaving as an individual contact point of instability. The interfacial crack initiation began at the cell boundaries around the outer periphery of the whole contact region and then the crack further spreads toward inner areas. A similar kind of detachment phenomenon has been reported by Jin et al., [START_REF] Jin | [END_REF] where adhesive contact was formed between a rigid sphere and a rippled elastic surface.

Different contrast during attachment and detachment phases:

Another interesting finding from a closer observation on the zoomed-in in-situ videos is the distinct colour contrast at the contact periphery for the attachment and detachment cycles (Figure 4). Continuous flowing-type of the contact advancement mechanism could be observed during attachment, as illustrated in Figure We believe, that the higher edge sharpness in the detachment sequence could be attributed to an increase in local strain and thus led to adhesion profiles on edges showing perpendicular connection fillet. Such a phenomenon has also been mentioned in a previous study by Charrault et al. [90] .

Effect of pre-load: Figure 5 shows adhesion force characteristics with increasing normal preload, for Hevea replicas. We investigated further the in-situ contact images to compute the normalized contact area (An) at the full loading condition, for all the FL values (Figure 5, blue data points). A clear increase in Fad from 0.384 mN to 0.490 mN was observed with increasing FL from 0.5 to 2.5 mN. However, adhesion force might get saturated with a further increase in FL above 2.5 mN. [14,22] The increase in Fad could be explained with the filling-up of fine microstructure pockets between the wrinkles (cuticular folds), while increasing FL. A similar phenomenon has been found also in previous studies. [14,15,24,109] Actually, at a low FL (0.5 mN) value, only partial contact occurs at the very top of cuticular folds, resulting in a small normalised real contact area, An = 0.44, reported in Figure 5. With further increasing FL, the tip material progressively fills up the small non-contact gaps between the wrinkles, hence resulting in an increase in adhesion force by a factor of about 30 %. One can also notice in Figure 5.I and Figure 5.II that there was an evidently larger true area still left in contact even when load was removed (at the absolute zero load), for the higher pre-load (FL = 3.5 mN) condition in comparison to the smaller pre-load FL = 0.5 mN. This supports the previous statement and confirms that the observed increase of Fad for Hevea replica is caused by its specific fine microstructuring.

Interestingly, An increases in the beginning with increasing FL until 2.5 mN is reached. Above this value, An appears to get saturated at higher pre-loads (3.0 mN and 3.5 mN).

An curve shows a similar dependency to FL as the adhesion force (Figure 5) does. This similar trend corroborates the filling of microstructures at high pre-loads, which leads to saturate at high FL values: An tends to saturate at 0.6, and could never reached 1 (the latter meaning complete full contact). The spherical shape of the tip contributes to this experimental fact since it causes a parabolic (non-flat) pressure distribution. [98,110] This also further implies that the full intimate contact arose in the middle region, partial contact slowly spreading at the outer edges with increasing FL. [13,19]

Ludisia Replica

As illustrated in Figure 6.a, the real contact (the bright circular areas) always occurred at the very top of the conical shaped micro-structures of the Ludisia replica, for the complete pre-load range (0.5-3.5 mN). Interestingly, one may note that, the real contact area was significantly lower than the apparent area, and thus is reducing An to 6.91% (at the full loading condition for FL = 1.5 mN), owing to the specific conical shaped topography. A closer examination of in-situ video (see supporting Video 3 in supporting data) for a whole test run revealed that the contact formation was continuous, and real and apparent contact grew locally and homogenously circular in shape. The microstructures in middle region contributed more in bearing the contact pressure, as confirmed by the larger size bright spots distributed in the middle area if compared to the outer side (Figure 6.b -threshold image). This can be attributed to the Hertzian-type parabolic pressure distribution given for two contacting bodies. [98,111] This holds likewise for the whole FL range investigated. As soon as the detachment step is initiated, each true contact periphery instantly started separating, without exhibiting any explicit contact hysteresis.

Further on, a local examination of the contact formation on Ludisia replica pointed out that, the attachment-detachment mechanism on an individual microstructure could be considered similar as to the contact of single asperities on a smooth surface. Indeed, it is meaningful to approximate this behaviour as an inverted case of a half-sphere pressed on a flat surface contact model: at a small scale, the top of each cell tip of Ludisia replica behaves like a spherical asperity, that is locally forming contact against the almost flat surface of the probe. By analysing the surface topographies, average real contact densities and contact formation for a given pre-load, one can assume that the coupling of elastic displacements between a cell microstructure and its neighbouring cells is negligible. [21,112] Thus, we used an approach previously introduced by [21,89] A standard contact under Hertzian conditions for a sphere in contact with a half-space flat surface [102] can be computed using equation 2.

𝑃 > = 𝐸 * ? 4 𝑎 B . 3 𝑅 B (2)
With E * is the effective elastic modulus, which can be here calculated using

𝐸 * = 𝐸 2(1 -𝑣 ) ) ⁄
; E is the PDMS Young's modulus and v is the Poisson's ratio (vPDMS = 0.5). [87,90] The E value for PDMS was calculated as 1.01 ± 0.01 MPa by using the classical JKR model fitting on smooth PDMS substrate; [102,103] Ri is the normalized radius of curvature of the Effect of pre-load: Here, the Fad values found were the lowest compared to all other surfaces investigated. For instance, at the pre-load condition of 1.5 mN, Fad (0.014 mN) was reduced by about 98% in comparison to a smooth PDMS surface (0.809 mN). This reduction in Fad holds for all FL conditions. Fad results as a function of FL are presented in Figure 8 and clearly illustrate no notable variation in Fad with increasing FL. This is consistent with the validation of the local Hertzian contact on each cell of the Ludisia replica surface and thus supports our assumption of non-adhesive contact at local scale. Our results also suggest that loading and unloading at small scale can still be considered as quasi-static since no viscous hysteresis was detected.

Litchi Replica

An in-situ video for a whole adhesion test cycle, on Litchi replica, at a FL of 1.5 mN is presented in Video 4 (supporting data). At first sight, contact mechanics appeared to be very complex, given the fact of the high complexity of surface microstructures of Litchi replica combined with the obvious video quality limitations. Indeed, one could point out that the in-situ videos on

Litchi replica lacked in-detail clarity in comparison to other three surfaces studied, pointing out the technical limitations of proposed technique for certain highly complex morphologies.

Nevertheless, it was visible that the real contacts (bright spots) were discretely distributed over the apparent contact zone, related to the heterogeneous and random surface structuring. To gain a better understanding of the attachment mechanism, zoomed-in videos have been checked and the local contact zones were analysed. As soon as the loading event started, fine cuticular folds began to form partial top-contact. With further increase in pre-load, the overhanging fine structures (cuticular folds) of an individual 'rose-flower-shaped' unit accumulated together and started forming a localised cluster of units, as could be seen as close bright spots appearing in the in-situ video, at the fully loaded condition (dashed domain in Figure 9.II). After the full loading phase, during the retraction cycle a particular feature was observed: the sudden fluctuating behaviour and profile of the force-displacement curve, as presented in Figure 9.I (like the point "C" for instance), which called for further analysis. We examined the in-situ video sequence corresponding to the position of major sudden fluctuations on the forcedisplacement curve. This became possible thanks to the real-time image-data point synchronization. A series of in-situ contact images corresponding to various points of interest during the retraction part of the force-displacement curve are illustrated in Figure 9.II. This fluctuation behaviour could be understood by taking into account two energy factors: (1) the bending of overhanging patterns leading to localized storage of the elastic strain energy, and

(2) the agglomeration of 'rose-flower-shaped' unit (micro-morphologies) gathered in a short range coupling style. During the unloading phase, the bended folds started popping out, and thus caused a sudden release of the stored elastic energy that acted against the adhesion force. [36] These real contact morphologies are discretely distributed over the apparent contact zone, thus implying the individual agglomerated cluster released the stored energy locally with the instable separation. This behaviour appeared to be more force sensitive, and therefore, it was more pronounced and clearly visible at higher pre-load (FL = 3.0 mN as compared to 0.5 mN), which can be seen in Figure 9.II. This might be attributed to similar force-dependence mechanism as found in Hevea replica (section 2.2.2).

Effect of pre-load: Figure 10 shows the results from the effect of FL on Fad characteristics for Litchi replicas. An initial increase in Fad with raising FL, until 3.5 mN is clearly visible. Since the previously described parameter range (i.e. FL between 0.5-3.5 mN) did not provide a particular transition, we therefore continued the tests up to FL = 5.0 mN. Thus, Fad appeared to get saturated with a minor variation from 4.0 mN onwards. This could be explained with the force sensitive phenomenon associated with the complex surface morphology of Litchi replicas previously discussed. At low FL values, true contact formed partially only on the very top of overhanging micro-structures, whereas with increasing in FL more real contact is formed leading to higher adhesion measured at the detachment. Actually, one could note in the Figure 9.a and Figure 9.A, that there was a larger true contact area for higher FL (3.0 mN) as compared to low FL (0.5 mN), at the absolute zero load condition during the unloading cycling.

Overall Comparison and Conclusions

In the present paper, a systematic study of adhesion contact mechanics at a low force range on high-precision transparent replicas of biological micro-structured surfaces is presented. In order to achieve such detailed investigation, we successfully developed an innovative technique for in-situ contact visualization, allowing the analysis of attachment-detachment mechanism on transparent replicas of complex biological surface morphologies. In this study we experimentally evaluated the pull-off adhesion force of four distinct surfaces and analysed its dependence to pre-load and surface morphology. Based on the key findings of this investigation the following conclusions can be drawn:

• A significant difference in Fad between the all four surfaces was found, as presented in Figure 11, for FL = 1.5 mN condition. The same trends appear for all other FL conditions tested as well.

• Apparently, the smooth PDMS surface exhibited the highest value of adhesion force (Fad ≃ 0.809 mN, at FL = 1.5mN) out of all the four tested surfaces (Figure 11). On the surface with fine wrinkle shaped micro-structuring (Hevea replica) Fad is reduced by 42.5 % (Fad ≃ 0.465 mN at FL = 1.5 mN) in comparison to the smooth PDMS surface. However, the adhesion force of the Hevea replica is still significantly higher in comparison to the other studied micro-structured surfaces. The surface showing coarse sized micro-structures (Ludisia replica) and the surface with complex hierarchical structuring (Litchi replica) both showed a markedly lower value of adhesion force, with Fad = 0.014 mN and 0.035 mN respectively.

The large reduction in adhesion force in the case of Ludisia replica could be attributed to the specific structure shape which favours local adhesion-free (Hertzian-like) contacts. This

differs to what has been demonstrated in some previous studies of adhesion enhancement by using soft and compliant geometries with high aspect ratio. [32,35,66,[113][114][115] Moreover, it is interesting to note in the Figure 11 that the adhesion force plot clearly follows a similar decreasing trends as the An plot. This could be interpreted as a confirmation of the correlation between the adhesion ability and the contact surface splitting, although we believe that local phenomena inducing large strain rate during detachment may overcome this rule.

Unexpectedly, further analysis consisting of the normalization of Fad by Ar did not permit to get additional insights in this respect.

• Contrary to smooth PDMS and Ludisia, the surface of Litchi and Hevea replicas demonstrated an increase in Fad with FL, which was attributed to the filling of fine microstructure pockets at high pre-loads.

• Thanks to the newly developed method of in-situ visualization, the distribution of real contact zones reveals a high dependency on size and morphologies of the surface structures and also exhibits distinct attachment and detachment phenomena.

The new in-situ contact visualisation method, its technical advancement and the results based thereon being presented in this study open up a promising direction in the understanding of contact formation on high-precision replicas of complex biological surface structures. The insights gained from this study, especially concerning force-dependent adhesion behaviours associated to morphology dependent attach-detach modes, may provide a valuable basis for designing novel bio-inspired functional surfaces with (fine-)tunable adhesion properties at a local scale. As prospects, the individual contribution of each level of hierarchical surface structuring on the adhesion force characteristics appears of prior importance.

Experimental Section

Investigated Biological Surfaces

Three different plant leaf surfaces were selected for this study, on the basis of different size, distinct morphology and complexity of their surface structuring. Rubber tree (Hevea brasiliensis; adaxial, i.e. upper leaf surface) shows two levels of structuring consisting of "puzzle piece-shaped" epidermal cells covered by fine cuticular fold microstructures (cf. Figure 1.a), with both height and width of about 0.5-1 µm and an intermediate spacing of 0.5-1.5

µm. [49] Jewel orchid (Ludisia discolor; adaxial) exhibits circular cone-like shaped microstructures (cf. Figure 1.d) with a diameter of about 50-100 µm and a height of about 50

µm. [80] Lychee (Litchi chinensis; abaxial, i.e. lower leaf surface) has a complex hierarchical surface structuring (cf. Figure 1.g) consisting of 'rose-flower-shaped' units inducing undercuts and overhanging substructures. [49] Additionally, previous work by Prüm et al. [50] showed that the leaf surfaces of H. brasiliensis and L. chinensis possess anti-adhesive properties for insects resulting from their specific micro structuring. For better morphological visualization, threedimensional scan animations for all three leaf surfaces are presented in supporting data (Video 7, Video 8 and Video 9). All leaves used in this investigation were freshly collected just before processing the replication. Corresponding plants were grown in the Botanic Garden of the University of Freiburg, Germany.

Preparation of PDMS Replicas and Smooth Surfaces

PDMS (Polydimethylsiloxane), a silicone based soft elastomer, offers various key advantages: easy handling, low cost, non-toxicity, low surface energy (γ = 22 mJ.m -2 ) and low elastic modulus (E = 0.5-4 MPa). [116] Because of its high γ/E ratio, it is suitable for adhesion mechanics studies and has been widely used in the past by various researchers. [21][22][23]87,90,106] This siliconbased elastomer has a very low glass transition temperature (-120 °C). It gets easily cross-linked into a very stable elastic network, showing high chemical stability at room temperature, and no explicit interaction with other material. [117] Importantly, PDMS exhibits high optical transparency, thus being a perfect applicant for achieving the in-situ contact visualization. [118] Surface microstructures from the original plant leaves were precisely transferred onto PDMS, using the two-step replication technique described elsewhere. [80] A simplified description of the replication process is schematically presented in Figure S1 (supporting data). At first, negative moulds were developed with two components epoxy resin (Epoxydharz HT 2, R&G Faserverbundwerkstoffe GmbH, Germany) directly from the original fresh plant leaves. Fresh plant leaves were cut into small pieces (approximately 4 cm × 4 cm), and immediately secured onto a plastic petri dish with a double-sided adhesive tape. Subsequently, the uniformly mixed bubble free epoxy mixture (resin to hardener ratio of 10: 4.8) was steadily poured onto the leaf sample surfaces. After curing for 15 h at ambient conditions (temperature = 20-25°C and relative humidity = 40-60%), plant leaves were separated from negative epoxy moulds. In the case of L. chinensis leaves, the demoulding was fulfilled by a special anti-stiction step comprising the treatment of negative replica sample in an aqueous solution (60 g/100 ml) of potassium hydroxide (KOH, ≥85%, p.a., Carl Roth GmbH & Co. KG, Karlsruhe, Germany) at with a JKR contact mechanics based apparatus, and described in-details in the following section. An ultra-nanoindentation tester (UNHT³, Anton Paar Tritec, Switzerland) with a highresolution load-control of 3 nN and a depth resolution down to 0.3 nm was modified to perform low range adhesion force measurements, together with in-situ contact visualisation, as illustrated in a simplified schematic in Figure 12. Indeed, a dedicated optical system was introduced into the equipment, permitting high-resolution differential contrast microscopy based on transmission light microscope principle: a light beam shines from the probe and propagates through the tested substrate. [123,124] In our investigations, since the size of the probe (half sphere of 1.5 mm radius) and the substrate patterning are small, it was delicate to incorporate the optical prerequisites associated to transmission light microscopy within the adhesion tester. White light from a fibre optic cold light illumination (KL 1600 LED, Schott AG, Germany) shines at the tip, via passing from a polarizer filter, and focused with a condenser lens onto the tip opening spot, with the purpose to get high-intensity illumination (Figure 12). A special tip (Figure 13) was fabricated with an internal micro-hole (radius = 0.5 mm) and with a micro-metal machining finished (alighted at an angle of ca. 45°) on top of the tip, so that the light beam is perpendicularly reflected and follows a path in the direction to the soft probe. An analyser along with an objective lens and a high definition colour camera (Basler acA3800-14uc, with a 10Mpix CMOS sensor, Germany) lie just under the substrate surface. This optical setup is mounted on a micromanipulator, benefiting to precisely focus on contact spot. The transmitted light beams through the contact junctions, where both surfaces (PDMS-PDMS) are in real-contact with matching refractive index, leading to high contrast bright spots. Light get randomly scattered over the area where no intimate contacts (PDMS-air interface) were established and thus appears as dark domains.

Polarizer and analyser filters facilitated to obtain sharp yielded contact edges. Furthermore, the apparatus is equipped with a custom-build dedicated electronic system that enables simultaneous recording of the video frame in real-time synchronization with the corresponding force data points.

Model Adhesive Tip

A particular caution was paid to the soft elastomeric adhesive tip which was fabricated using a two-step moulding process. In the first step, a thick Polymethyl methacrylate (PMMA) substrate was locally heated up till ca. 100 ± 5°C, with an air heating gun. A sapphire ball (Edmund Optics, United States) with a radius of 1.5 mm and surface roughness of Ra = 0.005 µm was gradually pressed down until the ball half subsided into the PMMA surface. After cooling for 30 min, the ball was separated out from the bulk with a vacuum suction gripper, and consequently provides a negative impression on the PMMA surface. This latter was then filled up with a PDMS mixture (prepared with same mixing protocol as described in section 4.2) and degassed in a vacuum desiccator for 20 min to remove any air bubble that could be trapped during pouring. The homebuilt tip holder evoked previously (Figure 13) consists of the assembly of an external screw thread and a nut attachment at its very end. Tip assembly was gradually put on top of the PDMS mixture which filled PMMA mould (Figure 13.b). During the PDMS curing process, the PDMS mixture filled up by capillarity the small gaps formed at the thread in between stud and bolt, as illustrated in Figure 13.a, thus ensuring a good anchorage of the PDMS tip with the holder.

Additionally, we observed that the PDMS level also raised up inside the drilled hole, consequently promoting a strong bulk attachment of PDMS tip with the tip holder (Figure S2 in supporting data). Tip's centre alignment with the centre of the PMMA mould was essential to get a uniform contrast all over the whole contact image during the in-situ visualization: the development of a perfectly aligned tip required several attempts (typically up to 4-5 tips) to become successful. After development, each tip assembly was checked under optical microscope and non-aligned tips were discarded. Furthermore, to examine any surface imperfection, tips were checked with a scanning electron microscope (Figure S2 in supporting data).

Surface Referencing

An important aspect during low force range tests is the accurate detection of the very-top point of contact i.e. surface referencing. [96] It becomes even more crucial, when both tip and substrate samples are made up of a soft matter polymer, where long-range adhesive forces in between tip and sample surface exist, and time-dependent creep of the polymer material during the referencing process itself could greatly influence the exactness of reference position and then further affect the accuracy of adhesion measurement. [97,125] In order to ensure a precise surface referencing, we used a separate parallel referencing tip-head made of metal against a metallic reference base, mounted with a high precision micrometre head, as shown in Figure 12. For the referencing process, at first both the reference tip and the adhesive tip are brought down together, in such a way so that the reference tip touched the reference base at first.

Experimental Protocol

All substrates were placed on rigid transparent glass plates and further fixed on the test platform.

Prior to each set of tests, surface referencing was performed for each surface type. Then, the

  , prove the high precision of PDMS replicas, made directly from original plant leaves by using Epoxy-PDMS replication. All three plant leaf surfaces, H. brasiliensis leaf with wrinkle-shaped fine microstructures (Figure 1.a and Figure 1.b); L. discolor surface with cone-shaped coarse microstructures (Figure 1.d and Figure 1.e); and L. chinensis with complex hierarchical microstructures (Figure 1.g and Figure

  4.a andFigure 4.b, while a sharp contrast on contact edges is exhibited during the detachment (Figure 4.c and Figure 4.d).

Figure 7 ,

 7 Figure7, and apparently loading and unloading data points found to be closely following a linear behaviour, with R-square values of 0.998 and 0.997, respectively.Moreover, Pc values for the unloading cycle trace back to nearly zero, and overlap quite well with the loading curve, thus validating the assumption of the Hertzian local contact model for the Ludisia replica. Accordingly, our results are supporting the assumption of a negligible elastic couplings in between the neighbouring cells so that at this scale each asperity in contact can be considered as an individual non-adhesive contact.

Figure 1 .

 1 Figure 1. Scanning electron microscope images of original plant leaf surfaces (left column; a, d and g), their PDMS replicas (middle column; b, e and h), and respective PDMS replicas after n-heptane treatment (right column; c, f and i). (a-c) Hevea brasiliensis (adaxial, upper side surface) exhibits a micron-size wrinkled shape folds. (d-f) Ludisia discolor (adaxial, upper side surface) presents circular cone-like surfaces structures. (g-i) Litchi chinensis (abaxial, lower side surface) surface shows highly complex hierarchical structures.

Figure 2 .

 2 Figure 2. Graph (A) showing the force-time (displacement) curve for a whole test cycle on smooth PDMS sample, at a pre-load (FL) of 1.5 mN. Fad is the adhesion (pull-off) force recorded during the retraction step. Different points of interest (a-f) are marked on the curve. Phase a-b corresponds the loading step, b-c represents the relaxation step, c-d is the unloading phase. Phase d-e-f represents the retraction phase recording the adhesion pull-off force. In (B), a set of in-situ contact images (a-f) corresponding to the points marked on the force-displacement curve is shown. For the full in-situ video corresponding to the above images, see the supporting video 1 in supporting data. The scale bar in image 'a' holds for all the six images.

Figure 3 .

 3 Figure 3. Plot of the variation of adhesive pull-off force (Fad) with change in applied pre-load (FL) for smooth PDMS sample. Data points are plotted as standard box-plot (25/75 th percentile range) representing mean and median for N=7.

Figure 4 .

 4 Figure 4. High magnification image sequence representing the contact formation behaviour during an attachment and detachment cycle. Bright areas represent the real area in contact. (a, b) shows the two consecutive (nth and (n+1)th) frames captured during the attachment (loading) cycle, and (c, d) during the detachment (unloading) cycle. The arrows are pointing to the areas of interest with distinct colour contacts at the edge for the attachment and detachment cycles. The scale bar in image b applies to all four images. A corresponding full video (Video 2) is provided in supporting data.

Figure 5 .

 5 Figure 5. Upper part of the figure: Plot (in red, left-side ordinate) of the variation of adhesion pull-off force (Fad) as a function of applied normal pre-load (FL) for Hevea replica samples (N=6). Advancement of the normalized contact area (An) with increase in the pre-load (FL), in blue on right-side ordinate. Lower part of the figure: Image I (for FL= 0.5 mN) and image II (for FL= 3.5 mN) show the in-situ contact areas at the absolute zero load condition, just before the retraction began.

Figure 6 .

 6 Figure 6. In-situ contact image for Ludisia replica at the full loading state for FL = 1.5mN, clearly exhibiting the spatial distribution of micro-contact spots (a). Bright circular shaped spots on top of the conical structures represent the real contact area. The inset image in the upper left corner of the image indicates how the diameter (2a) of a real contact spot on top of a conical tip was estimated. (b) Shows the threshold image after image processing.

Figure 7 .

 7 Figure 7. Pc versus F plot for the Ludisia replica surface. Linear slope and the overlapping of loadingunloading data points are validating the Hertzian contact theory locally. Pc= summation of inversely computed local normal loads, F= applied load.

Figure 8 .

 8 Figure 8. Variation in adhesion pull-off force (Fad) with increasing applied normal pre-load (FL) for Ludisia PDMS replicas, (N = 6).

Figure 9 .

 9 Figure 9. (I, upper part of the Figure) Diagram showing the retraction part of force-displacement curve for a Litchi replica sample, recorded at two different normal pre-loads, keeping all other parameters the

Figure 10 .

 10 Figure 10. Plot of the variation of adhesion force (Fad) with increasing applied normal pre-load (FL) for Litchi replica samples, (N = 5). Plot shows an initial increase in Fad which, however, gets saturated after 4.0 mN.

Figure 11 .

 11 Figure 11. Overall comparison of adhesion pull-off force (Fad) and normalized contact area (An) obtained for the four surfaces investigated, at a pre-load of 1.5 mN. Plot in blue (left-side ordinate) represents Fad and plot in red (right-side ordinate) stands for An. Inserts show the representative real contact images under loading condition, for each surface type.

Figure 12 .

 12 Figure12. Simplified drawing of the modified ultra-nanoindenter setup for low-range adhesion force measurement in synchronization with in-situ contact visualization. The pull-off force measurement head is additionally equipped with a vertical displacement sensor and a reference tip to perform surface referencing. A white light was shined at the tip opening, passing through a polarizer and condenser lens. The light beam, after being reflected from an internal mirror-like surface bend toward the soft tip, is further transmitted through the real-contact junctions, appearing as high contrast bright spots in the recorded videos.

Figure 13 .

 13 Figure 13. Simplified sketch of the tip assembly (a) showing that the PDMS tip is embedded within the tip assembly holder. (b) The PDMS liquid mixture filled up the hole due to capillary action, and a photograph of the resulting PDMS tip attached to the tip holder (in up-right corner of image b).

  Romero et al. 2014 and Yashima et al. 2015, to validate the Hertzian character of the contact at the local cell structures scale for Ludisia replica.

Polydimethylsiloxane (Bluesil ESA 7250 A & B kit, Bluestar Silicones GmbH, Germany) mixture (monomer to cross-linker ratio of 10: 1) and were kept in a vacuum chamber for one hour to remove air entrapped at the interface. After curing at 75°C for 3 h, cross-linked PDMS replicas were peeled off from the negative moulds. Each replica sample was quality inspected for any replication imperfection, using an optical stereo microscope. Four samples were developed for each investigated surface type, to ensure statistical representativeness during adhesion tests. Smooth PDMS samples were produced by curing the same PDMS mixture in a freshly opened flat bottom glass petri dish.

All samples (PDMS leaf replicas as well as the smooth PDMS samples) were swollen in a solution of n-heptane and 1-dodecanethiol (0.01 %) for overnight to remove the sol fraction (to extract the remaining unreacted free chains). After then, all the swollen samples were dried at ambient room conditions for at-least 24 h to restore back to their original state. [99,119,120]

Surface Characterization

Surface morphology visualization and characterization was done using scanning electron microscopy (Leo 435 vp, Leica, Wiesbaden, Germany and Hitachi SU8010, UHR FE-SEM, France). For the plant leaf samples, fresh leaves were dehydrated in methanol solution and dried by using critical point drier (LPD 030, Bal-Tec). [121,122] In order to avoid surface charging, all samples (plant leaves, PDMS replicas, and replicas after n-heptane solution treatment) were metalized with a thin (ca. 10 nm) coating of gold (Cressington Sputter Coater, 108 auto) after being mounted on aluminium stubs (Plano GmbH, Wetzlar, Germany) and side walls coated with conductive silver paint (Acheson Silver DAG 1415M, Plano GmbH, Wetzlar, Germany).

All SEM examinations were performed in the 30°-45° tilting angle range. In addition, threedimensional morphological scan of all fresh plant leaf surfaces was visualized and recorded with a confocal laser scanning microscope (LEXT OLS4000, Olympus Corporation, Japan).

Adhesion Testing

Description of the Modified Adhesion Force Tester

In 1971, Johnson, Kendall and Roberts (JKR) proposed a theory to describe the contact area of two elastic bodies in contact while considering adhesive forces acting between them. [102] The key characteristics of this theory include a finite non-zero contact at the zero normal load condition and a minimum negative (tensile) force (Fad) required to separate the contacting bodies during the detachment cycle. In this work, the adhesion investigation was performed adhesive tip slowly approaches near to the substrate surface. As the tip reaches in close proximity to the substrate surface, sudden snap-in (pull-in) takes place leading to a resulting pulling force. As the tip displacement increases, the normal load attains zero condition, corresponding to the initial zero load state in the force-displacement graph (Figure 2 -point a).

At this state, the adhesion test can start and the adhesive tip begins forming contact under quasistatic loading, at a constant loading rate of 0.083 mN/s, until the defined normal pre-load (FL) is reached. The adhesive tip is kept under constant FL for a set time and then the tip is retracted under displacement control motion, at a constant speed of 0.83 µm/s. A graph of a typical adhesion test and pull-off force measurement (named Fad) is shown in Figure 2. The range of FL was kept low enough and substrate thickness was chosen large enough, so that the ratio of substrate thickness to mean contact radius was higher than ca. 10, thus the underlying substrate (glass slide) effect could be neglected. [90,103,126] All measurements were conducted in a climate controlled room (temperature = 22 ± 3°C, relative humidity = 50% ± 10%).

Image Processing

All the recorded in-situ videos were processed and analysed with the digital image processing tool ImageJ (v. 1.51p, National Institutes of Health, USA) permitting initial homogenous filtering, thresholding, and estimating the real contact and apparent areas. [START_REF] Burger | Digital Image Processing: An Algorithmic Introduction Using Java[END_REF] The real contact area (Ar) was calculated by summing all the individual local real areas (𝐴 E F ). For the apparent area (Aap) estimation, multiple outmost peripheral point coordinates were sampled from all directions and fitted with a standard best-fit ellipse. The normalized contact area (An) is defined as:

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

Acknowledgements

The authors would like to express their gratitude to the gardeners of Botanic Garden of the University of Freiburg for cultivating the plants used in this study. We are thankful to the electron microscopy platform and to the micro-nano-mechanic platform of Institut Charles Sadron, Strasbourg. We would also like to thank the German Research Foundation (Deutsche

Conflict of Interest

The authors declare no conflict of interest.

Adhesion mechanics investigation was carried on elastomeric replicas of natural leaf surfaces (decorated with complex hierarchical microstructures) using a modified ultra-nanoindenter equipped with an original in-situ real-contact visualization system. The study presents original insights on force-dependent adhesion behaviours associated to morphology dependent attachment-detachment modes, which may offer a valuable basis for designing bio-inspired functional surfaces. Charchit Kumar 1,2,3 , Damien Favier 2 , Thomas Speck 1,3 , and Vincent Le Houérou 2,4,*

In-situ investigation of adhesion mechanisms on complex micro-structured biological surfaces