S Lessanibahri
email: saman.lessanibahri@irccyn.ec-nantes.fr

M Gouttefarde
email: marc.gouttefarde@lirmm.fr

S Caro

P Cardou
email: pcardou@gmc.ulaval.ca

Twist Feasibility Analysis of Cable-Driven Parallel Robots

Although several papers addressed the wrench capabilities of cable-driven parallel robots (CDPRs), few have tackled the dual question of their twist capabilities. In this paper, these twist capabilities are evaluated by means of the more specific concept of twist feasibility, which was defined by Gagliardini et al. in a previous work. A CDPR posture is called twist-feasible if all the twists (point-velocity and angular-velocity combinations), within a given set, can be produced at the CDPR mobile platform, within given actuator speed limits. Two problems are solved in this paper: (1) determining the set of required cable winding speeds at the CDPR winches being given a prescribed set of required mobile platform twists; and (2) determining the set of available twists at the CDPR mobile platform from the available cable winding speeds at its winches. The solutions to both problems can be used to determine the twist feasibility of n-degree-of-freedom (DOF) CDPRs driven by m ≥ n cables. An example is presented, where the twist-feasible workspace of a simple CDPR with n = 2 DOF and driven by m = 3 cables is computed to illustrate the proposed method.

Introduction

A cable-driven parallel robot (CDPR) consists of a base frame, a mobile platform, and a set of cables connecting in parallel the mobile platform to the base frame.

The cable lengths or tensions can be adjusted by means of winches and a number of pulleys may be used to route the cables from the winches to the mobile platform. Among other advantages, CDPRs with very large workspaces, e.g. [START_REF] Gouttefarde | A versatile tension distribution algorithm for n-DOF parallel robots driven by n+2 cables[END_REF][START_REF] Lambert | Implementation of an Aerostat Positioning System With Cable Control[END_REF], heavy payloads capabilities [START_REF] Albus | The NIST Robocrane[END_REF], or reconfiguration capabilities, e.g. [START_REF] Gagliardini | Discrete reconfiguration planning for cable-driven parallel robots[END_REF][START_REF] Rosati | On the design of adaptive cable-driven systems[END_REF] can be designed. Moreover, the moving parts of CDPRs being relatively light weight, fast motions of the mobile platform can be obtained, e.g. [START_REF] Kawamura | High-speed manipulation by using parallel wire-driven robots[END_REF].

The cables of a CDPR can only pull and not push on the mobile platform and their tension shall not become larger than some maximum admissible value. Hence, for a given mobile platform pose, the determination of the feasible wrenches at the platform is a fundamental issue, which has been the subject of several previous works, e.g. [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF][START_REF] Hassan | Analysis of Bounded Cable Tensions in Cable-Actuated Parallel Manipulators[END_REF]. A relevant issue is then to determine the set of wrench feasible poses, i.e., the so-called Wrench-Feasible Workspace (WFW) [START_REF] Bosscher | Wrench-feasible workspace generation for cable-driven robots[END_REF][START_REF] Riechel | Force-feasible workspace analysis for underconstrained point-mass cable robots[END_REF], since the shape and size of the latter highly depends on the cable tension bounds and on the CDPR geometry [START_REF] Verhoeven | Analysis of the Workspace of Tendon-Based Stewart-Platforms[END_REF]. Another issue which may strongly restrict the usable workspace of a CDPR or, divide it into several disjoint parts, are cable interferences. Therefore, software tools allowing the determination of the interference-free workspace and of the WFW have been proposed, e.g. [START_REF] Ruiz | Arachnis : Analysis of robots actuated by cables with handy and neat interface software[END_REF][START_REF] Perreault | Geometric determination of the interferencefree constant-orientation workspace of parallel cable-driven mechanisms[END_REF],. Besides, recently, a study on acceleration capabilities was proposed in [START_REF] Eden | Available acceleration set for the study of motion capabilities for cable-driven robots[END_REF][START_REF] Gagliardini | Determination of a dynamic feasible workspace for cable-driven parallel robots[END_REF].

As noted in [START_REF] Gagliardini | Dimensioning of cable-driven parallel robot actuators, gearboxes and winches according to the twist feasible workspace[END_REF] and as well known, in addition to wrench feasibility, the design of the winches of a CDPR also requires the consideration of cable and mobile platform velocities since the selection of the winch characteristics (motors, gearboxes, and drums) has to deal with a trade-off between torque and speed. Twist feasibility is then the study of the relationship between the feasible mobile platform twists (linear and angular velocities) and the admissible cable coiling/uncoiling speeds. In the following, the cable coiling/uncoiling speeds are loosely referred to as cable velocities. The main purpose of this paper is to clarify the analysis of twist feasibility and of the related twist-feasible workspace proposed in [START_REF] Gagliardini | Dimensioning of cable-driven parallel robot actuators, gearboxes and winches according to the twist feasible workspace[END_REF]. Contrary to [START_REF] Gagliardini | Dimensioning of cable-driven parallel robot actuators, gearboxes and winches according to the twist feasible workspace[END_REF], the twist feasibility analysis proposed here is based on the usual CDPR differential kinematics where the Jacobian matrix maps the mobile platform twist into the cable velocities. This approach is most important for redundantly actuated CDPRs, whose Jacobian matrix is rectangular.

A number of concepts in this paper are known, notably from manipulability ellipsoids of serial robots, e.g. [START_REF] Yoshikawa | Foundations of Robotics[END_REF], and from studies on the velocity performance of parallel robots, e.g. [START_REF] Krut | Velocity performance indices for parallel mechanisms with actuation redundancy[END_REF]. A review of these works is however out of the scope of the present paper whose contribution boils down to a synthetic twist feasibility analysis of n-degrees-of-freedom (DOF) CDPRs driven by m cables, with m ≥ n. The CDPR can be fully constrained or not, and the cable mass and elasticity are neglected.

The paper is organized as follows. The usual CDPR wrench and Jacobian matrices are defined in Section 2. Section 3 presents the twist feasibility analysis, which consists in solving two problems. The first one is the determination of the set of cable velocities corresponding to a given set of required mobile platform twists (Section 3.1). The second problem is the opposite since it is defined as the calculation of the set of mobile platform twists corresponding to a given set of cable velocities (Section 3.2). The twist and cable velocity sets considered in this paper are convex Fig. 1: Geometric description of a fully constrained CDPR polytopes. In Section 4, a 2-DOF point-mass CDPR driven by 3 cables is considered to illustrate the twist feasibility analysis. Section 5 concludes the paper.

Wrench and Jacobian Matrices

In this section, the well-known wrench matrix and Jacobian matrix of n-DOF mcable CDPRs are defined. The wrench matrix maps the cable tensions into the wrench applied by the cables on the CDPR mobile platform. The Jacobian matrix relates the time derivatives of the cable lengths to the twist of the mobile platform. These two matrices are essentially the same since one is minus the transpose of the other.

Some notations and definitions are first introduced. As illustrated in Fig. 1, let us consider a fixed reference frame, F b , of origin O b and axes x b , y b and z b . The coordinate vectors b a i , i = 1, . . . , m define the positions of the exit points, A i , i = 1, . . . , m, with respect to frame F b . A i is the point where the cable exits the base frame and extends toward the mobile platform. In this paper, the exit points A i are assumed to be fixed, i.e., the motion of the output pulleys is neglected. A frame F p , of origin O p and axes x p , y p and z p , is attached to the mobile platform. The vectors p b i , i = 1, . . . , m are the position vectors of the points B i in F p . The cables are attached to the mobile platform at points B i .

The vector b l i from B i to A i is given by

b l i = b a i -p -R p b i , i = 1, . . . , m (1)
where R is the rotation matrix defining the orientation of the mobile platform, i.e., the orientation of F p in F b , and p is the position vector of F p in F b . The length of the straight line segment

A i B i is l i = || b l i || 2 where || • || 2 is the Euclidean norm.
Neglecting the cable mass, l i corresponds to the length of the cable segment from point A i to point B i . Moreover, neglecting the cable elasticity, l i is the "active" length of the cable that should be unwound from the winch drum. The unit vectors along the cable segment A i B i is given by

b d i = b l i /l i , i = 1, . . . , m (2)
Since the cable mass is neglected in this paper, the force applied by the cable on the platform is equal to τ i b d i , τ i being the cable tension. The static equilibrium of the CDPR platform can then be written [START_REF] Hiller | Design, analysis and realization of tendon-based parallel manipulators[END_REF][START_REF] Roberts | On the Inverse Kinematics, Statics, and Fault Tolerance of Cable-Suspended Robots[END_REF]

Wτ τ τ + w e = 0 ( 3
)
where w e is the external wrench acting on the platform, τ τ τ = [τ 1 , . . . , τ m ] T is the vector of cable tensions, and W is the wrench matrix. The latter is an n × m matrix defined as

W = b d 1 b d 2 . . . b d m R p b 1 × b d 1 R p b 2 × b d 2 . . . R p b m × b d m (4)
The differential kinematics of the CDPR establishes the relationship between the twist t of the mobile platform and the time derivatives of the cable lengths l

Jt = l ( 5
)
where J is the m × n Jacobian matrix and l = l1 , . . . , lm T . The twist t = [ ṗ, ω ω ω] T is composed of the velocity ṗ of the origin of frame F p with respect to F b and of the angular velocity ω ω ω of the mobile platform with respect to F b . Moreover, the well-known kineto-statics duality leads to

J = -W T (6)
In the remainder of this paper, l is loosely referred to as cable velocities. The wrench and Jacobian matrices depend on the geometric parameters a i and b i of the CDPR and on the mobile platform pose, namely on R and p.

Twist Feasibility Analysis

This section contains the contribution of the paper, namely, a twist feasibility analysis which consists in solving the following two problems.

1. For a given pose of the mobile platform of a CDPR and being given a set [t] r of required mobile platform twists, determine the corresponding set of cable velocities l. The set of cable velocities to be determined is called the Required Cable Velocity Set (RCVS) and is denoted l r . The set [t] r is called the Required Twist Set (RTS). 2. For a given pose of the mobile platform of a CDPR and being given a set l a of available (admissible) cable velocities, determine the corresponding set of mobile platform twists t. The former set, l a , is called the Available Cable Velocity Set (ACVS) while the latter is denoted [t] a and called the Available Twist Set (ATS).

In this paper, the discussion is limited to the cases where both the RTS [t] r and the ACVS l a are convex polytopes. Solving the first problem provides the RCVS from which the maximum values of the cable velocities required to produce the given RTS [t] r can be directly deduced. If the winch characteristics are to be determined, the RCVS allows to determine the required speeds of the CDPR winches. If the winch characteristics are already known, the RCVS allows to test whether or not the given RTS is feasible.

Solving the second problem provides the ATS which is the set of twists that can be produced at the mobile platform. It is thus useful either to determine the velocity capabilities of a CDPR or to check whether or not a given RTS is feasible.

Note that the feasibility of a given RTS can be tested either in the cable velocity space, by solving the first problem, or in the space of platform twists, by solving the second problem. Besides, note also that the twist feasibility analysis described above does not account for the dynamics of the CDPR.

Problem 1: Required Cable Velocity Set (RCVS)

The relationship between the mobile platform twist t and the cable velocities l is the differential kinematics in [START_REF] Eden | Available acceleration set for the study of motion capabilities for cable-driven robots[END_REF]. According to this equation, the RCVS l r is defined as the image of the convex polytope [t] r under the linear map J. Consequently, l r is also a convex polytope [START_REF] Ziegler | Lectures on Polytopes[END_REF].

Moreover, if [t] r is a box, the RCVS l r is a particular type of polytope called a zonotope. Such a transformation of a box into a zonotope has previously been studied in CDPR wrench feasibility analysis [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF][START_REF] Gallina | 3-dof wire driven planar haptic interface[END_REF][START_REF] Gouttefarde | Characterization of Parallel Manipulator Available Wrench Set Facets[END_REF]. Indeed, a box of admissible cable tensions is mapped by the wrench matrix W into a zonotope in the space of platform wrenches. However, a difference lies in the dimensions of the matrices J and W, J being of dimensions m × n while W is an n × m matrix, where n ≤ m. When n < m, on the one hand, W maps the m-dimensional box of admissible cable tensions into the n-dimensional space of platform wrenches. On the other hand, J maps n-dimensional twists into its range space which is a linear subspace of the m-dimensional space of cable velocities l. Hence, when J is not singular, the n-dimensional box [t] r is mapped into the zonotope l r which lies into the ndimensional range space of J, as illustrated in Fig. 3 . When J is singular and has rank r, r < n, the n-dimensional box [t] r is mapped into a zonotope of dimension r.

When an ACVS l a is given, a pose of the mobile platform of a CDPR is twist feasible if

l r ⊆ l a (7)
Since l a is a convex polytope, ( 7) is verified whenever all the vertices of l r are included in l a . Moreover, it is not difficult to prove that l r is the convex hull of the images under J of the vertices of [t] r . Hence, a simple method to verify if a CDPR pose is twist feasible consists in verifying whether the images of the vertices of [t] r are all included into l a .

Problem 2: Available Twist Set (ATS)

The problem is to determine the ATS [t] a corresponding to a given ACVS l a . In the most general case considered in this paper, l a is a convex polytope. By the Minkowski-Weyl's Theorem, a polytope can be represented as the solution set of a finite set of linear inequalities, the so-called (halfspace) H-representation of the polytope [START_REF] Fukuda | Frequently asked questions in polyhedral computation[END_REF][START_REF] Ziegler | Lectures on Polytopes[END_REF], i.e.

l a = { l | C l ≤ d } (8)
where matrix C and vector d are assumed to be known. According to (5), the ATS is defined as

[t] a = { t | Jt ∈ l a } (9)
which, using (8), implies that

[t] a = { t | CJt ≤ d } (10)
The latter equation provides an H-representation of the ATS [t] a . In practice, when the characteristics of the winches of a CDPR are known, the motor maximum speeds limit the set of possible cable velocities as follows li,min ≤ li ≤ li,max [START_REF] Gouttefarde | Characterization of Parallel Manipulator Available Wrench Set Facets[END_REF] where li,min and li,max are the minimum and maximum cable velocities. Note that, usually, li,min = -li,max , l1,min = l2,min = . . . = lm,min , and l1,max = l2,max = . . . = lm,max . In other words, C and d in (8) are defined as

C = 1 -1
and d = l1,max , . . . , lm,max , -l1,min , . . . , -lm,min

T ( 12
)
where 1 is the m × m identity matrix. Eq. ( 10) can then be written as follows

[t] a = { t | lmin ≤ Jt ≤ lmax } (13)
where lmin = l1,min , . . . , lm,min T and lmax = l1,max , . . . , lm,max T .

When a RTS [t] r is given, a pose of the mobile platform of a CDPR is twist feasible if

[t] r ⊆ [t] a (14)
In this paper, [t] r is assumed to be a convex polytope. Hence, ( 14) is verified whenever all the vertices of [t] r are included in [t] a . With the H-representation of [t] a in (10) (or in ( 13)), testing if a pose is twist feasible amounts to verifying if all the vertices of [t] r satisfy the inequality system in (10) (or in ( 13)). Testing twist feasibility thereby becomes a simple task as soon as the vertices of [t] r are known. Finally, let the twist feasible workspace (TFW) of a CDPR be the set of twist feasible poses of its mobile platform. It is worth noting that the boundaries of the TFW are directly available in closed form from [START_REF] Gallina | 3-dof wire driven planar haptic interface[END_REF] or [START_REF] Hassan | Analysis of Bounded Cable Tensions in Cable-Actuated Parallel Manipulators[END_REF]. If the vertices of the (convex) RTS are denoted t j , j = 1, . . . , k, and the rows of the Jacobian matrix are -w T i , according to ( 13), the TFW is defined by li,min ≤ -w T i t j and -w T i t j ≤ li,max , for all possible combinations of i and j. Since w i contains the only variables in these inequalities that depend on the mobile platform pose, and because the closedform expression of w i as a function of the pose is known, the expressions of the boundaries of the TFW are directly obtained.

Case Study

This section deals with the twist feasibility analysis of the two-DOF point-mass planar CDPR driven by three cables shown in Fig. 2. The robot is 3.5 m long and 2.5 m high. The three exit points of the robot are named A 1 , A 2 are A 3 , respectively. The point-mass is denoted P. b d 1 , b d 2 and b d 3 are the unit vectors, expressed in frame F b , of the vectors pointing from point-mass P to cable exit points A 1 , A 2 are A 3 , respectively. The 3 × 2 Jacobian matrix J of this planar CDPR takes the form:

J = -   b d T 1 b d T 2 b d T 3   ( 15
)
Figure 3 is obtained by solving the Problem 1 formulated in Sec. 3. For the robot configuration depicted in Fig. 3a and the given RTS of the point-mass P represented in Fig. 3b, the RCVS for the three cables of the planar CDPR are illustrated in Figs. 3c to 3f. Note that the RTS is defined as:

-1 m.s -1 ≤ ẋP ≤ 1 m.s -1 (16) -1 m.s -1 ≤ ẏP ≤ 1 m.s -1 (17)
where [ ẋP , ẏP ] T is the velocity of P in the fixed reference frame F b . Figure 4 depicts the isocontours of the Maximum Required Cable Velocity (MRCV) for each cable through the Cartesian space and for the RTS shown in Fig. 3b. Those results are obtained by solving Problem 1 for all positions of point P. It is apparent

y b O b x b A 1 A 2 A 3 P d 1 d 2 d 3 F b p a 3 3.5 m 2.5 m
Fig. 2: A two-DOF point-mass planar cable-driven parallel robot driven by three cables that P RTS is satisfied through the Cartesian space as long as the maximum velocity of each cable is higher than √ 2 m.s -1 , namely, l1,max = l2,max = l3,max = √ 2 m.s -1 with li,min = -li,max , i = 1, 2, 3.

For the Available Cable Velocity Set (ACVS) defined by inequalities [START_REF] Gouttefarde | Characterization of Parallel Manipulator Available Wrench Set Facets[END_REF] with

li,max = 1.3 m.s -1 , i = 1, 2, 3 (18)
Fig. 5 is obtained by solving the Problem 2 formulated in Sec. 3. For the two robot configurations illustrated in Fig. 5a and5c, the Available Twist Set (ATS) associated to the foregoing ACVS is determined from Eq. ( 13). It is noteworthy that the ATS in each configuation in delimited by three pairs of lines normal to three cables, respectively. It turns out that the first robot configuration is twist feasible for the RTS defined by Eqs. ( 16) and ( 17) because the latter is included into the ATS as shown Fig. 5b. Conversely, the second robot configuration is not twist feasible as the RTS is partially outside the ATS as shown Fig. 5d.

Finally, Fig. 6 shows the TFW of the planar CDPR for four maximum cable velocity limits and for the RTS shown in Fig. 3b. It is apparent the all robot poses are twist feasible as soon as the cable velocity limits of the three cables are higher than √ 2 m.s -1 .

x [m]

Conclusion

In summary, this paper presents two methods of determining the twist-feasibility of a CDPR. The first method uses a set of required mobile platform twists to compute the corresponding required cable velocities, the latter corresponding to cable winding speeds at the winches. The second method takes the opposite route, i.e., it uses the available cable velocities to compute the corresponding set of available mobile platform twists. The second method can be applied to compute the twist-feasible workspace, i.e., to determine the set of mobile platform poses where a prescribed polyhedral required twist set is contained within the available twist set. This method can thus be used to analyze the CDPR speed capabilities over its workspace, which should prove useful in high-speed CDPR applications.

The proposed method can be seen as a dual to the one used to compute the wrench-feasible workspace of a CDPR, just as the velocity equations may be seen as dual to static equations. From a mathematical standpoint, however, the problem is much simpler in the case of the twist-feasible workspace, as the feasibility conditions can be obtained explicitly. Nevertheless, the authors believe that the present paper complements nicely the previous works on wrench feasibility. Finally, we should point out that the proposed method does not deal with the issue of guaranteeing the magnitudes of the mobile platform point-velocity or angular velocity. In such a case, the required twist set becomes a ball or an ellipsoid, and thus is no longer polyhedral. This ellipsoid could be approximated by a polytope in order to apply the method proposed in this paper. However, since the accuracy of the approximation would come at the expense of the number of conditions to be numerically verified, part of our future work will be dedicated to the problem of determining the twist-feasibility of CDPRs for ellipsoidal required twist sets. Fig. 6: TFW of the planar CDPR for four maximum cable velocity limits and for the RTS shown in Fig. 3b Manufacturing Technologies for Composite, Metallic and Hybrid Structures) and of the RFI AT-LANSTIC 2020 CREATOR project.

Fig. 3 :

 3 Fig. 3: Required Twist Set (RTS) of the point-mass P and corresponding Required Cable Velocity Sets for the three cables of the CDPR in a given robot configuration

Fig. 4 :

 4 Fig.4: Maximum Required Cable Velocity (MRCV) of each cable through the Cartesian space for the RTS shown in Fig.3b

Fig. 5 :

 5 Fig. 5: A feasible twist pose and an infeasible twist pose of the CDPR

1 x 1 x

 11 li,max = 1.12 m.s -li,max = 1.32 m.s -li,max = 1.414 m.s -1

Acknowledgements

The financial support of the ANR under grant ANR-15-CE10-0006-01 (Dex-terWide project) is greatly acknowledged. This research work was also part of the CAROCA project managed by IRT Jules Verne (French Institute in Research and Technology in Advanced


Tahir Rasheed
email: tahir.rasheed@ls2n.fr

Philip Long
email: philip.long@northeastern.edu

David Marquez-Gamez
email: david.marquez-gamez@irt-jules-verne.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Tension Distribution Algorithm for Planar Mobile Cable-Driven Parallel Robots

Keywords: Cable-Driven Parallel Robot, Mobile Robot, Reconfigurability, Tension Distribution Algorithm, Equilibrium

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

A Cable-Driven Parallel Robot (CDPR) is a type of parallel robot whose movingplatform is connected to the base with cables. The lightweight properties of the CDPR makes them suitable for multiple applications such as constructions [START_REF] Albus | The nist spider, a robot crane[END_REF], [START_REF] Pott | Large-scale assembly of solar power plants with parallel cable robots[END_REF], industrial operations [START_REF] Gagliardini | Discrete reconfiguration planning for cable-driven parallel robots[END_REF], rehabilitation [START_REF] Rosati | Performance of cable suspended robots for upper limb rehabilitation[END_REF] and haptic devices [START_REF] Gallina | 3-dof wire driven planar haptic interface[END_REF].

A general CDPR has a fixed cable layout, i.e. fixed exit points and cable configuration. This fixed geometric structure may limit the workspace size of the manipulator due to cable collisions and some extrernal wrenches that cannot be accepted due to the robot configuration. As there can be several configurations for the robot to perform the prescribed task, an optimized cable layout is required for each task considering an appropriate criterion. Cable robots with movable exit and/or anchor points are known as Reconfigurable Cable-Driven Parallel Robots (RCDPRs). By appropriately modifying the geometric architecture, the robot performance can be improved e.g. lower cable tensions, larger workspace and higher stiffness. The recent work on RCDPR [START_REF] Gagliardini | A reconfiguration strategy for reconfigurable cable-driven parallel robots[END_REF][START_REF] Gagliardini | Discrete reconfiguration planning for cable-driven parallel robots[END_REF][START_REF] Nguyen | On the analysis of largedimension reconfigurable suspended cable-driven parallel robots[END_REF][START_REF] Rosati | On the design of adaptive cable-driven systems[END_REF][START_REF] Zhou | Analysis framework for cooperating mobile cable robots[END_REF] proposed different design strategies and algorithms to compute optimized cable layout for the required task, while minimizing appropriate criteria such as the robot energy consumption, the robot workspace size and the robot stiffness. However, for most existing RCDPRs, the reconfigurability is performed either discrete and manually or continuously, but with bulky reconfigurable systems.

This paper deals with the concept of Mobile Cable-Driven Parallel Robots (MCDPRs). The idea for introducing MCDPRs is to overcome the manual and discrete reconfigurability of RCDPRs such that an autonomous reconfiguration can be achieved. A MCDPR is composed of a classical CDPR with m cables and a n degreeof-freedom (DoF) moving-platform mounted on p mobile bases. Mobile bases are four-wheeled planar robots with two-DoF translational motions and one-DoF rotational motion. A concept idea of a MCDPR is illustrated in Fig. 1 with m = 8, n = 6 and p = 4. The goal of such system is to provide a low cost and versatile robotic solution for logistics using a combination of mobile bases and CDPR. This system addresses an industrial need for fast pick and place operations while being easy to install, keeping existing infrastructures and covering large areas. The exit points for the cable robot is associated with the position of its respective mobile bases. Each mobile base can navigate in the environment thus allowing the system to alter the geometry of the CDPR. Contrary to classical CDPR, equilibrium for both the moving-platform and the mobile bases should be considered while analyzing the behaviour of the MCDPR.

A Planar Mobile Cable-Driven Parallel Robot with four cables (m = 4), a point mass (n = 2) and two mobile bases (p = 2), shown in Fig. 2, is considered throughout this paper as an illustrative example. This paper is organized as follows. Section 2 presents the static equilibrium conditions for mobile bases using the free body diagram method. Section 3 introduces a modified real time Tension Distribution Algorithm (TDA), which takes into account the dynamic equilibrium of the moving-platform and the static equilibrium of the mobile bases. Section 4 presents the comparison between the existing and modified TDA on the equilibrium of the

Static Equilibrium of Mobile Bases

This section aims at analyzing the static equilibrium of the mobile bases of MCD-PRs. As both the mobile bases should be in equilibrium during the motion of the end-effector, we need to compute the reaction forces generated between the ground and the wheels of the mobile bases. Figure 2 illustrates the free body diagram for the jth mobile base. u i j denotes the unit vector of the ith cable attached to the jth mobile base, i, j = 1, 2. u i j is defined from the point mass P of the MCDPR to the exit point A i j . Using classical equilibrium conditions for the jth mobile base p j , we can write:

∑ f = 0 ⇒ m j g + f 1 j + f 2 j + f r1 j + f r2 j = 0 (1)
All the vectors in Eq. ( 1) are associated with the superscript x and y for respective horizontal and vertical axes. Gravity vector is denoted as g = [0g] T where g = 9.8 m.s -2 , f 1 j = [ f x 1 j f y 1 j ] T and f 2 j = [ f x 2 j f y 2 j ] T are the reaction forces due to cable tensions onto the mobile base p j , C 1 j and C 2 j are the front and rear wheels con- tact points having ground reaction forces f r1 j = [ f x r1 j f y r1 j ] T and f r2 j = [ f x r2 j f y r2 j ] T , respectively. In this paper, wheels are assumed to be simple support points and the friction between those points and the ground is supposed to be high enough to prevent the mobile bases from sliding. The moment at a point O about z-axis for the mobile base to be in equilibrium is expressed as:

M z O = 0 ⇒ g T j E T m j g + a T 1 j E T f 1 j + a T 2 j E T f 2 j + c T 1 j E T f r1 j + c T 2 j E T f r2 j = 0 (2)
with

E = 0 -1 1 0 (3) a 1 j = [a x 1 j a y 1 j ] T and a 2 j = [a x 2 j a y 2 j ] T denote the Cartesian coordinate vectors of the exit points A 1 j and A 2 j , c 1 j = [c x 1 j c y 1 j ] T and c 2 j = [c x 2 j c y 2
j ] T denote the Cartesian coordinate vectors of the contact points C 1 j and C 2 j . g j = [g x j g y j ] T is the Cartesian coordinate vector for the center of gravity G j of the mobile base p j . The previous mentioned vector are all expressed in the base frame F B . Solving simultaneously Eqs. ( 1) and ( 2), the vertical components of the ground reaction forces take the form:

f y r1 j = m j g(c x 2 j -g x j ) + f y 1 j (a x 1 j -c x 2 j ) + f y 2 j (a x 2 j -c x 2 j ) -f x 1 j a y 1 j -f x 2 j a y 2 j c x 2 j -c x 1 j (4)
f y r2 j = m j g -f y 1 j -f y 2 j -f y r1 j (5)
Equations ( 4) and ( 5) illustrate the effect of increasing the external forces (cable tensions) onto the mobile base. Indeed, the external forces exerted onto the mobile base may push the latter towards frontal tipping. It is apparent that the higher the cable tensions, the higher the vertical ground reaction force f y r 1 j and the lower the ground reaction force f y r 2 j . There exists a combination of cable tensions such that f y r 2 j = 0. At this instant, the rear wheel of the jth mobile base will lose contact with the ground at point C 2 j , while generating a moment M C1 j about z-axis at point C 1 j :

M z C1 j = (g j -c 1 j ) T E T m j g + (a 1 j -c 1 j ) T E T f 1 j + (a 2 j -c 1 j ) T E T f 2 j (6)
Similarly for the rear tipping f y r 1 j = 0, the jth mobile base will lose the contact with the ground at C 1 j and will generate a moment M c2 j about z-axis at point C 2 j :

M z C2 j = (g j -c 2 j ) T E T m j g + (a 1 j -c 2 j ) T E T f 1 j + (a 2 j -c 2 j ) T E T f 2 j (7)
As a consequence, for the first mobile base p 1 to be always stable, the moments generated by the external forces should be counter clockwise at point C 11 while it should be clockwise at point C 21 . Therefore, the stability conditions for mobile base p 1 can be expressed as:

M z C11 ≥ 0 (8) M z C21 ≤ 0 (9)
Similarly, the stability constraint conditions for the second mobile base p 2 are expressed as:

M z C12 ≤ 0 (10) M z C22 ≥ 0 ( 11
)
where M z C12 and M z C22 are the moments of the mobile base p 2 about z-axis at the contact points C 12 and C 22 , respectively.

Real-time Tension Distribution Algorithm

In this section an existing Tension Distribution Algorithm (TDA) defined for classical CDPRs is adopted to Mobile Cable-driven Parallel Robots (MCDPRs). The existing algorithm, known as barycenter/centroid algorithm is presented in [START_REF] Lamaury | A tension distribution method with improved computational efficiency[END_REF][START_REF] Mikelsons | A real-time capable force calculation algorithm for redundant tendon-based parallel manipulators[END_REF]. Due to its geometric nature, the algorithm is efficient and appropriate for real time applications [START_REF] Gouttefarde | A versatile tension distribution algorithm for-dof parallel robots driven by cables[END_REF]. First, the classical Feasible Cable Tension Domain (FCTD) is defined for CDPRs based on the cable tension limits. Then, the stability (static equilibrium) conditions for the mobile bases are considered in order to define a modified FCTD for MCDPRs. Finally, a new TDA aiming at obtaining the centroid/barycenter of the modified FCTD is presented.

FCTD based on cable tension limits

The dynamic equilibrium equation of a point mass platform is expressed as:

Wt p + w e = 0 =⇒ t p = -W + w e ( 12
)
where W = [u 11 u 21 u 12 u 22 ] is n × m wrench matrix mapping the cable tension space defined in R m onto the available wrench space defined in R (m-n) . w e denotes the external wrench exerted onto the moving-platform. W + is the Moore Penrose pseudo inverse of the wrench matrix W. t p = [t p11 t p21 t p12 t p22 ] T is a particular solution (Minimum Norm Solution) of Eq. ( 12). Having redundancy r = mn = 2, a homogeneous solution t n can be added to the particular solution t p such that:

t = t p + t n =⇒ t = -W + w e + Nλ λ λ ( 13
)
where N is the m × (mn) null space of the wrench matrix W and λ λ λ = [λ 1 λ 2 ] T is a (mn) dimensional arbitrary vector that moves the particular solution into the feasible range of cable tensions. Note that the cable tension t i j associated with the ith cable mounted onto the jth mobile base should be bounded between a minimum tension t and a maximum tension t depending on the motor capacity and the transmission system at hand. According to [START_REF] Gouttefarde | A versatile tension distribution algorithm for-dof parallel robots driven by cables[END_REF][START_REF] Lamaury | A tension distribution method with improved computational efficiency[END_REF], there exists a 2-D affine space Σ defined by the solution of Eq. ( 12) and another m-dimensional hypercube Ω defined by the feasible cable tensions:

Σ = {t | Wt = w e } (14
)

Ω = {t | t ≤ t ≤ t} (15)
The intersection between these two spaces amounts to a 2-D convex polygon also known as feasible polygon. Such a polygon exists if and only if the tension distribution admits a solution at least that satisfies the cable tension limits as well as the equilibrium of the moving-platform defined by Eq. [START_REF] Rosati | On the design of adaptive cable-driven systems[END_REF]. Therefore, the feasible polygon is defined in the λ λ λ -space by the following linear inequalities:

t -t p ≤ Nλ λ λ ≤ t -t p (16)
The terms of the m × (mn) null space matrix N are defined as follows:

N =     n 11 n 21 n 12 n 22     (17)
where each component n i j of the null space N in Eq. ( 17) is a (1 × 2) row vector.

FCTD based on the stability of the mobile bases

This section aims at defining the FCTD while considering the cable tension limits and the stability conditions of the mobile bases. In order to consider the stability of the mobile bases, Eqs. (8 -11) must be expressed into the λ λ λ -space. The stability constraint at point C 11 from Eq. ( 8) can be expressed as:

0 ≤ (g 1 -c 11 ) T E T m 1 g + (a 11 -c 11 ) T E T f 11 + (a 21 -c 11 ) T E T f 21 (18)
f i j is the force applied by the ith cable attached onto the jth mobile base. As f i j is opposite to u i j (see Fig. 2), from Eq. ( 13) f i j can be expressed as:

f i j = -[t pi j + n i j λ λ λ ] u i j (19)
Substituting Eq. (19) in Eq. (18) yields: Term [n i j λ λ λ ]u i j is the mapping of homogeneous solution t ni j for the ith cable carried by the jth mobile base into the Cartesian space. M C11 represents the lower bound for the constraint (8) in the λ λ λ -space:

(c 11 -g 1 ) T E T m 1 g ≤ (c
M C11 = (c 11 -g 1 ) T E T m 1 g + (a 11 -c 11 ) T E T t p11 + (a 21 -c 11 ) T E T t p21 (22)
Simplifying Eq. ( 21) yields:

M C11 ≤ (c 11 -a 11 ) T E T u 11 (c 11 -a 21 ) T E T u 21 n 11 n 21 λ 1 λ 2 (23)
Equation ( 23) can be written as:

M C11 ≤ n C11 λ λ λ (24)
where n C11 is a 1 × 2 row vector. Similarly the stability constraint at point C 21 from Eq. ( 9) can be expressed as:

n C21 λ λ λ ≤ M C21 (25)
where:

M C21 = (c 21 -g 1 ) T E T m 1 g + (a 11 -c 21 ) T E T t p 11 + (a 21 -c 21 ) T E T t p 21 ( 26
)
n C21 = (c 21 -a 11 ) T E T u 11 (c 21 -a 21 ) T E T u 21 n 11 n 21 (27)
Equations ( 24) and ( 25) define the stability constraints of the mobile base p 1 in the λ λ λ -space for the static equilibrium about frontal and rear wheels. Similarly, the above procedure can be repeated to compute the stability constraints in the λ λ λ -space for mobile base p 2 . Constraint Eqs. ( 10) and ( 11) for point C 12 and C 22 can be expressed in the λ λ λ -space as:

n C12 λ λ λ ≤ M C12 (28) M C22 ≤ n C22 λ λ λ (29)
Considering the stability constraints related to each contact point (Eqs. ( 24), ( 25), ( 28) and ( 29)) with the cable tension limit constraints (Eq. ( 16)), the complete system of constraints to calculate the feasible tensions for MCDPR can be expressed as:

t -t p M ≤ N N c λ 1 λ 2 ≤ t -t p M (30)
where:

N c =     n C11 n C21 n C12 n C22     , M =     M C11 -∞ -∞ M C22     , M =     ∞ M C21 M C12 ∞     , (31)
The terms -∞ and ∞ are added for the sake of algorithm [START_REF] Gouttefarde | A versatile tension distribution algorithm for-dof parallel robots driven by cables[END_REF] as the latter requires bounds from both ends. The upper part of Eq. (30) defines the tension limit constraints while the lower part represents the stability constraints for both mobile bases.

Tracing FCTD into the λ λ λ -space

The inequality constraints from Eq. ( 30) are used to compute the feasible tension distribution among the cables using the algorithm in [START_REF] Gouttefarde | A versatile tension distribution algorithm for-dof parallel robots driven by cables[END_REF] for tracing the feasible polygon P I . Each constraint defines a line in the λ λ λ -space where the coefficients of λ λ λ define the slope of the corresponding lines. The intersections between these lines form a feasible polygon. The algorithm aims to find the feasible combination for λ 1 and λ 2 (if it exists), that satisfies all the inequality constraints. The algorithm can start with the intersection point v i j between any two lines L i and L j where

λ 1 λ 2 L 1, m ax L 1, m in L 4,min 4,max L 2, m in L L 3,m in L 3,m ax L 2, m ax v 2 v 3 v8 v 7 v 6 v v 4 v 5 f = v 1 init v = P I1
L 1, m ax L 1, m in L 4,min 4,max L2 ,m in L 2, m ax L 3,m in L 3,m ax L LC 2 1 ,m a x L C 1 2 ,m a x L C 1 1 ,m in L C 2 2 ,m in λ 1 λ 2 v = init v 1 v 2 v 5 v 4 v 3 v 6 v 7 v 8 v f = P I 2
Fig. 4: Feasible Polygon considering both tension limit and stability constraints each intersection point v corresponds to a specific value for λ λ λ . After reaching the intersection point v i j , the algorithm leaves the current line L j and follows the next line L i in order to find the next intersection point v ki between lines L k and L i .

The feasible polygon P I is associated with the feasible index set I, which contains the row indices in Eq. (30). At each intersection point, the feasible index set is unchanged or modified by adding the corresponding row index of Eq. (30). It means that for each intersection point, the number of rows from Eq. (30) satisfied at current intersection point should be greater than or equal to the number of rows satisfied at previous visited points. Accordingly, the algorithm makes sure to converge toward the solution. The algorithm keeps track of the intersection points and updates the first vertex v f of the feasible polygon, which depends on the update of feasible index set I. If the feasible index set is updated at intersection point v, the first vertex of the polygon is updated as v f = v. Let's consider that the algorithm has reached a point v ki by first following line L j , then following L i intersecting with line L k . The feasible index set I ki at v ki should be such that I i j ⊆ I ki . If index k is not available in I i j , then I ki = I i j ∪ k as the row k is now satisfied. At each update of the feasible index set I, a new feasible polygon is achieved and the first vertex v f of the polygon is replaced by the current intersection point. This procedure is repeated until a feasible polygon (if it exists) is found, which is determined by visiting v f more than once. After computing the feasible polygon, its centroid, namely the solution furthest away from all the constraints is calculated. The λ λ λ coordinates of the centroid is used to calculate the feasible tension distribution using Eq. [START_REF] Sardain | Forces acting on a biped robot. center of pressure-zero moment point[END_REF].

For the given end-effector position in static equilibrium (see Fig. 2), the feasible polygon P I1 based only on the tension limits is illustrated in Fig. 3 while the feasible polygon P I2 based on the cable tension limits and the stability of the mobile bases is illustrated in Fig. 4. It can be observed that P I2 is smaller than P I1 and, as a consequence, their centroids are different.

Case Study

The stability of the mobile bases is defined by the position of their Zero Moment Point (ZMP). This index is commonly used to determine the dynamic stability of the humanoid and wheeled robots [START_REF] Lafaye | Linear model predictive control of the locomotion of pepper, a humanoid robot with omnidirectional wheels[END_REF][START_REF] Sardain | Forces acting on a biped robot. center of pressure-zero moment point[END_REF][START_REF] Vukobratović | Zero-moment pointthirty five years of its life[END_REF]. It is the point where the moment of contact forces is reduced to the pivoting moment of friction forces about an axis normal to the ground. Here the ZMP amounts to the point where the sum of the moments due to frontal and rear ground reaction forces is null. Once the feasible cable tensions are computed using the constraints of the modified TDA, the ZMP d j of the mobile base p j is expressed by the equation:

M z d j = M z O -f y r j d j (32)
where f y r j is the sum of all the vertical ground reaction forces computed using Eqs. ( 4) and ( 5), M d j is the moment generated at ZMP for the jth mobile base such that M z d j = 0. M O is the moment due to external forces, i.e., weight and cable tensions, except the ground reaction forces at O given by the Eq. ( 2). As a result from Eq. (32), ZMP d j will take the form:

d j = M z O f y r j = g T j E T m j g + a T 1 j E T f 1 j + a T 2 j E T f 2 j f y r j (33)
For the mobile base p j to be in static equilibrium, ZMP d j must lie within the contact points of the wheels, namely, Modified Algorithm for MCDPRs is validated through simulation on a rectangular test trajectory (green path in Fig. 2) where each corner of the rectangle is a zero velocity point. A 8 kg point mass is used. Total trajectory time is 10 s having 3 s for 1-2 and 3-4 paths while 2 s for 2-3 and 4-1 paths. The size of each mobile base is 0.75 m × 0.64 m × 0.7 m. The distance between the two mobile bases is 5 m with exit points A 2 j located at the height of 3 m. The evolution of ZMP for mobile base p 1 is illustrated in Fig. 5a. ZMP must lie between 0 and 0.75, which corresponds to the normalized distance between the two contact points of the wheels, for the first mobile base to be stable. By considering only cable tension limit constraints in the TDA, the first mobile base will tip over the front wheels along the path 3-4 as ZMP goes out of the limit (blue in Fig. 5a). While considering both cable tension limits and stability constraints, the MCDPR will complete the required trajectory with the ZMP satisfying Eqs. (34) and (35). Figure 5b depicts positive cable tensions computed using modified FCTD for MCDPRs.

A video showing the evolution of the feasible polygon as a function of time considering only tension limit constraints and both tension limits and stability constraints can be downloaded at 1 . This video also shows the location the mobile base ZMP as well as some tipping configurations of the mobile cable-driven parallel robot under study.

Conclusion

This paper has introduced a new concept of Mobile Cable-Driven Parallel Robots (MCDPR). The idea is to autonomously navigate and reconfigure the geometric architecture of CDPR without any human interaction. A new real time Tension Distribution algorithm is introduced for MCDPRs that takes into account the stability of the mobile bases during the computation of feasible cable tensions. The proposed algorithm ensures the stability of the mobile bases while guaranteeing a feasible ca-ble tension distribution. Future work will deal with the extension of the algorithm to a 6-DoF MCDPR by taking into account frontal as well as sagittal tipping of the mobile bases and experimental validation thanks to a MCDPR prototype under construction in the framework of the European ECHORD++ "FASTKIT" project.

Fig. 1 :

 1 Fig. 1: Concept idea for Mobile Cable-Driven Parallel Robot (MCDPR) with eight cables (m = 8), a six degree-of-freedom moving-platform (n = 6) and four mobile bases (p = 4)

2 PFig. 2 :

 22 Fig. 2: Point mass Mobile Cable-Driven Parallel Robot with p = 2, n = 2 and m = 4

  11 -a 11 ) T E T [t p11 +n 11 λ λ λ ]u 11 +(c 11 -a 21 ) T E T [t p21 +n 21 λ λ λ ]u 21 (20) M C11 ≤ (c 11 -a 11 ) T E T [n 11 λ λ λ ]u 11 + (c 11 -a 21 ) T E T [n 21 λ λ λ ]u 21 (21)

Fig. 3 :

 3 Fig. 3: Feasible Polygon considering only tension limit constraints

Fig. 5 :

 5 Fig. 5: (a) Evolution of ZMP for mobile base p 1 (b) Cable tension profile

https://www.youtube.com/watch?v=XmwCoH6eejw

Acknowledgements This research work is part of the European Project ECHORD++ "FASTKIT" dealing with the development of collaborative and mobile cable-driven parallel robots for logistics.


Gabrielle Bouleau

Christine Argillier

Yves Souchon

Carole Barthelemy

Marc Babut

Carole Barthélémy

How ecological indicators construction reveals social changes -the case of lakes and rivers in France

come

How ecological indicators construction reveals social changes -the case of lakes and rivers in France

Introduction

For forty years, ecological concerns have spread beyond scientific spheres. Public interest and investment for ecosystems have been growing, notably for wetlands, rivers, and lakes. As a consequence, many agencies are developing so-called 'ecological restoration' projects the efficacy of which is not always obvious [START_REF] Kondolf | Two Decades of River Restoration in California: What Can We Learn?[END_REF]. Ecological indicators (EI) are meant to support decisions in order to set restoration priorities and/or to assess whether the proposed management will improve ecological conditions or not, or to appraise completed projects. Despite increasing developments in the field of EIs during the last thirty years [START_REF] Barbour | Measuring the attainment of biological integrity in the USA: A critical element of ecological integrity[END_REF][START_REF] Kallis | Evolution of EU water policy: A critical assessment and a hopeful perspective[END_REF][START_REF] Moog | Assessing the ecological integrity of rivers: walking the line among ecological, political and administrative interests[END_REF][START_REF] Wasson | Typologie des eaux courantes pour la directive cadre européenne sur l' eau : l' approche par hydro-écorégion. Mise en place de systèmes d' information à références spatiales[END_REF], many scientists complain that such indicators are hardly used to support decisions, management plans, and programs evaluations ( [START_REF] Dale | Challenges in the development and use of ecological indicators[END_REF][START_REF] Lenz | From data to decisions: Steps to an application-oriented landscape research[END_REF]. They usually attribute the gap separating the creation and the use of EI to social factors [START_REF] Turnhout | Ecological indicators: Between the two fires of science and policy[END_REF]. Therefore, research addressing the social perspective of EI has been developed recently. Yet such work focuses on short-term analyses. It has mainly addressed what social drivers are responsible for using or not using EI, once EI have been designed by scientists.

The recent literature on this topic falls in two categories: the market or the political arena as driving forces. In the first category, a market is presumed to exist for EI in which environmental scientists, the providers, must fulfil the expectations of decision makers, the buyers. In this perspective, authors recommend that EI be reliable, cheap, easy to use [START_REF] Cairns | A history of biological monitoring using benthic macroinvertebrates[END_REF][START_REF] Lenat | Using Benthic Macroinvertebrates Community Structure for Rapid, Cost-effective, Water Quality Monitoring: Rapid Bioassessment[END_REF]. They should provide users with the information they need in a form they can understand [START_REF] Shields | The role of values and objectives in communicating indicators of sustainability[END_REF][START_REF] Mcnie | Reconciling the supply of scientific information with user demands: an analysis of the problem and review of the literature[END_REF] according to their tasks, responsibilities, and values. Different problems in which scientists, politicians, and experts have different roles, may therefore require different indicators [START_REF] Turnhout | Ecological indicators: Between the two fires of science and policy[END_REF]. Even so, [START_REF] Ribaudo | Environmental indices and the politics of the Conservation Reserve Program[END_REF] noticed there is a need for generalised indicators which meet multiple objectives. In this market-like perspective, social demands challenge EI that were designed under purely scientific considerations. These authors suggest that defining indicators with stakeholders' participation improves their chance of success. In the second category, EI are not presumed to compete in a market but rather in a political arena. Each indicator is believed to promote a particular political point of view. Indeed, the ecological status of an ecosystem depends on the boundaries chosen for the system and on the characteristics chosen to be restored. Scholars adopting this perspective insist on the plurality of ecological objectives [START_REF] Higgs | Expanding the scope of restoration ecology[END_REF][START_REF] Jackson | Ecological restoration: a definition and comments[END_REF] and consider that defining restoration goals and objectives is a value-based activity [START_REF] Lackey | Values, policy, and ecosystem health[END_REF]. In this second perspective, economical and technical characteristics of indicators are hardly addressed. Emphasis is put on the expression of choices made within any indicator. Both categories may overlap since the frontier between science, market and policy is a fuzzy area [START_REF] Davis | The Science and Values of Restoration Ecology[END_REF][START_REF] Hobbs | Restoration ecology: the challenge of social values and expectations[END_REF][START_REF] Turnhout | Ecological indicators: Between the two fires of science and policy[END_REF]. In both perspectives, authors have addressed a small feedback loop between science and policy in which social factors act as selection forces for useful or compelling EI, which in turn are meant to supply data for adaptive management and policy changes if required (Fig. i).

To date, very little attention has been paid to the manner values or expectations of stakeholders influence the development of EI prior to selection. Determining the role the social context plays in EI development requires a historical perspective to study a larger feedback loop (Fig. ii). This paper develops a method to study long-term interactions between EI development and social factors. Social factors are understood here to designate all dynamics within human society including structural constraints and human agency. We elaborate an analytical framework to account for these interactions. Then we apply this approach to five case studies in France on lakes and rivers (saprobic index, test on minnows, biotic index, fish index and rapid diagnosis). Last we conclude on the interest of this new approach which provides new elements for explaining the gap between production and use of EI.

An analytical framework to study the long-term co-evolution of EI and society

Managing the environment and the human population is a recent concern of states. Foucault argues that western governments and scientists became interested in indicators only in the last 150 years, as governmental legitimacy shifted from expansionism to optimizing the well-being of domestic population. As political definitions of well-being evolved, they reshaped the scientific agenda [START_REF] Foucault | Naissance de la biopolitique[END_REF]. Scientific facts are not simply given, but selected by actors to make sense at one point [START_REF] Fleck | Genesis and Development of a Scientific Fact[END_REF][START_REF] Latour | Science in action[END_REF]). Access to nature influences data collecting [START_REF] Forsyth | Critical Political Ecology: The Politics of Environmental Science[END_REF]. Power structures influence the way experts define crises [START_REF] Trottier | Water crises: political construction or physical reality[END_REF]. In turn, experts' discourses spur new social demand. This longterm dynamic changes relationships between technology and society [START_REF] Callon | Some elements of a sociology of translation: domestication of the scallops and the fishermen of Saint Brieuc Bay. In Power, Action and Belief: A New Sociology of Knowledge? Sociological Review Monograph[END_REF][START_REF] Latour | Science in action[END_REF][START_REF] Star | Institutional Ecology, ' translations' and Boundary Objects: Amateurs and Professionals in Berkeley' s Museum of Vertebrate Zoology, 1907-39[END_REF]Sabatier and Jenkins-Smith 1993). This applies for EI as well. Exploring their history sheds light on the manner EI evolve, are selected and kept and how in turn, they influence knowledge and social representations (Fig. ii). This research identifies such long-term influences. For this purpose, we propose an analytical framework to sketch a new approach for social studies of EI, which is (1) interdisciplinary, (2) inductive, and (3) historical.

(1) A social study of EI requires a multidisciplinary dialogue between social and natural sciences. Social representations of nature are simultaneously influenced by the scientific state of the art, with which biologists are more familiar, and by the cultural and political context of the period, which social scientists know better. Both competencies are needed. They should not be separate; they should provide different assumptions for the same research questions. Once this is achieved this multidisciplinary approach has yielded an interdisciplinary framework [START_REF] Trottier | Managing Water Resources Past and Present[END_REF].

(2) An inductive and qualitative approach allows emerging concepts to be incorporated during the research [START_REF] Bryman | Qualitative Research Methodology -A review[END_REF][START_REF] Altheide | Ethnographic Content Analysis[END_REF]). In doing so, we identify causal relationships that may be significant even though not being always necessary nor sufficient. We advocate focusing on a small set of largely used indicators rather than studying many different ones. We look for factors that were significant at least in one case study.

(3) Without questioning the validity of scientific methods used, we explain how social factors historically influenced the choice of species and the finality of research. For this purpose, we define constitutive elements of the historical trajectory of an EI: the historical background of places where it was developed, the way data were collected and treated, what ecological knowledge was available, what belief concerning nature and society was spread, and what were the ecological management goals. We elaborate on the variation-selection-retention model of socio-environmental co-evolution developed by [START_REF] Kallis | Socio-environmental co-evolution: some ideas for an analytical approach[END_REF]: why different social groups promoted different representations of nature (variation), how a mainstream representation emerged (selection) and how it was maintained despite possible criticisms (retention). The variation step received very little attention in previous academic work. A second layer of evolution was also missing in previous short-term analyses. Scholars often considered society as a homogeneous entity one could represent in one box (Fig. This analytic framework enables to address cumulative effects, historical opportunities, technological gridlocks and path-dependence of EI that are otherwise ignored.

Case study background and method

The recent evolution of freshwater quality and its relationship with EI development are well documented. Experts generally agree that in western countries, during the 1960s, increasingly visible environmental catastrophes spurred public support for environmental legislation. This resulted in pieces of law as the Clean Water Act in the USA [START_REF] Barbour | Measuring the attainment of biological integrity in the USA: A critical element of ecological integrity[END_REF] and the first Council Directives in the European Community addressing water quality [START_REF] Aubin | European Water Policy. A path towards an integrated resource management[END_REF]. The adopted legislation has proven to be generally efficient in addressing point-source chemical pollution but is too narrowly focused to deal with habitat degradation that became more obvious once industrial and urban discharges into streams were reduced [START_REF] Karr | Defining and assessing ecological integrity beyond water quality[END_REF]. EI development appeared as a secondary social response to this initial improvement in chemical quality of freshwater. Yet the knowledge and the data used to design EI were developed much sooner in the last century. Therefore a study stretching back to the late nineteenth century was necessary to understand what data was available and what social representations framed (or were framed by) the emergence of EI in France.

Identifying consistent periods during which water management was uniform is difficult. Transitions emerge slowly from several, sometimes conflicting, influences. Yet, several-decade phases offer convenient time intervals to identify social changes affecting water management. Three main periods are presented within the scope of our study. In the first phase , the quality of surface water became a national issue for fishermen and hygienists as they developed specific sectors for drinking water and fishing. In the second phase economists and urban planners began to worry about the availability of water resources and they developed a sector to treat pollution. In the third phase (since 1980) a social demand for ecosystems management grew at the European level and promoted water body restoration.

Our research project was undertaken by two sociologists and three freshwater biologists. Sociologists undertook in depth interviews with actors having played different roles in different places (managers, scientists, activists, policy makers …). They conducted thirty-two 2-3 hour semi-structured interviews with people involved in the design or the use of five indicators in France -i.e. saprobic index [START_REF] Kolkwitz | Okologie des pflanzlichen saprobien[END_REF], test on minnows (Phoxinus phoxinus, L.1758), biotic index [START_REF] Woodiwiss | The biological system of stream classification used by the Trent River Board[END_REF][START_REF] Tufféry | Méthode de détermination de la qualité biologique des eaux courantes -exploitation codifiée des inventaires de la faune de fond[END_REF], fish index [START_REF] Oberdorff | Development and validation of a fishbased index for the assessment of 'river health' in France[END_REF][START_REF] Pont | Assessing the biotic integrity of rivers at the continental scale: a European approach[END_REF], and rapid diagnosis for lake [START_REF] Barbe | Diagnose rapide des plans d' eau[END_REF]). These indicators are the most currently used by water agencies and wildlife services. Sociologists attend some field work to get familiar with methods. Freshwater biologists compiled bibliographic and ethnographic work on published and non-published ecological research. Together we analysed the evolution of available knowledge and social justifications in each document. The snowballing method was used to identify actors involved in this process. Being affiliated to Cemagref and/or graduated from the National School of ingénieurs du GREF sometimes helped contacting officials and state engineers involved in the story. Most interviewees were contacted several times. We asked them to review their interviews and to send related personal archives. Historical studies were used for most ancient periods. We asked the actors to tell how they have been involved in the process and how they might explain choices done before they had been involved. We triangulated the results from the interviews using scientific, legal, and technical materials published at each stage to support or confront testimonies. We completed this material with bibliographical review of social works dealing with how water-related social and scientific practices and ideas have evolved during time. We compared different trajectories in order to assess whether common characteristics could be identified.

Results: the development of EI in France for lakes and rivers

In this section, we focus on three historical phases, 1900-1959, 1960-1980, and since 1980 to study five EI, namely the saprobic index, the test on minnows, the biotic index, the fish index and the rapid diagnosis for lake. All EI we studied were first set up by outsiders challenging mainstream water management. Following this phase of construction where adaptation prevailed, legal changes in law induced a shift by resting the burden of proof on other stakeholders and leading to a phase of accumulation.

1900-1959: Focus on fish and germs

At the end of the 19 th Century, two social groups alerted the public to the deterioration of surface water quality in France. The first group stemmed from a handful of physicians who experienced difficulty demonstrating that water could be pathogenic. Fishing clubs made up the second group as they complained about fish depletion. Unlike epidemiologists, they did not join the hygienist movement. They advocated restocking, reducing poaching, and punishing industrial discharges. These approaches led to two different tools: the saprobic index (developed in Germany at the beginning of the 20 th century) and a poorly reliable ecotoxicological test on minnows, perfected by French legal experts at the same period. While the hygiene act of 1902 compelled to test water quality systems systematically and secured funds for nationwide surveys, French law did not acknowledge accidental pollution as a misdemeanour before 1959 and fishermen kept struggling to obtain the recognition of its impact on fish population. This difference had a major effect on the fate of the variable promoted by each group.

Sanitary actors experienced hard time in the beginning. Freshwaters were long considered very healthy in France. The miasma theory predominated, which held that odours -rather than water -were malignant. Moreover, until the mid-nineteenth century, decayed matter was considered as beneficial by most people, including a large part of physicians. Despite the initial reluctance from the medical world, some outsiders started listing organisms living in contaminated waters. They succeeded in getting other stakeholders (urban planners, engineers, politicians) interested in their hygienist ideas. These sanitary reformers made rivers suspicious as they established more relations between contaminated places and water [START_REF] Goubert | La conquête de l' eau. L' avènement de la santé à l' âge industriel[END_REF]. It took time before hygienism spread out as a political doctrine with the scientific support of German physician Robert Koch (1843Koch ( -1910) ) and French chemist Louis Pasteur (1822-1895) pointing out the existence of germs. The French hygiene act of 1902 resulted from this evolution and paved the way for systematic chemical and biological assessments of diversions for water supply. The first EI got stabilized in this context. German naturalists [START_REF] Kolkwitz | Okologie des pflanzlichen saprobien[END_REF] developed the "saprobic" index. It consisted of an array of organisms ranked according to their response to organic matter. Presence and abundance of saprobic organisms, those that resisted best to faecal contamination, meant risks and corresponding waters were not to be used for drinking purposes. The saprobic index became a common "passage point" [START_REF] Star | Institutional Ecology, ' translations' and Boundary Objects: Amateurs and Professionals in Berkeley' s Museum of Vertebrate Zoology, 1907-39[END_REF] for many technologies addressing human well-being in cities. Sanitary authorities published in 1903 the first nationwide inventory of water addressing quality and updated periodically [START_REF] Goubert | La conquête de l' eau. L' avènement de la santé à l' âge industriel[END_REF]). Ironically, the sanitary perspective did not lead to any improvement of surface water quality. Since technical solutions to convey spring water or groundwater such as long aqueducts existed, urban rivers became stale. Large sewers were developed, more and more effluents were discharged in rivers [START_REF] Barles | The Nitrogen Question: Urbanization, Industrialization, and River Quality in Paris, 1830-1939[END_REF]. The development of the saprobic index is a success story. Outside the mainstream medical world, a few physicians lobbied to change waste water management and ended with standardization of the EI once the law institutionalized hygienist ideas.

Fishermen were not as successful. Fishing clubs had long complained about industrial impacts. But fish depletion in freshwaters was always attributed to poaching [START_REF] Barthélémy | Des rapports sociaux à la frontière des savoirs. Les pratiques populaires de pêche amateur au défi de la gestion environnementale du Rhône[END_REF] and French regulation targeted fishermen first. If ever acknowledged, pollution was considered as reversible and likely to be compensated by restocking. In case of fish mortality, claimants remained isolated outsiders. They had to provide evidence to pursue effluents dischargers. Fishing authorities would sample suspicious waters and perform tests on minnows to assess presence of toxic substances. Frequently samples were done after the toxic discharge stopped. Judges constantly considered that such cases were peculiar cases of poaching [START_REF] Corbin | L' avénement des loisirs (1850-1960[END_REF]. In the late 1880s, the state promoted fishing as an accessible supply of proteins to the poor. To prevent poaching and to maintain fish populations, regardless of water quality, fishing clubs and governmental authorities agreed to raise a tax on fishermen to increase control and develop hatcheries. Since the beginning of the twentieth century, fishermen helplessly asked for a specific legislation against pollution. The ecotoxicological test on minnows failed. Its story started with isolated actors claiming for a different water management. But fishermen hardly convinced larger coalitions of actors that pollution was a problem and the law did not change. No public fund was ever secured to apply the test at large scale. Since the burden of proving pollution rested on fishing clubs, data they accumulated remained scattered, and depending on their own activism. The legal acknowledgement of any ecological dysfunction is a crucial step for EI development. It changes the need for its monitoring.

This is what happened in 1959, when pollution became punishable in France.

1960-1980: resource management

Discontent fishermen suddenly found an opportunity to be listened to in 1959. As the colonial empire challenged French political influence overseas, President de Gaulle was trying to get legitimacy by paying attention to current domestic claims during emergency powers. By the edict of 1959, fishermen obtained that accidental pollution in rivers should be punished. Before the edict of 1959, only fishermen were trying to find evidence of pollution. After this edict, polluters too suddenly became concerned. This piece of law was the impetus of a major change in the French water management, although it was not discussed by parliament and adopted during a troublesome period [START_REF] Bouleau | La gestion française des rivières et ses indicateurs à l' épreuve de la directive cadre[END_REF]. In response, industrials and municipal councils, threatened by the new edict, asked for public funds to support treatment plants. The French post-war planned administration relied on the Commissariat Général au Plan for making prospects and public policies [START_REF] Colson | Revisiting a futures studies project--' Reflections on 1985[END_REF]. In this arena, many engineers were in favour of an economic and spatial approach of natural resources planning [START_REF] Rabinow | French Modern: Norms and Forms of the Social Environment[END_REF]. Influenced by [START_REF] Pigou | The Economics of Welfare[END_REF] and Kneese's ideas (1962), they promoted a "polluter-pays principle" at the scale of the watershed. It was enacted by the 1964 water law which established basin agencies and basin committee [START_REF] Nicolazo | Les agences de l' eau[END_REF]. To apply this principle, basin committees agreed on several conventional chemical parameters in order to make pollution commensurable from upstream to downstream, in rivers and lakes: suspended matter, dissolve salts, and a combined parameter including biological and chemical oxygen demand. Basin Agencies collected levies on water users and delivered grants to fund water supply and waste water treatment facilities. Needs were huge given the post-war demographic expansion, the repatriation after the end of colonisation, and industrial development fuelled by the reconstruction economy. The focus on chemistry helped build consensus on investment priorities, but it left habitat degradation aside. The biotic index was an attempt to fill the gap on rivers. Because this story is quite recent, we were able to interview actors who promoted this new EI [START_REF] Bouleau | La gestion française des rivières et ses indicateurs à l' épreuve de la directive cadre[END_REF].

The origin of the French biotic index dated back to the creation of a single state corps of engineers in 1965, ingénieurs du génie rural, des eaux et des forêts (IGREF), gathering "developers" of the corps of rural planners (ingénieurs du génie rural) and more "conservationist" engineers of the corps for water and forestry (Ingénieurs des eaux et des forêts). Developers promoted agricultural modernization. They allocated state grants to farmers and rural districts in order to drain wetlands, canalize streams, raise levees against frequent floods and build dams storing winter flows for irrigation. IGREF conservationists were critical of such projects they called "anti-tank ditches", and they felt marginalized. Basin agencies did not pay attention to upstream developments, focusing on downstream urbanized and industrialized rivers, more affected by pollution. As they shared the same concern for upstream rivers, conservationists built alliances with fishing clubs, getting them organized at the State level, supporting their claims against polluting effluents and sharing material and staff. They searched an EI that would perform better than the test on minnows. As sampling was often performed after the toxic discharge had stopped, they were looking for biological sentinels proving that pollution occurred. They hired freshwater biologists "who were anglers, interested in this stuff, because this was the issue". One IGREF said to a candidate: "your one and only job is to find some stuff we could present and standardize". Tuffery and Verneaux (1967) adapted the biotic index developed by the Trent River Authority [START_REF] Woodiwiss | The biological system of stream classification used by the Trent River Board[END_REF] to French rivers. [START_REF] Verneaux | Cours d' eau de Franche-Comté (Massif du Jura). Essai de biotypologie[END_REF] identified 10 river types on the Doubs basin based on a factorial correspondence analysis of the presence or abundance of Macrobenthic Invertebrates. In the foreword of his PhD dissertation Verneaux recounted: "This deals with a long story which started with my father twenty-five years ago on the banks of the Doubs River and is probably going to stop in few years as the river dies". He established reference conditions correlated to a combination of abiotic variables (slope, distance to source, width, and summer temperature). Then he extended his survey and adapted the biotic index to all French rivers and got it standardized (Indice Biologique Global Normalisé, IBGN). Conservationists enlisted the biotic index in the state regular monitoring (arrêté interministériel of September 2, 1969). Following the 1976 act on nature protection, IBGN became a common tool for environmental impact assessment. Because of its initial target on accidental pollution, IBGN remained poorly relevant to point out potential disturbance of trout habitat (channel simplication, increasing temperature) by hydraulic works. IBGN had little consideration to environmental variability, which would have required much more data collecting and processing. Albeit imperfect, such an option enabled rapid development of professional training for ecological monitoring. It secured funds for monitoring activities on a regular basis, and resulted in a today-available thirty-year database.

Analysing documents concerning the biotic index, we found the same pattern of EI development as the one induced from the saprobic index case. Outsiders promoted another focus on river management, made up a new EI, enrolled allies, and obtained a legal obligation of monitoring. Our interviews revealed other aspects of the social trajectory of the biotic index. Experts focused on what they valued in personal ties and collective strategies or practices. In the case of the biotic index their preference for upstream reaches and their targeting accidental pollution were embedded in the EI they developed.

to now: ecosystem management

Whereas post-war industrial fast development notably improved French workers conditions, it did not thoroughly fulfil middle class expectations. Graduate employees from the baby-boom generation did not adhere completely to the stalwart praise for industrial achievements. They complained of environmental degradation. Better equipped with computers to quantitatively address ecosystem relationships, ecological scientists supported this claim [START_REF] Fabiani | Sciences des écosystèmes et protection de la nature. In Protection de la nature[END_REF][START_REF] Aspe | Construction sociale des normes et mode de pensée environnemental[END_REF]. Data collection remained the weak spot. Developers had much more data to promote their projects than ecologists had to sustain environmental preservation. The creation of the Ministry of Environment (1971) and the 1976 act requiring environmental impact assessments, were critical steps securing funding for research. But the French state would not have extended its regular monitoring without European binding legislation. It spurred the development of two other EI, the rapid diagnosis for lakes and the fish index.

Although, lake dystrophy had long been ignored by the French state, freshwater biologists of the state field station in Thonon-les-Bains initiated a local eutrophication monitoring in the sixties, after algae proliferation was observed in fishermen's gillnets in nearby Lake Geneva, the largest natural lake in France and Switzerland. They chose chlorophyll a, a chemical surrogate of the abundance of the phytoplankton to demonstrate a causal relationship between nutrients and dystrophy. Similarly Swiss researchers started worm communities monitoring in the last 1970s [START_REF] Lang | Eutrophication of Lake Geneva indicated by the oligochaete communities of the profundal[END_REF]. This was the time when Northern American researchers supported by the "Soap and Detergent Association" published the article "We hung phosphates without a fair trial" (Legge and Dingeldein 1970 cited by [START_REF] Barroin | Phosphore, azote, carbone... du facteur limitant au facteur de maîtrise[END_REF]. Transatlantic controversies and French chemical lobbies hushed-up eutrophication alerts in France. Experts remained isolated. Managers of the Rhône Basin Agency in charge of financing restoration plans for alpine lakes finally agreed on supporting the development of a "rapid diagnosis" for lakes. Mouchel proposed to build on the trophic index developed in the United-States and Canada [START_REF] Dillon | The phosphorus-chlorophyll relathionship in lakes[END_REF][START_REF] Carlson | A trophic state index for lakes[END_REF][START_REF] Vollenweider | Synthesis report[END_REF][START_REF] Canfield | Prediction of chlorophyll a concentrations in Florida lakes: the importance of phosphorus and nitrogen[END_REF]. He correlated nutrient concentrations (total phosphorus, orthophosphates, total nitrogen, mineral nitrogen), transparency (Secchi depth) and chlorophyll a, and expanded results from the Alps to set up a national typology. Including then Oligochetae describing mineralization and assimilation of nutrients [START_REF] Lafont | Contribution à la gestion des eaux continentales : utilisation des oligochètes comme descripteurs de l' état biologique et du degré de pollution des eaux et des sédiments[END_REF][START_REF] Lafont | Un indice biologique lacustre basé sur l' examen des oligochètes[END_REF], and a Mollusc index responding to dissolved oxygen and organic matter content within sediment [START_REF] Mouthon | Un indice biologique lacustre basé sur l' examen des peuplements de mollusques[END_REF] the 'rapid diagnosis' was tested on about thirty lakes spread across different regions [START_REF] Mouchel | Essai de définition d' un protocole de diagnose rapide de la qualité des eaux des lacs. Utilisation d' indices trophiques[END_REF]1987;[START_REF] Barbe | Diagnose rapide des plans d' eau[END_REF]). Nevertheless this work did not lead to systematic lake monitoring in France. Resistance from lobbies defending chemical industries was strong [START_REF] Barroin | Phosphore, azote, carbone... du facteur limitant au facteur de maîtrise[END_REF]. Political engagement for ecological preservation of lakes was low. Such EI in the making were mainly used in the Alps.

In rivers, the 1976 act on nature protection encouraged further research on impacts beyond IBGN to address morphological changes on streams. The Research Centre for Agriculture, Forestry and Rural Environment (CERAFER, which later became Cemagref) devised new methods to determine minimum instream flows downstream hydroelectric impoundments [START_REF] Souchon | Peut-on rendre plus objective la détermination des débits réservés par une approche scientifique ?[END_REF] based on American existing works [START_REF] Tennant | Instream flow regimes for fish, wildlife, recreation and related environmental resources[END_REF][START_REF] Milhous | The PHABSIM system for instream flow studies[END_REF][START_REF] Bovee | A Guide to Stream Habitat Analysis Using Instream Flow Incremental Methodology[END_REF]. It resulted in the adoption of the 1984 act on fishing which required minimum instream flows. Anticipating the revival of hydropower development after the oil crises, the Ministry of Environment and the National Center for Scientific Research (CNRS) decided to support environmental research on the Rhône in 1978 (PIREN Rhône). Between 1979 and 1985, scientists of the PIREN assessed the environmental impact of hydropower projects in the upper-Rhône valley. They were politically supported by ecologist activists and urban recreational users of the valley who were calling into question the power facilities. Sampling floodplain fauna and flora, they related such inventories to physical and chemical analyses. They showed evidence of seasonal longitudinal, lateral and vertical transfers between the river, the floodplain and the groundwater [START_REF] Roux | Cartographie polythématique appliquée à la gestion écologique des eaux. Etude d' un hydrosytème fluvial : le Haut-Rhône français[END_REF][START_REF] Castella | Macroinvertebrates as 'describers' of morphological and hydrological types of aquatic ecosystems abandoned by the Rhône River[END_REF][START_REF] Amoros | A method for applied ecological studies of fluvial hydrosystems[END_REF][START_REF] Richardot-Coulet | Classification and succession of former channels of the French upper rhone alluvial plain using mollusca[END_REF]. They elaborated a general deterministic understanding of the "hydrosystem" which advocated for the preservation of natural hydrological patterns in rivers. Scientific evidence and political support led to the 1992 water act. It required permits and public hearings for all developments significantly affecting water. Yet, it did not result in a national change in monitoring activities. State authorities had already much invested in collecting biochemical, IBGN and fish population data. If ecological concerns were high in the Rhône valley, they were not priorities in the rest of the country. The list of currently monitored indicators appeared to be locked-in at that level. This reduced the scope of possible data accumulation. Instead, the adaptation phase extended.

Challenged by the European on-going research and inspired by Karr's work on biological integrity (1981;[START_REF] Karr | Biological Integrity: A Long-Neglected Aspect of Water Resource Management[END_REF], French freshwater biologists developed more theoretical work. They recomputed accumulated ecological data of the Rhône to test theories of functional ecology (Statzner, Resh et al. 1994;Statzner, Resh et al. 1994;[START_REF] Beisel | Stream community structure in relation to spatial variation: the influence of mesohabitat characteristics[END_REF][START_REF] Usseglio Polatera | Biological and ecological traits of benthic freshwater macroinvertebrates: relationships and definition of groups with similar traits[END_REF]. The functional approach spread out, notably at European level where it makes very different ecosystems comparable. The European political arena happens to be much more sensitive to ecologist concerns than the French one. In 2000, European Parliament and council enacted the Water Framework Directive (WFD) which has set an ecosystem-based regulation. It holds undisturbed conditions of freshwaters as references defined in different hydro-ecoregions and requires economic justification for any development resulting in a significant gap between the current status of a water body and its reference. Designed as a masterpiece of binding legislation for European waters, the WFD created favourable conditions to break down the previous locked-in fits of the French freshwater monitoring.

Freshwater biologists re-processed the thirty-year database of faunistic inventories used for IBGN to identify eco-regional references for rivers [START_REF] Khalanski | Quelles variables biologiques pour quels objectifs de gestion ? In Etat de santé des écosystèmes aquatiques -les variables biologiques comme indicateurs[END_REF][START_REF] Wasson | Typologie des eaux courantes pour la directive cadre européenne sur l' eau : l' approche par hydro-écorégion. Mise en place de systèmes d' information à références spatiales[END_REF]. Others compiled records on fish population to calibrate a "fish index" [START_REF] Oberdorff | Development and validation of a fishbased index for the assessment of 'river health' in France[END_REF]. Further research is currently underway to better understand how already monitored biological assemblages respond to contamination [START_REF] Archaimbault | Biological And Ecological Responses Of Macrobenthic Assemblages To Sediment Contamination In French Streams[END_REF]). More research is required to calibrate indicators for compartments that were previously little monitored such as diatoms and macrophytes. French lake specialists were spurred to update the 'rapid diagnosis' method according to WFD standards, i.e. better considering environmental and morphological heterogeneity, including missing biological information (abundance of taxa for example). Previous ecological research on Alpine lakes was reprocessed accordingly to propose new index WFD compliant (phytoplankton and macrobenthos index) Nevertheless, this integrative approach was mainly calibrated for the quality assessment of natural lakes of the Alpine area and remained less reliable on other lakes. Moreover, because fish experts and botanists hardly explored lakes in France before the WFD was enacted, fish and macrophytes are still to be studied in a bioindication perspective. Relationships between fish assemblages and environmental factors have been studied (Argillier, Pronier et al. 2002;Argillier, Pronier et al. 2002;[START_REF] Irz | Comparison between the fish communities of lakes, reservoirs and rivers: can natural systems help define the ecological potential of reservoirs?[END_REF], but EI development are still under development at the national level.

These two cases ('rapid diagnosis' and fish index) show how EI initially designed for a specific political arena can be reshaped to fit another one. Regional EI promoters may get institutional support above state level (European Community) to impose the legal framework favourable to the implementation of EI at lower level. This requires that EI promoters, who initially worked at regional scale, shall reprocess their indicators to meet international specifications. When no previous research was conducted at national level, data must be collected in the first place. To reduce this endeavour, experts promote their methods at European level. Yet few adaptations to regional features remain enshrined in EI and impair their universality.

Conclusion: a cumulative and adaptive model of social trajectories of EI

This study is meant to be inductive and allows us to derive a theoretical framework from observations. Bioindication tools we studied were never optimised for the present market or political arena but rather recycled. Their development was (a) adaptive, i.e. innovative to respond to new questions; (b) constrained by law; and (c) cumulative, i.e. built on what already existed.

(a) We observed innovative ecological monitoring stemming from regional outsiders who wanted to address a new water quality problem raised by a social minority. Interaction between regional scientists and the social minority framed the issue (double arrow in Fig. iii). Actors reluctant to admit the problem asked for evidence (Fig. iii shows no direct influence from innovation to implementation). The burden of the proof rested on EI developers, who adapted previously existing data and knowledge to their specific concern. It consisted of changing the format of data and including new information, very often without immediate recognition. From adapted and new gathered data, they induced ecological causal relationships. They refined protocols in order to address more specifically the problem. At this stage, ecological data were not framed as indicators -i.e. decision support tools -they were only variables. But variables could be mapped in space and monitored in time. This phase is called "adaptation" in Fig. iii. (b) New EI promoters experienced difficulty in convincing existing institutions and did not get much funding for implementing their monitoring. Some failed to convince other actors and went on adapting their tools at regional level. Others were able to raise larger public attention or to meet other stakeholders' concerns who joined in "advocacy coalitions" [START_REF] Sabatier | The Advocacy Coalition Framework. An Assessment[END_REF]. Negotiations required quantification and standardization. Variables that were previously only used by activists became "passage points" [START_REF] Star | Institutional Ecology, ' translations' and Boundary Objects: Amateurs and Professionals in Berkeley' s Museum of Vertebrate Zoology, 1907-39[END_REF] for many different actors and got standardized. People translated their interests using such variables, and spread their use. Successful EI promoters found a suitable political arena for changing law despite institutional resistance to change and got their EI institutionalized. We illustrate this phase with the box "institutions" in Fig. iii. (c) Evolutions we studied were also cumulative processes. Once enacted, a new law imposed that a successful EI had to be regularly monitored. It transferred the burden of the proof onto projects initiators whose plans might harm freshwaters or made the state responsible for collecting data. It spread the use of new EI outside the region where it was developed. Given the cost of developing an EI, recycling prevailed. Systematic monitoring got established and data became available for further research. We call this phase "accumulation" in Fig . iii. Yet regional fits of the EI were often kept in this process. Moreover the inertia of the routine worked as a locked-in fit preventing other EI to be adopted. Adaptive management is limited (dotted line in Fig. iii). Therefore new environmental problems could hardly be seen through mainstream monitoring and required another cycle of adaptation (a). This analytical model applies in the development of all the five EI we studied, (i.e. saprobic index, test on minnows, biotic index, 'rapid diagnosis', and fish index). It reveals the evolution of French water management in two ways. The emergence of an EI corresponds to the emergence of an ecological problem mainly ignored by authorities. Secondly groups of biota used for the construction of an EI reveal a converging interest of different stakeholders for the same type of information at one historical moment. We inherit from EI that fit the social context of their emergence. New phases of adaptation may be required to adjust them to our current concerns. It enlightens weaknesses of the models illustrated in Fig. This analytical framework helps us addressing the initial question: "why EI seem to be hardly used to support decisions, management plans, and programs evaluations?" In the adaptation phase, outsiders raise the alarm without bending the trend of the current management. They collect new data and reprocess available information in order to convince others. This phase can be long and very frustrating. Biologists get little support and often complain about indifferent decision-makers. Institutional recognition on the contrary is a sudden shift which challenges the existing water management. Authorities seek new solutions. EI are used to assess pilot restoration techniques. Soon best practices emerge and got standardized. Managers make plans and programs based on such standards. They used EI to define targets but EI do not influence everyday decisions. Bad records may raise the attention of the public and the decision makers. But if the problem is settled enough, social mobilization declines. In such cases EI are monitored by precaution rather than decision support. Hence two kinds of EI exist, the ones addressing an issue that is not yet acknowledged and the ones monitoring a problem already tackled. The period inbetween is too short to get noticed. One should not base the righteousness of EI according to their influence on everyday decisions. On the contrary, the virtue of environmental indicators is to put into question paradigms of management. And this may not happen everyday. On a regular basis EI are collected for the sake of accumulation of data and knowledge. They will constitute a valuable resource for the next outsiders tracking regional biases of existing EI and revealing unnoticed environmental problems.

Figures

  i and ii). To account for the social evolution, we propose to split the social component into two elements (Fig. iii), one representing what is established (institutions, law) and one representing more evolving social features (coalitions, representations).

  i and Fig. ii and supports the theoretical framework presented in Fig. iii. The adaptation phase accounts for innovations mainly due to outsiders who seek opportunities for institutionalization. Implementation is more an accumulation phase secured by regular funding than one of selection or adaptation.

Fig. 1 .

 1 Fig. 1. To date, authors have studied a small loop of feedback between social and political factors and EI implementation. They have not addressed the social and political influences on the EI development.

Fig. 2 .Fig. 3 .

 23 Fig.2. We propose to address a larger loop of social interactions in which EI development is also included. This approach enables to take into account influences of data availability, changes in social and scientific representations, opportunity and resistance to changes.

Acknowledgements

This work was financially supported by Cemagref call for research "MAITRISES". The authors are grateful to the scientists, managers and stakeholders interviewed for the purpose of this study. The paper benefited from valuable discussions with and wise advice of Tim Duane, Adina Merenlender, Stefania Barca, and Matt Kondolf from University of California at Berkeley. The authors would like to thank Delaine Sampaio, Christelle Gramaglia and Julie Trottier and two anonymous reviewers for their helpful comments and suggestions.