Maud Lemagnen
email: maud.lemagnen@hispano-suiza-sa.com

Fabrice Mathieux
email: fabrice.mathieux@g-scop.inpg.fr

Daniel Brissaud
email: daniel.brissaud@inpg.fr

How managing more efficiently substances in the design process of industrial products? An example from the aeronautics sector

Keywords: chemical substances risk management, design process, aeronautical product, REACh. Lemagnen, M., Mathieux, F., Brissaud

published or not. The documents may come

How managing more efficiently substances in the design process of industrial products? An example from the aeronautics sector.

Lemagnen, M., Mathieux, F., Brissaud, D.,-How managing efficiently substances in the design process of industrial products? An example from the aeronautics sector 2/10

I. INTRODUCTION: HELPING COMPANIES TO BETTER MANAGE SUBSTANCES 1. Context

As damages to the environment, related to plant activities and effluents, are nowadays quite well controlled through a set of regulation texts and clean technologies, those related to products are barely covered. However, the link between products and potential hazard due to their composition has increasingly been noticed for the past decade. At the European level, three past Directives started to focus on particular issues among environmental impacts generated by products. These texts, which are applied to three different families of products (Pack: packaging [START_REF]European Parliament and Council Directive 94/62/EC of 20 December 1994 on packaging and packaging waste[END_REF], RoHS: electronic equipments [START_REF]The European Parliament and the Council, Directive 2002/95/EC of the European Parliament and of the Council of 27[END_REF] and ELV: cars [START_REF]The European Parliament and the Council, Directive 2000/53/EC of the European Parliament and of the Council of 18 September 2000 on end-of life vehicles -Commission Statements[END_REF]), have in common to limit the use of some chemical substances (e.g. Pb, Hg, Cd) into the products. Pack, RoHS and ELV notably aim at reducing the toxicity risk at the end of life of the products.

Solutions had to be found to adapt industrial production activities to these new requirements. Examples of such adaptations can be illustrated by electronic and automotive sectors, through technologies evolution (lead-free brazing processes development) or substances management all along the supply chain (through for example the IMDS system).

REACh [START_REF]European Parliament and of the Council of 18 December 2006 concerning the Registration, Evaluation, Authorisation and Restriction of Chemicals (REACH)[END_REF], a European regulation adopted in 2006, gives new orientations to the European substances-orientated policy. Three aspects of REACh are of key importance:

• There should have complex mechanisms of registration and evaluation of chemical substances put in the market, which tends at last to eliminate the most toxic ones. • The products' chemical composition should be known through masses parameters.

• The potential toxic impact of products should be determined during its life-cycle.

REACh does not only require that industrial companies control their supply chains, but also that they are able to quantify the amount of toxic substance at each step of the product life. Chemical substance toxicity should therefore be taken into account by designers as a new criterion, together with any other environmental impact categories. This is in line with current LCA method development [START_REF] Jolliet | IMPACT 2002+: A new life cycle impact assesment methodology[END_REF].

Objectives

As suggested initially by Safran group [START_REF] Lartiguet | Impact of REACh on material and processes in Safran Group[END_REF], REACh brings a new complexity to industrial activities as companies should comply with its requirements from the early steps of product development. We think at Hispano-Suiza that one of the key answers to REACh compliance may lie in the capacity of carrying out a process of chemical risk assessment for substances, materials and production process at any step of the design. As recommended by [START_REF] Iso | Environmental management -Integrating environmental aspects into product design and development[END_REF], integrating environmental aspects during the design process means that an environmental assessment should be lead as a baseline on a reference product. Environmental impacts can be then quantified and classified into categories, and even prioritized. This approach would help the designers to choose the best solutions with an aim of decreasing environmental burden of the product life-cycle. The results of the assessment should be presented so that designers can operate a choice between, substances, mixtures, materials or production process [START_REF] Ljunberg | Material selection and design for development of sustainable products[END_REF].

This article presents a new method developed for the aeronautics sector that should help design decision making through chemical risk quantification. Section 2 introduces the method itself while Section 3 reports its application to an aeronautical product. Last Sections focus on discussion, conclusions and perspectives.

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II. PRESENTATION OF A NEW METHOD TO BETTER MANAGE SUBSTANCES 1. Industrial context of the method development

The proposed method is developed in an aeronautical company, Hispano-Suiza (Safran group). Hispano-Suiza designs and produces aeronautical engine equipments and engine regulation systems. Probably because noise and gases emissions are considered to be the main environmental impacts generated by engines, aeronautics sector is however still at the early stages of taking a multicriteria approach on environmental product management, especially in comparison with automotive or electronic sectors. Furthermore, until recently, aeronautics has not been much concerned with substances regulation, mainly because of flight safety requirements and of related exemptions. Because of the large scope of new European directives (RoHS, REACh), aeronautical industries begin to be impacted by substances requirements, in general indirectly through suppliers. Direct impacts can however not be ignored in short-term.

One of the first initiatives of chemicals traceability lays in the construction of substances' lists, defined through their CAS number, their risk-phrase and their legal status: "banned" / "exempted for particular use" / "authorized but critical". Several lists have been proposed since 2000 by Safran (ex-Snecma), Airbus and DGA (French General Direction of the Army). Some of these lists have then been retrieved in 2007 by the aeronautical and spatial industries French Federation (GIFAS), in order to propose a common list [START_REF] Lartiguet | Impact of REACh on material and processes in Safran Group[END_REF]. Lists have often been used as an argument toward ISO 14001 environmental auditors to illustrate the product-centered environmental approach (forbidden materials, materials specification from ordering party to their article suppliers). Even European aeronautics sector uses this type of list as one step to take into account REACh regulation requirement. However, it does not seem to entirely comply with the whole regulation requirements.

Specification of the method

As noticed above, the aeronautics sector seems to lack tools to efficiently manage toxicity-related issues. There is indeed a need to develop a method to do so. This method should in particular meet the three following objectives:

• lead to results that are usable by designers to orientate their choices: indeed, expertise on toxicity issues is usually the field of small communities, such as risk assessment public institutions. To be efficient from a design perspective, the method should not require any particular knowledge on toxicity, so that it can be used by designers. Moreover, results should be simple to interpret by toxicology non -experts [START_REF] Janin | Démarche d'éco-conception en entreprise. Un enjeu : construire la cohérence entre outils et processus[END_REF]; they should also properly discriminate several elements, from non hazardous to hazardous. • be implemented with data available in engineering departments [START_REF] Brissaud | Environmental-based negotiation throughout the design proces"s[END_REF]: data collected from any accurate design documentation should be sufficient to be used as entry data : material type, mixture commercial name, production process specification reference. Method should be considered by the designer as a "blackbox" that doesn't necessitate any intermediate calculation to be run; • answer REACh compliance approach defined in Section 1.1: in particular, toxicity level knowledge isn't accurate enough to answer REACh requirements. Substances amount data should also be available, in order to evaluate the product compliance with very hazardous substances thresholds.

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Global structure of the method

As toxicity is considered as an environmental impact, its treatment trough the design process could be made through:

• Hazard assessment of substance, mixtures, materials and production processes, • Evaluation of the amount of toxic substances contained in mixtures, materials and production processes.

In order to comply with these requirements, the proposed method is composed of 4 steps, described below from Step 1 to Step 4 a. Step 1: Technical data extraction step The necessary data to be used in the method concern all the materials, mixtures and production processes which are used to produce the article 1 . The data is usually found in the products' bill of materials or can be extracted from product management information systems. However, not all accurate data can be found, because sometimes an element is chosen for its function, and not for its composition. As the designer possesses the data needed to run the method describing the materials, mixtures and production process that make the product, he proceeds to the step of their labelisation. We suggest using a basis criterion that is the chemical substances list [START_REF] Lartiguet | Impact of REACh on material and processes in Safran Group[END_REF], built according to three levels. Materials, mixtures and production processes are also organized into the same three levels, according to the substances they are composed or realized with. We decided of some rules, which are very simple, to pass from substance to materials, mixture or production processes. The rules, which are not described in this paper, have notably been elaborated using the Directives 1999/45/EEC and 67/548/EEC. Figure 1 illustrates the link between substance and materials or mixtures labelisation. When any substance is under 0,1% w/w threshold, materials and mixtures are considered free labeled.

Database

In order to compile the label results for all materials, mixtures and production process of the product, a database as been built by Hispano-Suiza [START_REF] Lobreau | Intégration de l'environnement en phase de conception / Application aux équipements électriques et électroniques[END_REF].In a primary approach, it compiles all the different materials, mixtures and production process used specified by Lemagnen, M., Mathieux, F., Brissaud, D.,-How managing efficiently substances in the design process of industrial products? An example from the aeronautics sector 5/10 the company. That database is firstly bound for designers and will quicken further requests for new complete assessments. The database displays elements such as substance, materials, mixture or production process label, but also others data which have been chosen because of their relevance for the user (commercial mixture name or a design specification reference). The database presented above provides a function that permits to compile all the materials, mixtures or production processes that are requested for a product development. It is possible to create a figure (indicator) which summarizes the number of each materials, mixture or production process according to its label. However, at this stage of the method application, one can find numerous materials under the same label. In order to discriminate the results presented above, it was decided to use a quantitative chemical risk assessment method for exempted and critical materials, mixtures and production processes. This two steps (qualification + quantification) approach helps to filter the data and concentrate on the most critical identified values, as usually requested by traditional H&SE practitioners (cf. e.g. [START_REF] Balsat | A structured strategy for assessing chemical risks, suitable for small and medium-sized enterprises[END_REF]).

Substance

The method which is employed is inspirited from the French INRS chemical assessment risk methodology presented in [START_REF] Vincent | Evaluation du risque Chimique :'hiérarchisation des risques potentiels[END_REF]. This method has been chosen because of its ergonomics: indeed, only few parameters are needed as input; furthermore, these parameters are intuitive for any people in an industrial environment. This method is originally used in the context of production plant management in Small and Medium size Enterprises (SME), in order to help H&SE engineers to manage the chemical products risks (use, storage). Figure 3 indicates the structure of the method.

Lemagnen, M., Mathieux, F., Brissaud, D.,-How managing efficiently substances in the design process of industrial products? An example from the aeronautics sector

6/10

Hazard Level H Score from 1 to 4

Exposure Level E Score from 1 to 4

Quantity Level

Score from 1 to 5

Frequence Level

Score from 1 to 4

Potential Risk R POT Score : R POT =10 (D) x 3.16 (E-1)

Matrix

Figure 3: INRS chemical risk assessment method structure (adapted from [START_REF] Vincent | Evaluation du risque Chimique :'hiérarchisation des risques potentiels[END_REF]).

We adapted some of the parameters with the data extracted from the product design bill and determinate correspondences between the INRS scores leveling. In order to comply with the calculation system of the initial method, we adapt danger, quantity and exposure parameters to the same 4 or 5 levels scales used. We verified the correct equivalence between INRS method and the scale we chose.

d. Decision making step

The quantification step is crucial to guide the designer's team environmental choice. The INRS method suggests a way to prioritize the potential risk through the setting of thresholds for the scores obtained. According to [START_REF] Vincent | Evaluation du risque Chimique :'hiérarchisation des risques potentiels[END_REF], three levels have been defined (<100; 100 < … < 1000; > 1000), and for each of them, an orientation of action is delivered, as presented in the following figure. This approach was chosen for our method.

Banned label

Exempted critical label

No Label

Figure 4: Decision making guidance according to qualification and quantification assessment results

In order to file the assessment results and to justify the technical choices made during the design process, we completed the approach with a technical document called "compliance matrix". This document is simultaneously filled in as the assessment is conducted. The matrix is today required as any technical document during the design reviews and is examined.

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• electric/ electronic parts: 4 electronic printed wired boards, internal and external connectors (around 55% of the ECU mass). As numerous aeronautical equipments, which life lasts around twenty five years and due to the technical evolutions or customers requirements, the ECU is subjected to regular redesign processes. The method has been applied during the redesign phase of the product. Most of the Technical data have been extracted from the design specifications. The experience showed that some data was not directly available from the design plans, and calculations as well as measurements on previous equipment generation were necessary. As a first approximation and because of time constraints the assessment was made without considering the electronic components. However, all electronic components used are certified as RoHS compliant, critical chemicals should not be found in these parts. The assessment leads to the following results, represented on figure 7: • Two hazardous substances concentrations are higher than 0,1% weight/weight : lead in brazing and beryllium in a specific alloy. These substances should be notified to the customers in order to comply with REACh requirements. From this assessment, the design team took action to improve data collection: a questionnaire has been created to be distributed to suppliers, for furnished articles. So far, a few REACh conscious suppliers have already answered to our questionnaire.

The following Design recommendations have been notified by the design team in the documentation which accompanies the equipment: • Lead substitution: lead brazing processes are still used to produce the ECU. Even though free-lead brazing processes haven't been totally validated, aeronautical Research and Development activities still cover that field. We hope that the a brazing process change will be possible for the next generation of ECU, • Beryllium alloy risks management: this alloys presents particular elasticity properties, but its machining produces carcinogenic dust. Thanks to the current state of knowledge, no other material could perform these properties. In order to limit risks during processing phase, we explicitly require from our suppliers to reduce exposure factors.

IV. DISCUSSION

Thanks to the case study analysis, we can conclude that the method proposed in this paper seems to be:

• applicable during design as it uses information on materials, mixtures and production processes available at design stages; • usable by designers as assessment results are precise enough to guide designer's choice between different elements;

• Efficient to answer REACh requirements about substances mass thresholds: the method results point out the most hazardous substances mass ratio in the equipment.

The method presents also some limits, in particular: • The exposure step calculation has mainly been applied to the production and maintenance phased of the equipment: this should be extend in the future to other life-cycle stages, • the sources of input parameter should be explored further , so that uncertainties linked to measurement on former generation equipment are reduced.

V. CONCLUSION / PERSPECTIVES

The present paper aimed at presenting a new chemical risk assessment method, to be used during the design process of a product. This method has been (and is still being) developed in an aeronautical company, in order to answer its particular requirements. We however think that this method could be of use for other companies of the sector. The method is a combination of a qualitative and quantitative approach. The main elements are: • a substances list, which is a classification of various substances according to their toxicological and chemical intrinsic hazard, • a database which connects substances list to operational data used by designers: materials, mixtures and production processes, • a risk assessment which permits to guide the designer's choice among different materials, mixture or production processes, thanks to their environmental weight.

The application of the method on a case study has been lead and conclusions have been made: • entry data, necessary to run the method, have been validated, • the assessment results turns possible to formulate design recommendations, and to justify from substitution research orientation, • the assessment results permit to answer the REACh requirement concerning hazardous substance notification in products to be sold.

b. Step 2 :

 2 Hazard qualification step 1. Description

Figure 1 :

 1 Figure 1: Labelisation rules from substance to materials and mixture

Figure 2 :

 2 Figure 2: Database structure c. Hazard quantification stepThe database presented above provides a function that permits to compile all the materials, mixtures or production processes that are requested for a product development. It is possible to create a figure (indicator) which summarizes the number of each materials, mixture or production process according to its label. However, at this stage of the method application, one can find numerous materials under the same label. In order to discriminate the results presented above, it was decided to use a quantitative chemical risk assessment method for exempted and critical materials, mixtures and production processes. This two steps (qualification + quantification) approach helps to filter the data and concentrate on the most critical identified values, as usually requested by traditional H&SE practitioners (cf. e.g.[START_REF] Balsat | A structured strategy for assessing chemical risks, suitable for small and medium-sized enterprises[END_REF]).

Figure 6 :

 6 Figure 6: An Electronic Control Unit.

Figure 7 :

 7 Figure 7: Assessment test results -Materials label repartition quoted on total ECU massAfter analyzing Figure7, it can be concluded that:• Most of the ECU parts do not present any toxicological or chemical risk, according to the substances list standard, • About 3,6 % of the ECU mass could not be assessed, because of lack of data. This percentage corresponds to furnished articles, which are bought to answer technical requirements and function, and which structure is unknown for the customer (i.e. Hispano-Suiza), because of industrial property. • Two hazardous substances concentrations are higher than 0,1% weight/weight : lead in brazing and beryllium in a specific alloy. These substances should be notified to the customers in order to comply with REACh requirements. From this assessment, the design team took action to improve data collection: a questionnaire has been created to be distributed to suppliers, for furnished articles. So far, a few REACh conscious suppliers have already answered to our questionnaire.

  ., Mathieux, F., Brissaud, D.,-How managing efficiently substances in the design process of industrial products? An example from the aeronautics sector 9/10

"Article" stands for any object which shape and design is more relevant for its function than its chemical composition[REACh]

ACKNOWLEDGEMENT:

The authors would like to thank L.-F LOBREAU, R.POIRIER, S.COCHARD, C.GALLAIS, P.E. PLACHOT, P.BERHAULT and Q.MOLLET for their involvement.

Lemagnen, M., Mathieux, F., Brissaud, D.,-How managing efficiently substances in the design process of industrial products? An example from the aeronautics sector 7/10 e. Summary of the method

The method is summarized in figure 5:

Materials Production Processes

Product Design Bill

Hazard labels applied to materials and processes

Materials and processes database

Substances list

Hazard scores calculated for most Hazardous materials and processes

Chemical risk Assesment method

Substituting ?

Reducing ?

Compliance Matrix

III. CASE STUDY: APPLYING THE METHOD TO ELECTRONIC CONTROL UNIT FOR TURBOSHAFTS ENGINE

The method presented above has been applied on aeronautical equipment (cf. Figure 6), designed and produced by Hispano-Suiza ( [START_REF] Lartiguet | Impact of REACh on material and processes in Safran Group[END_REF], [START_REF] Lobreau | Intégration de l'environnement en phase de conception / Application aux équipements électriques et électroniques[END_REF]). The equipment is an Electronic Control Unit (ECU) for turboshafts engine for both civil and military helicopters. This equipment is connected to the engine and its main functions are to regulate the engine and to ensure its functional safety. Basically, the ECU is an intermediate between the pilot and the engine, and runs according to the 4 following operations:

• gathering data on engine, sent by sensors via wiring harnesses, • filtering data, due to severe environment (electromagnetic interferences …),

• analyzing data and calculating commands to be sent to the control devices • formatting commands to drive the actuators and other control devices.

The equipment was chosen because it is representative of the main technologies (materials and production processes) used at Hispano-Suiza. The parts of this equipment are:

• mechanical part: housing, armour plating, screws, various metallic parts which represent about 45% of total ECU mass,


Naim Zoghlami
email: naim.zoghlami@ibisc.univ-evry.fr

Lotfi Beji
email: lotfi.beji@ibisc.univ-evry.fr

Rhouma Mlayeh
email: rhouma.mlayeh@ipeit.rnu.tn

Azgal Abichou
email: azgal.abichou@ept.rnu.tn

Finite-time average consensus in networked dynamic systems

The paper tackles to the finite-time average consensus problem of networked dynamic systems. A dynamic behavior is described by a controlled first order differential equation with and without drift term. The objective of the analysis is to cover a large variety of autonomous systems without restoring their explicit form. For high dimensional multi-system, protocols are proposed to solve the average consensus problem in finite-time. Like an interaction topology, the agreement achievement is analyzed through an undirect fixed graph.The theoretical results are supported by simulations based on multi-unicycle and multi-pendulum systems.

I. INTRODUCTION

In the past few years, the cooperative control problem for a group of agents is a popular research topic in decentralized control. Many applications can be found in various area: rendezvous problem of multi-vehicle, control of training, flocking, attitude the synchronization, the fusion of sensors. Is one of the main challenges in cooperative design decentralized control systems, such as some objectives of the group can be achieved. The coherent movement in masses is called consensus. Thus, the problem of consensus plays a central role in study of multi-agent systems. Early works on the consensus problem of multi-agent systems can be found in [4], [5], [6], [7] and [10]. A special case of the consensus problem in multi-agent systems is the finite-time consensus problem, which is sufficiently studied in the literature ([8],[9],[11],[12],[15],[16].

Nevertheless, the finite-time consensus problem that has been solved so far is mostly only for agents with first or second order dynamics, in ([13] et al. and [14] et al.) the authors treated finite-time consensus for nonlinear networked systems where each systems is modeled by drift/driftless systems. An interesting topic in consensus problem is the average consensus problem means to design a networked interaction protocol such that the states of all the agents converge (asymptotically/ finite time) to the average of their states ([18], [19],[20] and [21]), to name just a few.

In this paper, we investigate the finite-time average consensus problem. Using an undirected fixed graph, the average consensus study for a nonlinear networked systems remains

a challenge problem. Further, the research is motivated by the fact that each dynamic system is taken highly nonlinear with/without drift term in the model. Inspired from finite-time stability results presented in [START_REF] Bhat | Finite Time Stability of Continuous Autonomous Systems[END_REF], [START_REF] Hong | Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties[END_REF] and the graph theory [START_REF] Cremean | Stability Analysis of Interconnected Nonlinear Systems Under Matrix Feedback[END_REF], nonlinear consensus protocols are proposed throughout the paper.

The paper is organized as follows. Some preliminaries results, the problem statement, and the finite-time average consensus protocol are formulated in section 2. In section 3 one solves a finite-time average consensus of multi-system without drift terms. The finite-time average consensus of multi-system with drift is detailed in section 4. Finally, illustrative examples are presented in section 5.

II. PRELIMINARIES AND PROBLEM FORMULATION

Throughout the paper, we use R to denote the set of real number. R n is the n-dimensional real vector space and . denotes the Euclidian norm. R n×n is the set of n × n matrices. diag{m 1 , m 2 , ..., m n } denotes a n × n diagonal matrix. I n ∈ R n×n is the identity matrix. The symbol ⊗ is the Kronecker product of matrices. We use sgn(.) to denote the signum function. For a scalar x, note that ϕ α (x) = sgn(x)|x| α . We use x = (x 1 , ..., x n ) T to denote the vector in R n . Let φ α (x) = (ϕ α (x 1 ), ..., ϕ α (x n )) T , and 1 n = (1, ..., 1) T . The exponent T is the transpose.

A. Graph theory

In this subsection, we introduce some basic concepts in algebraic graph theory for multi-agent networks. Let G = {V, E} be a directed graph, where V = {1, 2, ..., n} is the set of nodes, node i represents the ith agent, E is the set of edges, and an edge in G is denoted by an ordered pair (i, j). (i, j) ∈ E if and only if the ith agent can send information to the jth agent directly. A = [a ij ] ∈ R n×n is called the weighted adjacency matrix of G with nonnegative elements, where a ij > 0 if there is an edge between the ith agent and jth agent and a ij = 0 otherwise. Moreover, if A T = A, then G is also called an undirected graph. In this paper, we will refer to graphs whose weights take values in the set{0, 1} as binary and those graphs whose adjacency matrices are symmetric as symmetric. Let D = diag{d 1 , ..., d n } ∈ R n×n be a diagonal matrix, where d i = n j=1 a ij for i = 0, 1, ..., n. Hence, we define the Laplacian of the weighted graph

L = D -A ∈ R n×n 1
The undirected graph is called connected if there is a path between any two vertices of the graph. Note that time varying network topologies are not considered in this paper.

B. Some useful lemmas

In order to establish our main results, we need to recall the following Lemmas. Lemma 1: [START_REF] Bhat | Finite Time Stability of Continuous Autonomous Systems[END_REF]. Consider the system ẋ = f (x), f (0) = 0, x ∈ R n , there exist a positive definite continuous and differentiable function V (x) : U ⊂ R n → R, real numbers c > 0 and α ∈]0, 1[, and an open neighborhood U 0 ⊂ U of the origin such that V + c(V (x)) α ≤ 0, x ∈ U 0 \{0}. Then V (x) converges to zero in finite time. In addition, the finite

settling time T satisfies T ≤ V (x(0)) 1-α c(1 -α) . Lemma 2: [17] Consider the following system, x = [x 1 , ..., x n ] T ∈ R n ẋ = g(x) + g(x) (1)
where g(0) = 0 and g(x) is a continuous homogeneous vector field of degree d < 0 with respect to dilation [σ 1 , ..., σ n ], and g(x) = [g 1 (x), ..., gn (x)] T ∈ R n satisfies g(0) = 0. Assume that x = 0 is an asymptotically stable equilibrium of the system ẋ = g(x). Then x = 0 is a globally finite time stable equilibrium of system (1) if lim ε→0 g(ε σ1 x 1 , ..., ε σn x n ) ε d+σi = 0, i = 0, ..., n, ∀x = 0, and the stable equilibrium x = 0 of system (1) is globally asymptotically stable. Lemma 3: [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF]. For a connected undirected graph G, the Laplacian matrix L of G has the following properties,

x T Lx = 1 2 n i,j=1
a ij (x ix j ) 2 , which implies that L is positive semi-definite. 0 is a simple eigenvalue of L and 1 is the associated eigenvector. Assume that the eigenvalues of L are denoted by 0, λ 2 , ..., λ n satisfying 0 ≤ λ 2 ≤ ... ≤ λ n . Then the second smallest eigenvalue satisfies λ 2 > 0. Furthermore, if 1 T x = 0, then x T Lx ≥ λ 2 x T x. Lemma 4: [START_REF] Hardy | Inequalities[END_REF]. Let x 1 , x 2 , ..., x n ≥ 0 and 0 < p ≤ 1.

Then ( n i=1 x i ) p ≤ n i=1 x p i ≤ n 1-p ( n i=1 x i ) p .

C. Problem statements

We study the finite-time average consensus of two-types of networked systems. The first type is given by equation (2) which describes a controlled system without drift. The second type is represented by equation (3) which is clearly a controlled linear system with drift. One notes that the matrix B for the two models depends on the system's states.

Consider a group of N high-dimensional agents where each agent's behavior is described by a controlled nonlinear model without drift Σ 1 , as given by dynamic (2), and the system with drift Σ 2 , as shown by dynamic [START_REF] Bhat | Finite Time Stability of Continuous Autonomous Systems[END_REF]

, ∀i ∈ I = {1, ..., N } Σ 1 : ẋi = B(x i )u i (2)
and

Σ 2 : ẋi = f i (x i ) + B(x i )u i (3)
where x i ∈ R n , and for

1 ≤ i ≤ N x i = [x i 1 , x i 2 , ..., x i n ] T , B(x i ) ∈ R n×m , the continuous maps f i : R n → R n , u i ∈ R m is the input, and B(x i ) = [b kl ] for 1 ≤ k ≤ n and 1 ≤ l ≤ m.
Definition 1: Given a control-input u i , we say that systems in networks meet a finite-time average consensus if for any system's state initial conditions, there exists some finite time T such that:

lim t→T x i (t) -χ(t) = 0 (4)
for any i ∈ I, and where

χ(t) = 1 N N j=1
x j (t).

χ(t) can be interpreted as the instantaneous consent providing and serves the group objective. For multi-Σ 1 and multi-Σ 2 systems, one might analyze the following consensus protocol.

For i ∈ I, the consensus protocol candidate is given by,

u i = -C(x i ) N j=1 a ij φ α (x i -x j ) (5)
where the a ij elements are of the G adjacency matrix, α ∈ ]0, 1[, and φ α (.) is defined in section II. C(x i ) ∈ R m×n is such that the following assumption hold..

Assumption 1: C(x i ) is such that the matrix product B(x i )C(x i ) is positive semi-definite matrix.
Throughout the paper, one denotes by B(x i ) = B(x i )C(x i ).

Assumption 2: For a given control matrix C(x i ), for all x, y ∈ R n , we assume that

(φ α (x) -φ α (y)) T B(x i )φ α (x -y) ≥ (φ α (x) -φ α (y)) T B(x i ) (φ α (x) -φ α (y)) Assumption 3: Consider that g(x i ) = - N j=1 a ij B(x i )φ α (ξ i -ξ j ) (6)
is a locally homogeneous vector field of degree d with respect to dilation [σ 1 , ..., σ n ].

III. THE MULTI-Σ 1 FINITE-TIME AVERAGE CONSENSUS

For finite-time average consensus of multi-Σ 1 one considers each Σ 1 vector state to compute the average vector, and the protocol candidate (5) while the interaction topology is an undirected fixed graph. As the matrix B structure is taken identical for each Σ 1 , than one might think to networked homogeneous systems.

Proposition 1: Let G be an undirected and connected graph, under the protocol ( 5) and Assumptions 1-2-3 the multi-Σ 1 achieves a finite-time average consensus in the sense of (4).

Proof. We introduce ξ i (t) = x i (t)χ(t). Due to the fact that a ij = a ji for all 1 ≤ i, j ≤ N (undirected graph) and φ α is an odd function, we have,

χ(t) = 1 N N i=1 ẋi (t) = - 1 N N i=1 N j=1 a ij B(x i )φ α (x i -x j ) = - 1 2N N i=1 N j=1 a ij B(x i ) -B(x j ) φ α (x i -x j )
Introducing the protocol (5), we obtain

ξi (t) = ẋi (t) -χ(t) = - N j=1 a ij B(x i )φ α (x i -x j ) + 1 2N N i=1 N j=1 a ij B(x i ) -B(x j ) φ α (x i -x j ) = - N j=1 a ij B(x i )φ α (ξ i -ξ j ) + 1 2N N i=1 N j=1 a ij B(x i ) -B(x j ) φ α (ξ i -ξ j )
let ξ(t) = (ξ 1 , ..., ξ N ), we can write the last equation in the form ξi (t) = g(ξ i ) + g(ξ) [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF] where

g(ξ i ) = - N j=1 a ij B(x i )φ α (ξ i -ξ j ) and g(ξ) = 1 2N N i=1 N j=1 a ij B(x i ) -B(x j ) φ α (ξ i -ξ j ).
By now, it remains to prove that the equilibrium of ( 7) is finite-time stable, and this is achieved in the subsequent two steps.

Step1. First, the goal is to prove the finite time stability of system ξi (t) = g(ξ i )

Taking the Lyapunov function,

V (ξ(t)) = 1 α + 1 N i=1 (ξ i ) T φ α (ξ i ) (9)
The derivative of V along the solutions of system (8) yields

V (ξ(t)) = N i=1 φα(ξ i ) T ξi = - N i,j=1 aij φα(ξ i ) T B(x i )φα(ξ i -ξ j ) = -1 2 N i,j=1 aij φα(ξ i ) -φα(ξ j ) T B(x i )φα(ξ i -ξ j )
From Assumption 2, the following inequality holds

V ≤ -1 2 φ T α (ξ) L ⊗ B(x i ) φ α (ξ) = -1 4 φ T α (ξ)Θφ α (ξ) where Θ = 1 2 L ⊗ B(x i ) + L ⊗ BT (x i ) . Let D(x i ) = diag{0 n , γ 2 (x i ), ..., γ N (x i )}
where 0 n = diag{0, ..., 0} ∈ R n×n , and ∀j = 2, ..., N γ j

(x i ) = λ j (L)̺ n (x i ) with ̺ n (x i ) = diag{0, µ 2 (x i ), ..., µ n (x i )} ∈ R n×n ,
and where µ 2 (x i ), ..., µ n (x i ) are the eigenvalues of the matrix B(x i ) given in increasing order. λ j (L) denotes the j th eigenvalue of L. Let them be λ 2 (L), ..., λ N (L) in increasing order. Since G is connected (by Lemma 3) λ 2 (L) > 0. Therefore, ∀x i we have

λ 2 µ 2 (x i ) > 0.
Further, since Θ is symmetric matrix, then there exist an orthogonal matrix P ∈ R N n×N n such that Θ = P T D(x i )P .

Let

z α = P φ α (ξ), thus V ≤ -1 4 z T α Dz α ≤ -1 4 λ 2 µ 1 (x i ) z α 2 = -1 4 λ 2 µ 1 (ξ i ) φ α (ξ) 2 with λ 2 µ 1 (x i ) = min zα⊥1 N n z T α Dz α z T α z α . Let k = min x i ∈R N λ 2 µ 1 (x i ) > 0 and ξ = 1 N ⊗ ξ i = ( ξ1 , ..., ξNn ) T , consequently, V ≤ - k 4 N n i=1 |ϕα( ξi)| 2 ≤ - k 4 N n i=1 | ξi| 2α ≤ - k 4 ( N n i=1 | ξi| α+1 ) 2α α+1 (10)
which permits to write

V ≤ - k 4 (α + 1) 2α α+1 V 2α α+1 (11)
Since 0 < 2α α+1 < 1 and k 4 (α + 1) 2α α+1 > 0, by Lemma 1 the above differential equation [START_REF] Cortes | Finite-time convergent gradient flows with applications to network consensus[END_REF] shows that V reaches zero in finite time.

Step2. From Assumption 3, the vector field g(ξ i ) is homogeneous of degree d which is negative due to the fact that ξ = 0 is a finite time stable equilibrium of ξi = g(ξ i ). Moreover, it is straightforward to prove that lim ε→0 g(ε σ1 x 1 , ..., ε σn x n ) ε d+σi = 0. Then by Lemma 2, system (7) is finite time stable. Thus, as a result the multi-Σ 1 dynamic system with the protocol (5) solve a finite-time average consensus. This ends the proof.

IV. THE MULTI-Σ 2 FINITE-TIME AVERAGE CONSENSUS

The multi-Σ 2 in network is based on dynamic (3) while the consensus protocol candidate is given by [START_REF] Jadbabaie | Coordination of groups of mobile auronomous agents using nearest neighbor rules[END_REF]. Recall that the Σ 2 dynamic as given by ( 3) is currently present in controlled autonomous systems. Further, the drift term will be considered as linear with respect to the Σ 2 's states. Note that f i in (3) can be different for each Σ 2 dynamic systems. Than one might think to networked heterogeneous systems.

Case1: Linear drift term we consider equation (3) and let

f i (x i ) ≡ Ãx i , system (3) becomes Σ 2,1 : ẋi = Ãx i + B(x i )u i (12)
where à ∈ R n×n with à = [ã p,q ] 1≤p,q≤n .

Proposition 2: Let G be an undirected and connected graph. Under the protocol ( 5) and Assumption 1-2-3 the multi-Σ 2,1 achieves a finite-time average consensus in the sense of (4).

Proof. One introduces ξ(t) = x i (t)χ(t). The goal is to rewrite equation ( 12) in closed loop depending on ξ i and to prove that ξ converges to zero in finite time. Since a ij = a ji and φ α is an odd function, then we have

χ = 1 N N i=1 ( Ãx i + B(x i )u i ) = 1 N N i=1 Ãx i + 1 N N i=1 B(x i )u i = 1 N N i=1 Ãx i - 1 2N N i=1 N j=1 a ij B(x i ) -B(x j ) φ α (x i -x j ) Consequently, ξi = Ãξ i - N j=1 a ij B(x i )φ α (ξ i -ξ j ) + 1 2N N i=1 N j=1 a ij B(x i ) -B(x j ) φ α (ξ i -ξ j )
keeping the same steps of the previous proof, we introduce

ξi = h(ξ i ) + h(ξ) where h(ξ i ) = Ãξ i - N j=1 a ij B(x i )φ α (ξ i -ξ j ) and h(ξ) = 1 2N N i=1 N j=1 a ij B(x i ) -B(x j ) φ α (ξ i -ξ j ).
As h(ξ) = g(ξ), then it remains to prove the finite-time stability of the system

ξi = h(ξ i ) ( 13
)
Using the Lyapunov function ( 9), the time derivative of V (ξ) along the networked system trajectories [START_REF] Zoghlami | Finite-time consensus and stability of networked nonlinear systems[END_REF], is given by

V (ξ(t)) = N i=1 φ T α (ξ i ) ξi = N i=1 φ T α (ξ i ) Ãξ i - N i,j=1 aijφ T α (ξ i ) B(x i )φα(ξ i -ξ j ) ≤ Ã ∞ N i=1 φ T α (ξ i )ξ i - k 4 (α + 1) 2α α+1 V 2α α+1 ≤ Ã ∞V (ξ(t)) -k 4 (α + 1) 2α α+1 V 2α α+1 ≤ -V (ξ(t)) 2α α+1 [ k 4 (α + 1) 2α α+1 -Ã ∞(V (ξ(t))) 1-α α+1 ]
where

à ∞ = max 1≤p≤n n q=1
|ã pq | > 0. Since 1-α α+1 > 0 and V is continuous function which takes 0 at the origin (ξ = 0), there exists an open neighborhood Ω of the origin such that the last inequality ( 14) yields to

V (ξ(t)) ≤ - k 8 (α + 1) 2α α+1 V (ξ(t)) 2α α+1 (14)
by Lemma 1, V reaches zero in finite time. Therefore ξ i = 0 is a finite-time stable equilibrium of system [START_REF] Zoghlami | Finite-time consensus and stability of networked nonlinear systems[END_REF]. We may follow step 2 of the previous analysis to end the proof.

Case2: Nonlinear drift term In this case, we consider the system (3) and the drift term as nonlinear and we assume that f i is a convex vector field.

Proposition 3: Let G be an undirected graph. Under the protocol (5), a networked system multi-Σ 2 based on (3) meets a finite-time average consensus.

Proof. As f i is assumed to be convex, we have

f i (x i ) - 1 N N i=1 f i (x i ) ≤ f i (x i ) -f i ( 1 N N i=1 x i ) Moreover f i is locally Lipschitz function in an open set Ω ⊂ R n containing ξ. Therefore, f i (x i ) -1 N N i=1 f i (x i ) ≤ f i (x i ) -f i (χ) ≤ c ξ i
where c > 0 is the Lipschitz's constant. Now, for convenience the Lyapunov function is given by ( 9) and the following inequality is obtained

V (ξ(t)) = N i=1 (φ α (ξ i )) T ξi ≤ c N i=1 φ T α (ξ i )ξ i - k 4 (α + 1) 2α α+1 V 2α α+1 ≤ -V (ξ(t)) 2α α+1 [ k 4 (α + 1) 2α α+1 -c(V (ξ(t))) 1-α α+1 ]
Since 1-α α+1 > 0 and V is continuous function which takes 0 at the origin (ξ = 0), there exists an open neighborhood Ω of the origin such that

V (ξ(t)) ≤ - k 8 (α + 1) 2α α+1 V (ξ(t)) 2α α+1 (15)
Then, we may conclude that the multi-Σ 2 system issues from (3) associated to the protocol (5) leads to a finite-time average consensus. This ends the proof.

V. ILLUSTRATIVE EXAMPLES Two illustrative examples are considered where the multiunicycle represents a networked system modeled by (2) (driftless) and the multi-pendulum implies a networked multi-model of type (3) (with drift). Each associated protocol is deduced from [START_REF] Jadbabaie | Coordination of groups of mobile auronomous agents using nearest neighbor rules[END_REF]. The results of the finite-time average consensus are obtained under the undirected graph of figure

A. Finite-time average consensus for multi-unicycle

Consider N wheeled mobile robots (unicycles) where the i th nonholonomic kinematic model is as:

  ẋi ẏi θi   =   cos θ i 0 sin θ i 0 0 1   u i w i i = 1, ..., N (16)
where (x i , y i , θ i ) denotes the position and the orientation in an inertial frame. The inputs u i and w i are the linear and where ϕ α is defined in section II and a ij are associated to the graph in Fig. 1. The simulation results are limited to N = 4 that integrate the following initial conditions (x 1 , y 1 , θ 1 )(t = 0) = (14, 2, π), (x 2 , y 2 , θ 2 )(t = 0) = (-4, 2, -π 2 ), (x 3 , y 3 , θ 3 )(t = 0) = (10, 8, π 2 ) and (x 4 , y 4 , θ 4 )(t = 0) = (-10, -8, 0). The numerical simulations are performed using [START_REF] Ren | Finite-time Consensus for Second-order Multi-agent Networks with Inherent Nonlinear Dynamics Under Fixed Graph[END_REF] and protocols ( 17)- [START_REF] Zhu | Discrete-time dynamic average consensus[END_REF]. The results of figures Fig.

B. Finite-time average consensus in multi-pendulum dynamics

Consider a set of N pendulum with the following model where m i , g i , l i and ψ i are positive constants. For this system the drift term issues from the first order differential form (see

θi = - g l i sin(θ i ) - ψ i m i l i θi + u i (19)
)) is f i (θ i , θi ) = θi -g li sin(θ i ) -ψi mili θi (3
we can easily check the convexity condition for the drift term f i . Following to the subsequent theoretical analysis (see Proposition 3), taking C = (1 1), a protocol that solves the finite-time average consensus for multi-pendulum is as

u i = - N j=1 a ij (ϕ α (θ i -θ j ) + ϕ α ( θi -θj )) (20)
This set of N = 4 pendulums is analyzed. As heterogenous multi-system, the 4 pendulum parameters aren't similar. Thus, m 1 = 1, m 2 = 2, m 3 = 3 and m 4 = 4 (Kg).

The standard gravity vector is g = 9.8(m.s -2 ), the lengths l i = 1 (m) and the coefficient ψ i = 0.1 (Kg.m 2 .s -1 ). Initial conditions are such that θ i = (-0.8, 0.4, 1, 2, 1.6) (rad) and θi = (0, 0, 0, 0) (rad.s -1 ).

Clearly from figures in Fig. 5-6, the synchronization toward the average trajectory of 4 pendulums in angular positions and velocities are obtained. It is important to note that the average is time-varying and the multi-system of pendulums is heterogeneous with respect to the proposed physical parameters. This confirm the theoretical results of Proposition 3.

VI. CONCLUSION

In this work, a controlled dynamic model of networked autonomous systems is formulated by two-types of nonlinear and continuous first-order differential equations. Some protocols are proposed and sufficient conditions are achieved covering high dimensional networked homogeneous dynamics. The theoretical results solve finite-time average consensus problems of dynamic systems in the form multi-Σ 1 and multi-Σ 2 , and they are supported by simulations based on multi-unicycle and multi-pendulum systems. For the future, it is interesting to treat the case of networked heterogeneous multi-system as multi-Σ 1 -Σ 2 .

FigFig. 1 .

 1 Fig. 1. G for a system with 4 agents.

  cos θ i sin θ i 0 -sin θ i cos θ i 0 Based on Proposition 1, the finite-time average consensus problem can be achieved through the following protocol ui = -N j=1 aijϕα(xixj) cos θi -N j=1 aijϕα(yiyj) sin θi (17) wi = N j=1 aijϕα(xixj) sin θi -N j=1 aijϕα(yiyj) cos θi (18)

2

 2

  -3 evolve according to the developed theoretical results of multi-Σ 1 . The common value is also the average trajectory of the unicycles. The (x i , y i ) -(ave(x i ), ave(y i )) converges in finite-time to zero as show in figure Fig.4.

Fig. 2 . 1 Fig. 3 . 1 Fig. 4 .

 21314 Fig. 2. Average consensus of position x i for 4 unicycles as multi-Σ 1

Fig. 5 .

 5 Fig. 5. The time-varying average of angular positions consent by 4 pendulums.

Fig. 6 .

 6 Fig. 6.The time-varying average of angular velocities consent by 4 pendulums.


Emile Muller

Marianne Huchard
email: marianne.huchard@lirmm.fr

Pierre Martin
email: pierre.martin@cirad.fr

Pascal Poncelet
email: pascal.poncelet@lirmm.fr

Arnaud Sallaberry
email: arnaud.sallaberry@lirmm.fr

RCAviz: Visualizing and Exploring Relational Conceptual Structures

Keywords: Formal Concept Analysis, Relational Concept Analysis, Visualization, Navigation, Concept Lattice

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

Introduction

Formal Concept Analysis (FCA) aims at classifying a set of objects described by Boolean attributes as a concept lattice [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]. Other conceptual structures may also be produced, such as AOC-poset or Iceberg lattice. FCA and its various extensions enable the extraction of frequent patterns, attribute implications, mutual exclusions within attribute groups, and object similarities. The survey of [START_REF] Poelmans | Formal concept analysis in knowledge processing: A survey on applications[END_REF] shows the variety of applications where different knowledge discovery methods based on FCA are successfully used [START_REF] Poelmans | Formal concept analysis in knowledge processing: A survey on models and techniques[END_REF].

Among the extensions of FCA, Relational Concept Analysis (RCA) supports multiple binary relations between several sets of objects. As output, RCA builds conceptual structures (one per object set) interconnected by relational attributes, that embed the inter-object relations provided as input. Analyzing multi-relational datasets with RCA may benefit from exploring these interconnected structures by navigating from one concept to another. This navigation is therefore performed within the same conceptual structure, through the classification links, or from one concept of a conceptual structure to a concept of another one, through one of the relations linking the lattices.

FCA and RCA present a strong interest when querying a dataset using a set of attributes. In addition to the objects that respond to the query, they provide the classification of these objects [START_REF] Messai | Querying a bioinformatic data sources registry with concept lattices[END_REF][START_REF] Azmeh | Querying Relational Concept Lattices[END_REF][START_REF] Ferré | Reconciling Expressivity and Usability in Information Access[END_REF]. When some objects respond strictly to the query, these objects are part of the extent of the concept, whose intent is constituted by the query attributes. The sub-concepts of this concept then contain the objects having more attributes than those listed in the query. Conversely, the super-concepts contain the objects having less attributes, corresponding to a constraint release. This latter case is of particular interest to domain experts when the dataset is likely to be incomplete, as constraint release may create hypotheses worth to be investigated. Classification also helps to reformulate the query when the answer set is empty, as it shows the concepts highlighting the partially answering objects.

Several authors investigate the potential offered by the visualization for assisting exploring, querying, and analyzing a classification produced by FCA [START_REF] Carpineto | Concept data analysis -theory and appl[END_REF][START_REF] Ferré | Reconciling Expressivity and Usability in Information Access[END_REF][START_REF] Greene | Visualizing and exploring software version control repositories using interactive tag clouds over formal concept lattices[END_REF]. For RCA, the most advanced tools propose either a sophisticated visualization of the conceptual structure (e.g. Galicia5 ) or a simple visualization of the conceptual structure with the ability to jump from one structure to another through the relational links (e.g. RCAexplore6 ). These tools lack features enabling full visualization support for gradually navigating between concepts without being overwhelmed by the size of the structure. This paper addresses this need by introducing RCAviz7 , an online tool which aims to support such navigation.

Section 2 briefly introduces FCA and RCA basics, and a small RCA navigation example to illustrate the main features expected for a navigation tool. Section 3 presents the visualization and navigation features of RCAviz. in Section 4, we present related work on FCA and RCA visualization and exploration. Section 5 concludes with a summary of the work and a few perspectives.

RCA Navigation at a glance

This section briefly introduces FCA and RCA through an example inspired by the Knomana project [START_REF] Silvie | Projet knomana. knowledge management on pesticidal plants in africa for a safer food and a better environmental health. rapport d'extractions[END_REF] which motivated the tool presented in this paper. The Knomana dataset deals with the control of pests using plants in agriculture. We then develop the needs addressed by our proposed tool on a small example.

FCA in a nutshell FCA input is a formal context (FC) which includes a binary relation that connects each object with each of the attributes it has. E.g. the FC Plant presented in Table 1 describes four plants by their family and a few characteristics like being aromatic or comestible. FCA builds formal concepts on top of the FC and organizes these concepts as a conceptual structure. A formal concept is a maximal group of objects (extent) sharing a maximal group of attributes (intent). For instance, in Table 1, as all the plants of the group E 9 = {chromolaenaOdorata aspiliaAf ricana, ageratumConyzoide, } share all the attributes I 9 = { asteraceae, aromatic, toxic, evergreen, applicEssentialOil, applicExtract}, CP l 9 = (E 9 , I 9 ) is a concept. The conceptual structure reflects group inclusion and highlights concept specialization. The top right part of Figure 1 presents an example of specialization of plant concepts. Concept CP l 11 , which associates E 11 = {ageratumConyzoide} with its common attributes I 11 = I 9 ∪ {antidysenteric} is a subconcept of CP l 9 , with I 11 ⊇ I 9 and E 11 ⊆ E 9 . This figure also presents relational attributes, specific to RCA (i.e. initiated using the quantifier ∃), which are explained here after.

RCA in a nutshell RCA extends FCA to datasets that conform to the entityrelationship model [START_REF] Hacene | Relational concept analysis: mining concept lattices from multi-relational data[END_REF]. The top left part of Figure 1 shows such a model presented using the UML syntax: A plant treats (controls) a pest; A pest is found in a country; A country possesses plants. As input, RCA takes a set of FCs and a set of relational contexts (RC) connecting objects of two FCs. FCs correspond to the classes of the UML model (i.e. P lant, P est, Country), while RCs correspond to the associations (i.e. treats, isF oundIn, possesses). Each FC is computed as a conceptual structure, which has incorporated iteratively the object clustering of the other FCs thanks to the relational attributes. Object clustering propagation applies when a RC (denoted by r) connects objects of a source FC (denoted by F C s ) to objects of a target FC (denoted by F C t ). E.g. The RC r = treats connects objects of FC F C s = P lant to objects of FC F C t = P est. Once a concept C t is built in F C t , some objects in F C s may be connected to objects of C t using r. Different schemes of connection can be adopted, e.g. the connection of an object o s of F C s by r to at least one object of C t , the connection of o s to all objects of C t , or half of o s connections by r are with some objects of C t . For example in Figure 1, the concept CP e13 groups aspergillusOchraceus and aspergillusP arasiticus, two pests that attack peanuts. In addition, aspiliaAf ricana and chromolaenaOdorata control respectively aspergillusP arasiticus and aspergillusOchraceus. Thus both plants control at least one pest of CP e13. The selection of the scheme is captured in the relational attributes that include a quantifier (e.g. ∃, ⊇, ∀ ≥50% ), the relation r and the target concept C t . The use of the quantifier ∃ in Figure 1 conducts to express the relational attribute "treats at least one pest of CP e13" as follow: ∃treats(CP e13). Relational attributes are added to the description of the objects to respect FCA principle. E.g. aspiliaAf ricana and chromolaenaOdorata both receive ∃treats(CP e13) as an attribute. During the conceptual structure computation, existing concepts may be enriched by these relational attributes or new concepts may be built to group objects that share some of them. In our example CP l21 is built to factorize ∃treats(CP e13). The propagation thus follows the paths and cycles of the entity-relationship model until a fix-point is reached. RCA algorithm and theoretical details can be found in [START_REF] Hacene | Relational concept analysis: mining concept lattices from multi-relational data[END_REF]. Navigating within linked conceptual structures The above example is used to reveal the gradual navigation needs of the RCA outputs 8 . Let us suppose the user wants information about the pests that attack peanuts. Pest Concept CP e13 introduces attackP eanuts and is therefore the natural entry point. Pests of CP e13 extent are Aspergillus ochraceus and Aspergillus parasiticus. To know the localization of these pests, the user needs to follow the relational attribute ∃isF oundIn(CC20). This attribute leads to Country concept CC20 which groups Western Africa countries, i.e. Benin and Nigeria. In CC20, the relational attributes containing possesses point to plant concepts. Going towards the Plant concept CP l21 indicates that CC20 countries host Aspilia africana or Chromolaena odorata. CP l21 comports the relational attribute ∃treats(CP e13), meaning that its plants treat at least one of the Aspergillus pest species that attack peanuts. It can be interpreted that way: Aspergillus species that attack peanuts are observed in Western Africa countries and are treated by plants that can be found in the countries of this region. It can be interesting here, for the user, to go back by choosing Concept CC20 to look at relational attributes ∃possessesCP l6 or ∃possessesCP l9. Choosing ∃possessesCP l9 and going to CP l9, the user can see that these Western Africa countries possess plants that have characteristics that are close to the plants treating Aspergillus species attacking peanuts. This is the case of Ageratum conizoïdes which is analyzed by going to sub-concept CP l11. This plant is toxic and has an antidysenteric effect, it may have antibacterial properties. These characteristics may suggest that common chemical compounds could be shared between these 3 plant species. This navigation therefore brings a research hypothesis that domain experts may study through new experimental research: can Ageratum conizoïdes be used to control Aspergillus species (i.e. Aspergillus ochraceus and Aspergillus parasiticus) attacking peanuts? If the answer is true, this then would mean that farmers in Western Africa have an additional local plant that can be used for controlling Aspergillus ochraceus and Aspergillus parasiticus.

This brief navigation example illustrates the main needs of a user: (i) selecting an attribute (or an object) to initiate the navigation, (ii) visualizing the neighborhood of a concept to be able to choose the next step and do not end with an empty solution, (iii) moving from one concept to a sub-concept (or a super-concept) in the same conceptual structure, (iv) jumping to a neighboring conceptual structure using a relational attribute, and (v) backtracking easily.

Visualization

In the previous section, we presented the extracted data structure using an RCA classification. As can be seen in Figure 1, this structure can be modeled as a set of Directed Acyclic Graphs (DAG), each DAG corresponding to a context (here, one DAG for plants, one for pests and one for countries). There is an extensive literature on the visualization of DAG (see [START_REF] Healy | Hierarchical drawing algorithms[END_REF] for an introduction). Most articles present algorithms for positioning vertices and drawing edges to limit the number of edge crossings. This type of approach is particularly interesting for small datasets, like the one presented in Figure 1. However, when datasets have hundreds of vertices, visualizing the entire structure induces visual cluttering issues. Moreover, in our context, it can be useful to be able to switch from one DAG to another interactively via the relational attributes. This is why we opted for a local visualization system with a step-by-step navigation.

Requirement analysis Generally, visualization design is done by iterating a process consisting of (1) defining needs and suitable data structures, (2) proposing visual encodings and interactive features meeting these needs, (3) providing the results to the users so that they can refine their needs, etc. (see for instance [START_REF] Munzner | A nested process model for visualization design and validation[END_REF][START_REF] Sedlmair | Design study methodology: Reflections from the trenches and the stacks[END_REF]). In our case, we designed RCAviz by organizing regular meetings with the three visualization experts in charge of providing the visual platform, an RCA expert and a dataset expert. Here is the final list of the needs identified during these meetings: [R1] Selection of a concept. A step-by-step navigation requires a starting point. Our tool must therefore initially allow the selection of a concept, depending on the objects and attributes of interest for the user. [R2] Visualization of a concept and its neighbors. The user must be able to see a concept (with its objects and attributes) as well as its neighbors, i.e. the lower and higher level concepts in the DAG and the neighboring concepts in the other DAGs. [R3] Navigation step-by-step. The user must be able to explore neighboring concepts of the displayed concept (whether they belong to the same DAG or to another one). [R4] History. The users must be able to keep a history of their navigation, i.e. the list of the previously explored concepts. They must also be able to navigate through this history and to save a state to be able to resume their exploration later.

RCAviz is designed to meet these requirements. The visualization consists of an initial view allowing to select the starting concept, the Concept Selector, and two coordinated views: (1) the Explorer allows to navigate step-by-step and (2) the History allows to visualize the navigation history.

Concept Selector

The Concept Selector is the first view displayed when one launches RCAviz. It is designed to meet the requirement [R1]. It first allows to upload a data file. Once the file is uploaded, the user can choose a context, i.e. one of the DAGs of the dataset. Then appears the list of objects, the list of attributes and the list of concepts, as shown in Figure 2. The user can select one or more objects and one or more attributes. The selection of these elements updates the other lists according to the logical operators used, and in particular updates the list of the corresponding concepts on the right. One can for example display the concepts containing (object A and object B) and (attribute C or attribute D). When the list of objects and attributes is long, a search field allows the user to easily find the elements. The user also can select or deselect all the elements of each list. The list of selected items is displayed when hovering over Selected. When an object or an attribute is selected, the corresponding introducer concept number on the right panel appears in bold face. Once the user has found an interesting concept, they can select it from the list on the right and launch the Explorer and the History. It may seem critical to select a concept according to its index number and the selected associated objects and attributes. Thus, in a future work, we plan to show additional information on the listed concepts, such as their introduced objects and introduced attributes, or their whole extent and intent.

Explorer Figure 3 shows the History (top panel) and the Explorer (bottom). The Explorer consists of three sub-views. The central panel displays the selected concept in the center and its neighborhood above (children in the DAG) and below (parents in the DAG) [R2]. Each concept is represented as a colored rectangle composed of stacked boxes. The id is shown at the left corner of the top box. The introduced objects are presented in the box entitled "Objects". Simple (i.e. non relational) introduced attributes appear in another box. The introduced relational attributes, when they exist, appear in the box at the bottom of the concept. When there are several relational attributes with the same relation, they are grouped by relation. In addition, when there are many relational attributes with the same relation, only their number is shown in order to save space, and thus they are not clickable.

For example, the concept 20 in the country DAG in Figure 3 has links to the concepts 6, 9, and 21 in the plant DAG and 13 in the pest DAG. The user can navigate step-by-step by clicking on the concepts [R3]. For example, if they click on the concept 23, it is positioned in the middle of the view and its parents and children are displayed below and above. , for the top (resp. ⊥ for the bottom) at the top right of a concept means that it has no parents (resp. no children). To facilitate the navigation, an opposite link, e.g. "isFoundInopposite", is provided for each relational attribute. These opposite links do not correspond to a isFoundIn-opposite formal context which would be the reverse of isFoundIn. In a future work, additional information on each concept (intent, extent, etc.) will be provided to make the navigation easier.

Each DAG is represented by a color. We use a categorical color scale adapted to the representation of nominal variables [START_REF] Munzner | Interactive visualization of formal concept lattices[END_REF]. In our example, the country DAG is shown in blue, the plant DAG in red, and the pest DAG in green. When the user hovers over a link, the concept in the corresponding DAG is displayed on the right panel. In Figure 3, the mouse is positioned on the red link 21, the corresponding concept is displayed with its neighborhood on the right panel. When the user clicks on the link, the left DAG is positioned in the central panel [R3] and that of the central panel is positioned in the right panel. We thus keep track of the previous exploration step [R4].

The concepts that have already been explored appear with lower brightness than the other concepts [R4]. In Figure 3, we can see that all the concepts have already been explored except one of the red DAG at the bottom left.

Finally, it is common to have a high number of parents or children. Showing all of them induces visualization and navigation issues. Therefore, if the number exceeds 5, we display a single "super-concept". When the user clicks on it, a pop-up displays a view similar to the Concept Selector. The user can then select one or more concepts which, after validation, are displayed in the central panel.

In this release, we choose a top to bottom representation of the specialization relation as this is the common representation of partial orders by Hasse diagrams. Alternatively, we may explore a left to right representation, where a concept will be on the left of its sub-concepts, to show more sibling concepts as they will be vertically organized.

History

We have seen that a part of the path leading to the concept displayed is available in the left panel of the Explorer. The brightness of the concepts also makes it possible to know if a concept has already been explored. However, these functionalities do not fully meet the requirement [R4]. We therefore complete them with a view entirely dedicated to [R4]: the History. As shown in Figure 3, the history consists of two parts: a colored line at the top and a list of concepts below.

The colored line is made up of segments, each segment representing an explored concept. The color represents the DAG of the concept. The length of the line adapts to the width of the visualization so that all the concepts explored are visible. Therefore, the more concepts the user has explored, the smaller the segments representing those concepts are. The user can see if the current concept of the Explorer has already been explored with a small triangle placed above each segment representing this concept. For example, in Figure 3, we see two triangles, one on the last segment representing the concept displayed in the Explorer and one on a previous segment: this means that the current concept has already been explored once during the navigation.

The list of concepts explored with their id is displayed under the colored line. This list can be long, a horizontal slider is therefore activated when the available space is not enough to display all of them in the view. The concepts are represented by colored rectangles according to the DAG they belong to. An arrow connects two concepts if the user has moved from one to the other by navigating step-by-step. If the user clicks, in the History, on a previously explored concept, it is displayed in the Explorer and a vertical bar is used in the History to show that the sequence was not produced by a navigation stepby-step. When the user hovers over a concept, a small red cross appears at the corner. It can be used to remove this concept from the History. As selecting a concept only according to its id may be tricky, additional information (e.g. intent/extent) will be presented in a future release.

Finally, the user can save the state of their navigation as well as the history by clicking on the save button in the top bar. They can then load the file thus produced to resume their exploration.

Implementation RCAviz is a web platform developed in HTML, CSS and Java-Script. The panels and the visualizations are made using the D3.js 9 [6] library. Saving the history file is made possible with the FileSaver 10 library. The tool is available online 11 . An extensive documentation containing sample datasets is available to allow anyone to use the tool with the samples or with their own datasets.

Related Work

Visualizing and exploring in FCA tools There have been a lot of proposals to visualize and explore the conceptual structures. For space reasons, we only mention some of them to highlight the main followed directions. Some proposals visualize the structure itself such as Conexp [START_REF] Yevtushenko | Conexp[END_REF], Latviz [START_REF] Alam | Latviz: A new practical tool for performing interactive exploration over concept lattices[END_REF], Carve (based on a decomposition tree) and DAnCe (with concept enumeration guided by the user) [21], and the approach by expandable and collapsible concept trees [START_REF] Andrews | A tool for creating and visualising formal concept trees[END_REF]. Additional views can be provided, like in RV-xplorer [START_REF] Alam | RV-Xplorer: A Way to Navigate Lattice-Based Views over RDF Graphs[END_REF], with pie-charts showing local view on a concept, its content and its neighboring, and statistics on the next levels to guide the user. Other approaches do not disclose the DAG structure, such as CREDO [START_REF] Carpineto | Exploiting the Potential of Concept Lattices for Information Retrieval with CREDO[END_REF] or SearchSleuth [START_REF] Ducrou | SearchSleuth: The conceptual neighbourhood of an Web query[END_REF] for web queries which show links and terms that can be clicked to navigate on the underlying lattice, e.g. for specializing or generalizing the query; SORTeD [21] where terms can be added or removed; SPARKLIS [START_REF] Ferré | Sparklis: An expressive query builder for SPARQL endpoints with guidance in natural language[END_REF] for RDF data which uses lists of urls, types, operators; or ConceptCloud [START_REF] Greene | Visualizing and exploring software version control repositories using interactive tag clouds over formal concept lattices[END_REF] which represents a focus concept by a tag cloud showing objects and attributes (with size and colors for highlighting importance and categories) and tag selection to change the focus concept. Some approaches use objects representations other than terms, such as photos [START_REF] Ferré | CAMELIS: Organizing and Browsing a Personal Photo Collection with a Logical Information System[END_REF] or movie posters [START_REF] Crampes | Visualizing and Interacting with Concept Hierarchies[END_REF]. As presented in this paper, RCAviz shows the structure, enables selecting a focus concept from initial objects or attributes, enables going step-by-step towards sub-concepts and super-concepts, and backtracks to a previous step.

Visualizing in RCA tools Only four tools implement RCA: Galicia, eRCA 12 improved in Galatea 13 , RCAexplore, and a Cogui14 plugin. This last plugin was also converted as a Java library named FCA4J 15 . RCAexplore and Cogui have the largest functional features regarding the number of input/output and accepted quantifiers. Regarding the visualization, Galicia provides an advanced lattice representation (tridimensional, graph layout built with a force algorithm) but no specific operators to navigate within a structure or from one structure to another. RCAexplore provides a visualization tool relevant for navigating small-size structures. Compared to Galicia, eRCA/Galatea, and Cogui, RCAexplore provides a browser enabling the navigation between concepts textually described (no visualization), shows the whole structure but does not enable zooming. Concepts and relational attributes are clickable to go from one structure to a neighboring structure, but selecting a focus concept and going step-by-step using local views is not proposed. Finaly eRCA/Galatea, RCAexplore, and Cogui produce outputs in the dot file format, which can be visualized or transformed using Graphviz 16 . In RCAviz, several visualization panels show local views, and allow the user to know which neighboring structure has been previously accessed or can be accessed through relational attributes. The step-by-step navigation is done by clicking on the relational attributes. The colors help to distinguish the different sets of objects. Table 2 synthesizes the main features of RCA tools regarding our requirements.

Conclusion

This paper presents RCAviz, a web platform devoted to visualize and navigate within relational conceptual structures computed using RCA. First uses showed the relevance of the features adopted to navigate within and between conceptual structures. These uses also enabled to identify a need for additional features to be integrated, e.g. specifying the path length to be shown, presenting the complete conceptual structures when they are small, local computation as in [START_REF] Bazin | Ondemand relational concept analysis[END_REF], visualizing the ER model, showing a larger concept neighborhood, or expressing complex queries to initiate the exploration. Additional testings will be conducted to consolidate the reliability of the software. This release is a first step in the building of a general platform to support exploratory search in relational datasets. To this end, future development will also focus on considering implications visualization. Large user experiments will be carried out in the context of the Knomana project.

Fig. 2 .

 2 Fig. 2. The Concept Selector. The user can select Objects and Attributes of interest. Then, the list of the available Concepts is updated on the right, and the user can select one of them to launch the Explorer and the History. Concept 20 is in bold face as it introduces the selected attribute westernAfrica.

Fig. 3 .

 3 Fig. 3. The History. The top panel shows the History containing the list of the previous explored concepts. The Explorer. On the center panel, the view shows a concept and its neighborhood. The user can navigate through the concepts by clicking on them. The left panel shows the previous DAG displayed. When the user hovers over a link of another DAG, the last is displayed on the right panel.

Table 1 .

 1 Example of relational dataset made of the Formal Contexts (FC) plants (FC Plant), pests (FC Pest) and countries (FC Country ). Plants are described by their family and a few characteristics. Pests are described by their family and entities they attack. Countries are described by their place in a continent. Relational Contexts (RC) connect objects of two FCs. RC treats connects FC Plant and FC Pest. RC isFoundIn indicates in which country is found a pest. RC possesses indicates which country possesses a plant.

	FC Plant	lauraceae asteraceae aromatic comestible toxic evergreen contraceptive antidysenteric applicOil applicEssentialOil applicExtract
	cinnamomumZeylanicum	x	x	x	x		x
	chromolaenaOdorata	x	x		x	x		x	x
	aspiliaAfricana		x	x		x	x	x	x	x
	ageratumConyzoides	x	x		x	x		x	x	x
		Pest	attacksPeanuts attacksCheese attacksRice aspergillus	Country	australAfrica westernAfrica
	aspergillusFlavus		x		x		Namibia	x
	aspergillusParasiticus	x			x		SouthernAfrica	x
	aspergillusOchraceus	x			x		Nigeria	x
	aspergillusTamarii		x	x	x		Benin	x
			RC treats	aspergillusFlavus aspergillusParasiticus aspergillusOchraceus aspergillusTamarii
		cinnamomumZeylanicum	x				x
		chromolaenaOdorata					x
			aspiliaAfricana				x
		ageratumConyzoides	x
	RC isFoundIn	Namibia SouthernAfrica Nigeria Benin	RC possesses cinnamomumZeylanicum chromolaenaOdorata aspiliaAfricana ageratumConyzoides
	aspergillusFlavus	x				Namibia
	aspergillusParasiticus			x	SouthernAfrica
	aspergillusOchraceus			x		Nigeria			x
	aspergillusTamarii		x			Benin			x

Table 1 .

 1

	8 A	navigation	for	this	example	using	RCAviz	is	presented	at
	https://rcaviz.lirmm.fr/documentation.html#examples. A video on YouTube
	is also proposed.

Table 2 .

 2 Comparison with other RCA tools on the visualization requirements.

		R1	R2	R3	R4
		Initial concept	Local visu with	Step-by-step nav.	History
		selection by O/A only concept neighbors among concepts with clicking
	Galicia
	eRCA/Galatea
	RCAexplore		Textual descr. only	In concept browser and
			(concept browser)	Along relational attr.
				in global visu.
	RCAviz	x	x	x	x

http://www.iro.umontreal.ca/ galicia/

http://dataqual.engees.unistra.fr/logiciels/rcaExplore/

http://rcaviz.lirmm.fr/

https://d3js.org/

https://github.com/eligrey/FileSaver.js/

https://rcaviz.lirmm.fr/

https://code.google.com/archive/p/erca/

https://github.com/jrfaller/galatea

http://www.lirmm.fr/cogui/

https://www.lirmm.fr/fca4j/

Acknowledgments The authors wish to thank the anonymous reviewers for their valuable comments that helped improving the paper. They also warmly thank Alain Gutierrez for his strong support during this work and for enabling FCA4J to generate output files with the JSON data format as required by RCAviz. This work was supported by the French National Research Agency under the Investments for the Future Program, referred as ANR-16-CONV-0004.

This work was supported by the French National Research Agency under the Investments for the Future Program, referred as ANR-16-CONV-0004 4 https://rcaviz.lirmm.fr/