Fatima El

Myriam-Emilie Kessari

Giulia Palma

Leila Temri

Hadad Gauthier

Leïla Temri

Selma Tozanli

Towards a sustainable convention: values and practices in the French stone fruits'value chain

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

Introduction

As some authors point it out, the food systems are undergoing a "quality turn" [START_REF] Goodman | The quality 'turn' and alternative food practices: reflections and agenda[END_REF] or a "qualitative shift" [START_REF] Ilbery | Registering regional speciality food and drink products in the United Kingdom: the case of PDOs and PGIs[END_REF][START_REF] Marsden | Sustainable agriculture, food supply chains and regional development: editorial introduction[END_REF][START_REF] Murdoch | Quality, nature, and embeddedness: some theoretical considerations in the context of the food sector[END_REF][START_REF] Kirwan | The interpersonal world of direct marketing: examining conventions of quality at UK farmers' markets[END_REF]. We assist to the emergence of alternative initiative or alternative food networks (AFN), which have in common to try to overcome the unsustainability of the conventional food systems. The AFN gather "road embracing term to cover newly emerging networks of producers, consumers, and other actors that embody alternatives to the more standardized industrial mode of food supply [START_REF] Murdoch | Quality, nature, and embeddedness: some theoretical considerations in the context of the food sector[END_REF] » [START_REF] Renting | Understanding alternative food networks: exploring the role of short food supply chains in rural development[END_REF]. They also pretend to have "an ecological and social vision and discourse embracing environmental awareness and progressive social goals [START_REF] Watts | Making reconnections in agro-food geography: alternative systems of food provision[END_REF][START_REF] Hassanein | Practicing food demo-cracy: a pragmatic politics of transformation[END_REF][START_REF] Goodman | The quality 'turn' and alternative food practices: reflections and agenda[END_REF][START_REF] Renting | Understanding alternative food networks: exploring the role of short food supply chains in rural development[END_REF]Hendrickson and Heffernan, 2002;[START_REF] Feenstra | Local food systems and sustainable communities[END_REF]" [START_REF] Kirwan | The interpersonal world of direct marketing: examining conventions of quality at UK farmers' markets[END_REF]. The sustainability outcomes of the AFN are quite hard to catch and their effectiveness too: as Treagear (2011) points it out, it is quite dependent of the goals of each initiative and the border of each initiative can be blurred. Thus, [START_REF] Watts | Making reconnections in agro-food geography: alternative systems of food provision[END_REF] distinguish weak and strong AFN: the strong AFN are those that are totally independent of the conventional systems (such as the Short Food Supply Chains) and the weaker initiatives are those that are still submitted to the conventional system (such as the organic certification chains). If [START_REF] Watts | Making reconnections in agro-food geography: alternative systems of food provision[END_REF] states the possibility of hybrid systems, as well as [START_REF] Sonnino | Beyond the divide: rethinking relationships between alternative and conventional food networks in Europe[END_REF], the crossing from a degree to another is not very clear and as [START_REF] Dupuis | Alternative modes of governance: organic as civic engagement[END_REF] point it out, we suggest that "the dynamics of these alternative markets are poorly understood" and we can add that to our knowledge, the dynamics between conventional and alternative markets could also be further explored.

Yet, for the food industry, the "sustainable" turn seems essential: "Future developments in food markets depend considerably on the differentiated capacity of food circuits to regain consumer trust and establish new institutional arrangements guaranteeing food quality in credible ways. » [START_REF] Renting | Understanding alternative food networks: exploring the role of short food supply chains in rural development[END_REF]p396). The objective of this paper is to explore this sustainable turn and thus to contribute to the understanding of the dynamics between the conventional and alternative markets. Relying on exploratory data, we place our work in the conventions theory (CT) framework to analyze the restraints to integrate some sustainable dimensions of the AFN for the fruit industry. We treat of the specific case of the stone fruits chain, by studying the peach and apricot chains. In this article, we first present the framework of the CT and the sustainable criteria of the AFN. Then, we present our results chain's actor by chain's actor (processors, supermarkets, corner grocery stores and producers). We indicate their consideration for the local and organic food products and their values around these themes. Finally we discuss these results within the CT conceptual framework showing that the producers seems anchored in the industrial world while the other actors are in majority coordinated by the market world.

Thus, we suggest that the levers will be different for the producers and for the other actors of the peach and apricot chains and yet some common values have to been found within the chain.

Theory:

Alternative food network as a framework to find the drivers of a sustainable convention.

Food systems and convention theory

In addition to the classical analysis of coordination by the market, several other streams have incorporated the role of institutions into the New Institutional Economy framework, such as the convention theory. The concept of convention, as used in economics, was first exposed by Durkheim. (Dupuy et al., 1989). He criticized the contract that seems too limited to understand the coordination. He argued that contractual relationships require a prior common framework. A convention can thus be understood as an unfixed framework of rules that will guide the actions of actors. Boltansky and Thévenot developed this theory and the concept of convention in their major opus, "On Justification: Economies of Worth" published in 1991. They proposed six worlds (also called worth) in which actors are able to interact. Coordination will be possible if the actors share some common values or worth. In the domestic world: the reference is the family; hierarchy is very important and the place of the "father" is the most important. Respect for tradition and paternalism are part of this world. In the civic world: the notions of group and democracy are the primary elements. Law, rules and voting are the elements that guide actors. In the industrial world: the reference is the measurement, which means that the engineer is the most important person. The final goal is technical progress. The coordination occurs around quantitative aspect or around technical consideration. In the market world, the final goal is to earn money. The price and the market are the references for the coordination. Originally, in this world, the most important person is the accountant. The world of fame is guided by the public reputation. "The worth is nothing but the results of the other people's opinion" ( Boltonski and Thévenot, 1999:371). And finally, in the world of inspiration, the creation is predominant; it's the artistic world. Dequech (2008) argues that "worth in the world of inspiration does not depend on prestige and critical success, or on market price and public success"...but "reflects one's self-esteem and not the opinion of others".

It is important to posit that actors are not locked into one world but may of course share values from several worlds and that the proposition of the six worlds is not fixed. Thus, [START_REF] Evans | Consuming conventions: sustainable consumption, ecological citizenship and the worlds of worth[END_REF]p111) remembers the possibility of new proposal and specifically here in the sustainable domain: « pursuing fully the ramifications of is the emerging possibility of a green order of worth (Boltanski and Thévenot, 1999: 369). The analysis that follows works with the possibility that a green order of worth -underpinned by environmental conventions -might be emerging through which practices of sustainable consumption might be legitimated ». Following this example, it seems interesting to try to design what could be some of the characteristics of a "sustainable" convention. The possibility of a sustainable or green convention has been the object of debates (Boltanski and Thevenot, 1999;[START_REF] Godard | De la pluralité des ordres -Les problèmes d'environnement et de développement durable à la lumière de la théorie de la justification[END_REF][START_REF] Bruno | Socio-économie de l'environnement et du développement durable : état des lieux et perspectives[END_REF]. To considerate the sustainability, do we have refer to a new order of worth or do we have to focus on the six "classical" orders of worth? In this exploratory work, we will above all consider that the AFN are a reference for a more sustainable food system and that we need to remind the sustainable characteristics of the AFN to guide the definition a sustainable convention or to consider the compromises existing from the "classical" worlds.

Sustainability drivers in the food sector

We suggest not only considering a "green order of worth" but a 360 degrees sustainable convention. Sustainability in the food systems is generally considered, as in other fields, according to three axes: the economic one, the environmental one and the social one [START_REF] Forssell | The sustainability promise of alternative food networks: an examination through "alternative" characteristics[END_REF]. By linking the sustainable expected outcomes to the specificity of the AFN, we propose to highlight the main themes of the sustainability in the food systems. It is important to precise that the sustainability of the AFN is discussed in the literature, and we assume in this work, that some of the following elements are the object of discussions [START_REF] Born | Avoiding the local trap. Scale and food systems in planning research[END_REF][START_REF] Tregear | Progressing knowledge in alternative and local food networks: critical reflections and a research agenda[END_REF][START_REF] Forssell | The sustainability promise of alternative food networks: an examination through "alternative" characteristics[END_REF].

Environmental outcomes

Following the suggestion of Forsell and Lankoski (2015), we will consider that the environmental impact of the AFN will take into account the environmental aspects of food production, processing, packaging, distribution, consumption and the use and damages of resources (water, air, animal welfare, etc.).

The local attribute of the AFN has linked them to the concept of foodmiles (Marsden et Murdoch, 2006;Noberg-Hodge, 2002cited in Edwards-Jones et al., 2008). The foodmiles is a concept considering the distance traveled by a product before reaching to the final consumer [START_REF] Smith | The validity of food miles as an indicator of sustainable development[END_REF]cited in Edwards-Jones et al., 2008).). It is often considered that the reduction of the food miles reduces the greenhouse gas emissions, and the AFN, by being local will reduce the food miles. We cannot consider the environmental impact of the AFN only by the local aspect: it is also important to consider the assumption that the AFN production systems could be more environmentally friendly.

It is considered that we will not find in the AFN intensive systems of production that will require high levels of chemicals input or drugs for breeding. According to some authors, the organic farming is considered as an agro-alternative system (Qazi and Selfa, 2008); [START_REF] Watts | Making reconnections in agro-food geography: alternative systems of food provision[END_REF].

Two elements are key in the AFN literature to assure the environment sustainability: the local attribute of the systems and the green practices (as the organic one).

Economic outcomes

Forsell and Lankoski (2015) consider AFN economic outcomes including the incomes and livelihoods of producers and the local development. Most of the literature is turned on three economic aspects: the income/livelihoods of the producers and the impact on territories (that can include social aspects, such as the food accessibility).

First of all, at an individual scale, it has been showed that the AFN can have a direct positive impact on the producers 'income. The reduction of the chain seems to confer "a significant amount of value added (…) at the farm level" (Marsden et al., 2000 p 435), the reduction of the informational distance with the consumer can confer a better price for the producer (Marsden, 2000;[START_REF] Renting | Understanding alternative food networks: exploring the role of short food supply chains in rural development[END_REF], elements confirmed by the producers explaining there motivation in participating to local food chains by the greater sale prices they can obtain in such type of selling (Miller and Buller, 2003).

At the territory level, some other economic outcomes of the AFN have been pointed out: it seems that the AFN can support the maintaining of the rural economic activity contributing to the creating jobs but also permitting that the money spent on food contributes to the local economy [START_REF] Follett | Choosing a food future: differentiating among alternative food options[END_REF]Forsell and Lankoski, 2015).

Social outcomes

Forsell and Lankoski (2015) integrate in the social outcomes the "labor rights, and the safety of the workers, consumer health, food culture, and the accessibility, availability,, and affordability of nutritious food (food security)." The consumers seem to benefit the AFN first by the health dimension: several academics point out the consumers' interest for a fresher food, which includes more nutrients [START_REF] Born | Avoiding the local trap. Scale and food systems in planning research[END_REF]La Trobe et Little cited in Treager, 2011). If we consider the "greener" production practices in the AFN, we can suppose that the products will be healthier (less chemicals, etc.) (Forssel and Lankoski, 2015) but we can also do the parallelism for the producers that will be less confronted to the chemicals, and thus protect their own health. The second dimension for the consumers is the affordability that could be considered as a socio-economic one with the objective to permit the access to fresh food for low-income consumers.

The relations between the producers are less studied, and this can be explained by the fact that the AFN involving producers collective are less numerous. However, [START_REF] Chiffoleau | From Politics to Cooperation : The Dynamics of Embeddedness in Alternative Food Supply Chains[END_REF] shows how producers involved in AFN will reinforce their links, especially exchanging information, cooperating technically and developing friendship. She also argues how this type of configuration can finally lead to innovation.

Finally, the empowerment and shared governance among the chain can be expected in the social outcomes but also in the economic one. Indeed, if the democracy is expected in an AFN, the control of the chain and thus of the prices among the chain is another key element.

We can see here that in the three sustainable dimensions, several themes seem fundamental to assure this sustainability. In the environment dimension, the local characteristic and the green practices seem essential. The proximity encompasses the local situation and closed relationship and can be associated to the domestic world [START_REF] Raynolds | Fairtrade, certification, and labor: global and local tensions in improving conditions for agricultural workers[END_REF]. Nevertheless, the references to technical practices and to some certifications permit to include an industrial side to the environmental dimension. The expected economic outcomes are focused on two elements: the income of the producers and the affordability of food for consumers. Beyond the market value, the concern to renew the market power and to get a more collective governance among the chain bring a civic taint to the market one. Finally, the social outcomes are associated to health preoccupation and relationship elements. This permits to consider civic and domestic worth together in the case of the social side of the sustainability.

According to the expectations and these values, which restraints could be found that prohibit the way to the sustainability for the stone fruits chains?

Material and methods

Case presentation

According to 2010 Agreste statistics, in the EU 28, 11% of the peach and 31% of the apricot productions were realized in France, mostly concentrated in the Rhône-Alpes and Languedoc-Roussillon regions of the country. These two value chains are now economically "vulnerable because of a drop in acreage, aging orchards and the high cost of salaried labor that pull down continuously the competitiveness of France face to other competing producer countries and against the strong import competition (in particular from Italy and Spain). The introduction of more sustainable pathways to meet the growing demand for "sustainable" fruits and vegetables could be an asset to withstand international competition and thus maintain farm activity. However, organic production represents a very marginal share of the production of peaches (2.5%) while it is slightly higher for apricots (7.3%). Moreover, peach and apricot value chains are complex because of their high seasonability, perishability and high price fluctuations. On the other hand, implementing sustainable practices is not so evident. Thus, it seems important to examine current practices and values of the actors in these value chains as to better identify drivers and obstacles to sustainable development.

Data collection

A semi-structured survey, integrating a set of themes concerning aside general data such as the turnover, the volume produced, collected or processed, integrating specific data on organic fruits, has been developed. Although several questions are common, different questionnaires were created to better target different actors: for super/hypermarkets, corner grocery stores, industrial processors and producers. So, we investigated 16 fruit processing companies, 6 large retail chains, 4 specialized stores (among which 3 coops and 2 organic stores), 4 wholesalers and 51 producers. The interviews have been done by phone or face to face and have lasted between 30 minutes and 2 hours. The geographical whole area of Languedoc-Roussillon region.

Diagram 1: Actors and linkages in the peach and apricot value chain

Source : Elaborated by the authors

Data analysis

In this exploratory work, we did not consider all the elements of the surveys but only some specific elements that could permit to address the specific dimensions of the During each interview, the answer linked to these themes have been the object of a content analysis permitting us to highlight the different position of the chain actors concerning the proximity, the organic certification, the governance and the economic aspects. It has to be specified that these themes were not present in their totality in each group of actors.

Results

Processors (P)

Organic

Considering the fruit processing SMEs, we have found that 10 out of 16 are working with the organic certification. This figure is quite important considering the market share of the organic fruits production in France (7.7% for total fruits and 2.3% for apricots and peaches) 1 . Nevertheless, it is important to consider that this share of organic products in the processing industries is highly variable from one industry to another: from less than 20% to 100%. Moreover, it is important to mention that each of these SMEs may have a different reason to include organic products in their industry. For example, market worth prevails when the processors declare that they opt for organic production to better respond to the expectations of their clients (P1, P3, P14, P16) or because they expect to increase their market shares by including organic in their product line (P1, P14). The industrial worth is also mentioned by other processors who consider that the organic certification is above all a set of specifications to follow (P1, P5, P13, P14). Finally, civic worth arises when the processors mention their "values" (P8, P9, P11, P14) or say that they consider above all the welfare of the consumers and the producers (P9) or that they are committed to the organic way of production.

However, some of the interviewed processors did not have organic production in their product line. For those, the market and volume availability (industrial worth) appear to be essential obstacles that restrain them from going organic. One of those processors declared: "we need to achieve economies of scale by working on large and stable volumes. As the customers do not ask it, the volumes [of organic production] stay too small and hazardous" (P2) or another processor said: "the problem with the organic products is the availability of the raw material [fresh organic fruits]: the sourcing is neither satisfactory in quantity, nor stable" (P4).

Importance of the localization of the production

The processors are directly linked to the producers (or the co-operatives, i.e. group of producers) for their fresh peach and apricot sourcing. When they are choosing their suppliers, most of them pay attention to the farm prices (P1, P4, P6, P9, P10, P13, P15) but some others also consider local sourcing as a priority (P4, P6, P9, P11, P16), while for some others (P1, P2, P3, P4) availability and stability of the quantities are the most important factors that shape their sourcing policy.

Consideration of the other actors of the value chain

A quarter of the processors interviewed consider that the most important actors in the chain are the consumers (P1, P2, P4, P16). Nevertheless, for some, all the actors operating on the upstream and the downstream activities of the chain have equal importance (P11, P13) while some others express that the consumers and the producers are the two main group of actors to take into consideration (P14, P15).

Wholesalers (WH)

Six wholesalers of which 4 are co-operatives were interviewed within the framework of our research. WH1 and WH2 are operating in the Min de Montpellier and deal with great volumes of fruits and vegetables. All the produce that they handle is conventional. Both indicated that they are not selling organic produce because of the lack of market demand.

Their sourcing pattern depends entirely on the seasonality of the produce. Most of the volumes that they handle are imported from Spain and other southern Mediterranean countries at the early season, but then they deal with local products. These wholesalers are entirely driven by market worth and do not have any initiative to move towards sustainable chain or alternative food networks. The four other wholesalers (co-ops) are entirely dedicated to the organic fruit chain. They source organic grocery stores as well as large retailers in organic produce. They are certified organic and WH3 and WH6 have also DOP certified produce (apricots). WH5 is also certified EcoCert. Concerning the sourcing patterns, WH3 and WH4 resort to imports at early season but then turn to local procurement at regional level while WH6 sources only locally. Concerning the economic aspects, these co-operative wholesalers are worried about the price and the margins that they make. Another concern pointed out by them was the diminishing number of agricultural producers and the difficulties that they have to continue the local sourcing.

The four co-operatives are engaged in the development of sustainable chain and invested (or foresee to do so) in planting solar and/or photovoltaic panels. They can be placed in a hybrid market-civic world.

Supermarkets (SM)

The supermarkets' organic fruit sourcing volumes are quite low and change between 1% (SM1) and 5% for the SM2 and SM3. This can be explained by this quote "But in general the organic consumers go mostly to specialty grocery stores". The three interviewed supermarkets had a real concern about the local dimension as to better respond to their clients' demand. In fact, consumers have high expectations concerning the local criterion:

"the proximity criterion gains of importance. This takes precedence over the environment" (SM1). SM3 bases its strategy on the proximity, in order to respond to its clients' demand. Thus, its first criterion to choose its suppliers is the proximity criterion as "to support the local producers" (SM3). However, this proximity criterion has its limits. For example, the sourcing structure of SM2 does not permit its purchasing manager to choose its suppliers, as this supermarket is part of a general buying group. For all these three interviewed supermarkets, the focal point is to "satisfy the clients' demand". Nevertheless, we observe that the market worth is tainted by some domestic consideration with the strong local discourse and by some civic values as in the case of SM3 and to a lesser extend, in the case of SM1.

Corner grocery stores (CGS)

For the corner grocery stores, two main factors taken into consideration are the quality of the produce and the proximity to the producers. One of them is specialized in organic products and 100% of the products sold are certified organic. Three out of four of the interviewed corner grocery sell organic products in parallel with conventional products and one of these has half of its turnover realized by the sale of organic products half of its turnover (CGS4). CGS1 limits its organic products' line based on vulnerable conservation issues and on the scarce sourcing possibilities at the regional level.

The criteria that the corner grocery stores take into account in choosing their suppliers are variable but are not closely linked to environmental consideration: for example, for CFS3, the taste is the first choice element: "I go to the market in Alpilles and taste directly the product, I ask from where it comes and then, if the taste pleases me, I get in touch with its producer". Further, CGS1 also mention the price as one of the choice criteria.

With regard to the vision on the clients' demand, the answers collected are very homogeneous despite the fact that the proximity and the taste step out as the two main choice criteria. However, the "proximity" is defined differently from one shop to other.

For CGS1, proximity is there to show the quality of the produce while CGS3 exposes the proximity as to justify the price of the fruits. Finally, all of these corner grocery shops have the same priories for the future, as they all want to sell local products. However, CGS1 points out the seasonality, CGS2 mentions the quality and the quantity and also the difficulty to provide organic produce, while CGS3 and CGS4 indicate the competitiveness of local produce.

The market worth is present in the discourses of the interviewed corner grocery shops.

However the market worth is in one way or in another, always associated with the proximity and the local characteristic of the products. The environment is present but is not predominant and finally the way of growing the produce seems not to be a worry.

The "local side" of their values could be seen here more from the domestic perspective than from the civic one.

Producers

Near to half (24) of the 51 interviewed producers have a part or the totality of their field under the organic certification. The share of the organic area varies from 7% to 100% according to the producers. They are more to sell their products in short supply chains (36/51), and the turnover concerned by the short supply chain are from 2% to 100%. It is important to notice that the producers that are producing organic fruits are not necessary the ones that are selling through the short supply chains.

We suggest to the producers to define the ideal fruit fort them: first of all, the producers (26/44) are looking for fruits that tastes good, has sugar, which is juicy, etc. until some hedonistic consideration "a peach that gives pleasure when we eat it" (F32) or "that please the consumers"(F48). Nevertheless, it is important to consider that 15/51 producers mentioned technical elements linked to the difficulties to produce the fruits and thus would have a fruit provided by an exceptional yield or fruits that can resist against the different diseases. Some producers express (4/51) their interest for the environment considering for example an ideal fruit that could be "healthy and not filled with products"(F49). Finally, four of the producers hope a fruit that will provide the "largest net margin" (F42) or "a fair income to the producer" (F29).

Finally the organic certified producers are motivated because they prefer this way of producing and they don't refer to the clients demand or to the market: "the organic production allows to have a more complex environmental and technical approach thus it is more interesting" (F29). In the same way, the restrains to convert into an organic production mentioned by 20 producers are the technical difficulties and then the fear of the decrease of the yield they associated to "it is almost impossible to have a decent income in organic production because of the technical constraints" (F12). Nine producers mention the lack of clients or the lack of market for this type of production but it is never presented as the first argument.

For the producers, the most important actors in the chain are the consumers (22/51) of replies and the producer in (17/51) of responses. Intermediaries are only mentioned 5 times (5/51).

Discussion

Our results let appear that most of the downstream actors (GMS, small retailers and processors) are concerned about market consideration. Motivations to evolve to more sustainable practices are generally driven by market worth. Nevertheless, it is important to consider the local element, that is much more present in the discourse of the retailers (large retailers and corner grocery stores) and the wholesalers (when the produce available locally). We are facing here some market-domestic considerations that are not present in the group of processors. Finally, the producers are specifically concerned about industrial considerations: the main restraints to evolve towards more sustainable practices are technical reasons and the fear to lose their level of income if they if they take the challenge to change their conventional practices.

We first point out that there are not common values among actors of the stone fruits chain considering the evolution to a more sustainable chain. Secondly, it seems that the market (clients) could be the main driver for the actors operating in downstream activities of the chain. The fruit processing SMEs consider the organic production and the localization as the equally important factors that lead the way to a more sustainable value chain. For the large and small retailers, the way to a more sustainable chain seems rather to pass through the proximity element, i.e. the localization. The producers, operating in the upstream of the chain, consider the market and the client. But they are also influenced by industrial worth, as they think that it is really difficult to overcome the technical difficulties. Thus, the positioning of the retailers in favor of a more sustainable chain will have a direct and positive effect on the choice and the conduct of the producers. Whilst, the processors will continue to have difficulties for their sourcing in organic and local stone fruits and will continue (for some time) to rely on the imports from neighbor countries (Spain and Italy).

Concerning the debate of the pertinence of a sustainable order of worth, we can suggest the necessity to refer to a sustainable convention. If our results permit to understand some restraints to the sustainability in a classical chain, it doesn't offer a complete view of the levers and differences between a classical and a sustainable system. Indeed to understand, with a more complete view, these differences it would be interesting to consider all the elements present in a sustainable system, starting with their common base. The AFN share their will to renew the food system that could be seen as the "superior common principle". This let us suggest that even if some elements of the "repertoire" of the "classical" worlds can be present in a sustainable convention (such as the collective / solidarity aspects of the civic world) they are not complete to understand the schemes of action of a sustainable food system. And thus, the reference to the classical worlds is perhaps not sufficient to analyze the differences between the sustainable systems and the not sustainable ones.

  the chain, that is considered by the vision of the other actors of the chain the economic aspect in the case when the interviewees are linked to the first two themes, i.e. to the proximity and the organic certification.

Estimations based on the figures of

given by FranceAgriMer for total fruits production in volume and by Interfel for organic fruits production


S Briot

P Martinet

F Chaumette
email: francois.chaumette@inria.fr

Determining the Singularities for the Observation of Three Image Lines Sébastien Briot 1 , Philippe Martinet 2 and Franc ¸ois Chaumette 3 Abstract-The determination of the singularity cases in the observation of image features is a complicated problem which is still open, apart from image points. For the first time, we provide the singularity cases in the observation of three image lines. We show that a concept named the "hidden robot", which was formerly used for understanding the singularities of a visionbased controller dedicated to parallel robots and, more recently which proved to be efficient for finding the singularity cases in the observation of image points, can be used for interpreting the singularities in the observation of three image lines. The hidden robot concept considerably simplifies the analysis by using geometric interpretations of the mapping degeneracy and tools provided by the mechanical engineering community. We prove that in the most complicated case where three general lines in space are observed, singularities appear when the origin of the observed object frame is either on a quadric or a cubic surface whose location in space depends on the configuration of the observed lines. In simpler cases where at least two lines belong to the same plane, these two surfaces can degenerate into simpler geometrical loci (e.g. planes, cylinders, lines).

Index Terms-Visual Servoing, Kinematics

I. INTRODUCTION

F INDING the singularity cases in the observation of image features is crucial in visual servoing and pose estimation [START_REF] Hutchinson | A tutorial on visual servo control[END_REF]. However, this problem is still unsolved for most of the cases due to its considerable complexity. The singularities were found in the case of three image points after rather complicated mathematical computations [START_REF] Michel | Singularities in the determination of the situation of a robot effector from the perspective view of 3 points[END_REF], but they are still unknown for other image features.

In order to avoid the controllability issues near singularities, authors usually propose to observe additional features. However this leads to a non-minimal representation of the interaction between the system and the camera, resulting in the potential appearance of local minima [START_REF] Chaumette | Visual Servoing and Visual Tracking[END_REF] which are also difficult to determine. Additionally, even adding visual features may not ensure the absence of singularity cases [START_REF] Rosenzveig | A method for simplifying the analysis of leg-based visual servoing of parallel robots[END_REF]. Therefore, finding the singularities is crucial but it is usually prevented by the complexity of the mathematical problem to solve.

Recently, a concept named the "hidden robot" was introduced in [START_REF] Briot | Minimal representation for the control of Gough-Stewart platforms via leg observation considering a hidden robot model[END_REF], [START_REF] Briot | The hidden robot: an efficient concept contributing to the analysis of the controllability of parallel robots in advanced visual servoing techniques[END_REF]. This concept was first used to determine the singularity cases of a vision-based controller dedicated to parallel robots [START_REF] Andreff | Vision-based control of a Gough-Stewart parallel mechanism using legs observation[END_REF] but it was recently proven that it can be extended to more general cases, such as the observation of n image points [START_REF] Briot | Revisiting the determination of the singularity cases in the visual servoing of image points through the concept of "hidden robot[END_REF].

The basic idea shown in [START_REF] Briot | The hidden robot: an efficient concept contributing to the analysis of the controllability of parallel robots in advanced visual servoing techniques[END_REF], [START_REF] Briot | Revisiting the determination of the singularity cases in the visual servoing of image points through the concept of "hidden robot[END_REF] is that the singularity cases of the mappings involved in the visual observation of geometric primitives for estimating the pose of an object or for visual servoing are equivalent to mappings representing the geometric and kinematic properties of given hidden parallel robots. By geometric property, we mean that the solutions of the Forward Geometric Model (FGM) of the parallel robots under consideration are also the solutions of the 3-D localization problem associated with the considered observations. By kinematic property, we mean that the singularities of the inverse kinematic Jacobian matrix of the robot are the same as the singularities of the interaction matrix that defines the link between the time variation of the visual features and the camera velocity. The hidden robot is indeed a tangible visualization of the mapping between the observation space and the Cartesian space. A methodology is proposed in [START_REF] Briot | The hidden robot: an efficient concept contributing to the analysis of the controllability of parallel robots in advanced visual servoing techniques[END_REF] in order to define the hidden robot models associated with visual servoing dedicated to parallel robots while [START_REF] Briot | Revisiting the determination of the singularity cases in the visual servoing of image points through the concept of "hidden robot[END_REF] presents an approach for finding the virtual robot models associated with more general cases based on the observation of geometric visual primitives.

By finding this correlation, it is then possible to study the singularities of the interaction matrix, by using advanced tools coming from the mechanical engineering community (e.g. the Grassmann-Cayley algebra [START_REF] Ben-Horin | Singularity analysis of a class of parallel robots based on Grassmann-Cayley algebra[END_REF] and/or the Grassmann geometry [START_REF] Merlet | Parallel Robots[END_REF]). The interest in using these tools is that they are (most of the time) able to provide simple geometric interpretations of the singularity cases. However, it should be noted that these tools still require an experienced user.

In the present paper, we show that a virtual parallel robot is hidden within the mapping used in the observation of image lines [START_REF] Andreff | Visual servoing from lines[END_REF] and, thanks to this correlation, we analyze the singularity cases when three lines are observed. To the best of our knowledge, this is the first time that this problem is solved, except for obvious cases such as when the three 3-D lines are parallel (leading to an unconstrained translation along the direction of the lines).

As a result, the paper is organized as follows. Next section shows that the geometric / kinematic mapping involved in the observation of image lines is the same as the mapping

z x y i U i u i l i L image plane i L i ℓ i m i M i P O Fig. 1. Observation of a line
required to control a particular 3-UPRC parallel robot 1 . Then, in Section III, the singularities of the mapping are analyzed and validated through simulations. Finally, conclusions are drawn in Section IV.

II. THE ROBOT MODEL HIDDEN IN THE OBSERVATION OF LINES

Before presenting the architecture of robot hidden in the observation of three lines, we make some brief recalls on the computation of the related interaction matrix.

A. Interaction matrix of three image lines

In the following section of the paper, we use the standard pin hole model with focal length equal to 1 for the representation of the camera model. However, any other model based on projective geometry could be used.

A 3-D line L i of Plücker coordinates [U T i L T i ] T in the camera frame (U i being a unit vector characterizing its direction while L i = U i × X, X being the coordinates of any point M i belonging to L i ) is projected in the image plane on a 2-D line i of Plücker coordinates [u T i l T i ] T (u i being a unit vector characterizing its direction while l i = u i × x, x being the coordinates of any point m i belonging to i ) (Fig. 1). From [START_REF] Chaumette | La relation vision-commande : théorie et application à des tâches robotiques[END_REF], we know that

u i =   u xi u yi u zi   =   L yi /∆ -L xi /∆ 0   , l i =   l xi l yi l zi   =   L xi /∆ L yi /∆ L zi /∆   (1)
where

L i = [L xi L yi L zi ] T and ∆ = L 2 xi + L 2 yi .
By differentiating these equations, the classical equations linking the velocities ˙li (and as a result the velocities ui ) to the twist τ T c = [υ T c ω T c ] of the camera in its relative motion with respect to the observed object frame (υ c being the translational velocity and ω c the rotational velocity expressed in the camera frame) are ˙li = M i τ c where

M i = [M T i1 M T i2 M T i3 ] T with M i1 =         l xi l yi U zi /∆ l 2 yi U zi /∆ -l yi (l xi U xi + l yi U yi )/∆ l xi l yi l zi l 2 yi l zi -l yi         T (3)
M i2 =         -l 2 xi U zi /∆ -l xi l yi U zi /∆ l xi (l xi U xi + l yi U yi )/∆ -l 2 xi l zi -l xi l yi l zi l xi         T (4)
M i3 =         (U yi + l yi l zi U zi )/∆ -(U xi + l xi l zi U zi )/∆ l zi (l xi U yi -l yi U xi )/∆ l yi (l 2 zi + 1) -l xi (l 2 zi + 1) 0         T (5)
In order to fully servo the relative motion between an object and a camera, at least three lines fixed on the object must be observed [START_REF] Andreff | Visual servoing from lines[END_REF] (Fig. 2). Thus, considering the observation of three lines L 1 , L 2 and L 3 , the interaction matrix linking the velocities ˙li of the lines i (i = 1, 2, 3) grouped in the vector ṡ = [ ˙lT

1 ˙lT 2 ˙lT 3 ] T to the camera twist τ c by the relation ṡ = Mτ c (6)
is thus given by

M = M T 1 M T 2 M T 3 T (7)
Singularities appear when the matrix M is rank deficient. Looking at the analytical form of M, it is clear that determining the singularities of M through the determinant of M T M seems to be out of reach. Note that we have used above the Plücker representation of lines, but following results are not dependent of the choice of the representation (Cartesian, cylindrical) [START_REF] Chaumette | La relation vision-commande : théorie et application à des tâches robotiques[END_REF].

B. Hidden robot model

In this section, we show that a virtual parallel robot is hidden within the mapping used in the observation of image lines by considering what follows. First, from the single measure of the location of a line i in the image plane, it is impossible to know the location of its corresponding 3-D line L i . The only information that we can extract from this measure is that the 3-D line L i lies on the interpretation plane P i passing through i and the optical center O. As a result, the measure of i gives the location of the plane P i . Furthermore, from the same single measure, the orientation of the observed body B cannot be deduced (we need the three measures of the positions of lines 1 , 2 and 3 , see Fig. 2).

From a mechanical engineer point of view, these geometric properties can be obtained by the kinematic architecture depicted in Fig. 3. This architecture is made of an actuated cardan (or universal (U )) joint rotating around point O and fixed on the camera frame at that point. The U joint is followed by a passive prismatic (P ) joint whose direction z i is reciprocal to the axes x i and y i of the cardan joint. Then, the passive P joint is attached at its other extremity to a passive revolute (R) joint whose axis is directed along y i . Finally, the last link (moving platform, equivalent to the observed body B) is connected to the R joint by a passive cylindrical (C) joint whose axis a i is orthogonal to the axis of the R joint. Thus, we have a leg with a U P RC architecture linking the camera frame to the observed object frame. To this leg, we associate a vector q i = [α i β i ] T , where α i and β i represent the rotation angles of each revolute joints composing the U joint (Fig. 3). With such a leg, the observed body B is constrained to translate along L i and to rotate around it. As a result, the vectors U i and L i associated with the Plücker coordinates of the line L i are collinear with the axes a i and y i , respectively (Fig. 3).

It should be mentioned that the rotational component of the C joint in this leg is necessary because we know that line L i is attached to a rigid element (the observed body B) whose orientation cannot be defined by considering a single measure i . Without taking into account this information, the displacement of the line L i can be obtained by a leg with a U P R architecture, i.e. a kinematic chain with two passive degrees of freedom (instead of four for the U P RC leg).

Considering now that:

• the three observed 3-D lines L 1 , L 2 and L 3 have motions whose geometric properties can be parameterized, for each of them, by a U P RC mechanical architecture depicted above, • L 1 , L 2 and L 3 are fixed on the same body B,

• there is a global diffeomorphism between the configuration of the line i and the U joint rotations q i ; as a result, l i is a singularity-free observation of the U joint motions q i , then the relative motion between the body B and the camera frame has the same geometric and kinematic properties as the motion of a 3-U P RC parallel robot (Fig. 4), i.e.

• the solutions of the FGM of the 3-U P RC parallel robot are also the solutions of the 3-D localization problem when three 3-D lines are observed by a perspective camera.

O P i L i α i β i x i z i ξ i1 ξ i2 a i , U i y i y i , L i x i i γ
• the singularities of the inverse kinematic Jacobian matrix of the robot are the same as the singularities of the interaction matrix [START_REF] Andreff | Vision-based control of a Gough-Stewart parallel mechanism using legs observation[END_REF].

To prove the last item, let us consider what follows. The inverse kinematic Jacobian matrix J inv of the robot links the actuators velocities q = [ qT

1 qT 2 qT
3 ] T to the platform twist τ p through the relation

q = J inv τ p ( 8
)
J inv is of dimension (6 × 6). Moreover, as mentioned above, there is a global diffeomorphism between the measure of the image line configuration i and the U joint rotations q i . As a result, q = Gṡ (9

)
where G is a (6 × 9) block-diagonal matrix which cannot be rank-deficient. Finally, by combining ( 6) and ( 9) with the screw transformation relation linking the camera twist τ c (expressed with respect to the object frame) to the observed object twist τ p (expressed with respect to the camera frame) by [START_REF] Chaumette | Visual Servoing and Visual Tracking[END_REF]

τ c = H τ p , where H = -I 3 [ c t p ] × 0 3×3 -I 3 (10)
(in which I 3 is the

(3 × 3) identity matrix, 0 3×3 is a (3 × 3)
zero matrix, and [ c t p ] × is the cross-product matrix associated to the 3-D vector c t p characterizing the translation between the camera and the observed object frames), we found:

J inv = GMH (11)
Then, as matrices G and H are never rank-deficient, we obligatorily have a loss of rank of M if and only if J inv is rank-deficient. Therefore, the conditions of singularity of the hidden robot inverse kinematic Jacobian matrix J inv are the same as the singularity conditions of the interaction matrix M.

III. SINGULARITY ANALYSIS

A. General conditions of singularity

Singularities on the inverse Jacobian matrix J inv of a parallel robot (also called Type 2 or parallel singularities [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]) appear when at least two solutions of the FGM are identical [START_REF] Merlet | Parallel Robots[END_REF]. As mentioned in Section II, these singularities are analogous to the singularities of the interaction matrix.

In Type 2 singularities, parallel robots gain one (or more) uncontrollable motion. Kinematically speaking, there exists a non-null vector t s defined such that J inv t s = 0 while q = 0, i.e. the actuators are fixed (which means that t s is in the null space of J inv ). As known in mechanics, if a rigid body got an uncontrollable motion, this means that it is not fully constrained by the system of wrenches applied on it, i.e. the static equilibrium is not ensured. As this uncontrollable motion appears only in a singularity, this means that locally the system of actuation wrenches, i.e. wrenches transmitted from the actuators to the platform by the legs, is degenerated [START_REF] Merlet | Parallel Robots[END_REF].

For a given leg i, any actuation wrench denoted by ξ ij is reciprocal to the unit twists denoted ζ ik characterizing the displacements of the passive joints [START_REF] Bonev | Geometric analysis of parallel mechanisms[END_REF], i.e. ξ T ij ζ ik = 0 for any j and k. This means that the virtual power developed by the wrench ξ ij along the direction of motion ζ ik is null; in other words, the actuator j of the leg i cannot transmit a wrench ξ ij to the platform along the direction ζ ik .

Let us consider a U P RC leg belonging to our 3-U P RC hidden robot. In the frame F i : (M i , x i , y i , z i ) (M i being the point of intersection between the axis of the prismatic and revolute joints) attached to the leg, the unit twist defining the motion of the passive P joint is expressed as (see Fig. 3):

ζ i1 = 0 0 1 0 0 0 T (12) while ζ i2 = 0 0 0 0 1 0 T ( 13
)
is the unit twist defining the motion of the passive R joint and

ζ i3 = cos γ i 0 -sin γ i 0 0 0 T ( 14
)
ζ i4 = 0 0 0 cos γ i 0 -sin γ i T (15)
are the unit twists defining the motions of the passive C joint, γ i being the angle between x i and a i (and thus U i ), both axes being contained in P i (Fig. 3). In these twists, the three first components represent the direction of the translation velocity while the three last components represent the direction of the rotational velocity.

As a result, the unit actuation wrenches

ξ ij = [f T ij m T ij ] T expressed in the frame F i are ξ i1 : f i1 = [0 1 0] T , m i1 = 0 3×1 (16)
ξ i2 : f i2 = 0 3×1 , m i2 = [sin γ i 0 cos γ i ] T (17)
in which f ij represents the direction of the force exerted on the platform and m ij the direction of the moment. As a result, ξ i1 is a pure force directed along y i and ξ i2 is a pure moment reciprocal to ξ i1 and U i which is included in the plane P i (Fig. 3). Thus, for the three robot legs, the system of actuation wrenches is given by ξ = [ξ 11 ξ 21 ξ 31 ξ 12 ξ 22 ξ 32 ]. There exists some tools able to define the conditions of degeneracy of a wrench system among which are the Grassmann geometry [START_REF] Merlet | Parallel Robots[END_REF] and the Grassmann-Cayley algebra [START_REF] Ben-Horin | Singularity analysis of a class of parallel robots based on Grassmann-Cayley algebra[END_REF], [START_REF] Ben-Horin | Application of Grassmann-Cayley algebra to geometrical interpretation of parallel robot singularities[END_REF]- [START_REF] Kanaan | Singularity analysis of lower mobility parallel manipulators using Grassmann-Cayley algebra[END_REF].

In what follows, we use the Grassmann-Cayley algebra in order to find the singularity conditions related to our problem. Indeed, as any wrench of the wrench system ξ can be seen as the Plücker representation of a line2 [START_REF] Merlet | Parallel Robots[END_REF], the Grassmann-Cayley algebra makes it possible to compute the determinant of the wrench system ξ as an expression involving twelve points selected on the six lines (corresponding to the six wrenches). This expression is a linear combination of 24 monomials [START_REF] Ben-Horin | Singularity analysis of a class of parallel robots based on Grassmann-Cayley algebra[END_REF], each monomial representing the volume of a tetrahedron whose extremities correspond to four of the considered twelve points. By a smart selection of the twelve points, due to the experience of the user, it is possible to vanish a large number of the monomials and thus to simplify the expression of the determinant of the wrench system.

Regarding our particular case, the Grassmann-Cayley algebra was used in [START_REF] Kanaan | Singularity analysis of lower mobility parallel manipulators using Grassmann-Cayley algebra[END_REF] to prove that, if the system of wrenches is composed of a pair of three forces ξ 11 , ξ 21 , ξ 31 and of three moments ξ 12 , ξ 22 , ξ 32 , conditions of singularities appear if and only if :

f 1 = f T 11 (f 21 × f 31 ) = 0 or f 2 = m T 12 (m 22 × m 32 ) = 0 (
18) which means that the vectors f 11 , f 21 and f 31 (m 11 , m 21 and m 31 , resp.) lie in the same plane (or are collinear to the same vector c). As a result, the mechanism is not able to resist to forces f (moments m, resp.) orthogonal to this plane (or in any directions orthogonal to c) and thus translations (rotations, resp.) are gained along the direction of f (around the axis of m, resp.).

From these two conditions, it is possible to find the configurations of the end-effector (the observed body) in the camera frame leading to singularities. Examples of singular configurations, depending on the arrangement of the lines on the observed body, are detailed below.

B. Singularity loci

Thanks to the particular geometric properties of the system to analyze (invariance of robot leg configurations for any rotation around O), it is possible to simplify the singularity analysis by fixing the platform orientation. Thus, all expressions below will be given for the "zero" platform orientation. For another given platform orientation (parameterized by the rotation matrix R), singularity loci will be found by parameterizing the variables X, Y and Z characterizing the position of the origin of the object frame F b in the camera frame (see Figs. 5 and6) when considering the "zero" platform orientation thanks to new variables X , Y and Z representing the position of the origin of the object frame for the considered "non-zero" platform orientation such that

X Y Z T = R X Y Z T (19)
1) Three coplanar lines with no common intersection point: In such a case, the three lines intersect in points M 1 , M 2 and M 3 (Fig. 5(a)) (we disregard the case where at least two lines are parallel, which is considered in Section III-B5). We define the frame F b : (Q, x b , y b , z b ) attached to the observed body B such that Q is the center of the circumcircle of the triangle ∆M 1 M 2 M 3 and x b is collinear to ---→ QM 1 . Moreover, for the analysis, we fix the orientation of the body so that F b can be obtained from the camera frame by a translation of vector --→ OQ. As a result,

--→ OQ = [X Y Z] T , ---→ QM 1 = [ρ 0 0] T , ---→ QM 2 = ρ[cos φ sin φ 0] T , ---→ QM 3 = ρ[cos ψ sin ψ 0] T (20)
where φ, ψ are two angles defined in Fig. 5(a) and ρ is the radius of the circle. Then, from section III-A, we know that

f i1 ∝ U i × --→ OM i , m i2 ∝ U i × -→ f i1 (21)
leading to, from (18), and after some tedious developments,

f 1 = 0⇔Z = 0 f 2 = 0⇔Z(X 2 + Y 2 -ρ 2 ) = 0 (22)
Condition Z = 0 means that the plane containing the lines L 1 , L 2 and L 3 also contains the camera center O, which is not critical in practice. In such a case, all three lines project in a single line in the image plane and the pose of the object cannot be determined. Condition X 2 +Y 2 = ρ 2 means that the camera center O lies on the cylinder whose axis is perpendicular to the plane containing all three points M 1 , M 2 and M 3 and which includes the three points. These conditions are the same as the singularity cases where three image points are observed [START_REF] Michel | Singularities in the determination of the situation of a robot effector from the perspective view of 3 points[END_REF], which is indeed not a surprise due to the equivalence of both problems.

1

M 2 M 3 M b x b y 1 U 2 U 3 U 1 L 3 L 2 L ) X Y Z ( Q φ ψ ρ (a) with no intersection point b x b y ) X Y Z ( Q 1 U 2 U 1 L 3 L 2 L φ ψ
(b) with a common intersection point Fig. 5. Observation of three coplanar lines 2) Three coplanar lines with a common intersection point: We define the frame F b : (Q, x b , y b , z b ) attached to the observed body B such that Q is the intersection point of all three lines and x b is collinear to U 1 (Fig. 5(b)). Again, we fix the orientation of the body so that F b can be obtained from the camera frame by a translation of vector --→ OQ. Thus,

--→ OQ = [X Y Z] T , U 1 = [1 0 0] T , U 2 = [cos φ sin φ 0] T , U 3 = [cos ψ sin ψ 0] T (23)
where φ, ψ are two angles defined in Fig. 5(b).

Then, from section III-A,

f i1 ∝ U i × --→ OQ, m i2 ∝ U i × -→ f i1 (24)
leading to, from (18),

f 1 = 0 for any object configuration f 2 = 0⇔Z(X 2 + Y 2 ) = 0 (25)
Thus, if three coplanar lines intersect in a single point, the rank of the interaction matrix is never full, i.e. at least one degree of freedom (a translation towards the intersection point) of the object cannot be controlled, as already known [START_REF] Andreff | Visual servoing from lines[END_REF]. Moreover, additional degrees of freedom become uncontrollable if

• Z = 0, which means as previously that the plane containing the lines L 1 , L 2 and L 3 also contains the camera center O. • X 2 + Y 2 = 0, which means that the camera center O lies on the line which passes through Q and which is perpendicular to all vectors U i , which is the same condition as when f 1 = 0.

3) Three lines in space with a common intersection point: Here, the three lines are spatial, not orthogonal and intersect in a common point. We define the frame F b : (Q, x b , y b , z b ) attached to the observed body B such that x b is collinear to U 1 , y b is in the plane containing U 1 and U 2 , while Q is the intersection point of the lines. As a result,

--→ OQ = [X Y Z] T , U 1 = [1 0 0] T , U 2 = [a b 0] T , U 3 = [c d e] T (26)
where a, b, c, d and e are variables parameterizing the direction of the lines.

) X Y Z ( Q 1 U 2 U 3 U 1 L 3 L 2 L (a) no common intersection point ) X Y Z ( Q b x b y 1 U 2 U 3 U b z a b 1 L 3 L 2 L
(b) two lines are parallel Fig. 6. Observation of lines in space f i1 and m i2 can be found thanks to expression (24), and from (18) we obtain, f 1 = 0 for any object configuration

f 2 = 0⇔b(adeY 3 + ((-ad 2 + bcd + ae 2 )Z + (ac -bd)eX)Y 2 -e(bcX 2 + (ad -bc)Z 2 + 2beXZ)Y + ((-ad 2 + bcd -ae 2 )X 2 Z + (bd + ac)eXZ 2 )) = 0 (27)
Thus, if three spatial lines intersect in a single point, the rank of the interaction matrix is never full, i.e. at least one degree of freedom (a translation) of the object cannot be controlled, as already known [START_REF] Andreff | Visual servoing from lines[END_REF]. Moreover, additional degrees of freedom become uncontrollable if the origin of the body frame belongs to a cubic surface parameterized by f 2 = 0.

Note also that in the case where the three lines are orthogonal, the equation f 2 = 0 degenerates into XY Z = 0.

4) Three orthogonal lines in space: Here, the three lines are orthogonal and do not intersect. We define the frame F b : (Q, x b , y b , z b ) attached to the observed body B such that x b is collinear U 1 , y b is collinear to U 2 and z b is collinear to U 3 . Moreover, the distance between L 1 and L 2 is equal to 2a, between L 2 and L 3 is equal to 2b, and between L 1 and L 3 is equal to 2c, while Q is the barycenter of the lines (Fig. 6(a)). As a result,

--→ OP 1 = [X (Y -c) (Z -a)] T , U 1 = [1 0 0] T --→ OP 2 = [(X -b) Y (Z + a)] T , U 2 = [0 1 0] T --→ OP 3 = [(X + b) (Y + c) Z] T , U 3 = [0 0 1] T (28)
where P i is a point belonging to L i . Then, we have

f i1 ∝ U i × --→ OP i , m i2 ∝ U i × -→ f i1 (29)
which leads to, from (18),

f 1 = 0⇔aXY + bY Z -cXZ -abc = 0 f 2 = 0⇔acX -abY + bcZ -XY Z = 0 (30)
Expression f 1 represents a quadric surface while expression f 2 is a cubic surface (Fig. 7). 5) Three lines, two of them being parallel: Here, the lines Q is at the intersection of P and the line L 3 . From Fig. 6(b), we define

--→ OP 1 = [X (Y -a) Z] T , U 1 = [1 0 0] T --→ OP 2 = [X (Y + b) Z] T , U 2 = [1 0 0] T --→ OP 3 = [X Y Z] T , U 3 = [c d e] T (31)
where c, d, e are variables parameterizing the direction of the line L 3 .

Then, f i1 and m i2 being found from (29), by using the singularity conditions (18), we obtain

f 1 = 0⇔Z(dZ -eY ) = 0 f 2 = 0⇔Z(X(d 2 + e 2 ) -cY d -cZe) = 0 (32)
Here, singularities are equations of planes:

• Z = 0, which occur when the plane P containing L 1 and L 2 also contains the optical center, • eY -dZ = 0 is the plane containing U 1 , U 3 and the optical center, • X(d 2 + e 2 ) -cdY -ceZ = 0 is the plane containing (U 1 × U 3 ), U 3 and the optical center.

The two first cases are degenerated cases where two 3-D lines are contained in the same interpretation plane (i.e. the plane passing through the camera center and the image line), leading to the same image line for the two 3-D lines. 6) Three general lines in space: This is the most general case. We do not provide here the expressions which are more complex than in the previous cases but they are still exploitable and can be found in a technical report [18].

As in the case of the three perpendicular lines, condition f 1 = 0 provides the expression of a quadric surface while f 2 = 0 leads to a cubic surface.

C. Illustrative examples

1) Singularities when observing three general lines in space: In order to show the exactness of our results, we first perform simulations in the case where the three lines have a general configuration (Section III-B6). The three general lines are parameterized as follows. We define the frame F b : (Q, x b , y b , z b ) attached to the observed body B such that x b is collinear to U 1 , y b is lying in the plane P containing L 1 and L 2 . Then, the line L i is parameterized by its direction U i and a point P i lying on it by taking the following expressions:

- Then, a relative motion between the object frame and camera frame origins is imposed and is parameterized by the following functions characterizing the location of Q in the camera frame: lying on the quadric surface whose expression can be found in the technical report [18]. As a result, the rank-deficiency of the matrix M appears for s = 0, which is shown on Fig. 9 in which the inverse of the condition number is null only for s = 0.

-→ OP 1 = [(X -b) (Y -c) (Z -a)] T , U 1 = [1 0 0] T --→ OP 2 = [(X -b) (Y -c) (Z + a)] T , U 2 = [d e 0] T --→ OP 3 = [(X + b) (Y + c) Z] T , U 3 = [f g h] T (
X = 1-0.5s, Y = 1 -0.5s, Z = 0.75 -2.25s (
= -1 ⇒ 1 L 2 L 3 L X O [m] Y O [m] Z O [m]
2) Singularities when observing three orthogonal lines in space: A second simulation is performed in order to show now the singularities when observing three orthogonal lines in space (see Section III-B4). For simulation purpose, we take the following values of the parameters a, b and c in (28): a = b = 0.5 m, c = 1.5 m. Then, a relative motion between the object frame and camera frame origins is imposed and is parameterized by the following functions characterizing the location of Q in the camera frame: As a result, the rank-deficiency of the matrix M appears for s = 0, which is shown on Fig. 11 in which the inverse of the condition number is null only for s = 0.

IV. CONCLUSION

In this paper, we determined the singularity cases for the observation of three image lines thanks to a tool named the "hidden robot concept". We showed that the hidden robot concept allows for considerable simplification of the analysis, leading to the computation of new singularity cases. Indeed, the hidden robot is a tangible visualization of the mapping between the observation space and the Cartesian space. As a result, the singular configurations of the hidden robot corresponded to the singularities of the interaction matrix.

Indeed, the concept of hidden robot allowed to change the way we defined the problem. By doing so, we were able to replace the degeneracy analysis of the velocity transmission between inputs (velocity of the observed features) and outputs (camera twist), by its dual but fully equivalent problem which was to analyze the degeneracy in the transmission of wrenches between the inputs of a virtual mechanical system (virtual actuators of the hidden robot whose displacement was linked to the motions of the observed features) and its outputs (wrenches exerted on the virtual platform, i.e. the observed object).

Then, by using geometric interpretations of the mapping degeneracy and tools provided by the mechanical engineering community such as the Grassmann-Cayley algebra, we were able to find rather simple geometric interpretation of the interaction matrix degeneracy. We proved that in the most complicated cases where three general lines in space are observed, singularities appear when the origin of the observed object frame is either on a quadric or a cubic surface. In simpler cases where at least two lines belong to the same plane, these two surfaces can degenerate into simpler geometrical loci (e.g. planes, cylinders, lines).

Future works concern the analysis of the singularity cases in the observation of n lines, and research about the singularities when observing other primitives (spheres, circles, and eventually of combination of different primitives), or when using several cameras.

Fig. 2 .

 2 Fig. 2. Observation of three lines lying on a body B

Fig. 3 . 2 L 3 LFig. 4 .

 3234 Fig. 3. A U RC kinematic chain

1 and 2

 2 are parallel. We define the frame F b : (Q, x b , y b , z b ) attached to the observed body B such that x b is collinear to U 1 = U 2 , y b is lying in the plane P containing L 1 and L 2 .

Fig. 7 .

 7 Fig. 7. Example of singularity loci when three perpendicular lines are observed (computed for a = 3.13, b = 3.13, c = -3.13)

Fig. 8 .Fig. 9 .

 89 Fig. 8. Relative motion of the camera with respect to the observed lines in a general configuration.

  33) where d, e, f , g and h are variables parameterizing the direction of the lines L 2 and L 3 , and X, Y , Z are the coordinates of Q in the camera frame. For simulation purpose, we set a = b = c = 1 m and d = e = 1, f = -1, g = h = 1.

  34) with s ∈ [-.1 0.05] a linearly increasing function. The relative motion of the camera with respect to the observed lines is shown in Fig. 8. In such simulation, when s = 0, the point O reaches the configuration X O = 0.05 m, Y O = 0.05 m and Z s = 0.225 m in the object frame (see Fig. 8) which is a point

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Relative motion of the camera with respect to the three observed orthogonal lines.

X = 1

 1 -0.5s, Y = 1 -0.5s, Z = 2s (35) with s ∈ [-1 1] a linearly increasing function. The relative motion of the camera with respect to the observed lines is shown in Fig. 10. In such simulation, when s = 0, the point O reaches the configuration X O = 0.5 m, Y O = 0.5 m and Z O = 1 m in the object frame (see Fig. 10) which is a point lying on the cubic surface whose expression is given in (30).

In the following of the paper, P, U, R, C will stand for passive prismatic, universal, revolute and cylindrical joints, respectively. If the letter is underlined, the joint is considered active.

A pure force will be represented by a line in the

3-D space while a pure moment will be a line in the projective space at infinity.

This paper was recommended for publication by Editor Jana Kosecka upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the project RobEcolo funded by the French Région Pays de la Loire (Convention No. 2015-10773).