Laura Le Du
email: laura.le_du@mines--paristech.fr

Sophie Hooge
email: sophie.hooge@mines--paristech.fr

Pascal Le Masson
email: pascal.le_masson@mines--paristech.fr

THE TECHNICAL OBJECT AS A RESOURCE FOR IMAGINARIES STIMULATION: THE CASE OF TWIZY IN THE AUTOMOTIVE INDUSTRY

 of new imaginaries non previously determined. How to evaluate the creation of new imaginaries? What are the potential advantages for an industrial firm in its innovation strategic management?

. We suggest in this paper to extend the design theory hypothesis that technical objects can be generative by stimulating socio--technical imaginaries. Our empirical setting is particularly suitable for the purpose because after being highly evolving for decades, the automotive sector is currently perceived as fixed and locked. In this context we studied: Twizy, a disruptive object for the domain since it does not answer the traditional performance criteria but generates a lot of emulation between actors in the market.

Indeed, Twizy is an electric quadricycle only for two passengers who are sitting one behind the other. We have used a longitudinal collaborative research with the French car manufacturer Renault since 2007 in order to well understand the design of the object. Thanks to interactions with clinic clients, the analysis of secondary data and semi--structured interviews of the main characters concerned by the development of the object in 2014, it has enabled us to understand the impact of the object. The findings show an example of an imaginary stimulating object and highlight the potential resource that it could be for the strategic design. Whereas the expected reaction from the professional actors and the market would be a rejection of the object, our article demonstrates why the design of imaginary generative object has a potential value and the conditions to catch it. The paper contributes to the management innovation theory by going further in the generative product notion with the proposition that a product is able to generate new imaginaries. In this way, the paper also contributes to improve that imaginaries are not totally fixed but able to evolve (in less than decades). The paper also contributes in the practice to integrate a new resource, which enables firms to integrate a design product & design management pro--active strategy.

II. OBJECTIVE AND RESEARCH QUESTION : characterising an object, which generate a potential of imaginaries. This article aims to deepen the theoretical and management knowledge about an imaginaries stimulating object. How characterize the value of a disruptive object, which does not answer the traditional performance criteria? How can we estimate the stimulation of new imaginaries? We focus our literature review on two theoretical frameworks: the first one about the imaginaries often mobilized in philosophy and sociology and the second one in management strategic innovation in management sciences. [START_REF] Bachelard | La formation de l'esprit scientifique[END_REF], one of the famous reference authors about imaginaries shows the interest in stimulation of imaginary process. Indeed, without activation the mechanism it would be sterile. In this context, technical object is often considered as an output of the imaginary mechanism [START_REF] Balandier | Le grand système[END_REF]. In the contemporary debate, we think that imaginaries are evolving and that an impact on them is possible [START_REF] Jasanoff | Sociotechnical Imaginaries and Science and Technology Policy: A Cross--National Comparison[END_REF]. By considering technical object as the embodiment of socio--technical imaginaries, the literature does not show how it could be an input of the imaginary process or a part of this mechanism. Simultaneously, management sciences use imaginaries as a design constraint but the literature does not expose how they could be also a resource for the management. Following works on the generative product uses, we will try to fill this gap by showing that imaginaries could be stimulated thanks to a technical objects. And that it represent a potential value for the design but also for the strategic innovation management. III. LITERATURE REVIEW Employed in transdisciplinary academic fields, as in mathematics, sociology, philosophy, psychoanalysis, anthropology as in poetic, artistic, cinematic, religious, gastronomic or policy, the imaginary term is mainly discussed in the literature of philosophical, sociological or anthropological. Reference authors as [START_REF] Jung | L'homme à la découverte de son âme: structure et fonctionnement de l'inconscient[END_REF], [START_REF] Bachelard | La formation de l'esprit scientifique[END_REF], [START_REF] Durand | Les structures anthropologiques de l'imaginaire[END_REF], [START_REF] Wunenburger | L'imagination mode d'emploi ?[END_REF] brought a precious lighting about the nature and the working of imaginary by trying to define it and to understand its mechanic and its logic. While other authors as [START_REF] Jasanoff | Sociotechnical Imaginaries and Science and Technology Policy: A Cross--National Comparison[END_REF] highlight its power by describing the capacity of sociotechnical imaginaries to support scientific and technological national projects. [START_REF] Wunenburger | L'imagination mode d'emploi ?[END_REF] describes imaginaries as "sets of productions, mental or materialized in works, based on visual images and language, which forming coherent and dynamic sets". Firstly, as confirm [START_REF] Musso | Innover avec et par les imaginaires[END_REF], the semantic part is important "language made of stories, images and world of dynamic forms with coherence and grammar structures that can be formalized" Secondly, imaginary involve a collective dimension that Giust--Desprairies ( 2003) defines as "a set of elements, which in a group is organized in a meaningful unity". So there would be as many imaginaries as cohesive units. [START_REF] Chamoiseau | L'imaginaire de la diversité[END_REF] describes the imaginary diversity by having a "mosaic character composed of several traces, imaginary perfumes where the relationship is not harmonious, but rather paradoxical and confrontational". Thirdly, mental representations are connected with reality and adopt a certain form of materiality. In this way, reality is transformed in representations and representations in a social or individual reality [START_REF] Musso | L'évolution des cultures numériques -- Usages et imaginaires des TIC : la friction de fiction dans l'ouvrage[END_REF]. Users--designers have theirs own way to evolve in industrial built universe, so they modify the object by a "transforming", "hacking" or "creeping function", depending on the possibilities offered by the product [START_REF] De Certeau | L'invention du quotidien --Arts de faire[END_REF]. Now, we propose to detail what are the relations between imaginary and technical object. Through the [START_REF] Balandier | Le grand système[END_REF] definition, "the technical object is a cultural and social construction as an imaginaries genealogy". In this way, management sciences usually consider technical object as imaginaries embodiment:

by the existing or latent imaginaries embodiment in the expectation approach [START_REF] Borup | The sociology of expectations in science and technology[END_REF][START_REF] Godet | The art of scenarios and strategic planning: tools and pitfalls[END_REF][START_REF] Gordon | Frontiers of futures research: what's next?[END_REF]. Here, the activity analyses current or latent expectations (van Lente, 2012) in order to propose a product, which meet them (Abernathy and Utterback, 1978). Usually, the new product adoption is facilitated by the marketing diffusion strategy used to pre--interpret the innovation [START_REF] Rogers | Diffusion of innovations[END_REF]. Moreover, to anticipate this phase, firms integrate users earlier and earlier in the design process [START_REF] Piller | Mass customization[END_REF][START_REF] Hippel | Lead users: a source of novel product concepts[END_REF] or modify their design process toward an open innovation strategy [START_REF] Chesbrough | Open innovation: The new imperative for creating and profiting from technology[END_REF]. --by the new imaginaries embodiment. It is the designing of a new valuable proposition [START_REF] Verganti | Design driven innovation: changing the rules of competition by radically innovating what things mean[END_REF], derived from the product and imaginaries but different from existing imaginaries and, possibly, different from the product itself [START_REF] Forty | Objects of Desire: Designs and Society 1750--1980[END_REF]: the product itself does not embody an existing imaginary, it contributes to the design of a new social--technical imaginary. This model is claimed by famous design schools such as Bauhaus, where teachers and students aim at creating new styles that would help create new society and values [START_REF] Hatchuel | Deconstructing meaning: Industrial design as Adornment and Wit[END_REF][START_REF] Le Masson | The interplay between creativity issues and design theories: a new perspective for design management studies?[END_REF]. Imaginary can be also viewed as a mechanic system. Indeed, [START_REF] Jung | L'homme à la découverte de son âme: structure et fonctionnement de l'inconscient[END_REF] qualifies it as "A system, an organizer dynamic images, which give them a depth notion by linking them" [START_REF] Jung | L'homme à la découverte de son âme: structure et fonctionnement de l'inconscient[END_REF], while [START_REF] Bachelard | La formation de l'esprit scientifique[END_REF] see it as "a transformer and deformer system of images which produce sens beyond the scientific understanding". [START_REF] Durand | Les structures anthropologiques de l'imaginaire[END_REF] links these two aspects by characterizing it as well as "a museum of all images: past, possible, produced and to be produce" and as "the procedure of them production, transmission and reception". Supported by the interest of stimulation of an imaginary mechanism highlighted by [START_REF] Bachelard | La formation de l'esprit scientifique[END_REF], could the technical object be a stimulator of this mechanism? As [START_REF] Brown | Collaborative disruptive innovation success: The critical role of domain--challenging common meaning[END_REF] speak about generative products uses, here the technical object could be an imaginary stimulator. Note that these two models have very different practical implications: in Model 1, designers should embody existing or new imaginaries in the product; in model 2, the success depends on how the product stimulates the design of new uses and values. The Model 1 is quite well known, thus in this paper, we aim to improve model 2 with empirical data in order to understand the role of the new product in the process of designing new uses. [START_REF] Shani | Handbook of collaborative management research[END_REF] with the French car manufacturer Renault since 2007. The paper is based on the analysis of the industrial case [START_REF] Yin | Case study research: Design and methods[END_REF] of Twizy, an electric disruptive quadricycle only for two passengers who are sitting one behind the other. The research follows the main principles of intervention research [START_REF] Hatchuel | Collaborating for management research: from action research to intervention research in management[END_REF] [START_REF] Radaelli | Intervention research as management research in practice: learning from a case in the fashion design industry[END_REF], that aims to produce actionable knowledge for practitioners through the strategic management application and to enrich the scientific knowledge about imaginaries (Argyris, 1993). Our close partnership enabled us to collect a rich empirical data through the analysis of secondary data, semi--interviews of Program strategy manager and different stakeholders intern in the firm but also through the participation of lead users focus groups. Please, refer to the following table for the sum up of data collection and treatment. To be able to offer a comparison of the technical object model that we describe in the article, we choose comparable products, which come from the same domain and share the same initial function in the same historical period but which impact the imaginaries in a different way. For these reason we have choosen: Twizy and Clio 4. 1. The heterogeneity of semantic mobilized by early adopters and the trade press We will start to observe the semantic used firstly by the early adopters about Twizy and secondly by the automotive trade press about Twizy and Clio 4.

Relation between technical object and imaginaries

Perception of imaginary's evolution

• Lexical fields used by early adopters about Twizy We propose in this part, to look into the vocabulary employed by French and German customers in a feedback survey and the vocabulary employed by French early adopters comes from the Twizy Internet social network community during a meeting. We identified some recurrent words that we organised in several categories, which draw four lexical fields: "future", "human", "game" and "animals". So we observe a large variety of terms to speak about the friendly aspect of the object and the last verbatims leads us directly to discuss about the "future" field. Here as well it exists a lot of different words to express the science fiction feeling generated by the object. As said a 32 years old men from Reims "It looks a bit like a lunar car, as vehicles they send into space" and there is no limit to qualify the space openness: Twizy is as "Buggy of the 3th millennium", "lunar vehicle", "Soldier of Star Wars", "It is a glimpse of Dark Vador's garage", "Cyclops", "a human oddity", "space capsule". To sum up this object "It has a really futurist (men, 55 years old, Frankfurt). The "Human" semantic field is very rich with numerous terms as "friendly", "chic", "sexy", "seduction power", "cute", "cute mug", "teasing", "humoristic", "and endearing" to speak about the object. They explain the human aspect by these terms: "sympathy capital". "The Twizy is small, cute, and looks happy" (woman, 35 years old, Wolfsburg). "We have a more sympathetic side" (men, 37 years old, Valencienne), "it is a sympathy capital with customers" (men, with the Twizy car sharing system). By its original exterior aspect, it is easier to be recognized "that's it! I am known", when another said, "I'm not better known than before, but people no more forget me". The "game" lexical field is also divers with "kart", "fun", "playful" "enjoyable" "humorous" Example they say "my funny small car", "it is like a toy", (men, 36 years old, Villeneuve--Loubet) "it is like a lego car" (men, 37 years old, Valencienne). Furthermore, users employ an additional lexical field about: animals. They qualify the object as "carapace" ("shell"), "mosquito", "strange bird", "dragonfly", "bug", "creature", "cockroach", "cockchafer", and "rabbit ears". "The form looks like a bug. The wheels and the king of shell above it gives the feeling, as a big cockroach, something like a cockchafer" (men, 50 years old, Gaillac).

• A semantic comparison in the trade press between Twizy and Clio 4 The previous part took into account the oral vocabulary used by users, now we are going to look into terms written in the automotive press by professionals. For Twizy, the database contains the press reactions from eight countries (with 62 % French articles and 99% European articles) in April 2012, a month after the "Twizy' launching" and for Clio 4, the database contains the press reactions from two countries (with 98% French articles and 100% European articles) in September 2012 at the beginning of school year after the Clio 4 launching two months earlier. After a first reading we identified recurrent words relative to the traditional automobile vocabulary. These terms about the technical aspect of the product match the traditional criteria used to evaluate the performance of an automotive product by professional actors. It is for example: "powerful", "comfort", "speed", "dimension", "weight", "engine", "suspension", "facility", "simplicity", "security", "pleasur". We observe these terms both in Clio 4 and Twizy articles, but with a density up to three times more for Clio 4 documents. It is important to notice that only these terms are used to describe both positive and negative aspects of the object. Of course we check if early adopters qualifying words were also present in trade press articles. It appears that only "modern" in the future semantic field and "friendly", "chic", "sexy" in the human field are used both for Twizy and Clio 4. So while the future and human field exist also for Clio 4 but it is poor, game and animal semantics do not exist for Clio 4 and are only assigned to the Twizy object. The diversity to qualify Twizy is clear compared to a traditional car. And the reality of the field variety is confirmed by the density of the terms that we find in the trade press. Above all for futuristic terms that are used up to around three times more and for game terms up to fifteen times more with these words: "fun", "kart" and "playful". To sum up the comparison of data press reactions between Twizy and Clio 4, we observe a richer variety of lexical terms to speak about Twizy than Clio 4. And, we can see in the trade press that the qualifying for Clio 4 is almost only focused on the automotive field. Moreover, we observe a richer diversity and intensity of words with the early adopters than with the special medias.

Automotive

2. The heterogeneity of profiles touched by the Twizy Whereas the Clio 4 meets with the classic customer target of B segment, Twizy touches multiple profiles in France and around the world, despite its low sales. Indeed, while around 15 000 Twizy have been sold since its launching, Clio 4 counts around 100 000 sales per year. The heterogeneity of Twizy customers is very important: urban drivers, urban trendy people, young businessmen, enthusiasts of beautiful and powerful cars, provincial drivers, drivers without driving licence for a while, younger buyers than for a standard car, handymen, private individuals, professional fleets, institutional fleets, anti--Renault customers, non traditional Renault's sales countries. As said the Program strategy manager « Twizy clients are not at all traditional: it is for example wealthy clients in the Gulf countries, but also cities for development of car sharing and attract new residents". The territorial dimension is interesting indeed, Twizy is able to touch European people as well as Gulf countries (first for the local Policy in Dubaï) and the American continent (for trendy people in Florida or rich American citizens who lived in secure quarters), whereas the culture is different. And it is also interesting to notice the Twizy variety perception of institutions. Twizy is viewed as a "relational connector" by this rich American businessman who wants to connect people in the part of Las Vegas city that he bought. In this way he bought some Twizy precisely because it is not a vehicle. More modestly, European cities perceive the object more like a mobility answer to their town planning (eg. environmentally aspect for Switzerland).

3. The heterogeneity of uses by the Twizy users Then we will observe the (non exhaustive) variety of uses generated by the Twizy all around the world thanks to the Product Planning benchmarking and early adopters testimonies. Are uses as diverse and rich as the lexical fields? What kinds of uses appear with the object? That is the point that we propose to highlight in the following part. Initially designed for an urban utility for short moving transport, we are going to see that it generates an unusual important variety of uses. In order to present uses in a structured way, we propose to organise practices in three parts: at the stationed position, at the moving position and around the hacking aspect of the object. Designed to move from a point to another, can we predict that people find a particular utility of the object at the stop position? Not really, however that is the first surprise that we observe. Indeed, Twizy became an artistic support for Pierre Hermé, a well--known pastry chef in the world, who made a "Twizy macaroon" at the 18 th Chocolate exhibition in 2012 in Paris. We could see the object totally covered by an artist macaroon creation. It was also an artistic support for a photograph who made a serie of pictures of animals in an urban context. But the park position can also have a functional utility, for instance, with this agile mobile coffee shop application launched by Van Dyck Rösterei a german from Frankfurt. His concept, with a coffee maker at the extern back of the Twizy has been named "Espressizy". On the moving application a lot of diversity of uses also appeared, that we can classified in different purposes as: quick intervention, communication application or material transport function. These three categories are not exclusive of each other. The most famous moving application is the quick intervention by the appropriation of different domains as, for instance, the medical profession with the firefighter in Deutschland and in Paris and the Red Cross in Spain. But a quick intervention interests also the police in France and the Château de Versailles for security supervision in the park. The communication application is easily guessable, it consists in considering the Twizy as an ad support for a sticker covering. A lot of firms, public organisations, carsharing system, use the Twizy as an advertising object for example: Marionnaud, Domino's pizza, Twizy way carsharing system, Police de Paris…etc. Because the categories are not exclusives Domino's pizza can also be integrated in our third category: material transport function. Addition to an advertising support, the firm use the object to quick deliver their customers. Finally, the Twizy is also used for symbolic event: the wedding in France. Paradoxically, it is used both as car married and as a broom wagon. To allow these different applications or functions in the Twizy, it is a necessity to modify the object. The "hacking" of the vehicle started with the customers and then with Renault. The automotive industrial proposes two new versions with the transformation of the second part of the vehicle that is the passenger seat. The "Twizy cargo" version offers a larger boot to answer the material transport needs while the VELUD (the electric vehicle for an sustainable urban logistic) totally switches the second seat to transform it in a modularity platform where you can install, for example, a quite bigger storage box than proposed by the original version. But most of the transformations of the Twizy are not made by the industrial maker but rather by the customers themselvesthe industrial maker but rather by the customers themselves does not make most of the transformations of the Twizy. Indeed, firefighters in Deutschland and the Spanish Red Cross developped a kind of open luggage rack specified for their activity where there are able to transport a first aid luggage. Furthermore a "normal" additional rack is also created to receive a small close box in the Twizy's back for a similar boot using. In mountain, for the winter season, people also created specific applications. Example: to transport ski materials at the Twizy's back they designed in Alpes d'Huez in France a special rack and to drive on ice they designed in Turkey a kind of ice skate to add on the first two wheels. Almost all these previous transformations are available to anyone who wants to modify its Twizy. And it also exists particular changes / evolutions designed by customers who then have a single version of their object. It is the case of this handyman in the south of France who changed different part of its Twizy to adapt it to his specific needs. For example, he cut out the glove compartment to transform it into an electric meter box. He also cut out a piece of plastic part inside the vehicle to install two electric plugs. Moreover he also transformed the seat function in a storage place with a generator installation. Thanks to all of these evolutions he is able to know how much electricity he charges in his friends house, he is able to charge any electric device inside his vehicle and he increases his energy independence. We notice, as it is quite classic for some vehicle, the Renault automaker propose a Twizy sport version.

B. ANALYSIS: THE TWIZY OBJECT AS AN IMAGINARIES STIMULTION

To interpret this original and large variety of lexical fields and uses generate by Twizy compared to a traditional car as Clio 4, we propose a model with two status of the technical object: 1) the object suits current reference imaginaries and embodies them; 2) the object turns upside down the classic imaginaries reference and generate new one. • Model 1: the technical object embodies imaginaries In the first status, the interpretation and the adoption of the product is quite easy and very similar to the on that we use to see with the anterior object of the same domain. In such case, for instance with the Clio 4, it is easy for users, professional press actors and industrial car manufacturers to judge the performance of the product. Indeed, by the traditional technic criteria, the object respects most of the classic codes, where we expect for: motor performance, powerful, dimensions, weight, security and comfort. As expected in September 2012, in the press release, the Clio 4 find its market and customers targeted in the B segment, the multipurpose city's vehicle segment. Moreover, the product does not reveal some particularly uses. Because the product matches properly the expectations, we think we can say that the object does not modify the representations and the imaginaries of the domain because it totally matches the current imaginaries. In this case, there is no evolution of socio-technical imaginaries involved. Whereas Clio 4 embodies the current or latent imaginaries, that is not the case for Twizy. Because if we apply on it the traditional evaluation criteria, the risk as we see in the Twizy prospect survey, is to see in this object all the disadvantages of a car and a motorcycle. The product answers neither to a precise customer target. The expected clients were not so clear for Twizy, in March 2012, in the press release. The car manufacturer Renault said: "there is no real type of customers target" […] "it is for people who look for a nonstandard vehicle and a lot of sensations which is economic and environmentally" even if the Twizy Program said "all initial Twizy customer targeted are wrong: young fashion Parisian and stressed housewife". And as we see there is not a convergence to a defined use.

• Model 2: the technical object stimulate imaginaries To be able to see the impact of the object on their imaginaries, a solution consists in showing the evolution between the initial references and the current situation in these three following dimensions: semantic fields, collective aspect and action classes. Indeed, we propose to translate the rich diversity of the semantic fields as the number of potential representations, imaginaries that this object is able to generate. Because there is adoption, the object embodies at least a small part of current or latent imaginaries but above all it enable the projection of new imaginaries totally detach from the reference universe. Compared to its low sales, the object is curiously able to touch small but diverse populations on different territories. It is important to notice here, that the culture of these users are sometimes very different and the motivation to adopt the product it also heterogeneous. To summarize, in addition to generate a large variety of the initial function for which it is designed, Twizy is also able to question its original function and generate a variety of others and a diversity of applications of them. The different new functions of the object can be analysed and structured as: a means of mobility, an applications platform and also at the end as a symbolic object. C. MANAGERIAL APPLICATION How a firm should receive the design of stimulator imaginaries object? Does it represent a bug of the design process? Or does it represent an opportunity for the design and the strategic innovation? Finally, how to catch the potential value of it if it is really an opportunity? According the empirical case study, the first essential point for the strategic management is to take into account the value criteria. Secondly, to feel the potential of the added value of the generative imaginaries product and take the advantages of it, the firm have to be in a pro--active position by integrating them. So that is on these two axes that we propose to perceive the potential for the firm. By an evolution of value criteria, it has become possible for the firm to catch and understand the value of new imaginaries that was impossible with the standard view. So this first step allows to perceive the potential value. The second step, consist in transforming it into a real value. In the medium term it consists in integrating the understanding of new imaginaries and embodies it in the next product. The activity refers to the Bauhaus and design--driven innovation. In the long term, it consists in perceiving the appearance of new imaginaries as an innovation field. In this way, it is the role of the strategic innovation to manage it. Management of value criteria To sum up, the stimulator of imaginary potentials can be considered as a design resource for the next products but also as a resource for the strategic innovation management.

VI.

LIMITS AND PERSPECTIVES Our case shows that technical object can play an active role and stimulate imaginaries. The article tries to show the potential value creation and how it can be a resource for the organisation. Through this empirical case study, we hope to contribute to the imaginary academic field by showing that it is possible to move the current imaginary, even these that appear stable, locked and difficult to break. By taking the imaginary references promoted in the philosophy and sociology, which is the original literature on subject, we tried to demonstrate the nature of the impact of the different two different technical objects on the imaginaries. Moreover, we would like to bring to the strategic innovation management a positive sign about the designing of disruptive object. The relative perception of loss of control could be a resource for the management by becoming a new exploration space. So the designing of generative object imaginaries can viewed as a middle term or long term strategy input. In order to strengthen the case study and the hypothesis brought in this article, it would welcome to look into other empirical cases, which have the capacity to generate new imaginaries. So, we will be able identify the intrinsic characteristics that give a generative function. Like this, we could provide factual inputs to designers for the designing of future objects.

VII. CONCLUSION

The study in this article shows that it is possible to impact imaginaries thanks to a technical object. The empirical case showed that a technical object could generate a potential of imaginaries. It means that the imaginaries revealed are not previously determined, defined or decided by the designer. In this way, it avoids the risk to think that the generation of new imaginaries come under a manipulation action. We perceive it rather as resource of the exploration of new innovation space.

Δ

  . J. & Utterback, J. M. (1978) 'Patterns of innovation in technology', Technology review, 80, 40--47. Argyris, C. (1993) Knowledge for action: A guide to overcoming barriers to organizational change, ERIC.

Designer's imaginary on the object Management sciences Reference authors

		Object
	Output	embodies existing and/or latent	Stables, any possible influence	Determined	Expectations,	(Borup et al., 2006)
		imaginaries
						(Verganti,
	Input	Object embodies new imaginaries	Progressives, possible influence	Determined	Design--driven innovation, Bauhaus	2013), (Hatchuel, 2013), (Le Masson et al.,
						2011)
	Mechanic	Object as a stimulator of imaginary potential	Progressives, possible influence	No pre--determined	Current article	Current article

Table 1 :

 1 Synthesis of literature about the relation between imaginaries and management sciences IV. METHODS AND MATERIALSTo address this problematic of generative imaginaries we conducted a longitudinal collaborative research

  Thanks to the empirical materials and related to imaginaries characteristics presented earlier, we propose to expose the variety of: lexical fields, profiles touched by the product and the class of actions between two different automotive objects: Clio 4 and Twizy.

	Clio 4 Trade press feedback at the launching product	10/04/2014	Document	Press feedback synthesis 2 countries
	Twizy Product manager	06/02/2014	Document
	Electric vehicle
	competitive analysis	07/02/2014	Document
	manager
	Twizy Program manager	21/01/14 & 15/04/2014	Two semi--structured interviews + notes
	Twizy's designer	20/11/2014	Semi--structured interview + notes
	Table 2: Synthesis of methods and data collection
	V. CASE STUDY
	What	Date of data collection	Nature of data	Methods of data collection
				People who made a reservation,
				People registered to ZE
	Twizy Prospects (France)	26/06/2012	Document	newsletter, Curious who were not involved havebeen screened out. Online deliberative methodology
				(debate®): qualitative
				methodology.
				2 interview periods : same
				customer interviewed twice and
	TWIZY -Launching			fresh additional customers to be
	customer feedback survey (France &	26/06/2012	Document	included in the second period.Same questionnaire for
	Germany)			France & Germany by phone.
				Target respondents: Private &
				fleet customers
	Meeting with lead users	05/03/2014	Audio recording + notes + retranscription
	Twizy Trade press feedback at the launching product	28/02/2014	Document	Press feedback synthesis 8 countries

A. THE EFFECT OF TWO AUTOMOTIVE PRODUCTS ON USERS, NON USERS, PROFESSIONAL AND NON--PROFESSIONAL ACTORS

Table 3 :

 3 The following table summarizes the number of articles analysed in this study. Number of trade press articles

		Twizy	Clio 4
	France	58	40
	UK	4	1
	Suisse	1	0
	Espagne	12	0
	Allemagne	9	0
	Italie	5	0
	Belgique	3	0
	USA	1	0
	Total	93	41

Table 4 :

 4 Synthesis of terms characteristics for Twizy and Clio 4

			Future	Human	Game	Animals
	Density	Clio 4	Twizy
	Variety		Twizy	Twizy
	New				Twizy	Twizy

The Twizy heterogeneity functions and applications

CONCEPT OBJECT OF MOBILITY CONCEPT OF A PLATFORM OBJECT CONCEPT OF A SYMBOLIC OBJECT

A device to go from a physical point to another

A support of service applications

A support of energy applications Table 5: The heterogeneity of Twizy functions and applications The object's properties allow the generative concepts which themselves promote the development of new concepts. And the reconfiguration of new functions allows people to think about what are their own vision and perception of the object. In the same way, the heterogeneity perceptions that people have to others about this vehicle, (eg. that citizens have to users), modify and/or simulate the own perception of the object and increase the variety potential perception that people can have to others. We think we can say that Twizy does not embody a new imaginary but generates a potential on new ones. The technical object would be a generative potential imaginaries.


Christophe Dano
email: christophe.dano@ec-nantes.fr

Pierre-Yves Hicher
email: pierre-yves.hicher@ec-nantes.fr

Evolution of elastic shear modulus in granular materials along isotropic and deviatoric stress paths

Keywords: small strains, shear modulus, granular materials

INTRODUCTION

The "bender elements" technique has known an ongoing progress in the last two decades. It is now frequently associated with the triaxial apparatus in order to determine the elastic properties of soils such as sands, gravels, clays, etc. For instance, many theoretical works [START_REF] Bates | Dynamic soil property measurements during triaxial testing[END_REF][START_REF] Viggiani | Interpretation of bender element tests[END_REF][START_REF] Brignoli | Measurement of shear waves in laboratory specimens by means of piezoelectric transducers[END_REF][START_REF] Jovicic | Objective criteria for determining G max from bender element tests[END_REF][START_REF] Arulnathan | Analysis of bender element tests[END_REF][START_REF] Blewett | Phase and amplitude responses associated with the measurement of shear-wave velocity in sand by bender element[END_REF] have led to a more accurate measurement of the velocity of the shear waves generated by these piezoelectric transducers. A large quantity of experimental results was also published in the literature. This additional contribution demonstrates a typical evolution of the shear modulus for various granular materials along both isotropic and deviatoric stress paths.

MATERIALS

Three granular materials were used in this experimental study:

•= A fine and siliceous sand named Fontainebleau Sand NE34; •= A first class of natural alluvial deposits of the Seine River (France) noted AD1; •= A second class of coarser alluvial deposits of the Seine River noted AD2.

The basic properties of the soils (mean grain size d 50 , Hazen's ratio C u = d 60 /d 10 , minimal and maximal void ratio e min and e max ) are given in Table 1. The maximal grain size d max of alluvial deposits AD1 was also reduced to 3.15 mm (material noted AD3) in order to examine the effect of decreasing the maximal grain size on the stiffness properties of granular materials.

EXPERIMENTAL SET-UP

Consolidated drained triaxial tests were carried out in the laboratory. Cylindrical specimens were 100 mm in diameter and 200 mm in height. Dense samples of Fontainebleau Sand were reconstituted through air pluviation. Dense alluvial deposits were deposited in five layers and were compacted by tamping. Tests were performed on dry samples of alluvial deposits and on fully saturated samples of Fontainebleau Sand. The saturation process comprised three stages: gaseous carbon dioxide percolation, deaired water percolation and application of back pressures up to 350 kN/m 2 which guaranteed B-values of at least 0.95.

Volumetric strains were measured either in a burette or through the volume changes of the cell water recorded on a pressure/volume controller which applied the confining pressure σ' 3 . Axial strains were determined from an external displacement transducer. As stated by [START_REF] Scholey | A review of instrumentation for measuring small strains during triaxial testing of soil specimens[END_REF], such a measurement overestimates the true axial displacements (measured by local transducers located on soil specimens for instance) because of bedding errors and system compliance. The error made on the axial displacements involves an error, although low, on the relative position of the bender elements during the test and therefore on the absolute value of the shear modulus but has no effect on its general evolution.

The preparation procedure was not changed by the pair of bender elements embedded in the lower and upper pedestals of the triaxial cell. These bender elements, which are piezoelectric transducers, provide a non destructive and punctual measure of the shear modulus. Application of a square impulse is converted into tangential vibrations of the transmitter, which cause a shear wave to be propagated through the sample. This phenomenon is reversible, and then a mechanical force on the receiver produces an electric field. The input and output electric signals were recorded on an oscilloscope on which the propagation time T w of the shear wave could be determined, according to the instructions by [START_REF] Viggiani | Interpretation of bender element tests[END_REF] and [START_REF] Jovicic | Objective criteria for determining G max from bender element tests[END_REF]. The distance of propagation L w is the length between the tips of the bender elements. The velocity of the shear wave is therefore:

w w w T L V = (1)
Owing to the direction and the polarization of the shear wave, only the component G vhmax of the small strain stiffness matrix can be identified, assuming a linear cross-anisotropic elastic behavior [START_REF] Kuwano | Shear stiffness anisotropy measured by multi-directional bender element transducers[END_REF]. The shear modulus is related to the unit mass ρ of the material and to the shear wave velocity V w :

[ ] 2 w w 2 w max vh T L V G × = × = ρ ρ
(2)

EVOLUTION OF THE SHEAR MODULUS ALONG ISOTROPIC STRESS PATHS

A critical review of the literature shows that the shear modulus of unbound granular materials mainly depends on the void ratio e and on the effective principal stresses σ' 1 and σ' 3 . However, during triaxial tests, the minor principal stress σ' 3 is kept constant. Therefore, the shear modulus G vhmax is expressed as a function of the void ratio and of the mean effective stress p' such as [START_REF] Hardin | Elastic wave velocities in granular soils[END_REF][START_REF] Iwasaki | Effects of grain size and grading on dynamic shear moduli of sands[END_REF]Hicher, 1996;[START_REF] Lo Presti | Shear modulus and damping of soils[END_REF]:

( ) ( ) ( ) 2 n 2 max vh m / kN ' p e f K m / MN G × = (3)
where K and n are material constants and f(e) a function of the void ratio (Table 2). The values of the two parameters K and n proposed by the authors previously mentioned are clarified in Table 2. Fig. 1 represents a non-exhaustive compilation of experimental data relative to sands and gravels found in the literature: all the references can be found in [START_REF] Dano | Comportement mécanique des sols injectés[END_REF]. The correlation by [START_REF] Biarez | Elementary mechanics of soil behaviour -saturated and remoulded soils[END_REF] constitutes a good approximation for many experimental data (Fig. 1). 3. e 0 is the void ratio for a confining pressure equal to 100 kN/m 2 . The power n, which characterizes the quantity of grain-to-grain contacts, was close to 0.5. The deviation from the predictions of the Hertz's theory (n = 1/3) can be explained by the nature of the contacts, not necessarily punctual.

EVOLUTION OF THE SHEAR MODULUS ALONG DEVIATORIC STRESS PATHS

The shearing phase induced an obvious fabric change. In particular, the values of the shear modulus were systematically lower after shearing than before for the same isotropic stress state. Whatever the maximal axial strain might be, the reduction of G vhmax was located between 15 and 25%. Few publications tackle the problem relative to the evolution of the shear modulus over a large domain of strains. By performing tests on Hime gravel samples (d 50 ≈ 2 mm), [START_REF] Shibuya | Elastic deformation properties of geomaterials[END_REF] experimentally found a non-monotonic evolution of the elastic modulus determined on small unloading/reloading loops using local measurements of the axial strains. They first noted that the power laws, usually accepted in the geotechnical community, were suitable only on a restricted stress domain, without bringing any precision as for the size of this domain. [START_REF] Kuwano | Shear stiffness anisotropy measured by multi-directional bender element transducers[END_REF] performed tests on Ham River Sand (d 50 = 0.27 mm) using multi-directional bender element transducers. Their experiments showed a decrease of the shear modulus G vhmax which coincides with the onset of dilation at the characteristic state as defined by [START_REF] Luong | Phénomènes cycliques dans les sols pulvérulents[END_REF]. It means that the evolution of the shear modulus is mainly controlled by the volumetric behavior. However, the variations of the void ratio can not by themselves explain this particular feature which has consequently to be related to the fabric change.

Fig. 3 represents the evolution of the shear modulus G vhmax corrected by the void ratio e as a function of the axial strain ε 1 in the case of a dense Fontainebleau Sand (Fig. 3a) and in the case of three triaxial tests on dense alluvial deposits AD3 (Fig. 3b). Fig. 3 reveals a slight decrease of the shear moduli almost simultaneously to the crossing of the characteristic state whereas, in the contracting domain, the evolution of the shear modulus still follows the power laws deduced from the isotropic stress paths. Similar trends were observed for alluvial deposits AD1 and AD2.

ADDITIONAL TESTS

Additional tests were carried out in order to confirm these experimental observations:

•= Triaxial test on loose alluvial deposits AD3 (e 0 = 0.641 ; σ' 3 = 200 kN/m 2 ) (Fig. 4);

•= Test at a constant mean effective stress p' = 200 kN/m 2 (e 0 = 0.461) (Fig. 5).

Triaxial test on loose alluvial deposits

Loose alluvial deposits AD3 were first isotropically consolidated up to a mean effective stress p' equal to 200 kN/m 2 . The power law was then: Fig. 4 represents the evolution of the shear modulus (multiplied by the void ratio e) during shearing. As expected, the values increased according to the power law (6) until the volumetric strain rate became equal to zero. The shear modulus then remained almost constant for a small dilation rate. Also note that the shear modulus after unloading was very close to its initial value whereas previous tests carried out at larger strains in the dilating domain had shown a decrease of about 15 to 25 % of the stiffness. In the previous tests, the effect of the shear stress was found to be negligible over a large domain of strains. In order to examine this effect, a triaxial test at a constant mean effective stress p' = 200 kN/m 2 was carried out on alluvial deposits AD3 (e 0 = 0.461). The equipment used in the laboratory did not allow a stress servo control. The test was therefore performed by gradually decreasing the minor stress σ' 3 and by increasing the major stress σ' 1 . Fig. 5 shows that the value of the shear modulus was constant in the contracting domain. Once the characteristic state was reached, the shear modulus gradually decreased while increasing the deviatoric strain.

CONCLUSIONS

Triaxial tests were carried out on three unbound granular materials and the shear modulus was continuously measured using piezoelectric transducers. This experimental study shows that the power laws, usually considered to describe the effect of the mean effective stress on the shear modulus, are suitable only along isotropic stress paths and along deviatoric stress paths as long as the volumetric behavior is contracting. Indeed, when dilation appears, the shear modulus G vhmax gradually decreases during shearing. Further developments will use compression wave transmitter/receiver transducers associated to the bender elements. Numerical simulations using discrete element methods would have to be included to these developments in order to better understand the experimental observations described in this paper.

FIG. 2 .

 2 FIG. 1. Compilation of experimental data

  Fig.3a

Fig. 3b FIG

 3b Fig.3b

FIG. 4 .

 4 FIG. 4. Evolution of G vhmax along deviatoric stress paths for loose alluvia

  FIG. 5. Evolution of G vhmax for a test at a constant p' = 200 kN/m 2

TABLE 1 . Properties of the granular materials

 1

	Granular material	d 50 (mm)	C u	e min	e max
	Fontainebleau Sand	0.22	1.4	0.606	0.893
	Alluvial deposits AD1	0.50	2.4	0.480	0.779
	Alluvial deposits AD2	1.30	5.9	0.366	0.616
	Alluvial deposits AD3	0.47	2.3	-	-

TABLE 2 . Relations between G max , e and p'

 2

	Reference	K	n							f(e)	Comments
	Hardin & Richart (1963) Iwasaki & Tatsuoka (1977) 14.10 6.93	0.5 0.4	( 2	.	17	-	e	) ( ) e 1 2 +	Rounded grains C u < 1.8
	Biarez & Hicher (1994)	5.69	0.5	1	e			All soils
	Lo Presti et al. (1997)	9.25	0.45	1		e	. 1	3	Sands

TABLE 3 . Experimental values of K and n

 3

	Granular material	e 0	First isotropic loading Isotropic unloading after shearing
			K	n	K	n
	Fontainebleau Sand	0.598	4.38	0.56	5.84	0.49
	Alluvial deposits AD1	0.480	6.98	0.48	2.76	0.62
	Alluvial deposits AD2	0.399	5.65	0.51	6.02	0.46
	Alluvial deposits AD3	0.490	5.57	0.50	3.72	0.56

ACKNOWLEDGMENTS

The authors would like to acknowledge the companies RATP and INTRAFOR (France) for their financial support.


domain: spi sde

text:
The design and optimisation process combines the use of parametric modelling with dynamic solar simulation and evolutionary algorithm for solar urban planning. Thanks to their flexibility, it is possible to maximise the solar potential of a district at various scales with simple transformations while limiting solar availability reduction over the existing buildings by taking the complex overshadowing effects and mutual reflections into account.

Due to its combined rugged topography, limited land available and population growth, Reunion must inevitably address the new challenge of urban densification. Therefore, new urban forms need to be designed in a climate and context-sensitive way in order to provide low energy, sustainable and comfortable places for people.

Currently in Reunion, most buildings are designed following French building codes with very small and even no consideration of the context, the climatic conditions or the natural resources available. This leads to uncomfortable conditions for occupants, need for air conditioning and high energy consumption.

In this context, solar urban planning is a first step towards more urban sustainability and climate-sensitiveness by integrating solar active and passive solutions in the urban morphology. Indeed, this unlimited but also under used source of energy can be harvested for PV electric generation, domestic hot water, daylighting and plant growing at the urban scale.

But when looking at densification and solar urban planning together, the complexity of the 3d interplay between solar irradiation and the urban form, the multiple design constraints and parameters and the impact of new buildings on the existing ones make solar integration a real challenge for designers and urban planners.

They need flexible and performative tools for studying such phenomena, exploring and optimising design scenarios.

DESIGN & OPTIMISATION

CONTEXT

APPROACH In order to fully exploit the solar energy potential of urban settlements, the proposed approach is to maximise the solar irradiation received by the urban geometry, by both new and existing buildings. This is a way of limiting the adverse solar availability reduction over the existing environment due to the integration of new buildings. To do so, the geometry-based solar optimisation method needs to be applied following the hierarchy of scales, to the urban layout first and second to the building form.

Thesis : Building zero energy neighbourhoods in the tropics. Development of design tools for morphology/comfort/energy coupled optimization.

The applicability of the method is tested in two district case studies situated in extreme climate conditions. They are both under a significant densification development and comprise new building projects that were planned following similar text books with no consideration for climate conditions or solar integration. Therefore, in order to inform these urban rules, it has been decided to study one urban scenario in both climates.

CONCLUSION

°°°°

This study is a first step towards more urban climate-sensitiveness. This flexible and multi-scale approach will be will be further developed by integrating thermal and airflow dynamic simulations in the workflow. The aim is at the end to support the design of optimised bioclimatic buildings in dense tropical urban areas using parametric modelling and climate-based tools.

  Solar Potential in Extreme Climate Conditions Comparative analysis of two district case studies in Norway and Reunion Island AYMERIC DELMAS PhD candidate at PIMENT Research Engineer at IMAGEEN aymeric.delmas@imageen.re

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Jean-Marie Mirebeau

Johann Dreo

Automatic differentiation of non-holonomic fast marching for computing most threatening trajectories under sensors surveillance

Keywords: Anisotropic fast marching, motion planning, sensor placement, optimization

We consider a two player game, where a first player has to install a surveillance system within an admissible region. The second player needs to enter the the monitored area, visit a target region, and then leave the area, while minimizing his overall probability of detection. Both players know the target region, and the second player knows the surveillance installation details.

Optimal trajectories for the second player are computed using a recently developed variant of the fast marching algorithm, which takes into account curvature constraints modeling the second player vehicle maneuverability. The surveillance system optimization leverages a reverse-mode semi-automatic differentiation procedure, estimating the gradient of the value function related to the sensor location in time O(N ln N ).

Introduction

This paper presents a proof of concept numerical implementation of a motion planning algorithm related to a two player game. A first player selects, within a admissible class Ξ, an integral cost function on paths, which takes into account their position, orientation, and possibly curvature. The second player selects a path, within an admissible class Γ, with prescribed endpoints and an intermediate keypoint. The players objective is respectively to maximize and minimize the path cost C(Ξ, Γ) := sup C ξ (γ(t), γ (t), γ (t)) dt, [START_REF] Barbaresco | Minimal geodesics bundles by active contours: Radar application for computation of most threathening trajectories areas & corridors[END_REF] where the path γ is parametrized at unit euclidean speed, and the final time T (γ) is free. From a game theoretic point of view, this is a non-cooperative zero-sum game, where player Ξ has no information and player Γ has full information over the opponent's strategy.

The game (1) typically models a surveillance problem [START_REF] Strode | Optimising multistatic sensor locations using path planning and game theory[END_REF], and exp(-C(Ξ, Γ)) is the probability for player Γ to visit the intermediate keypoint without being detected by player Ξ (by convention, players are identified with their sets of strategies). For instance player Γ could be a robber trying to steal an object from a museum, and player Ξ would be responsible for the placement of video surveillance cameras. In another instance, player Ξ is responsible for the installation of radar [START_REF] Barbaresco | Minimal geodesics bundles by active contours: Radar application for computation of most threathening trajectories areas & corridors[END_REF] or sonar detection systems [START_REF] Strode | Optimising multistatic sensor locations using path planning and game theory[END_REF], and would like to prevent vehicles sent by player Γ from spying on some objectives without being detected.

The dependence of the cost C ξ w.r.t. the path tangent γ (t) models the variation of a measure of how detectable the target is (radar cross section, directivity index, etc.) w.r.t. the relative positions and orientations of the target and sensor. The dependence of C ξ on the path curvature γ (t) models the airplane maneuverability constraints, such as the need to slow down in tight turns [START_REF] Duits | Optimal Paths for Variants of the 2D and 3D Reeds-Shepp Car with Applications in Image Analysis[END_REF], or even a hard bound on the path curvature [START_REF] Dubins | On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents[END_REF].

Strode [START_REF] Strode | Optimising multistatic sensor locations using path planning and game theory[END_REF] have shown the interplay of motion planning and game theory in a similar setting, on a multistatic sonar network use case, but using isotropic graph-based path planning. The same year, Barbaresco [START_REF] Barbaresco | Computation of most threatening radar trajectories areas and corridors based on fast-marching & Level Sets[END_REF] used fast-marching for computing threatening paths toward a single radar, but without taking into account curvature constraints and without considering a game setting.

The main contributions of this paper are as follows:

1. Anisotropy and curvature penalization: Strategy optimization for player Γ is an optimal motion planning problem, with a known cost function. This is addressed by numerically solving a generalized eikonal PDE posed on a two or three dimensional domain, and which is strongly anisotropic in the presence of a curvature penalty and a detection measurement that depends on orientation. A Fast-Marching algorithm, relying on recent adaptive stencils constructions, based on tools from lattice geometry, is used for that purpose [START_REF] Duits | Optimal Paths for Variants of the 2D and 3D Reeds-Shepp Car with Applications in Image Analysis[END_REF][START_REF] Mirebeau | Anisotropic fast-marching on cartesian grids using Voronois first reduction of quadratic forms[END_REF][START_REF] Mirebeau | Fast Marching methods for Curvature Penalized Shortest Paths[END_REF]. In contrast, the classical fast marching method [START_REF] Rouy | A Viscosity Solutions Approach to Shape-From-Shading[END_REF] used in [START_REF] Benmansour | Derivatives with respect to metrics and applications: subgradient marching algorithm[END_REF] is limited to cost functions C ξ (γ(t)) independent of the path orientation γ (t) and curvature γ (t).

2. Gradient computation for sensors placement: Strategy optimization for player Ξ is typically a non-convex problem, to which various strategies can be applied, yet gradient information w.r.t. the variable ξ ∈ Ξ is usually of help. For that purpose, we implement efficient differentiation algorithms, forward and reverse, for estimating the gradient of the value function of player Ξ

∇ ξ C(ξ, Γ), where C(ξ, Γ) := inf γ∈Γ C(ξ, γ). (2)
Reverse mode differentiation reduced the computation cost of ∇ ξ C(ξ, Γ) from O(N 2 ), as used in [START_REF] Benmansour | Derivatives with respect to metrics and applications: subgradient marching algorithm[END_REF], to O(N ln N ), where N denotes the number of ization points of the domain. As a result, we can reproduce examples from [START_REF] Benmansour | Derivatives with respect to metrics and applications: subgradient marching algorithm[END_REF] with computation times reduced by several orders of magnitude, and address complex three dimensional problems.

Due to space constraints, this paper is focused on problem modeling, ization and numerical experiments, rather than on mathematical aspects of wellposedness and convergence analysis. Free and open source codes for reproducing (part of) the presented numerical experiments are available on the first author's webpage1 .

Mathematical background of trajectory optimization

We describe in this section the PDE formalism, based on generalized eikonal equations, used to compute the value function min γ∈Γ C(ξ, γ) of the second player, where ξ is known and fixed. Their discretization is discussed in §3. We distinguish two cases, depending on wether the path cost function C ξ (x, ẋ, ẍ) appearing in (1) depends on the last entry ẍ, i.e. on path curvature.

Curvature independent cost

Let Ω ⊆ E := R 2 be a bounded domain, and let the source set Υ and target set Θ be subsets of Ω. For each x ∈ Ω, let Γ x denote the set of all paths γ ∈ C 1 ([0, T ], Ω), where T = T (γ) is free, such that

γ(0) ∈ Υ, γ(T ) = x, and ∀t ∈ [0, T ], γ (t) = 1.
The problem description states that the first player needs to go from Υ to Θ and back, hence its set of strategies is Γ

= x∈Θ Γ x × Γ x . Therefore C(ξ, Γ) = 2 inf x∈Θ u ξ (x), where u ξ (x) := inf γ∈Γx C(ξ, γ). (3)
Define the 1-homogenous metric

F ξ : Ω × E → [0, ∞],
the Lagrangian L ξ and the Hamiltonian H ξ by

F ξ (x, ẋ) := ẋ C ξ (x, ẋ/ ẋ ), L ξ := 1 2 F 2 ξ , H ξ (x, x) := sup v∈E x, ẋ -L ξ (x, ẋ).
Under mild assumptions [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF], the function u ξ : Ω → R is the unique viscosity solution to a generalized eikonal equation

∀x ∈ Ω \ Υ, H ξ (x, ∇ x u ξ (x)) = 1/2, ∀x ∈ Υ, u ξ (x) = 0,
with outflow boundary conditions on ∂Ω. The discretization of this PDE is discussed in §3. We limit in practice our attention to Isotropic costs C ξ (x), and Riemannian costs C ξ (x, ẋ) = ẋ, M ξ (x) ẋ where M ξ (x) is symmetric positive definite, for which efficient numerical strategies have been developed [START_REF] Mirebeau | Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction[END_REF][START_REF] Mirebeau | Anisotropic fast-marching on cartesian grids using Voronois first reduction of quadratic forms[END_REF].

Curvature dependent cost

Let Ω ⊆ R 2 × S 1 be a bounded domain, within the three dimensional space of all positions and orientations. As before, let Υ, Θ ⊆ Ω. For all x ∈ Ω let Γ ±

x be the collection of all γ ∈ C 2 ([0, T ], Ω), such that η := (γ, ±γ ) satisfies

η(0) ∈ Υ, η(T ) = x, and ∀t ∈ [0, T ], γ (t) = 1.
Here and below, the symbol "±" must be successively replaced with "+" and then "-". Since the first player needs to go from Υ to Θ and back, its

set of strategies is Γ = x∈Θ Γ + x × Γ - x . Therefore C(ξ, Γ) = inf x∈Θ u + ξ (x) + u - ξ (x), where u ± ξ (x) := inf γ∈Γx C ± (ξ, γ). (4)
We denoted C ± the path cost defined in terms of the local cost C ξ (p, ± ṗ, p). Consider the 1-homogeneous metric F ± ξ : TΩ → [0, ∞], defined on the tangent bundle to Ω ⊆ R 2 × S 1 by

F ± ξ ((p, n), ( ṗ, ṅ)) := +∞ if ṗ = ṗ n, ṗ C ξ (p, ±n, ṅ/ ṗ ) else,
where p ∈ R 2 , n ∈ S 1 is a unit vector, and the tangent vector satisfies ṗ ∈ R 2 , ṅ ⊥ n.

Introducing the Lagrangian L ± ξ = 1 2 (F ± ξ ) 2 on TΩ, and its Legendre-Fenchel dual the Hamiltonian H ± ξ , one can again under mild assumptions characterize u ± ξ as the unique viscosity solution to the generalized eikonal PDE H ± ξ (x, ∇u ± ξ (x)) = 1/2 with appropriate boundary conditions [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF]. In practice, we choose cost functions of the form

C ξ (p, ṗ, p) = C • ξ (p, ṗ)C * (|p|)
, where C * is the Reeds-Shepp car or Dubins car [START_REF] Dubins | On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents[END_REF] curvature penalty, namely

C RS (κ) := 1 + ρ 2 κ 2 , C D (κ) := 1 if |ρκ| ≤ 1, +∞ otherwise,
where ρ > 0 is a parameter which has the dimension of a curvature radius. The Dubins car can only follows path which curvature radius is ≤ ρ, whereas the Reeds-Shape car, in the sense of [START_REF] Duits | Optimal Paths for Variants of the 2D and 3D Reeds-Shepp Car with Applications in Image Analysis[END_REF] and without reverse gear), can rotate into place if needed. The Hamiltonian then has the explicit expression

H((p, n), (p, n)) = 1 2 C 0 ξ (p, n) -2 H * (n, (p, n))
where

H RS = 1 2 ( p, n 2 + + n/ρ 2 ) H D = 1 2 max{0, p, n + n/ρ } 2 .

Discretization of generalized eikonal equations

We construct a discrete domain X by intersecting the computational domain with an orthogonal grid of scale h > 0

X = Ω ∩ (hZ) d ,
where d = 2 for the curvature independent models, d = 3 for the other models which are posed on R 2 × (R/2πZ) -using the angular parametrization S 1 ∼ = R/2πZ (in the latter periodic case, and 2π/h must be an integer). We design weights c ξ (x, y), x, y ∈ X such that for any tangent vector ẋ at x one has where a + := max{0, a}. (Expression [START_REF] Dubins | On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents[END_REF], is typical although some models require a slight generalization.) The weights c ξ (x, y) are non-zero for only few (x, y) ∈ X at distance x -y = O(h), see Figure 1. Their construction exploits the additive structure of the discretization grid X and relies on techniques from lattice geometry [START_REF] Schürmann | Computational geometry of positive definite quadratic forms[END_REF], see [START_REF] Duits | Optimal Paths for Variants of the 2D and 3D Reeds-Shepp Car with Applications in Image Analysis[END_REF][START_REF] Mirebeau | Anisotropic fast-marching on cartesian grids using Voronois first reduction of quadratic forms[END_REF][START_REF] Mirebeau | Fast Marching methods for Curvature Penalized Shortest Paths[END_REF] for details. The generalized eikonal PDE

H ξ (x, ẋ) ≈ h -2 y∈X c 2 ξ (x, y) x -y, ẋ 2 + , (5)
H ξ (x, ∇ x u ξ (x)) = 1/2 is discretized as y∈X c 2 ξ (x, y)(U ξ (x) -U ξ (y)) 2 + = h 2 /2 (6)
with adequate boundary conditions. The solution U ε : X → R to this system of equations is computed in a single pass with O(N ln N ) complexity using the Fast-Marching algorithm [START_REF] Rouy | A Viscosity Solutions Approach to Shape-From-Shading[END_REF].

To be able to use the gradient to solve the problem (1), we need to differentiate the cost C(ξ, Γ) w.r.t. the first player strategy ξ ∈ Ξ. In view of ( 3) and ( 4), this only requires the sensitivity of the discrete solution values U ε (x * ) at the few points points x * ∈ X ∩Θ, w.r.t to variations in the weights c ε (x, y), x, y ∈ X. For that purpose we differentiate (6) w.r.t. ξ and obtain

y∈X ω ξ (x, y) (dU ξ (x) -dU ξ (y) + (U ξ (x) -U ξ (y))d ln c ξ (x, y)) = 0, where ω ξ (x, y) := c 2 ξ (x, y)(U ξ (x) -U ξ (y)) + .
This equation ties dU ε (x) to those dU ξ (y), y ∈ X, such that ω ξ (x, y) > 0, hence satisfying U ξ (x) < U ξ (y) by construction. In the spirit of automatic differentiation by reverse accumulation [START_REF] Griewank | Evaluating derivatives: principles and techniques of algorithmic differentiation[END_REF], these relations are back-propagated from any Θ keypoint x * to the Υ seeds, at a modest cost O(N ).

Numerical results

The chosen physical domain R is the rectangle [0, 2] × [0, 1] minus some obstacles, as illustrated on Figure 2. Source point is (0.2, 0.5) and target No intervention from the first player. The cost function is

C ξ (p, ṗ, p) = C * (|p|), (7)
where C * (κ) is respectively 1, 1 + ρ 2 κ 2 and 1 + ∞ × χ ρκ>1 , with ρ := 0.3. The differences between the three models are apparent: the curvature independent model uses the same path forward and back; the Reeds-Shepp car spreads some curvature along the way but still makes an angle at the target point; the Dubins car maintains the radius of curvature below the bound ρ, and its trajectory is a succession of straight and circular segments.

Fresh paint based detection. In this toy model, see Figure 3, the first player spreads some fresh paint over the domain, and the second player is regarded as detected if he comes back covered in it from his visit to the keypoint. The cost function is

C ξ (p, ṗ, p) = ξ(p)C * (|p|) (8)
where ξ : R → R + is the fresh paint density, decided by the first player, and C * (κ) is as above. For wellposedness, we impose a bounds on the density, namely 0.1 ≤ ξ ≤ 1, and subtract the paint supply cost R ξ(p)dp to [START_REF] Barbaresco | Minimal geodesics bundles by active contours: Radar application for computation of most threathening trajectories areas & corridors[END_REF].

The main interest of this specific game, also considered in [START_REF] Benmansour | Derivatives with respect to metrics and applications: subgradient marching algorithm[END_REF], is that C(ξ, Γ) is concave w.r.t. ξ ∈ Ξ. The optimal strategy for player Ξ is: in the curvature independent case to make some "fences" of paint between close obstacles, and in the curvature penalized models to deposit paint at the edges of obstacles, as well as along specific circular arcs for the Dubins model. At the optimum for player Ξ, a continuum of paths have equal cost for player Γ, covering a large portion of the domain with a uniform density as illustrated on Figure 3 (bottom), see [START_REF] Benmansour | Derivatives with respect to metrics and applications: subgradient marching algorithm[END_REF] for details. Visual detection. The first player places some cameras, which efficiency at detecting the second player decreases with distance and is blocked by obstacles, see Figure 4. The cost function is

C ξ (p, ṗ, p) = C * (κ) q∈ξ [p,q]⊆R 1 q -p 2 , (9)
where ξ ∈ Ξ is a subset of R with prescribed cardinality, two in out experiments. The green arrows on Fig 4 originate from the current (non optimal) camera position, and point in the direction of greatest growth ∇C(ξ, Γ) for the first player objective function.

Radar based detection. The first player places some radars on the domain R = [0, 2] × [0, 1], here devoid of obstacles, and the second player has to fly by undetected. The cost function is

C ξ (p, ṗ, p) = C * (|p|) q∈ξ ṗ, n pq 2 + δ 2 ṗ, n ⊥ pq 2 p -q 4 (10)
where n pq := (q -p)/ q -p , and ξ is a three element subset of [0.4, 1.6]. The parameter δ is set to 1 for an isotropic radar cross section (RCS), or to 0.2 for an anisotropic RCS. In the latter where a plane showing its side to radar is five times less likely to be detected than a plane showing its nose or back, at the same position. Green arrows on Figure 5 point from the original position to the (locally) optimized position for player Ξ. At the equilibrium, several paths are optimal for player Γ, shown in red on Fig 5. Computational cost On a standard Laptop computer (2.7Ghz, 16GB ram), optimizing the second player objective, by solving a generalized eikonal equation, takes ≈ 1s in the curvature dependent case, and ≈ 60 times less in the curvature independent case thanks to the absence of angular discretization of the domain. Optimizing the first player objective takes ≈ 100 L -BF GS iterations, each one taking at most 8s. For the stability of the minimization procedure, the problems considered were slightly regularized by the use of soft-minimum functions and by "blurring" the target keypoint over the 3 × 3 box of adjacent pixels.

Conclusion

We have modeled a motion planning problem that take into account navigation constraints cand minimizing an anisotropic probability of detection, while computing the gradient of the value function related to the sensors location. This model is thus useful for surveillance applications modeled as a two-player zero-sum game involving a target that tries to avoid detection.

  , γ), where C(ξ, γ) := T (γ) 0

Figure 1 :

 1 Figure 1: Examples of adaptive stencils used by the Fast-Marching solver

Figure 2 :

 2 Figure 2: No intervention from the first player. Shortest path from the blue point (left) to the red keypoint (right) and back.

Figure 3 :

 3 Figure 3: Fresh paint based detection. Top: Optimal distribution of paint, to mark a path from the blue point (left) to the red keypoint (right) and back. Bottom: Geodesic density at the equilibrium.

Figure 4 :

 4 Figure 4: Visual detection. Field of view of the cameras (black gradients), optimal furtive paths (red lines), local direction of improvement of the camera position (green arrows).

Figure 5 :

 5 Figure 5: Radar based detection. Optimal radar placement with an isotropic (top) or anisotropic (bottom) radar cross section.

github.com/Mirebeau/HamiltonianFastMarching