Ramón Martí Solano

Maria Kolarova

Phraseological loan translations in Bulgarian and in French: a crosslinguistic and cross-cultural study

Phraseological loan translations can be described as a special type of Anglicisms (Picone 1996: 3), less evident to the naked eye than loanwords or other direct or semi-direct borrowings. Nevertheless they are pervasive and extremely widespread in all European language families and constitute a major sign of globalization, in general, and of linguistic Anglicization, in particular. French (Fr) and Bulgarian (Bu), and their sister languages for that matter, are no exception to this global trend. An important number of these loan translations have been detected in corpora, the media and on the Internet such as Fr. franchir la ligne rouge and Bu. пресичам червената линия (to cross the red line), Fr. être sur le même bateau and Bu. в една и съща лодка сме (be in the same boat) or Fr. au milieu de nulle part and Bu.

в средата на нищото (in the middle of nowhere). Lexicographical, typographical, contextual and multilingual methodological criteria have been used in selecting potential loan translations. Even though most of the strings are calques in both languages, a few of them can be said to be specific to one language or the other, even though specificity should be regarded rather as frequency of use and consequently degree of integration in the recipient language. Some instances of language-specific calques are Bu. имаш пеперуди в стомаха (have butterflies in the stomach) and слон в стаята (an elephant in the room) or Fr. ce n'est pas ma tasse de thé (it's not my cup of tea) and été indien (Indian summer).

intensifying function in a text (Gläser 1998: 125). Hence, following Gläser's definition and Vinogradov's approach to word-like combinations, we could say that all the examples cited above could be subsumed under the general term phraseological unit.

Characteristics of multi-word units

Within the general class of phraseological units, Vinogradov distinguished between phraseological fusions, also known as idioms, phraseological unities and phraseological combinations. The subcategory of the fusions was made up of structurally fixed but semantically opaque combinations. An example that can easily fit this category is let the cat out of the bag. By contrast, the phraseological unities were seen as partially motivated, i.e. Vinogradov described their meaning as "a metaphorical extension of some original neutral sense" (Cowie 1998: 5). A good example here is the expression break the ice. Vinogradov's third subcategory, the phraseological combinations, is considered to be the most interesting and the most difficult to delimit (Cowie 1998: 5). These combinations consist of two components that are open-class words. One of them is used in a literal sense, whereas the other is used figuratively. Cowie's example of phraseological combination is meet the demand, where it is the verb meet that is used figuratively (Cowie 1998: 5).

On the whole, the most important characteristic features of phraseological units can be summarized as follows (Кювлиева-Мишайкова 1986: 10-13;Fiedler 2012: 241): a) Each phraseological unit is made up of at least two lexical words, i.e. it has a polylexemic structure. A single word cannot have a phraseological meaning. b) Phraseological units are, in principle, stable word groups both in terms of lexical meaning and syntactic structure. Yet, their stability should be conceived of as something relative because they may often undergo certain changes. Hence, one and the same phraseological unit can have different variations. c) Their relative stability allows for their natural reproducibility in the process of communication as ready-made units. d) Phraseological units have a single specific meaning which is not a sum total of the literal meanings of their components. In other words, they are characterized by idiomaticity, which is their prototypical, yet potential feature. e) Last but not least, phraseological units may perform various pragmatic functions in discourse due to their optional connotative features such as expressiveness and emotional colouring.

Phraseological units can be classified in different ways. According to Fiedler's classification, which is based on structural-semantic criteria, there are six relevant types of phraseological units. These are: a) phraseological nominations, e.g. hot potato or the elephant in the room; b) binominals, e.g. ifs and buts or back and forth; c) stereotyped comparisons, e.g. as good as gold or sleep like a log; d) proverbs, e.g. blood is thicker than water or cut your coat according to your cloth; e) winged words/catch phrases, e.g. speak softly and carry a big stick or Big Brother is watching you and f) routine formulae, e.g. don't mention it or so long (Fiedler 2012: 242).

Idiomatic vs. non-idiomatic multi-word units

The idiomatic multi-word units are highly lexicalized in the sense that their meaning is not deducible from the meanings of their components. That is why idioms are seen as semantically opaque or 'unmotivated'. By contrast, the non-idiomatic multi-word units have a low degree of lexicalization and a high degree of compositionality, i.e. the meaning of the whole unit can be deduced from the meanings of its components. Hence, the phraseological meaning is quite predictable. Yet, although it is motivated by the meanings of the constituent elements, it is not confined to them.

Choice of items under investigation

The items selected for the purposes of the present research are English-induced phraseological calques taken out from French and Bulgarian electronic sources such as corpora, newspapers, magazines, forums, websites, blogs, TV and radio programmes. They are idiomatic multiword units that differ in terms of their stylistic features and degree of idiomaticity.

Taking into account Fiedler's classification suggested above, we can say that almost half of the selected idioms are phraseological nominations like hot potato, white elephant or the tip of the iceberg, and the only routine formulaic sequence have a nice day. Some of the items are common for both French and Bulgarian, others occur in French only and a third group are specific to Bulgarian. The selected items and their degrees of integration are discussed in detail in sections 4, 5, 6 and 7 respectively.

Contrastive phraseology

Research in phraseology has traditionally been either theoretical on matters such as idiomaticity [START_REF] Wulff | Rethinking idiomaticity. A usage-based approach[END_REF] or centred around specific languages. The interface between phraseology and other linguistic areas or sub-areas has also been developed in the last decades especially in cognitive linguistics, corpus linguistics and cultural studies [START_REF] Dobrovol'skij | Figurative Language: Cross-Cultural and Cross-Linguistic Perspectives[END_REF][START_REF] Boyadzhieva | Language Change and Social Practices -the Mirror Effect[END_REF]. However, the study of phraseological units from a cross-linguistic perspective has yet to be developed although some very important research has already been done (Dobrovol'skij 2000;[START_REF] Colson | Cross-linguistic phraseological studies. An overview[END_REF]. It is our intention to push even further the analysis of phraseological units across languages by focusing on the influence of Anglo-American phraseology in French and Bulgarian as "the influence of English idioms on other languages has almost never been the topic of systematic and multilingual research" (Piirainen 2012: 516). This influence has, nevertheless, been observed and analysed across the board (Görlach 2001[START_REF] Görlach | English in Europe[END_REF](Görlach , 2003) ) but also in the framework of individual languages such as Spanish [START_REF] Oncins-Martínez | Using Corpora for Exploring and Assessing the Influence of English in Contemporary Spanish[END_REF], 2012[START_REF] Oncins-Martínez | English Idioms Borrowed and Reshaped: The Emergence of a Hybrid Metaphor in Spanish[END_REF], German [START_REF] Fiedler | Der Elefant im Raum … The influence of English on German Phraseology[END_REF] or Latvian [START_REF] Veisbergs | Phraseological borrowing[END_REF], among others. Our aim is to look at this influence from a contrastive perspective by observing and analysing common influence as well as specific cases found in one language and not in the other and therefore our theoretical approach can be described as cross-linguistic corpus linguistics. Although it is sometimes hard to trace the origin of a phraseological unit and to demonstrate the influence exerted by its English counterpart, we have used the following methodological criteria for our purposes:

a) The first attested examples are found in the media, especially in an Anglo-American context. b) Their 'foreigness' is signalled by the use of inverted commas, italics or metalinguistical comments such as "As the English say".

c) The expression is not found in general dictionaries or in dictionaries of idioms in French or Bulgarian or has been incorporated only in their last editions. d) Other Romance languages, as regards French, or other Slavic languages, as regards Bulgarian, have adopted the same expression.

All the above criteria cannot be applied to every item of our findings. For instance ce n'est pas ma tasse de thé (a loan translation from the English phrase it's not my cup of tea) does not conform to the fourth criterion: neither Spanish nor Italian nor Portuguese have calqued this phrase1 , which in any case does not invalidate the English origin of the French phrase (Martí Solano 2012: 200).

We have opted for idiomatic expressions rather than transparent or non-idiomatic ones: grammatical phrases such as as long as or by virtue of or other phrases of the like have a grammatical function rather than a semantic function. We have avoided transparent noun phrases and have selected idiomatic ones such as hot potato or red line. An important part of our selection is made up of predicate idioms, such as give the green light or throw the baby out with the bath-water, which are colourful and very expressive idioms and which lend themselves to calquing in the target languages.

Contrastive or comparative linguistics has traditionally focused on grammar rather than on lexis and it can be stated that the development of studies on languages in contact in the second half of the 20 th century, begun by Weinreich's seminal work (1953)

Definition of a borrowing

Despite the rich abundance of theoretical research produced since the mid-20 th century, modern theory of language contact still fails to provide a consensus categorization of the types of lexical borrowing (Furiassi et al., 2012: 5). In very broad terms, borrowing proper, which is justifiably considered to be the most frequent form of lexical influence of one language on another, is defined as "taking over of the form and (part of) the content of foreign words" (Alexieva 2002: 256). In studies focused specifically on the influence of English on other languages, and today English is undoubtedly "a major source of language influence worldwide" (Fiedler 2012: 239), it is crucial to provide a precise and yet exhaustive definition of English borrowings or Anglicisms. In doing so, scholars should take into account the relevant fact that while Anglicisms usually preserve their English-looking form in the vocabulary of the receptor language, they also often occur in less-readily identifiable forms, disguised as adapted loans, false Anglicisms, calques or semantic loans. In addition, the influence of English has gone beyond the level of the individual lexical item so that the English-induced forms of borrowing today encompass a wider range of phenomena at levels higher than that of the individual lexeme, namely the levels of syntax and phraseology, including also the semantic, pragmatic, stylistic and cultural level (Furiassi et al. 2012: 5).

Hence, Görlach's definition of an Anglicism as "a word or idiom that is recognizably English in its form (spelling, pronunciation, morphology, or at least one of the three), but is accepted as an item in the vocabulary of the receptor language" seems to be convenient mainly for lexicographic purposes but fails to account for other existing or emerging phenomena of Anglicization. For this reason, some scholars propose that the term Anglicism should be taken in a much wider sense so that it could cover any instance of interference ascribable to the influence of English (Furiassi et al. 2012: 5). Gottlieb provides a very flexible and comprehensive definition of Anglicisms, one that could readily suit the purposes of any scientific research concerned with the phenomenon of Anglicization. In his definition, Gottlieb identifies an Anglicism as "any individual or systemic language feature adapted or adopted from English, or inspired or boosted by English models, used in intralingual communication in a language other than English" (Gottlieb 2012: 175).

Based on different scholarly views of language contact, the typology of lexical borrowing, as it is presented today in the linguistic literature, is quite detailed but still tentative.

Borrowings are generally discussed as direct and indirect. Direct borrowings into a receptor language are easily detectable, bearing some prominent formal features of the source language. They include adapted or non-adapted loanwords, false loans and hybrids. Indirect borrowings, by contrast, are not immediately recognizable because the source language concept is rendered into the receptor language by native means. Indirect borrowings include calques and semantic loans (Furiassi et al. 2012: 6). This type of lexical borrowing is given due attention in the following subsection.

Definition of a calque or a loan-translation.

Calquing, also referred to as replacement (Alexieva 2002: 256), is "a language contact phenomenon that is either conscious or spontaneous in nature and is realized through native elements on a foreign model" 2 (Благоева 2005: 12). Calques are generally subdivided into four types: loan translations, loan renditions, loan creations and semantic loans (Alexieva, 2002: 256;Görlach 2003: 96). Some scholars, however, treat semantic loans as a separate type of indirect lexical borrowing, which is outside the category of calques proper (Furiassi et al. 2012: 6).

Loan translations are single words or multi-word units that render as closely as possible the source language concept into the receptor language. Such examples from English into Bulgarian are върхът на айсберга, горещ картоф or слонът в стаята, which are one-toone translation equivalents of the etymons the tip of the iceberg, hot potato and the elephant in the room, respectively. A French example is le début de la fin from the English the beginning of the end. In many cases the calquing process is not complete and only one part is translated, which results in the formation of semi-calques. Bulgarian examples are екшънгерой from action hero and кавърверсия from cover-version.

Loan renditions are units which translate part of the foreign item and provide a looser equivalent for the other part. Thus, with loan renditions the equivalence involves partial difference in terms of form or meaning. This can be illustrated by the following examples: in Bulgarian изтичане на мозъци is the equivalent of brain drain and in French boîte de nuit stands for night club.

Loan creations are coinages prompted by foreign items but formally independent, i.e. they freely render the source item model in the receptor language. Examples of English-induced loan creations in Bulgarian are зърнени ядки (cornflakes) and връзки с обществеността (public relations). A French example is planche à voile prompted by the English windsurfing.

Semantic loans are already existing items in the receptor language, whether native or previously borrowed, which take over one of the meanings of the foreign word. Good examples in Bulgarian are мишка (mouse) and памет (memory). In French a semantic loan is papillon in the sense of the swimming style 'butterfly' (Alexieva 2002: 256-257;Görlach 2003: 96;Furiassi et al. 2012: 7-8). Semantic loans will not be further discussed because they fall outside the scope and purposes of the present research.

As Görlach points out, calques "provide an excellent alternative to loanwords", especially in language-conscious communities, whose members and/or authorities are against the excessive influx of foreign words, trying to keep their native language as intact as possible.

Thus, purism appears to be a strong determining factor behind calquing (Görlach 2003: 96-97). In the case of French and Bulgarian under the influence of English, it appears that Bulgarian is now as open to Anglicisms as it has never been before and what is more, it is the direct borrowing of English words that wins the upper hand. To some extent, this could be explained by the fact that Bulgarian society went through a long period of relative isolation and is now trying to keep up and identify with the western world. French, by contrast, is noteworthy for its tendency to purism so that calquing in French seems to be stronger than direct borrowing. This is also true for Icelandic, Russian and Greek but even these languages have recently quite conspicuously shifted to "the easy takeover of an English word rather than the more cumbersome search for a replacement", to quote Görlach (Görlach 2003: 98).

Phraseological loan translations versus internationalisms or widespread idioms

A phraseological loan translation is the translation, word for word, of a foreign expression or, to a lesser extent, its lexico-grammatical adaptation so that the new expression may conform to the genius of the language and to its own linguistic system. The resulting expression may, in some cases, develop a new sense unknown in the donor language, acquire new connotations, by amelioration or pejoration, or even change its grammatical category.

The most detailed account of the common phraseological stock in most European languages, and several other languages worldwide, is to be found in Piirainen 2012 (see

references). Widespread Idioms in Europe and Beyond: Toward a Lexicon of Common

Figurative Units is an impressive piece of research and an extraordinary academic achievement as far as cross-linguistic phraseology is concerned. In the preface Piirainen argues against the use of the term internationalisms and introduces and defines the newlycoined term Widespread idioms (WIs) as idioms that "have the same or a similar lexical structure and the same figurative core meaning in various different languages" (Piirainen 2012: xx).

Going through all the phraseological units that constitute the bulk of the book, we have detected four idioms that can be classified as phraseological calques from English: the light at the end of the tunnel, the tip of the iceberg, throw the baby out with the bath-water and have a skeleton in the closet/cupboard.

The first occurrence in the Oxford English Dictionary (OED) of the tip of the iceberg is from 1916. Concerning the light at the end of the tunnel the first attested occurrence in the OED, although not in this exact lexical frame, is from 1879. The first three occurrences in the OED of throw the baby out with the bath-water are from the period between 1840 and 1853 and all of them make reference to its German origin. As regards have a skeleton in the closet/cupboard the first attested occurrence is from Thackeray and dates back to 1855. What one can learn about all this is that these idioms have all been used in the English language for more than 100 years and that their counterparts in all the European languages are relatively recent: for instance, the first occurrence in the French general corpus Frantext of un cadavre dans le placard (a skeleton in the closet/cupboard) is from 1974. There is only one occurrence of refiler le bébé avec l'eau du bain (a variant form of throw the baby out with the bath-water) and it is from 2006.

The fact that an idiom may be widespread and that both its lexico-grammatical structure and meaning are shared by many different languages does not necessarily imply that the idiom in question cannot be a phraseological loan translation from English. Let us take the example of to be in the same boat, found not only in French and in Bulgarian (see section 4) but also in Spanish as estar en el mismo barco (Lorenzo 1996: 633) or in German as im selben Boot sitzen (Görlach 2003: 62). Both scholars claim that there is strong evidence that the respective phrases are loan translations from English. Even though the metaphor was used by Cicero and was later included in Erasmus's Adagia (Piirainen 2012: 137), we believe that a distinction should be made between the metaphor and the idiom itself as "The metaphorical source conjures up the original image rendered in whatever lexico-grammatical form whereas the phraseological source constitutes the exact or quasi-exact rendering of the phrase as found in the donor language." (Martí Solano 2012: 204)

Bulgarian and French common calques

Even though one might think that contrasting a Romance language with a Slavic one would not yield as many cases of common multi-word loan translations as if Romance languages or Slavic languages were compared between them, this proves to be wrong. The influence of English in general, and of English phraseololgy in particular, is extremely widespread in all European languages (Görlach 2003[START_REF] Pulcini | The Lexical influence of English on European Languages: From words to phraseology[END_REF]) regardless of language family, language contact or even number of speakers or international linguistic weight. However, it should be remembered that different historic and social events, mainly in the second half of the 20th century, have encouraged or hampered the flood of borrowings, calques, renderings and other hybrid types of lexical material of English origin in either language. The number of common calques exceeds all expectations and cannot obviously be analysed in the context and limits of a paper like this. These include, among many others, the French and Bulgarian loan translations of a safety net, a skeleton in the cupboard, an elephant in the room, below the belt, brain drain, brain washing, cross the red line, cross your fingers, give sb/sth the green light, glass ceiling, grey area, grey market, grey economy, Have a nice day!, have butterflies in your stomach, hot potato, in the middle of nowhere, Indian summer, keep a low profile, lame duck, love handles, not be sb's cup of tea, on the world scene, open-door day, open-door policy, paper tiger, pink elephants, point of no return, the ball is in sb's court, the best of both worlds, the eleventh hour, the tip of the iceberg, throw the baby out with the bathwater and white elephant.

Linguists and sociolinguists are unanimous that language reflects the political, economic, social and cultural developments in a given society, together with the beliefs, attitudes and modes of thinking of the people who belong to that society and share the same language [START_REF] Boyadzhieva | Language Change and Social Practices -the Mirror Effect[END_REF]. Bulgarian society was long isolated from the Western world and now it is still regarded as being in a process of transition from a rather confined ex-soviet-like type of society to a more open multicultural one [START_REF] Boyadzhieva | Reflections of Social Stereotypes in Modern Bulgarian and English Phraseology[END_REF]. During this transition period Bulgarian has been heavily influenced by English. This influence is part of the tendency to internationalization, which has become a characteristic feature of the development of the Slavic languages for the past decades (Благоева 2005: 56). The tendency to internationalization is triggered mainly by extralinguistic factors such as the intensity of intercultural communication, the increase in the economic, scientific and cultural relations, the vast exchange of information, the integration and globalization processes in the sociopolitical sphere (Благоева 2005: 57). Hence, the nature of internationalisms is again of particular interest to many scholars today. The issue has been thoroughly discussed in section 3 above.

Another active tendency in the development of the Slavic languages today is the tendency to nationalization, which is in opposition to the process of internationalization. Reproducing foreign items and patterns through native means, calquing represents the interface between the two opposing tendencies and can be regarded as a compensatory mechanism for achieving some balance between national and international elements in a given language (Благоева, 2005: 58). Some authors also speak of the so called serial calquing (серийно калкиране), which occurs when a linguistic unit from one language is reproduced in a number of foreign languages (Благоева 2005: 58;[START_REF] Obara | Teoretyczne problemy kalkowania[END_REF]). Thus, taking into account the fact that for the past decades, English has been the main source for the internationalization of Slavic lexis through calquing, it should come as no surprise that most of the English phrases discussed in this paper occur as calques not only in Bulgarian but also in other Slavic languages as well as in French and other Romance languages as illustrated below. The OED clearly specifies that the phrase throw the baby out with the bath-water is "After German das Kind mit dem Bade ausschütten (early 16th cent.)." The first three examples make direct reference to its German origin: "Schüttet das Kind mit dem Bade aus; literally 'empties out the baby with the bath'" (1840), "The Germans say, 'You must empty out the bathing-tub, but not the baby along with it.'" (1853) and "The defect of the above article seems to us to be that, in the language of a homely German proverb, it throws out the baby with the bath." (1860).

These are two very different idioms in the sense that the former is American and relatively recent whereas the latter is British-although a loan translation from a German phrase-and with a longer history. In any case, all of the common idioms listed above seem to have grabbed the attention of journalists and other media professionals and therefore they have been integrated in the phraseology of each language, although their degree of integration may differ from idiom to idiom and from language to language.

Calques specific to Bulgarian

For the past few decades, calquing from English has been a very productive means of expanding the lexical stock of Bulgarian, although, as mentioned above, the items that have come into Bulgarian through direct borrowing vastly outnumber the items that have resulted from calquing (Благоева 2005: 63). What is particularly interesting is that some of the English-induced phraseological calques in Bulgarian display a high degree of variability in the sense that they often occur in various forms as regards word order, the omission of an item or the addition of a new one.

English-induced calquing is also a widely used means of filling certain slots in specific thematic fields in the Bulgarian language such as computer technologies and the Internet, finances and banking, politics and diplomacy, military science, ecology and others. The phraseological calques in particular are prevalent in the sphere of computer technologies, as well as in the socio-political, economic, military, medical, ecological, sports and everyday life lexis (Благоева 2005: 64).

The following paragraphs discuss the meaning, use and formal structure of a number of phraseological calques that are specific to Bulgarian. They belong to the everyday life lexis and display a high degree of lexicalization.

The first phrase to begin with is (имам) пеперуди в стомаха, which is a one-to-one translation of the English idiom (have) butterflies in one's stomach, used to express a feeling of nervous anticipation about something you are going to do. The calque is very popular with young people but to many adults its meaning is still unclear. An important point to make here is that in Bulgarian the calque is frequently used to express a fluttering sensation caused by a love feeling. In other words, there is a certain degree of narrowing of the meaning of the calque, whereas its English etymon has a considerably wider sense, not being restricted so much to romantic feelings. Furthermore, the expression пеперуди в стомаха is registered in an online dictionary of Bulgarian phrases (frazite.com) precisely with the definition "a love feeling".

The calque occurs not only with the verb имам, which is the Bulgarian equivalent of the English have, but also with other verbs such as усещам (feel) and хвърча (fly). This can be seen in the following examples: Мисля, че е вълнуващо да имаш пеперуди в стомаха, защото това означава, че все още ти пука много -каза Серина (I think it is exciting to have butterflies in your stomach because this means that you still care a lot, said Serena); Усещам пеперудите в стомаха (I feel the butterflies in my stomach) and Пеперудите, които хвърчат в стомаха ти (The butterflies that are flying in your stomach). The last two example sentences display the degree of variability of this phraseological calque in Bulgarian.

On the one hand, the Bulgarian equivalent of butterflies may take the full article inflection (пеперуди-те) and on the other hand, it may be followed by a relative clause as in Пеперудите, които хвърчат в стомаха ти. In addition, this calque is often used simply as a noun phrase without a verb, i.e. пеперуди в стомаха (butterflies in one's stomach). It should be noted here that the English phrase itself is also characterized by such a degree of variability in the source language.

The second phrase to consider is слон в стаята. It is again a literal translation of its English etymon an elephant in the room and is frequently used by Bulgarian politicians, journalists and public figures to indicate that there is an obvious problem that people do not want to talk about. In writing, however, the expression слон в стаята often occurs enclosed within quotation marks, printed in italics or is accompanied by an explanation. This can be illustrated by the following citation: Ромите и "слонът в стаята" (The Roma people and "the elephant in the room'') Какво означава "слон в стаята"? Това е англо-саксонски израз, който означава, че има някакъв огромен проблем, голям колкото "слон в стаята", но ние не искаме да го видим. (What does the expression an elephant in the room mean? This is an Anglo-Saxon expression which means that there is a big problem, as big as an elephant in the room but we don't want to see it.)

The explanation and the metalinguistic elements that are often an invariable part of the use of this expression are indicative of its opacity and low degree of integration in Bulgarian. Yet, the calque is gaining more and more ground and can be heard on various TV and radio programmes, in addition to its use in the printed media.

Another English-induced calque specific to Bulgarian is бял слон, which literally translates its English etymon white elephant. The term derives from the white elephants kept by Southeast Asian monarchs in ancient times but its first occurrence in English as an idiomatic expression was in the middle of the 19 th century. The white elephants were considered sacred and laws protected them from labour. Thus, receiving a white elephant as a gift from a monarch was both a blessing and a curse. It was a blessing because the animal was seen as a sign of the monarch's favour, and a curse because the recipient now had an animal that was very expensive to maintain but he could neither give it away, nor use it for practical purposes.

Today the expression white elephant is generally used to refer to a burdensome possession that creates more trouble than it is worth.

The calque бял слон is still new and rarely used in Bulgarian. It is occasionally found in newspaper articles, surrounded by quotation marks and followed by an adequate definition as in "Бял слон" е етикет за скъп и безсмислен проект… Съвременна България е огромен развъдник за "бели слонове". Явно страната ни е много богата, щом въпреки това още не е фалирала ("White elephant" is a label for an expensive but meaningless project…

Modern Bulgaria is an enormous breeding place for "white elephants". Obviously our country is very rich because otherwise, it would have gone bankrupt by now).

The low frequency of occurrence of the calque бял слон in Bulgarian, the quotation marks and the provided definition prove better than anything else the foreign origin of this phrase and its extremely low level of integration in the receptor language. Nevertheless, its use is quite justifiable and effective because it contributes to the colourfuless and sensational tone of the journalistic texts in which the phrase occurs.

The last item to consider in this section is the routine formulaic sequence имайте хубав ден.

It is a direct, word-for-word translation of the English imperative structure have a nice day and may be frequently heard at the end of TV and radio broadcasts. The typically Bulgarian formulaic expression is приятен ден (nice day)a noun phrase in which the head noun ден (day) is pre-modified by the adjective хубав (nice). The use of the verb имам (have) in its imperative form имайте followed by the noun phrase хубав ден (nice day) is a direct transfer from English. Thus, the whole expression имайте хубав ден is considered by many Bulgarians as rather awkward, unnatural and superfluous because it does not fill an empty slot in Bulgarian lexis but simply reproduces a foreign model alongside an already existing native one. This particular calque could also be discussed as an instance of syntactic interference, which is due to word-for-word translation. Interference, for its part, is discussed by some authors as negative transfer (Danchev 1982: 41, 45), which could be seen as a reasonable explanation why the expression имайте хубав ден is not wholeheartedly accepted by many Bulgarians. This negative attitude could be illustrated by the following example: Имайте Another point to make concerning the phrase имайте хубав ден is that interference in this case does not affect the source language but the receptor language. Furthermore, although the calque arouses some negative feelings and is not fully integrated into Bulgarian, it is quite transparent and is often heard in everyday conversations, probably because it is more emphatic and produces a much stronger effect on the listener.

The analysis of the calques in this section shows again that Bulgarian is quite open to the influx of English items and patterns and it readily borrows them through calquing. In addition, it proves that English-induced calquing is not only a powerful and efficient means of enriching the Bulgarian lexical stock but it is also a very convenient way of achieving unexpectedness, sensationalism and a strong, dramatic impact.

Calques specific to French

Even though éléphant blanc (white elephant) has been classified as specific to Bulgarian, it is always possible to find examples of the French version in the press, and obviously on the Internet. We consider that cases like these, thinly-represented and usually found in italics and/or inverted commas, represent "the lowest degree of the continuum or cline of adaptation and institutionalisation of phraseological calques" (Martí Solano 2012: 211) and therefore have been excluded as common calques.

Most of the loan translations found in French are also found in the other three main Romance languages, i.e. Italian, Portuguese and Spanish. However, there is one that really constitutes 'the French exception', namely ce n'est pas ma tasse de thé (it's not my cup of tea). Together with été indien (Indian summer), poignées d'amour (love handles) et adopter un profil bas (keep a low profile), these are the four idioms that have been selected as specific to French and which are loan translations from English. It should be noted here that the English Indian summer has two Bulgarian equivalents which are not, however, literal, word-for-word translations of the English phrase. These are сиромашко лято (paupers' summer) and циганско лято (gypsy summer). Another point to make is that love handles has been calqued into Bulgarian as well, but the calque любовни дръжки, which is a literal translation of the English phrase, is popular mainly with young people who have some knowledge of English or are interested in fitness, weight loss, dieting, etc. Furthermore, the calque is usually enclosed within quotation marks or is accompanied by a definition, which is yet another sign of its very low degree of integration in Bulgarian, e.g. Може би сте пробвали различни стратегии да се отървете от "любовните дръжки", но не сте успели. (Probably you have tried various strategies to get rid of your love handles but all your efforts have been in vain).

Ce n'est pas ma tasse de thé is a well-known phrase in French with a high degree of institutionalisation but which tends to be used more in speech than in written form (Martí Solano 2012: 200). The following example illustrates its use: Le Génie du christianisme n'a jamais été la tasse de thé de l'athée inflexible qui refusa d'entrer dans l'église le jour des obsèques de son maître et ami Mitterrand. (Libération 12/03/2010) (The Génie du christianisme has never been the cup of tea of the inflexible atheist who refused to walk inside the church on the day of the funeral of Mitterrand, his master and friend). Not be sb's cup of tea is a typical case of a negative polarity idiom in English. Obviously the polarity of its French loan translation is equally negative as the phrase would not function pragmatically and discursively otherwise.

Keep a low profile and Indian summer are two phrases of American origin. The former is relatively recent whereas the latter dates back to the 18 th century. The OED provides a first example of low profile from 1967: "We made a series of right moves: ... adopting a policy of 'masterly inactivity' when the struggle was reaching its crucial stages, and maintaining a 'low profile' in Jakarta ever since." The first example recorded in the OED of Indian summer is from 1790: "Fine weather-Indian summer."

The paradigm of a multi-word loan translation seems to be epitomised by adopter un profil bas as it has beeb mentioned by scholars working in the field of languages in contact or Anglicisms (Picone 1996: 4 & 101;Görlach 2003: 62) Likewise, the phrase Love handles is from American origin and the first example in the OED is from 1970: "Love handles, the fat on one's sides." This first occurrence is actually a definition of the phrase, which shows the high level of opacity when the phrase started to be used in English. Because of its specialised meaning and denominational status, the phrase, both in English and French, is only found in specialised contexts as illustrated by the No occurrences of un éléphant blanc have been found in Frantext, which is not surprising as this phrase has been classified as specific to Bulgarian. By the same token, there are only 2 occurrences of papillons dans le ventre, which is really extremely low. The loan translation of an elephant in the room, although classified as specific to Bulgarian, seems to be gaining ground lately in French, but not sufficiently enough to be taken into consideration, as 7 occurrences of un elephant dans la pièce have been found in the newspaper archives only from 2010 to the date of the actual search 3 .

We have found 15 occurrences of au milieu de nulle part in Frantext, the first one being from 1985: J'appelle Lorraine d'une cabine plantée au milieu de nulle part (I ring up Lorraine from a phone box right in the middle of nowhere). However, only 3 occurrences of jeter le bébé avec l'eau du bain have been found in the same corpus, which again highlights the fact that even some common calques can be thinly-represented in one language or the other as their degree of integration and institutionalisation, as well as their frequency for that matter, is rather low.

Inverted commas and metalinguistic elements

The most straightforward, and the most widespread, way of signalling the presence of an A second method of signalling the presence of a phraseological loan translation consists in the inclusion of a metalinguistical comment such as comme disent les Anglais (as the English say) as in the next four examples: Si vous aviez besoin de plaire, je comprendrais, bien que ce ne soit pas, comme disent les Anglais, ma tasse de thé, la fonction de vos ruses (If you needed to be liked, I would understand it, even though the purpose of your trickeries are not, as the English say, my cup of tea). This is the first non-literal occurrence of ce n'est pas ma tasse de thé in Frantext, which corroborates its English origin. The second example is Gardons-nous donc, comme disent les Anglais, de jeter le bébé avec l'eau du bain (Let's avoid, then, throwing the baby with the bath-water, as the English say). The third one is: Bref, adhérer à un groupe de lecture n'est pas forcément un "lit de roses", comme disent les Anglais, (Anyway, to join a reading group is not a bed of roses, as the English say.) And finally, Attention ! Sans frontière dressée par nous, rédacteurs en chef, le ciel est la limite 4 , comme disent les Anglais. (Look out! Without walls being built by ourselves, chief editors, the sky is the limit, as the English say.)

In the example mais les observateurs sont unanimes pour constater que jamais un président « lame duck » (canard boiteux), c'est-à-dire condamné à la retraite, n'a manifesté autant de vigueur. (but the observers unanimously certify that never has a lame duck-that is a president condemned to retirement-shown so much vigour), a very special and rather infrequent case of metalinguistical comment is put into place. In fact the phrase lame duck could be seen as a culturally-bound compound word specific to American politics. The journalist not only uses italics and inverted commas but also the literal translation into French (canard boiteux) and a contextual explanation of its meaning (underlined in the example).

Information from an Anglo-American setting.

Phraseological loan translations are sometimes found in news and information concerning the United Kingdom, the United States or any other English-speaking country. Their presence is rather conspicuous because they tend to be the literal translation of a foreign phrase unknown in the recipient language. These 'weird' phrases are usually marked by the use of italics or inverted commas, or even both as in the following example: « C'est un message au président

Conclusions

Common research on phraseological calques in two languages from different families such as French and Bulgarian has proved to be very revealing of the global linguistic phenomenon of Anglicization. The impact that English idioms, phrases, formulas and other types of fixed word combinations has on other languages has not still been measured to its full extent. This influence is considerable and widespread but is just the tip of the iceberg of a much larger linguistic phenomenon that covers morphology, semantics and other linguistic subfields.

All in all we can say that recent phraseological calques from English are widespread both in Bulgarian and in French although some of them tend to occur in certain contexts or genres in one language or another. However, some calques appear to be language-specific although shared with other languages. Our results show that transnational intertextuality among various types of media is at the core of some of the novel or even well-established calques from English into French and Bulgarian. A search for occurrences of these multi-word units in the archives of newspapers, general corpora and the Internet casts light on their degree of institutionalisation and lexicalisation. Moreover, it shows actual instantiations in different contexts allowing a lexico-grammatical and pragmatic reanalysis of these loan phrases or loan translations. Phraseological calques tend to appear in the media, especially in the written press due to the massive influence on journalists of the English-speaking media.

  give the green light -(Bg) давам зелена светлина, (Polish) dać zielone światło, (Czech) dát zelenou, (Russian) давать зелёный свет; (Fr) donner le feu vert, (Sp and Po) dar luz verde, (It) dare luce verde. cross the red line -(Bg) пресичам червената линия, (Polish) przekroczyć czerwoną linię, (Czech) překročit červenou čáru, (Russian) пересечь красную линию; (Fr) franchir la ligne rouge, (Po) cruzar a linha vermelha, (It) varcare la linea rossa. in the middle of nowhere -(Bg) в средата на нищото, (Polish) pośrodku niczego, (Czech) uprostřed ničeho; (Fr) au milieu de nulle part, (Sp) en medio de ninguna parte. the tip of the iceberg -(Bg) върхът на айсберга, (Polish) wierzchołek góry lodowej, (Czech) jen špička ledovce, (Russian) вершина айсберга; Fr. la partie visible de l'iceberg, (Sp) la punta del iceberg, (Po) a ponta do iceberg. be in the same boat -(Bg) в една и съща лодка сме, (Polish) siedzimy w tej samej łódce, (Czech) být na jedné lodi, (Russian) быть /сидеть/ в одной лодке; (Fr) être sur le même bateau, (Sp) estar en el mismo barco, (It) essere sulla stessa barca, (Po) estar no mesmo barco. throw the baby out with the bath-water -(Bg) изхвърлям бебето (заедно) с мръсната вода, (Czech) vylít s vaničkou i dítě / vyhodit dítě ven s vaničkou, (Russian) вместе с водой выплеснуть и ребенка; (Fr) jeter le bébé avec l'eau du bain, (Sp) tirar el bebé/niño con el agua del baño/de la bañera. hot potato -(Bg) горещ картоф, (Polish) gorący kartofel, (Czech) horký brambor; (Fr) patate chaude, (Sp) papa/patata caliente, (It) patata bollente. The first occurrence of hot potato in its figurative sense of "[S]omething difficult or painful to deal with; a controversial or awkward matter or situation.", as defined in OED, is from the novel The Groves of Academe by the American writer Mary McCarthy: "It was a very hot potato…I chose to ignore the question." (1952).

  хубав ден! Скоро прочетох някъде това пожелание и се стъписах. Стандартните английски фрази в моя речник са ограничени на брой, но все пак направих асоциация с песента Have a Nice Day на Bon Jovi и с традиционното Have a nice evening, което казвам на преподавателката си след края на часовете по английски. (Have a nice day! Recently I read this greeting somewhere and I was shocked. The standard English phrases in my vocabulary are limited in number, but still I associated the expression with Bon Jovi's song Have a nice day and with the traditional Have a nice evening, which I say to my teacher at the end of our English class).

  . In the Dictionnaire des Expression et Locutions (DEL) there is an entry for [Adopter] un profil bas. After a brief definition, the authors add that this phrase is widely used in the political press and that it could be a loan translation form English. Even the French general dictionary Le Petit Robert signals the phrase with the label ANGLIC. and dates its use back to 1970. There are 11 tokens of profil bas in the French general corpus Frantext. The first occurrence is from 1986: "Là, adoptant un profil bas, il attendait qu'on lui demande à quel étage il désirait se rendre..." (Keeping a low profile, he was there waiting to be asked what floor he wanted to go to…). A much more recent example shows the hesitation of the journalist in using the idiom by surrounding the noun phrase with inverted commas: Les ambassades recommandent donc la prudence à leurs ressortissants, certaines les incitent à ne pas sortir de chez eux, tandis que d'autres conseillent d'adopter "un profil bas" en cas de déplacement indispensable (Le Nouvel Observateur (30/05/2006) (Embassies therefore recommend their nationals to be careful, some of them even encourage them not to leave their homes whereas others advise them to keep a low profile if they absolutely need to go outside.). Even though adopter (keep) is the verb most widely used in the French phrase other variants are also common, such as faire (do) or jouer (play). The former is as commonly used as adopter whether the latter is less frequent. Here are two examples: Patrick Drahi est un homme discret. Il aurait voulu faire profil bas (Le Monde, 09/01/2015) (Patrick Drahi is a discreet man. He would have liked to keep a low profile) and Tout en acceptant, cette fois-ci, de jouer profil bas (Le Monde 13/05/2002) (While accepting, this time, to keep a low profile). In the case of English the variant forms maintain a low profile and adopt a low profile are also found in corpora although their frequency is extremely low. Interestingly enough the first occurrence in Frantext of été indien is to be found in the following example from 1865 by the great French writer Baudelaire: Un brouillard fumeux, ressemblant à celui de l'été indien, enveloppait toutes choses (A smoky mist, like that of an Indian summer, shrouded everything). Another more recent example taken from Frantext is the following: Toute la journée, la douceur de l'été indien a fait chanter les arbres fauves dans le ciel vert (All day long the softness of the Indian summer has made the fawn trees sing in the green sky). The phrase has got two distinct senses in English: on the one hand, "A period of calm warm weather which sometimes happens in the early autumn" (CALD3) and on the other hand, "A pleasant or successful time nearly at the end of someone's life, job or other period." (CALD3). It is only the first sense that is calqued in French as can be observed in the examples above.

  following example: Ils sont intéressés par les opérations des paupières et des poignées d'amour (Le Monde, 11/12/2007) (They are specially interested in eye-lids and love handles We have used mainly Frantext and Le Monde to check on the degree of integration and the strategies used by writers and journalists to signal the presence of a phraseological loan translation in French. Most of the items selected are well represented in the French newspaper archives from 1944 to date. Phraseological loan translations are more likely to be present in the newspaper archives than in the general corpus for obvious reasons related to size, genre and contemporariness.

  unusual word or expression is the use of inverted commas as shown in the following examples: Dans le domaine scientifique, la coopération doit être intensifiée, affirme l'une des3 This research was carried out on October 7 th 2013.

  Obama qu'il ne peut pas jeter impunément Israël sous le bus. », […] Les républicains reprochent à l'administration Obama d'avoir « attendu la 11 e heure » pour réagir. (Le Monde 20/09/2011) (This is a message for president Obama that he cannot throw Israel under the bus with impunity […] The Republicans blame the Obama administration for having waited until the 11 th hour to react.). Un éléphant dans la chambre has yielded only two article results in the French newspaper Libération and both articles are about the USA as illustrated in the following examples: L'administration Bush est tel «un éléphant dans la chambre» dans laquelle il est incapable «de se déplacer», assure Gore. (Libération, 14/12/2007) (The Bush administration is like an elephant in the room where it is incapable to move about). Son appel [Obama's] à l'unité n'aura pas empêché l'«éléphant dans la chambre» de barrir (Libération, 17/10/2008) (His call [Obama's] for unity wouldn't have hampered the elephant in the room to trumpet.

2. Calquing vs. borrowing (phraseological)

  , has contributed to the growing interest among scholars in this field. Both the European Society for the Study of English (ESSE) and The European Society of Phraseology (EUROPHRAS) play a pivotal role in contrastive lexicology and phraseology among European languages. The ESSE 2010 conference, University of Turin, Italy, included a seminar on the "Anglicization of European

	Lexis" and at the EUROPHRAS 2012 Conference, University of Maribor, Slovenia, a
	workshop was organised entitled "The influence of English on the phraseology of European
	languages". Finally, the ESSE 2014 conference, University of Košice, Slovakia, had also a
	seminar entitled "Comparative studies of English Idioms."

  résolutions, afin de prévenir la « fuite des cerveaux » hors d'Europe, […] (Le Monde, 14/10/1966) (As stated in one of the resolutions, cooperation in science must be intensified, sate in order to prevent brain drain outside Europe) and M. Allon est resté très vague en ce qui concerne la « ligne rouge » que les Syriens sont censés ne pas franchir. (Le Monde, 03/06/1976) (Mr Allon remained vague concerning the red line that Syrians are supposed not to cross).

This phrase is unknown in these three Romance languages that use instead no es algo que me vuelva loco (Sp)., não é a minha praia (Po) and non è il mio pane (It) that translate literally and respectively as it's not something that makes me crazy, it's not my beach and it's not my bread.


Antonio Paolillo

Anastasia Bolotnikova

Kévin Chappellet

Abderrahmane Kheddar

Visual estimation of articulated objects configuration during manipulation with a humanoid

Robotic manipulation tasks require on-line knowledge of the operated objects' configuration. Thus, we need to estimate online the state of the (articulated) objects that are not equipped with positioning sensors. This estimated state w.r.t the robot control frame is required by our controller to update the model and close the loop. Indeed, in the controller we use the models of the (articulated) objects as additional 'robots' so that it computes the overall 'robotsobjects' augmented system's motion and contact interaction forces that fulfill all the limitation constraints together with the physics. Because of the uncertainties due to the floating-base nature of humanoids, we address the problem of estimating the configuration of articulated objects using a virtual visual servoing-based approach. Experimental results obtained with the humanoid robot HRP-4 manipulating the paper drawer of a printer show the effectiveness of the approach.

I. INTRODUCTION

Recently, we have devised a multi-objective task space robot controller formulated as a quadratic program (QP) that includes manipulated objects modeled as an augmentation of the robot structure [START_REF] Vaillant | Multi-character physical and behavioural interactions controller[END_REF] [START_REF] Bouyarmane | Multirobot and task-space force control with quadratic programming[END_REF]. By doing so, robots and manipulated objects are integrated in a "multi-robot system" that is controlled with a single QP when they come to interact. We have illustrated the use of this multi-robot QP (MQP) control framework in various challenging scenarios in computer graphics animation [START_REF] Vaillant | Multi-character physical and behavioural interactions controller[END_REF] and robotics [START_REF] Bouyarmane | Multirobot and task-space force control with quadratic programming[END_REF]. We have also used the MQP to make a humanoid robot drive a car [START_REF] Paolillo | Autonomous car driving by a humanoid robot[END_REF]. The driving wheel is (kinematically and dynamically) modeled as a "robot" with a fixed base and one rotational joint. It is then integrated to the MQP to form the multi-robot "humanoid plus driving wheel". The MQP computes desired state acceleration for the overall system that is coherent with both the kinematic constraints of the driving wheel and the contact forces to steer it. In this example, the position of the driving wheel matches exactly that of the rotation of the humanoid's wrist, as the grasping was particularly designed to meet such a request. However, the manipulation task would have been complicated if the robot has to grasp the driving wheel laterally or if it has to re-grasp. In fact, the driving wheel does not have encoders measuring its state. Doors, drawers, valves, switchers and other articulated tools can also be manipulated by a robot, and modeled as passive robotic structures to be integrated in the MQP. The user models all these object as separate urdf files to be This work is supported by the EU H2020 COMANOID project (www.comanoid.eu), and by the ROMEO 2 project, (www.projetromeo.com), bpifrance in the framework of the Structuring Projects of Competitiveness Clusters (PSPC)

1 CNRS-University of Montpellier, Laboratoire d'Informatique de Robotique et de Microéectronique de Montpellier (LIRMM), IDH Group, France. loaded by our controller and integrated as a single multirobot system as well described in [START_REF] Bouyarmane | Multirobot and task-space force control with quadratic programming[END_REF]. To know the state of the objects, that are not instrumented with sensors, we need to devise external observer to estimate them. Let us consider the example of a humanoid robot opening the drawer of a piece of furniture (see Fig. 1). The main link of the object is modeled as the base of a robot, and the drawer is considered as a link connected to the base through a prismatic joint. In order to control the opening motion, we can use the position task of our MQP that considers the error between the current position of the drawer p drawer (q), and its desired position. To achieve this task we need the measurement of the vector q, i.e. the configuration of the coupled "humanoid-piece of furniture" multi-robot system.

While the configuration of the humanoid is provided by the encoders mounted at each joint, an estimator has to be designed to reconstruct the configuration of the piece of furniture. We could think of using the robot forward kinematics for such estimation, assuming that the drawer is firmly grasped. However, in practice, this does not work for two main reasons: (i) the humanoid is a floating base system and we may have uncertainties to localize it, and (ii) the fingers that pull the printer have no encoders. Furthermore, some humanoid structures (such as HRP-4) have flexibilities in the ankles as well as in the fingers. As a results, we cannot rely much on the pure kinematics of the humanoid to estimate the drawer's joint value. One could think of improving the kinematic measurements integrating other sensors information (such as vision) that would inform about the grasp slipping or the backlash between the robot hand the drawer handle. In a nutshell, one needs to provide the robot with the capability of reconstructing the object configuration as required by the MQP (see Sect. II for the related work).

Through this example, we can extend the reasoning to the necessity of having the reconstruction of articulated objects configuration for the control of this "augmented" system. Knowing the configuration of the objects to be operated, allows to effectively close the loop of the MQP.

This paper addresses this problem:

• formulating (Section III) the estimation of articulated objects configuration as a virtual visual servoing problem (Section IV); • enabling closed-loop experiments with a humanoid operating the drawer of a printer, using the estimation of the object configuration as feedback in our MQP control framework (Section V).

II. BACKGROUND

Reconstruction of the configuration of articulated objects is a well studied problem in both the computer vision community (tracking of human motion, hands, etc.), and in robotics (tracking of robotic systems).

Several methods to track the motion of the articulated bodies have been proposed over the years. A kinematic tree based parametrization of articulated objects was used in [START_REF] Lowe | Fitting parameterized three-dimensional models to images[END_REF]. In this work the tracking was formulated as a tree parameters fitting problem, assuming full geometric model of an object to be known. In [START_REF] Drummond | Real-time visual tracking of complex structures[END_REF] the articulated structure tracking is handled as an extension of rigid object tracking. Independent trackers are used to compute motions of every link, then constraints between the links are imposed to find optimal set of motions that satisfy constraints. Multiple hypothesis using derivative-or gradient-free optimization techniques have been studied in the line of works (e.g. [START_REF] Oikonomidis | Efficient modelbased 3D tracking of hand articulations using kinect[END_REF][7] [START_REF] Park | Efficient 3D hand tracking in articulation subspaces for the manipulation of virtual objects[END_REF]). Such approaches are quite helpful to avoid local minima in optimization. Recently, these methods have also been further studied for estimating force from vision [START_REF] Pham | Towards force sensing from vision: Observing hand-object interactions to infer manipulation forces[END_REF] [START_REF] Pham | Handobject contact force estimation from markerless visual tracking[END_REF]. Yet, estimating the configuration of articulated structures with multiple hypothesis methods, can lack frame-to-frame consistency, due to occasional ambiguous observations which cause false hypotheses temporarily to have hight matching scores. This issue is not critical in many applications areas (surveillance, computer graphics, etc.), but in robotic closedloop control schemes, such estimation inconsistency could result in a bad behavior, such as jumps and jerky motions.

The motion of articulated objects can also be estimated using depth information in a GPU-based implementation of an Extended Kalman Filter (EKF) in [START_REF] Schmidt | DART: Dense articulated real-time tracking[END_REF]. This method has been extended further to consider physical constraints in tracker objective function by using contact information in robotic object manipulation scenario [START_REF] Schmidt | Depth-based tracking with physical constraints for robot manipulation[END_REF]. Another depth based method for estimating the state of a robotic arm was proposed in [START_REF] Klingensmith | Closed-loop servoing using real-time markerless arm tracking[END_REF]. It demonstrated robustness to calibration errors in a closed-loop manipulation task. Combination of depth and joint encoders data was used in [START_REF] Krainin | Manipulator and object tracking for in-hand 3D object modeling[END_REF] to track the state of robotic arm. For some robotic platforms, especially humanoids, end-effector distance from the camera may not exceed the minimum distance for depth data acquisition by a standard range sensor. In such case, RGB data processing is the only reliable source of visual information. Articulation tracking can also be done using images collected by a multicamera system and then processed by a particle filter [START_REF] Deutscher | Articulated body motion capture by stochastic search[END_REF]. These methods are yet computationally expensive to be applied at the low-level robot control.

Model-based approaches using monocular cameras are interesting techniques to achieve fast and accurate estimate of articulated objects configuration. In [START_REF] Nickels | Model-based tracking of complex articulated objects[END_REF] a Kalman filterbased tracking, using multiple models (such as the geometric and the appearance model of the object), is proposed to recover values of joint position and velocity, but not that of the floating base. Another method [START_REF] Comport | Kinematic sets for real-time robust articulated object tracking[END_REF] uses a virtual visual servoing-based approach [START_REF] Comport | Realtime markerless tracking for augmented reality: the virtual visual servoing framework[END_REF] providing the configuration of the object, but not expressed with the classical generalized coordinates. Further computations should be added to retrieve the joint variables.

An advantage of the model-based approaches is that the object and the visual features trackers can work together cooperatively: knowing the model of the object, the tracking of features leads to the reconstruction of the object configuration and vice-versa. Extending this concept, the features motion can also reveal geometric information of the observed object, that in turn is used to better track the features. This idea is exploited in [START_REF] Katz | Manipulating articulated objects with interactive perception[END_REF], that estimates the kinematic structure of the observed object combining the manipulation task with the perception algorithm. With the same principle, color and depth (RGB-D) information are processed by an EKF to provide also a measurement of the joint values in [START_REF] Martín Martín | Online interactive perception of articulated objects with multi-level recursive estimation based on taskspecific priors[END_REF]. In [START_REF] Martín Martín | An integrated approach to visual perception of articulated objects[END_REF], RGB-D data is processed in a unified framework able to estimate the pose, the shape and the structure of the observed object. These approaches do not need the model of the object, being itself estimated, and have been validated with simple articulated objects. Furthermore, they are not guaranteed to converge fast or to be reliable in all circumstances. Some are computationally expensive.

In our attempt to find the articulation tracking solution, whose formalism can be defined and used as a part of MQP in the closed-loop object manipulation control, no existing method could fit our requirements. Therefore, we took inspiration from previous works to devise a tracker whose formalism suits the MQP requirements and is presented in the remaining of the paper.

III. PROBLEM FORMULATION

To track articulated objects for robotic manipulation, we propose a method based on the so-called virtual visual servoing. To be self-contained, we briefly recall the basics of this technique and formulate our tracking problem.

Visual Servoing (VS) achieves a cartesian task using visual feedback [START_REF] Chaumette | Visual Servo Control, Part I: Basic Approaches[END_REF]: the control provides the camera velocity v c in order to zero the error between the measured and the desired value of visual features, denoted with s and s * , respectively.

VS can be exploited in a dual way: let a virtual camera moving in the cartesian space, whose unknown pose p is defined w.r.t. an observed object, i.e. in correspondence of some virtual visual features, collected in the vector s(p) 1 . The real pose of the camera p * is defined in correspondence of some measured visual features, collected in s * . The convergence of s(p) to s * implies the convergence of p to p * . Thus, in this context, the VS control law is used as "estimator" of the camera pose2 . This methodology, known as Virtual Visual Servoing (VVS), was introduced in the framework of augmented reality [START_REF] Marchand | Virtual Visual Servoing: a framework for real-time augmented reality[END_REF] [START_REF] Comport | Realtime markerless tracking for augmented reality: the virtual visual servoing framework[END_REF]. The same technique can be used to estimate the pose of the object moving in the camera scene, and extended to articulated object tracking as in [START_REF] Comport | Kinematic sets for real-time robust articulated object tracking[END_REF] that inspired our work. In particular, we share with [START_REF] Comport | Kinematic sets for real-time robust articulated object tracking[END_REF] the rationale of the approach, but we use a lighter formalism allowing an easier implementation of the method.

Let us consider the case of a camera looking at an articulated object, as in the depicted scenario of Fig. 1, where a humanoid robot is opening the drawer of a piece of furniture. The camera frame of the robot F c is used as reference in the algorithm. Its origin is placed at the focal point and the z-axis is aligned with the focal axis; the yaxis points downwards and the x-axis completes the righthanded coordinates system. Another frame of interest, F o , is arbitrarily defined at the floating base (FB) of the object.

The real configuration of the articulated object (the piece of furniture in the presented example) is denoted by

q * =   p o σ o q j   ∈ R 7+n (1)
where p o ∈ R 3 and σ o ∈ S 3 ⊂ R 4 are the position and orientation (expressed with a quaternion) of F o w.r.t. F c , respectively. Quaternions prevent the orientation representation from being singular. The vector q j ∈ R n in (1) is the vector of the generalized coordinates describing the internal configuration of the object, composed of n joints. Let us define a vector ρ ∈ R 3p of p points of interest (PoI) on the articulated object, whose projection on the image plane of the camera provides p corresponding visual features.

The real motion of the articulated object (1) produces the motion of real visual features collected in the vector s * ∈ R 2p . These features are actually observable and measurable on the image plane of the camera.

Whereas, the virtual configuration of the articulated object, denoted with q, affects the motion of virtual visual features, gathered in the vector s(q) ∈ R 2p . Note that the virtual visual features depend on (i) the geometric model of the object, assumed to be known and used to calculate the location of the virtual PoI in F c , and (ii) the camera intrinsic parameters, used to project the virtual PoI on the image plane. Since F c is the reference frame, we do not need the camera extrinsic parameters to build the projection model.

Assuming to known ρ and measuring s * , the aim of the tracking is to estimate q * . Only rigid objects are considered.

IV. VVS-BASED TRACKING OF ARTICULATED OBJECTS

The approach used to address the articulated object tracking problem relies on the VVS paradigm, i.e. uses a visual controller to estimate the derivative of the configuration vector. Thus, the estimate q is reconstructed through integration.

The error of the proposed control scheme is the difference between s(q) and s * defined in the previous section, i.e., e = s(q)-s * . The dynamics of this error says how the visual features change over time, and can be written as follows:

ė = ṡ(q) -ṡ * (2)
where the dot over the variables denotes the time derivative. The motion of the virtual visual features is a consequence of the motion of the virtual object (i.e. its PoI) w.r.t. the camera. In F c , the 6D velocity of each PoI can be expressed using the geometry of the object, i.e., v i = J i (q) q where J i is the 6×(7+n) Jacobian of the i-th PoI, i = 1, . . . , p. Thus, the dynamics of each virtual visual feature can be written as

ṡi (q) = -L i J i (q) q = A i q (3)
where L i is the image Jacobian associated to the i-th virtual visual feature [START_REF] Chaumette | Visual Servo Control, Part I: Basic Approaches[END_REF]; it depends on the depth of the corresponding PoI, that is available in the estimation process. The minus in (3) explains that the apparent motion of the features s(q) is due to the motion of the observed object (actually, its PoI) w.r.t. the camera (typical of hand-to-eye systems [START_REF] Chaumette | Visual Servo Control, Part II: Advances Approaches[END_REF]). The dynamics of the error can be now explicitly written:

ė = A q -ṡ * (4)
where we define, following the nomenclature in [START_REF] Comport | Kinematic sets for real-time robust articulated object tracking[END_REF], A = (A 1 , . . . , A p ) T as the 2p × (7 + n) articulation matrix, that relates the FB and joints velocity of the articulated object to the velocity of the visual features. Imposing a stable dynamics of the error ( ė = -λe, λ > 0), we derive the following "control law"

q = -λA # e + A # ṡ * (5)
where the term depending on ṡ * introduces an anticipatory action, improving the tracker performance. The bars below the variables denote approximations. The controller's local stability is ensured if A and its approximation are full-rank, and we use a good approximation of A and ṡ * . If (n+7) > p (as in the presented results), it has to be ensured that the configuration of the object is not singular [START_REF] Chaumette | Visual Servo Control, Part II: Advances Approaches[END_REF]. Note that we refer to (5) as a control law, being the term borrowed from the VS nomenclature. However, it actually is an estimator providing, at steady state, an estimate of the object's FB and joint velocities qk at each time kT s (T s being the sampling time of the algorithm and k the loop iterator). From this, q k , an estimate of the real object configuration, is obtained by numerical integration. To avoid the error introduced by brute-force normalization-based methods, the derivative of the quaternion is integrated using a closed-form exponential map method [START_REF] Zhao | A novel quaternion integration approach for describing the behaviour of non-spherical particles[END_REF]. The other elements of q k are Algorithm 1 VVS-based tracking of articulated objects for each new image frame I do s * ← DETECT FEATURES(I) ρ ← UPDATE MODEL(q) for each visual feature i do

J i ← COMPUTE JACOBIAN(q, ρ i ) s i (q) ← PROJECT(ρ i ) L i ← COMPUTE IMAGE JACOBIAN(ρ i,z , s i (q)) A i = -L i J i end for q = -λA # e + A # ṡ * q ← INTEGRATION( q)
end for obtained with Euler explicit integration:

q k =    p o,k-1 + ( ṗo,k + ṗo,k-1 ) Ts 2 exp(Ω k-1 T s )σ o,k-1 q j,k-1 + ( qj,k + qj,k-1 ) Ts 2    ( 6
)
where Ω is the 4×4 skew matrix of the FB angular velocity. Algorithm 1 presents the complete procedure to compute the vector q. Each new image is processed to detect the visual features filling s * . Using the current estimation q, the model of the object is updated, so that the positions of the PoI ρ are also estimated and available for the subsequent computations. Then, for each visual features i = 1, . . . , p these steps are performed:

• the Jacobian J i of the corresponding PoI is computed using the current q and the estimated ρ i ; • the virtual visual feature s i (q) is obtained projecting ρ i on the image plane; • the interaction matrix L i is computed using the estimated depth of the point (i.e., the z-coordinate of the point ρ i ) and the coordinates of the visual feature; • finally, the articulation matrix A i is obtained. Once these operations are repeated for all the features, the articulation matrix is fully composed. Finally, q is computed and q is obtained by numerical integration.

The method can reconstruct also other information about the object such as the length of the links, the PoI position in the presented study or even the composition of the CAD model, that is assumed to be known. This can be achieved modeling the distances as virtual prismatic joints. We exemplify this functionality in the next section.

V. EXPERIMENTAL RESULTS

For our experiments we used the humanoid robot HRP-4, that is equipped with a Xtion PRO LIVE RGB-D sensor. The Xtion is used as monocular camera, providing images with a resolution of 640 × 480 pixels at 30 Hz. A calibration procedure provided the intrinsic parameters used in the projection model of the algorithm. The images are processed by WhyCon [START_REF] Krajník | A practical multirobot localization system[END_REF], a vision-based localization library that detects proper markers placed in the field-of-view of the camera, and also gives an estimate of their position in the camera frame. In particular, p WhyCon markers are placed at known positions on the articulated object to track. These markers represent the p PoI in our algorithm, and the detection of their corresponding visual features fills the vector s * . For each new set of detected visual features, the algorithm described in Section IV provides an estimate of the articulated object configuration. The implementation of the algorithm is based on the Robotics and Vision Control toolbox [START_REF] Corke | Robotics, vision and control[END_REF], while the communication and the control of the robot have been managed by using the ROS framework.

The approach has been used to make HRP-4 manipulate the paper drawer of a printer (Fig. 2a). The printer has been structured with six WhyCon markers, placed at known positions, four on the FB and two on the drawer, as shown in Fig. 2b. The configuration vector q ∈ R 9 that we provide is composed of (i) the pose (position vector and quaternion) of the printer FB, (ii) the value of the prismatic joint of the drawer, q 1 , and (iii) the distance between the two markers on the drawer, q 2 , modeled as a virtual prismatic joint. The configuration vector has been initialized to q 0 = (0.0, 0.1, 0.6, 1.0, 0.0, 0.0, 0.0, 0.0, 0.1) T . To have a smooth transient phase and good tracking performance, we designed a profile of the gain λ dependent on the VVS error (high when the norm of the error is low and vice versa). The maximum and minimum values of the gain were set to λ max = 5 and λ min = 2, respectively. Since the computation of the visual features derivative was noisy and the camera did not move excessively during the execution of the experiment, we disabled the derivative action in [START_REF] Drummond | Real-time visual tracking of complex structures[END_REF].

To validate our approach, we compare the results of the proposed VVS-based tracking algorithm with a Singular Value Decomposition (SVD) based method for rigid motion reconstruction. It uses two sets of points: m PoI on the FB expressed in camera frame, c ρ, and the same points expressed in the object frame, o ρ. The first set is provided by WhyCon, the latter is known, manually measured. The reconstruction of the object pose (position vector p o and rotation matrix R o ) is formulated as a least squares error problem: To find an optimal combination of R o and p o that satisfies the minimization problem in [START_REF]Tracking the articulated motion of two strongly interacting hands[END_REF], we apply an SVD on the cross-covariance matrix of the two points distributions ( c ρ and o ρ), that results in the decomposition U ΣV T . The sum in ( 7) is minimized when R o = V U T . Then, the translation is computed as

(R o , p o ) = arg min Ro,p o m i (R o o ρ i + p o ) -c ρ i 2 . (7)
p o = µ c -R o µ o
, where µ c and µ o are the centroids of the sets c ρ and o ρ, respectively. For the validation of the drawer prismatic joint, we consider the 3D position of a marker on the drawer in F o reconstructed using R o and p o . The amount of drawer opening is equal to the position of the drawer marker on z-axis of F o (Fig. 1). For the validation of estimated distance between two markers on the drawer, Euclidean distance between corresponding points is computed using coordinates in F c provided by WhyCon.

As discussed in the Sect. I, the robot is controlled with our MQP [START_REF] Bouyarmane | Multirobot and task-space force control with quadratic programming[END_REF]. Indeed, in order to achieve the closed-loop behavior, it is possible to define the error between the current value of the drawer's joint and a desired joint target: τ q = (q 1 -q 1,d ). Here, q 1,d is specified by the user (it could be defined by a higher level planning), while q 1 is provided by our method, see Sect. IV. This term is actually added to the cost function of the MQP after the robot grasps the drawer

w q τq + 2 k p τq + k p τ q , (8)
where k p is a positive gain and τq = J q q1 + J q q1 ; J q is the Jacobian of the task and w q a given weight. The experiment starts with the robot already at the operational configuration, at stand position, with the left hand grasping the drawer of the printer. During the experiment, a user sends opening/closing commands to the MQP.

Figure 3 shows the norm of the VVS error. After a transient time required to make the virtual visual features converge on their real counterparts, the error decreases exponentially and remains below a threshold.

The position of the FB as estimated by the VVS (blue continuous lines with triangle markers) and by the SVDbased method (red dashed line) is shown in the plots of Fig. 4. Again, after an initial transient required to recover the bad initialization, the signals provided by the VVS converge to the position of the object; the curves provided by the SVDbased method validate the estimation results. Similarly, plots of the FB orientation (transformed in roll-pitch-yaw angles) are presented in Fig. 5. The effectiveness of the VVS at estimating the pose of the FB is validated by the comparison with SVD-based method. Furthermore, it appears to be less noisy and more appropriate to be used as control feedback. The plots in Fig. 6 refer to the joint of the printer drawer. The black dash-dot line shows the desired command sent to the MQP, that is followed by the estimation of the joint position, as shown by the curves. The control has been enabled around time 15 s. A phase can be observed in tracking. This is due to two main factors: the backlash in the humanoid-drawer system, especially when there are two consecutive opening/closing commands, and the task error decrease rate of the MQP that controls the robot. Note also that the estimation algorithm computes an output for each set of detected features. It could happen that the detection process fails creating instant lack of estimation.

Finally, the plot in Fig. 7 shows the effectiveness in estimating the distance between the markers placed on the printer drawer, modeled as a prismatic joint in our algorithm.

The experiment is also shown in the video available at https://youtu.be/Vr2LUEovof8.

VI. CONCLUSIONS AND FUTURE WORK

In this paper we addressed the problem of estimating the configuration of an articulated object to be manipulated by a humanoid robot. The approach is based on the virtual visual servoing paradigm, and uses visual information coming from the robot onboard camera to reconstruct the pose of the object floating base and its joints configuration. The output of the estimator is fed back to a multi-robot quadratic program framework for controlling the humanoid HRP-4. Experimental results showed the effectiveness of the approach in manipulating the paper drawer of a printer.

Future work will investigate the use of lines instead of points as features, to avoid the structuring the object with known markers. Robotic structures and more complex articulated objects can also be tracked with this method. Furthermore, future perspectives can be traced in the field of safe physical human-robot interaction, where the detection and prediction of a human partner motion is very important. In fact, the proposed algorithm can be extended to the tracking of human bodies.

Fig. 1 .

 1 Fig. 1. A humanoid robot manipulating the drawer of a furniture.

Fig. 2 .

 2 Fig. 2. Experimental setup: (a) HRP-4 manipulating the paper drawer of a printer and (b) the corresponding image acquired by the onboard camera.

Fig. 3 .

 3 Fig. 3. Experimental results: norm of the VVS error.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Experimental results: position of the object w.r.t. the camera frame. From top to bottom: x, y and z-coordinate.

Fig. 6 .

 6 Fig. 6. Experimental results: position of the drawer prismatic joint.

Fig. 7 .

 7 Fig. 7. Experimental results: distance between the markers of the drawer.

Note that s(p) is reconstructed by using the projection model of the camera. Thus, to be rigorous, it is dependent also on the camera intrinsic parameters, assumed to be known, that here are omitted to highlight the dependence on the unknown variable p.

For example, the camera pose can be reconstructed using p = exp(vc).


H Boutayeb
email: halim.boutayeb@univ-rennes1.fr

K Mahdjoubi

A C Tarot

EBG-ANTENNA ANALYSIS : UNIFICATION OF FREQUENCY AND ANGULAR DEPENDENCIES

In this paper, we present an original method to study planar EBG based antennas. From this method, which unifies the frequency and angular dependencies of the EBG structure, we obtain an analytical expression of the directivity of the structure versus frequency. To our knowledge, these results concerning EBG structures have never been presented.

I. INTRODUCTION

Several recent works [START_REF] Thevenot | Directive Photonic Band-Gap Antennas[END_REF][2][3] [START_REF] Enoch | A Metamatarial for Directive Emission[END_REF] deal with directive antennas based on EBG (Electronic Band Gap) structures. In these works the directivity characteristic is generally studied numerically. The purpose of this presentation is to establish an analytical expression and to show that the frequency and spatial behaviors of EBG structures are tightly interrelated. To this end we propose a new simple method for analyzing EBG-Antenna. Inspired from the antenna array theory [START_REF] Balanis | Antenna Theory : Analysis and Design[END_REF], this method can allow, for example, to obtain the radiation pattern of an EBG-Antenna from the frequency curves. We will consider first a point source inside multiple Partially Reflecting Surfaces (PRS). Then we will study in more details the case of one PRS on each side of the point source, estimating the directivity and the performance of this EBG-Antenna. Finally, we will focus on a structure composed of a ground plane in the same plane of the source.

II. GENERAL EQUATIONS

Let us consider an ideal point source inside a periodic structure composed of 2n identical partially reflecting surfaces (Figure 1). In practice, the source can be a dipole as in ref. [2][3]. The source creates plane waves uniformly propagating in all θ directions. Let (r) and (t) be respectively the complex reflection and transmission coefficients of a single layer (r=|r|exp(jϕ r ), t=|t|exp(jϕ t )). We have calculated the transmission coefficient T 2n (ψ) of the structure due to two symmetrical plane waves at a given angle θ (Figure 1). We have shown [START_REF] Boutayeb | Etude des structures périodiques planaires et conformes associées aux antennes. Application aux communications mobiles[END_REF] that the transmission coefficient T 2n (ψ) can be written in a simple closed form :

( ) ( ) ( ) ( ) ( ) ψ - ψ - ψ - ψ = ψ j exp n r 1 2 / j exp n t n 2 T (1)
t n and r n are evaluated by recursive equations from (r, t) [START_REF] Boutayeb | Etude des structures périodiques planaires et conformes associées aux antennes. Application aux communications mobiles[END_REF] :

( ) ( ) ( ) ( ) ( ) ψ - ψ - - ψ - ψ - = ψ 2 j exp . r . 1 n r 1 j exp . t . 1 n t n t (2) ( ) ( ) ( ) ( ) ( ) ( ) ψ - ψ - - ψ - ψ - + ψ - = ψ 2 j exp . r . 1 n r 1 2 j exp . r . 1 n t 1 n r n r 2 ( ) t t = ψ (3) , 1 1 , (4) ( ) r r = ψ ( ) θ = ψ cos kD and (5)
k is the free space wave number and D is the distance between two consecutive partially reflecting surfaces (Figure 1). It is important to note that the expression (1) unifies the frequency and incidence angle dependencies in a unique variable ψ=kDcos(θ). Another point is that T 2n (ψ) is a periodic function of ψ with period 2π. The knowledge of one period (0≤ψ≤2π) is sufficient to extract the frequency curves (0≤f<∞) and/or the radiation pattern (-π/2≤θ≤π/2). In Figure 2, we give an example of |T 2n (ψ)| for |r|=0,7.exp(j 2.35), |t|=0,71.exp(j0.78) [START_REF] Boutayeb | Etude des structures périodiques planaires et conformes associées aux antennes. Application aux communications mobiles[END_REF] (for n=1,2, 3). To simplify the study, we consider that (r, t) don't depend on frequency or on incidence angle. T 2n (ψ)| contains two types of pass-bands : pass-bands with high picks, and pass-bands with picks limited to one. The values of |T 2n (ψ)| higher than one are explained physically in ref. [6][7]. It does not correspond directly to antenna directivity as it was used in [START_REF] Feresidis | High gain planar antenna using optimized partially reflective surfaces[END_REF] Now it's easy to pass from the ψ domain to the frequency domain (for a given angle θ) or to the angular domain (for a given frequency), by using the classical method of antenna array theory [START_REF] Balanis | Antenna Theory : Analysis and Design[END_REF].

III. STRUCTURE WITH ONE LAYER ON EACH SIDE OF THE SOURCE

III. 1 Extraction of frequency curve and radiation pattern

We consider one PRS on each side of the source (Figure 3). For this simple structure, we will show how to pass from the ψ domain to the frequency or angular domains. We will also evaluate the directivity of this structure and calculate the frequency which gives the maximum of directivity. To pass from the ψ domain to the angular domain we use

( ) ( ) ( ) ( ) ( ) ( ) r j exp . r 1 2 / j exp . t j exp . r 1 2 / j exp . t 2 ϕ - ψ - - ψ - = ψ - - T ψ - = ψ is given in
        ψ ± = kD arccos θ . For f=f 0 ,         ϕ ψ ± = θ r arccos
.We use a graphical method [START_REF] Balanis | Antenna Theory : Analysis and Design[END_REF] as it is shown in Figure 4. Note that the well known "visible region" in the array theory starts from the 0 frequency and finishes at the frequency considered. It's interesting to note that the lower and upper envelops of the transmission coefficient |T 2 | are given by [START_REF] Boutayeb | Etude des structures périodiques planaires et conformes associées aux antennes. Application aux communications mobiles[END_REF] :

max 2 T 1 r 1 r 1 r 1 2 r 1 r 1 t min 2 T r 1 r 1 r 1 2 r 1 r 1 t max 2 T = + - = + - = + = - + = - - = - = , ( 6
)
Note that |T 2 | max should not be considered as the gain enhancement of the structure as it is proposed in ref. [8][9]. We will see later how to calculate the directivity and then the true performance of the structure. Figure 5 gives |T 2 (θ)| for a frequency higher than f 0 : in this case the maximum is not at θ=0° and multiple lobes appear. We call Q the quality factor of |T 2 | which is obtained calculating the inverse of the bandwidth at -3dB of |T 2 | multiplied by f 0 . Let us also call Q x the quality factor of T 2 obtained for a bandwidth corresponding to |T 2 | 2 =|T 2 | max 2 /x (then Q 2 =Q) (Figure 6). In the Appendix, Q is calculated in function of r and Q x is calculated in function of Q. 0.4 0.6 0.8

0 π/2 π 3π/2 2π |T 2 (ψ)| ϕ r ψ 1/Q x |T 2 (ψ)| max |T 2 (ψ)| 2 / |T 2 (ψ)| 2 max =1/x
1 1.2 1.4 1.6 1.8 2 2.2 2.4 0 π/2 π 3π/2 2π |T 2 (ψ)| ϕ r ψ 1/Q 2x 1/Q x       θ ∆ 2 cos dB 3 ( ) θ cos 1 0 Visible region (a) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0 π/2 π 3π/2 2π |T 2 (ψ)| ϕ r ψ 1/Q       θ ∆ 2 cos dB 3 ( ) θ cos 1 0 Visible region 1/Q x f' ψ' ψ'' (b)
Figure 7 -Link between quality factor and beam width (a) for a frequency smaller than f 0 (b) for a frequency greater than f 0 and smaller than f'. Now we will use the results presented in the Appendix to estimate the beam width ∆θ 3dB versus frequency. For a frequency smaller than f 0 (see Figure 7a) we have to resolve this following equation :

                ≤ θ ∆ - = - 2 f f , dB 3 cos 1 2 x Q 1 x 2 Q 1 0 (7)
Considering that ∆θ 3dB is small we can write :

x Q 1 x 2 Q 1 2 2 f f , dB 3 0 - ≈         ≤ θ ∆ (8)
Then (using (22)): Q

1 x 1 x 2 2 f f , dB 3 0 - - - ≈ ≤ θ ∆ (9)
For a frequency greater than f 0 but smaller than f' (see Figure 7b), this is written :

                ≤ ≤ θ ∆ - = + 2 ' f f f , dB 3 cos 1 2 Q 1 x Q 1 0 (10) Then : Q 1 1 x 2 ' f f f , dB 3 0 + - ≈ ≤ ≤ θ ∆ (11)
Note that we obtain easily the beam width at f 0 in function of Q, by putting x=1 in (11), we find

Q 2 f , dB 3 0 ≈ θ ∆
which is in accordance with the result in ref. [3][10]. We call this relation, the relation of performance of the structure.

This study gives a more complete result than [3][10], as all frequencies are considered. Figure 8a gives the beam width ∆θ 3dB versus the frequency (( 9) and ( 11)) from 0 to f' and Figure 8b gives π/∆θ 3dB which we consider as an elegant evaluation of the directivity D 0 at θ=0° of the structure. Note that the minimum of ∆θ 3dB and so the maximum of directivity is not obtained at f 0 but at a frequency slightly smaller than f 0 . To obtain this frequency we calculate the value of x which gives the maximum of the following function

: 1 x 1 x 2 ) x ( g - - - =
(see equation ( 9)).

after derivation :

( )( )

1 x 1 x 2 1 x 2 1 x 2 ) x ( ' g - - - - - = ; g'(x)=0 ⇔ x=1,5
The minimum beam width is then :

Q 2 min , dB 3 ≈ θ ∆ (12)
We obtain the corresponding frequency by resolving Q

2 1 Q 5 . 0 min Q 1 0 f min f 0 f 2 = ≈ θ ∆ = θ ∆ - Finally,         - ≈ θ ∆ Q 2 2 1 1 0 f min f (13)

IV. STRUCTURES WITH A GROUND PLANE

Let us consider now that a ground plane is placed in the same plane as the point source (in a manner that there is no radiation in the left side of the source) and there is only n PRS layers placed at a distance D/2 from the source. The PRS layers are spaced from each other by a distance D. We call T n,gp the transmission coefficient of the structure, whose expression becomes [START_REF] Boutayeb | Etude des structures périodiques planaires et conformes associées aux antennes. Application aux communications mobiles[END_REF] :

( ) ( ) ( ) ( ) ( ) ( ) π + ψ - ψ - ψ - ψ = ψ j exp n r 1 2 / j exp n t gp , n T (14)
Note that ( ) ψ n t and are periodic with period π (see equations ( 2) and ( 3)). Therefore :

( ) ψ n r ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) π + ψ = π + ψ - π + ψ - π + ψ = π + ψ - ψ - ψ = ψ n 2 T j exp n r 1 n t j exp n r 1 n t gp , n T (15)
We deduce that the |T n,gp (ψ)| curve can be obtained from the |T 2n (ψ)| one by π shifting it to the right as we can see in Figure 9. This is equivalent to add π to the phase of the reflection coefficient ϕ r . Note that in our example ϕ r is positive. We can imagine that for ϕ r negative the first |T n,gp (ψ)| pick will be higher than one.

V. CONCLUSION

We have presented a new and effective method to analyze 1-D Electromagnetic Band Gap based antennas. Considering Partially Reflecting Surfaces having characteristics independent of frequency and angle, we have estimate the directivity versus frequency and the frequency of maximum directivity for the simple structure composed of one layer on each side of the source. We have shown that EBG structures composed of a ground plane can be analyzed using the study of structures composed of multiple layers on each side of the source.

Figure 1 -

 1 Figure 1 -Point source inside multiple (2n) Partially Reflecting Surfaces (PRS).

Figure 2 -

 2 Figure 2 -Transmission coefficient versus ψ for n=1, 2, 3 (|r|=0,7.exp(j 2.35), |t|=0,71.exp(j0.78)).

Figure 3 -

 3 Figure 3 -Point source inside two partially reflecting surfaces

Figure 4 .

 4 Figure 4. To pass from the ψ domain to the frequency domain we use the following transformation :

Figure 4 -

 4 Figure 4 -|T 2 |, transformations from the ψ domain to the frequency domain (at θ=0°) and to the angular domain (at f=f 0 )

Figure 5 -

 5 Figure 5 -|T 2 (ψ))| and |T 2 (θ)| at f=f 1 >f 0. .

Figure 6 -

 6 Figure 6 -Quality factor at 1/x.

Figure 8 -

 8 Figure 8 -(a) ∆θ 3dB and (b) π/∆θ 3dB versus frequency.

Figure 9 -

 9 Figure 9 -Transmission coefficients |T n,pm |, for n= 1,2 et 3,in one period [0,2π].

APPENDIX

To calculate Q we resolve the following equation :

which gives us ψ' and ψ'' the values of ψ corresponding to the half power (see Figure 7b) :

and then

This can be approximated for high value of |r| by

By the same way, we can evaluate Q x , resolving the following equation :

We find

For high value of |r|, we can write :