Rodríguez-Noriega Lucía

Guillén

Dealing with all types of quotations (and their parallels) in a closed corpus: The methodology of the Project The literary tradition in the third and fourth centuries CE: Grammarians, rhetoricians and sophists as sources of Graeco-Roman literature

Keywords: Intertextuality, fragmentary literature, Greco-Roman scholars of the Empire

    Dealing with all types of quotations (and their parallels) in a closed corpus: The methodology of the Project The literary tradition in the third and fourth centuries CE: Grammarians, rhetoricians and sophists as sources of Graeco-Roman literature Lucía Rodríguez-Noriega

INTRODUCTION. THE PROJECT AND ITS MAIN AIMS

The Project The literary tradition in the third and fourth centuries CE: Grammarians, rhetoricians and sophists as sources of Graeco-Roman literature II (http://www.lnoriega.es/Fuentes.html) is funded by the Spanish Ministry of Economy and Competitiveness (reference FFI2014-52808-C2-1-P). At the moment, it involves eight researchers from the University of Oviedo, Spain (Lucía Rodríguez-Noriega Guillén, main researcher; Manuel González Suárez; Luis Alfonso Llera Fueyo; Virginia Muñoz Llamosas; Abigail Torre Beivide; Javier Verdejo Manchado), the University of the Basque Country, Spain (María José García Soler), and the University of Bari Aldo Moro, Italy (Piero Totaro). The Project also has an external collaborator, the computer engineer Patricia Conde Clemente, currently a PhD Research Fellow at the University of Oviedo, who is responsible for advising and developing computing tasks. A coordinated project, based at the University of Zaragoza and headed by Professor Javier Uría Varela, studies quotations in the Latin grammarians of the same period. Both projects have been granted funds for a second period of three years, finishing 31/12/2017. The aim of the Project is to examine the Graeco-Roman literary tradition in the third and fourth centuries CE by means of the compilation, analysis of and commenting on the quotations in the works of the grammarians (i.e. the authors of grammars, lexicographers and writers on literary and linguistic subjects in general) and rhetoricians of the time, and of those contemporary prose writers traditionally called "Greek Sophists". Until now, such a comprehensive study of this corpus has never been attempted. The Project deals with all types of quotations, and besides analysing and commenting on the quotes, tries to trace their parallels in both previous and later authors. This can help us to determine whether we are dealing with first hand or mediate quotations, and whether or not our authors were, in turn, the intermediate sources through which the same passage was known and later quoted.

The selection of authors and period is not random. Grammarians, rhetoricians and sophists very frequently quote other authors; hence, in their works there is always an abundance of testimonies and fragments from philosophers, poets, grammarians, orators, historians, etc. In addition, the Greek writers of the third and fourth centuries CE were, at that time, still able to access many works which have not been handed down to the present day by direct manuscript transmission. In addition, they constitute a group of 28 authors in all, making them a suitable corpus to be studied in a realistic time frame by a team of eight researches like ours. This is their list:

I STATUS QUAESTIONIS

One of the richest fields within literary studies is that of intertextuality, understood as the interrelationship between a certain literary work and others, whether contemporary, previous or subsequent, and whether an explicit (literal or allusive) or implicit reference is implied. The detection and study of these relationships is essential, for instance, to reach sound conclusions on the influence exerted by their cultural tradition on every writer in every generation, and to achieve comprehensive knowledge of the survival and circulation of the texts and the ideas they transmit throughout history.

In the particular case of Ancient Greek literature, there is in addition a special feature, which increases the interest of studies of intertextuality. In fact, up to the present times Greek literary texts have suffered a long and hazardous transmission, as a consequence of which only a very small part of them has come down to us by direct manuscript tradition, while many more have been lost through the centuries. This type of approach is thus especially important to recover quotations and testimonies of works that are only known by indirect transmission.

Classical scholars have traditionally paid considerable attention to the mutual relationships between authors, especially poets, and dedicated numerous studies to the sources of many writers. In the particular case of the 28 authors studied by our Project we might mention, for instance, though the list is far from exhaustive, the contributions by [START_REF] Rudolph | De fontibus quibus Aelianus in varia historia componenda usus sit[END_REF], [Cohn, 1886], [Rudolf 1894], [START_REF] Stefani | La fonte delle epistole III e VI di Eliano[END_REF]], [START_REF] Glombiowski | Fragments de Ctésias de Cnide chez Diodore et chez Élien non cités par Jacoby (FGrHist. 688)[END_REF], [START_REF] Schettino | Il passato e il presente di Roma nell'opera di Eliano. Troiani L. and Zecchini G. La cultura storica nei primi due secoli dell'impero Romano[END_REF] and [START_REF] Prandi | Memorie storiche dei Greci in Claudio Eliano[END_REF] on Aelian's sources; those on the sources of Athenaeus of Naucratis by [START_REF] Bapp | De fontibus quibus Athenaeus in rebus musicis lyricisque enarrandis usus sit[END_REF], [Bapp, 1888], [Rudolf, 1891], [START_REF] Cohn | Quellen des Aelian und Athenaeus[END_REF], [Rudolf, 1894], [START_REF] Ambaglio | I Deipnosofisti di Ateneo e la tradizione storica frammentaria[END_REF], [START_REF] Zecchini | La cultura storica di Ateneo. Vita e pensiero[END_REF], [START_REF] Hill | The Sources and Sauces of Athenaeus[END_REF], [START_REF] Villari | Une hypothèse sur les sources d'Athénée (Deipn. 120e-f) et de la «Vita Sophoclis[END_REF], [START_REF] Bowie | Knowledge of Early Greek Elegiac and Iambic Poetry[END_REF], [START_REF] Sidwell | and his World[END_REF], [START_REF] Quaglia | Citazioni da Aristofane e dalla Commedia Antica in Ateneo[END_REF], [Jacob, 2004], [START_REF] Cipolla | Le citazioni dei tragici in Ateneo[END_REF] and the papers gathered in [START_REF] Lenfant | Athénée et les fragments d'historiens[END_REF]; those on Philostratus' sources by [START_REF] Breloer | Drei unbenannte Megasthenesfragmente über die pravrajya[END_REF], [START_REF] Norman | Philostratus and Libanius[END_REF], [START_REF] Keil | Vertreter der zweiten Sophistik in Ephesos[END_REF], [START_REF] Szarmach | Essai de reconstitution du Palamède d'Euripide[END_REF], [START_REF] Cassin | Le plaisir de parler: Études de sophistique comparée[END_REF], [Suárez de la Torre, 1991], [START_REF] Dickie | Philostratus and Pindar's eighth Paean[END_REF], [START_REF] Mangia | Quattro nuove testimonianze su Callistene di Olinto[END_REF], [START_REF] Zeitlin | Visions and revisions of Homer. Goldhill S. Being Greek under Rome[END_REF], [START_REF] Moreno | Philostr. Her. XXXV 9-10: ¿un testimonio de Eurísaces de Sófocles?[END_REF], and [START_REF] Moreno | Philostr. Her. 20.2 : una posible alusión a El juicio de las armas de Esquilo[END_REF]; those on Philostratus Iunior's sources by [START_REF] Noack-Hilgers | Zur Rezeption der homerischen Gerichtsszene im 3. Jh. n. Chr. und zu ihrer juristischen Umdeutung[END_REF], [START_REF] Cejudo | La exégesis homérica en la paráfrasis filostratea del escudo de Aquiles[END_REF], and [START_REF] Cejudo | El escudo de Neoptólemo: la paráfrasis filostratea del Escudo de Aquiles[END_REF]; those on Himerius' sources by [START_REF] Mesk | Sappho und Theocrit in der ersten Rede des Himerios[END_REF], [START_REF] Meerwaldt | Epithalamica I: De Himerio Sapphus imitatore[END_REF], [Papadimitriou, 1971-72], [START_REF] Cuffari | I riferimenti poetici di Imerio[END_REF], [START_REF] Colonna | Una testimonianza eschilea in Imerio: T 150 Radt[END_REF], and [START_REF] Lazzeri | Imerio e la «Palinodia» di Stesicoro[END_REF]; those on Libanius' sources by [Middleton, 1919[Middleton, -1928]], [START_REF] Norman | The library of Libanius[END_REF], [START_REF] Schouler | La tradition hellénique chez Libanios[END_REF], [START_REF] Tiberga | Ideologia e comunicazione: l'impiego della citazione classica in Libanio. Garzya A. Metodologie della ricerca sulla tarda antichità[END_REF]], [START_REF] Soler | Reflejos de la comedia ática en la declamación XXVI de Libanio[END_REF], [García Soler, 1991-92], and [START_REF] Eire | Las citas homéricas en las Epístolas de Libanio[END_REF]; those on Longino's sources by [START_REF] Veit Apfel | Literary quotation and allusion in Demetrius Περὶ ἑρμηνείας and Longinus Περὶ ὑψοῦς[END_REF], [START_REF] Ruckh | Longinus' criticism of Theocritus (Περι ῦφους XXXIII 4)[END_REF], [START_REF] Sedgwick | Sappho in Longinus (X 2, line 13)[END_REF], [START_REF] Mazzuchi | Tre citazioni adespote nel Sublime[END_REF], [START_REF] Dyck | New light on Greek authors from grammatical texts[END_REF][START_REF] Talens | Los textos a través de los textos: Esquilo en el «De sublimitate[END_REF]; those on Themistius' sources by [START_REF] Cacciatore | Historical and biographical values of Plutarch's works: Studies devoted to Professor Philip A[END_REF]] and [Maisano, 2006]; and those on Porphyry's sources by [Sodano, 1965], [START_REF] Pépin | exégète d'Homère. Porphyre: huit exposés suivis de discussions[END_REF] and [Sodano, 1965-66].

Nevertheless, many of these studies are partial and most of them have been carried out without a clear, well stablished methodology based on modern research on intertextuality (some are in fact too old to have benefitted from the new approaches to the subject). Among the most important works on intertextuality since the end of the 20th century to the present day which affect the theoretical background of our Project are [START_REF] Compagnon | La seconde main ou le travail de la citation[END_REF], [START_REF] Reyes | Polifonía textual. La citación en el relato literario[END_REF], [START_REF] Reyes | Los procedimientos de cita: estilo directo y estilo indirecto[END_REF], [START_REF] Reyes | Los procedimientos de cita: citas encubiertas y ecos[END_REF], [START_REF] Fraser | An approach to discourse markers[END_REF], [START_REF] Schiffrin | Discourse markers[END_REF], [START_REF] Coulmas | Direct and Indirect Speech[END_REF], [START_REF] Most | Collecting Fragments -Fragmente sammeln[END_REF], [START_REF] Rosier | Le discours rapporté. Histoire, théories, pratiques. Duculot[END_REF], [START_REF] Récanati | oratio recta. An essay on Metarepresentation[END_REF], [START_REF] Darbo-Peschanski | La citation dans l'Antiquité, Actes du colloque du PARSA[END_REF], and [START_REF] Nicolas | Hôs ephat', dixerit quispiam, comme disait l'autre[END_REF].

New approaches to the study of intertextuality based on ICT have also flourished in recent years. But, as the international workshop held in Lyon in 2014 (see http://biblindex.hypotheses.org/1686) and this volume itself make clear, despite the increasing interest in sharing experiences to date no methodology or electronic tool able to meet the needs of all the groups working on corpuses written in ancient languages has been developed. Thus, our team is carrying out its own experiment, following a methodology we will expound on in the next section.

II THE METHODOLOGY OF THE PROJECT

Introductory remarks

As our Project was conceived of as long term, and taking into account the huge quantity and variety of data we had to deal with, the need to stablish a uniform and systematic work procedure was clear from the beginning. Our methodology involves first the gathering of all the quotes that can be traced in a given work, and then their analysis in accordance with a detailed and carefully planned protocol, with the help of an electronic card, which has been designed and improved as the work has progressed and the team has gained experience.

The lists of quotes

The task of searching for quotations is preceded by a thorough study of the quoting author and his work, and the history of the text. For every work we principally use the canonical printed edition, but often consult others, and we also take into account all the relevant bibliography.

Every member of the team deals with one or more quoting authors or works, depending on their personal tastes and interests, but also on the length of the work to be studied, and the time each person has at their disposal. Every researcher first makes a complete list of the quotations found in the assigned work, following the order of the passages in the quoting text. Quotes are searched for by direct reading of the quoting text (even if the edition has a final list of quoted authors or works), but we also take into account all previous studies on the quoting author's sources, and material such as the lists of sources included in the editions of fragmentary authors of all types. This list is then published on our website (Figure 1). Active links in the list of quoting authors (http://www.lnoriega.es/Autorescitantes.html) indicate that we have published a complete list of quotations for at least one of that author's works. Each time we indicate the name of the researcher in charge of the quotes for that particular work or author, the edition we base the study on, the exact number of quotations found, and various other data. Thus far the Project has listed 6623 quotes in all, but there are still 7 authors whose quotes have not yet been gathered, and for some of these (specifically Aelian, Libanius, Philostratus and Porphyry) not all works have a list of quotes already prepared. We expect to publish the complete list of quotes of the first 5 books of Athenaeus Deipnosophist in the first semester of 2016.

In the second phase, the original list is rearranged according to the quoted authors, in alphabetical order of author and work. Then these data are combined with those coming from the rest of the team, in order to make separate lists of quotations according to the quoted authors. These lists are also published on line in the corresponding section; see http://www.lnoriega.es/AutorescitadosA_C.html; http://www.lnoriega.es/AutorescitadosD_I.html; http://www.lnoriega.es/AutorescitadosL_Z.html. Unlike in the case of the quoting authors, the catalogue of quoted authors is still open and provisional, and so, of course, are the corresponding lists of quotations. Up to now, we have found quotations from more than 400 authors, both Greek and Roman.

The study of the quotes

As we make the lists of quotes, we also proceed to the analysis of the individual quotations. Over the first two years of work (2011)(2012), the team, based on the researchers' experience, developed a working protocol, according to which every single quote originates an electronic card, with several key-fields. As we will see, our analysis of every quote is very thorough, and every card runs to several pages. As a result, the publication of the cards advances at a much slower pace than that of the lists of quotes. The results of this part of our study are also published on line in single PDF files; the number of cards published thus far (February 2016) is 158.

The electronic card key-fields are the following: First of all, we record the data concerning the quoting author (name, century, work and passage where the quotation is found) and then, those corresponding to the quoted author (also name, century, work and passage). If the quoted author does not have an edition of his or her own, we mark the passage as "fragmentum nondum editum" or "nondum excerptum", and as "fragmentum novum" if it has passed unnoticed by editors up to the present. Mentions and testimonies regarding the authors and their lives are usually impossible to refer to any edition when they concern authors known by direct manuscript transmission, because in such cases, unlike in that of the fragmentary ones, editions do not usually include a section of "testimonies". In these cases, the reference of the quotation on the website may include a brief description of the content of the quotation, if appropriate (for instance "de vita", "de fama", etc.).


Next we indicate 1) whether or not the quotation is explicitly assigned to an author in the quoting text, 2) if the assignment is right, wrong or doubtful, and 3) whether or not the quote has an intermediary source also mentioned in the text, or whether it can be identified by other means (in which case we refer to the "parallel mentions" section). The latter is, in fact, one of the questions the Project is most interested in. To try to determine whether we are dealing with a secondary (i.e. second-hand) quotation when the author does not openly say so, we take into account, besides what we can deduce from the reading of the text itself, what is known about the author, his literary education and the libraries he might have had access to, and also about the preservation and transmission of the quoted text in the third and fourth centuries CE. The search for quotations of the same passage in previous authors can also, with appropriate analysis, be helpful to find intermediate sources from which the quotation has been taken, if that be the case.

Next on the card we classify the quotation, indicating if it is 1) a literal quotation; 2) a paraphrase; 3) an imitation; 4) a parody; 5) a lax reference; 6) a mere mention of an author's name or of a work's title; or 7) if the quoting author is using previous material hiding the fact that it is a quote. This classification has been developed by the members of the team in line with their practical experience, but of course also taking into account many previous studies (see section I above), of which [D'Ippolito, 1983] has been especially inspiring.

A literal quotation tries to reproduce the original text verbatim, but does not need to be exact. Thus, we also consider it a literal quotation if the quoting author chose to shorten some intermediate parts of the original text for some particular reason, being accurate for the rest. Or if he made some slip-up when quoting from memory, but still believed himself to be reproducing the quote in the right form.

A paraphrase implies a re-wording of the original text, while keeping its essential content. A very common type of paraphrase consists in rendering poetry texts in prose, a procedure that some ancient rhetoricians recommended in order to adapt the quote to the style of the quoting author.

Imitations, parodies, allusions and hidden quotations in general are sometimes difficult to distinguish from what [D'Ippolito, 1983: 327-328] calls "generic references", that is, literary allusions which belong to Greek cultural heritage in general, and cannot be considered references to any concrete work or author. On determining this, the researcher's personal judgment plays a very important, and irreplaceable, role. For the identification of hidden quotations, we rely on the direct reading of the original text, taking into account the apparatus of loci similes if the edition has one, but we also consult previous studies of the author's sources, if there are any available. Good annotated translations and commentaries are often also useful for this task, as too are the searches for key-words by means of the [Thesaurus Lingua Graecae® Digital Library] (hence TLG). As for explicit quotations, they have often already been identified in the editions of the quoting authors, but the references and indexes in those editions are sometimes incomplete, outdated or faulty, so even in these cases we proceed as in those of hidden quotations.

to quotes by reading, etc.). In addition, we think that a stock list of introductory expressions can, in the long run, be useful to trace new quotations in a given corpus by computational means.

The next two fields in the electronic card are the text of the quotation, and the critical apparatus. The text includes the quotation's context, which is essential to the correct understanding and assessment of the quote. The textual analysis is made from the point of view of the quoting author, so what we try to reproduce is the text of the quotation in the way it was known to the quoting author, or in the way he chose to quote it; therefore, our text of the quote may differ from the one that can be read in the quoted author's edition. Through all this process, we use one or more editions of both the quoting and the quoted author. More specifically, when a literal quotation can be compared with other sources, either direct or indirect, and there are differences between them, it is important to determine if the divergence is due to:

 A scribal error arising in the course of the manuscript transmission of the quoting text, which must, therefore, be corrected (errors due to iotacism, confusion of similarly shaped letters, errors of accentuation, incorrect word division, etc.).  The author being quoting from memory.  A deliberate change made by the quoting author for whatever purpose.  The existence of an ancient variant reading in the text consulted by the quoting author. In the three latter cases, we reproduce the author's version of the quote as it is, without emending it.

In addition, when dealing with fragmentary authors the problem sometimes arises of determining the boundaries between quoting and quoted text, which is not always obvious. Here we can resort to syntactic, stylistic and content criteria, and also to comparisons with parallel quotations in both previous and later writers.

Next, we offer a Spanish translation of the quotation and its context. As passages taken out of their original context are often difficult to understand, the translation may provide a helpful starting point for the reader, especially if they do not have the complete original text to hand.

Another aspect we are interested in is the function of the quote within the quoting text, so we have included on the card another field where we analyse the reasons why that particular passage was quoted at that particular point of the work. Some quotations prove to be merely ornamental, and others are mainly means to show the quoting author's culture or erudition. However, many others are essential for the purposes of the work; for instance, in discourses and argumentative texts in general quotes are often used to support theses or arguments, and in many grammatical treatises they are adduced because they provide the examples to illustrate the point under discussion. Some quotes are also adduced with a humorous purpose (thus the parodies, but not only them).

In another section of the card we indicate if the same quotation appears in other works by the same author, or in other passages of the same work. If that is the case, we refer the reader to the corresponding card.

The next three sections are very important to us, because in them we record occurrences of the same quotation in other previous, contemporary or subsequent authors. We register, first, the author, century, work and passage where the same quotation is found. Then we comment on the similarities and differences between the quotations. Finally we record the relationships between the respective authors, indicating if both 1) read the quotation directly in the original work, 2) share the same indirect source, 3) follow independent traditions, or if 4) there is a direct relationship between them, that is, if one author is the indirect source from which the other took the quotation.

The search for those parallel quotations can be aided by the apparatuses of loci similes if the editions have any, and by previous studies on the author's sources, but the Project also carries out searches by means of the [TLG], through searches for keywords. In the searches we take into account the possible variants of the inflected words, because our experience has demonstrated that, among the many alterations quotes may suffer, one of the most common is the alteration of their syntax to accommodate them to the new context, or simply to turn them into more rounded sentences.

The study of every quotation is completed with a commentary, where we analyse and discuss all relevant information concerning the quotation, its context, its parallels, etc. The length of this field varies a lot, depending on the interest of the passage studied.

Finally, there is a section of conclusions on the position and relevance of our author regarding the transmission of the quoted text.

Every card includes a bibliographical note, the name and affiliation of its author, and the date when it was last updated.

The "e_Fichas" application

To give support to the methodology we have just described, the team has designed a desktop application called "e_Fichas" (see Figure 2) in C# language, under the .Net 4.0 Framework. The application allows the creation and modification of the electronic cards (one for each quotation), which are stored individually in XML 1.0 format, and eventually exported to PDF format in order to publish them on our website. The Project members store the generated cards in a shared folder on the network by using a file transfer utility.

To upload and store the XML data the application uses the DataSet component, which performs these operations in a transparent way. The various fields on the record are stored in Rich Text Format (RTF) or plain text. RTF is used in those fields where it is necessary to give a special format to a piece of text, such as writing in Greek, place text in italic or bold, or include a superscript.

As we have already mentioned, after the insertion of the data corresponding to every card, the application allows the user to export them to a PDF file (a Word-file is also automatically generated together with the PDF file), which is then published in the static website http://www.lnoriega.es/Fuentes.html, which is open source.

III CONCLUSIONS

Through the analysis of this corpus in search of all type of quotations and their subsequent study we can:

 Determine which authors were still directly read in the third and fourth centuries CE, and which works were still quoted, but no longer read, and find networks of relations among writers.  Discover new testimonies and fragments, which may have passed unnoticed by editors for different reasons, or call fresh attention to them, if the author is only quoted by his sources, and has not an edition of his/her own.  Contribute a different viewpoint to the study of fragmentary texts, on analysing them from the perspective of the quoting author.  Help to a better appraisal of the textual variants in fragmentary texts transmitted by several sources, if relations of interdependency among these sources can be traced.  Help to improve the text of the quoting authors.  Establish an overall view, which, based on the study of the sources and using the passages of each specific author, provides in-depth information and shows us how much these intellectuals as a group knew of their own cultural history, and how important this material -especially the fragments -is in the preservation of Graeco-Latin literature.  Place the results of this research in the hands of the scientific community as open source on-line reference material, thus allowing those scholars interested not only in the transmission of every type of fragmentary text, but also in Grammar, Rhetoric, Learning and Graeco-Latin Philology, access to all the details and the main conclusions of the work carried out by the members of the research team.

Using the web site and the e_Fichas application, we cover three main objectives. Namely to:

 Provide Project members with an application that will guide them and facilitate the analysis process and the management of data (create, open, save, edit and export).  Store information in a way that makes its management easier.  Give all users access, via the website, to the lists of quotations (according to both the quoting and quoted authors) and to the respective cards (currently in PDF format).

IV THINKING AHEAD

As we have already said, for now the Project publishes the lists of quotations and the corresponding cards on a static website. In the medium term, however, we mean to develop an electronic tool able to store, handle and analyse the data in an automatic, dynamic and efficient way. In what follows, we will identify the step-wise leading to achieve that final aim:

 Digitisation.
o Establish a working method for the digitisation of the field material. o Create a database with electronic cards using the standard TEI. Cards will include metadata to facilitate the semantic searches typical of Web2.0. o Convert the present static electronic cards to the new format.  Innovative use of electronic tools and statistics for the treatment of the digitised data.

o Create a dynamic website, which allow access to the digital content. o Create a smart search engine for the cards.

o Create an analyser module, which shows in a graphic and intuitive way the relationships between authors and quotes, by applying advanced methods of social network analysis.

The graphical analysis of the information would be based on the methodology for bibliometric studies proposed by [START_REF] Anegón | A new technique for building maps of large scientific domains based on the cocitation of classes and categories[END_REF], [START_REF] Anegón | Visualizing the marrow of science[END_REF] and [START_REF] Vargas Quesada | Visualizing the structure of science[END_REF]. These authors have developed a new technique for building maps of large scientific domains based on the co-citation of works [START_REF] Salton | A citation study of computer science literature[END_REF]].

Figure 1 .

 1 Figure 1. The Project web site.

  Quoting author -Century -Quoting work -Passage  Quoted author -Century -Quoted work -Passage  Typology of the quotation o Direct/ Indirect quote (with or without explicit indication of the source; right/wrong/dubious adscription) o Literal quote/ Paraphrase/ Imitation/ Parody/ Lax reference/ Mere mention of the author's name or of the work's title /Hidden quote  Introductory expression  Greek text  Critical apparatus  Reason for the quotation  Spanish translation  Parallel mentions in the same author/ in previous authors/ in contemporary authors/ in subsequent authors. In all three cases the following fields are also included: o Quoting author -Century -Quoting work -Passage o Greek text o Critical apparatus o Reason for the quote

Figure 2 .

 2 Figure 2. e_Fichas desktop application.

http://jdmdh.episciences.org

We also record the expression that introduces the quotation, which can be useful to establish the author's quoting habits (certain expressions, for instance, can point to mnemonic quotations, others,

Journal of Data Mining and Digital Humanities http://jdmdh.episciences.org

We would like to adapt the proposed methodology to our field of study, by developing several metrics of co-citation based on the electronic cards stored in the database. Each metric would allow us to obtain a matrix of co-citations, which would link quoting and quoted authors, works, quotes, etc. All the content of the matrix would be represented in a map, which could be analyzed automatically by using techniques for social networks analysis [Scott, 2000] and [Wasserman and Faust, 1994]. As a result, dynamic graphical representations would be generated, according to the criteria set by the user. This would allow to see the conclusions derived from a detailed bibliometric analysis in a fast and intuitive way. It would also offer an innovative and complementary view to conventional statistical analysis.


Athena Picarelli
email: athena.picarelli@maths.ox.ac.uk

Christoph Reisinger
email: christoph.reisinger@maths.ox.ac.uk

Julen Rotaetxe Arto

Boundary Mesh Refinement for Semi-Lagrangian Schemes

We study semi-Lagrangian schemes for the Dirichlet problem for second-order degenerate elliptic PDEs. Like other wide stencil schemes, these schemes have to be truncated near the boundaries to avoid "over-stepping". The various modifications proposed in the literature lead to either reduced consistency orders for those points, or even a loss of consistency with the differential operator in the usual sense. We propose a local mesh refinement strategy near domain boundaries which achieves a uniform order of consistency up to the boundary in the first case, and in both cases reduces the width of the region where overstepping occurs, so that the practically observed convergence order is unaffected by overstepping. We demonstrate this numerically for a linear parabolic equation and a second order HJB equation.

Introduction

In this paper, we investigate the convergence behaviour of semi-Lagrangian schemes for second order degenerate elliptic equations on bounded domains.

For concreteness, we consider the Hamilton-Jacobi-Bellman (HJB) equation

u t (t, x) + sup α∈A {-L α [u](t, x) -c α (t, x)u(t, x) -f α (t, x)} = 0, (t, x) ∈ (0, T ] × Ω, (1)
u(0, x) = ψ(0, x), x ∈ Ω, (2)
u(t, x) = ψ(t, x), (t, x) ∈ (0, T ] × ∂Ω,

where

Q T := (0, T ] × Ω with Ω := Ω ∪ ∂Ω ⊆ R d , A is a compact set, L α [u](t, x) = tr[a α (t, x)D 2 u(t, x)] + b α (t, x)Du(t, x) (4)
is a second order differential operator, and the known function ψ contains the initial and spatial boundary values. Linear parabolic equations are a special case where |A| = 1, while fully-nonlinear equations of Isaacs-type are written as min max problems. The construction of the scheme in this paper is analogous in both of these cases.

The coefficients a α = 1 2 σ α σ α,T , b α , c α , f α , and the data ψ in (1) take their values, respectively, in S d , the space of d × d symmetric matrices, R d , R, R, and R, σ α ∈ R d×P , such that a α is positive semi-definite. We assume the usual well-posedness conditions, i.e. Lipschitz continuity of the coefficients in x uniformly in α, Hölder continuity with exponent 1 2 in time and continuity in α for each (t, x) ∈ Q T [START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations part 2: viscosity solutions and uniqueness[END_REF]. This guarantees existence and uniqueness of the solution in the viscosity sense [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] as well as a comparison principle.

In this paper, we focus on semi-Lagrangian schemes for the approximation of ( 1)-(3) as introduced in [START_REF] Camilli | An approximation scheme for the optimal control of diffusion processes[END_REF][START_REF] Menaldi | Some estimates for finite difference approximations[END_REF] and analysed more recently in [START_REF] Bonaventura | Semi-Lagrangian methods for parabolic problems in divergence form[END_REF][START_REF] Debrabant | Semi-Lagrangian schemes for linear and fully nonlinear diffusion equations[END_REF][START_REF] Falcone | Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations[END_REF][START_REF] Ferretti | On the relationship between Semi-Lagrangian and Lagrange-Galerkin schemes[END_REF]. For convenience of the reader and to introduce the notation, we briefly describe the specific scheme used here.

Following loosely [START_REF] Debrabant | Semi-Lagrangian schemes for linear and fully nonlinear diffusion equations[END_REF], we define a non-degenerate polyhedral coverage C = {C j } j∈J of Ω, where J is the index set of the cells. The elements of the set C are chosen to satisfy

int(C j ∩ C i ) = ∅, ∀ i = j, j∈J C j ⊇ Ω, ν∆x ≤ sup j∈J {diam B C j } ≤ sup j∈J {diam C j } ≤ ∆x,
for some ν ∈ (0, 1), where 'int' and 'diam' are the interior and the diameter, and B C j is the greatest ball contained in C j .

We define the subset of interior cells C = {C ∈ C : C ⊆ Ω} as the ones fully contained in the domain, and the nodes N := {x i } i∈I as the set of all the vertices of elements in C, where we denote by I the index set of the nodes with N x := |N |.

We assume non-negative basis functions {w i (•) : i ∈ I} associated with the mesh nodes, such that for any continuous function φ : Ω → R

[I ∆x φ](x) = i∈I φ(x i )w i (x), (5)
for all x ∈ Ω, x i ∈ N . For simplicity, we focus our attention occasionally on cuboid meshes and multilinear interpolants, defined by the standard piecewise multilinear basis. The interpolation error is then O ∆x 2 for sufficiently smooth functions, and this is the only property we will use for the consistency analysis. The non-negativity of the basis is required only for the monotonicity of the interpolation operation and subsequently for convergence of the scheme to the viscosity solution.

Writing

σ α = (σ α 1 , σ α 2 , . . . , σ α P ) ∈ R d×P , where σ α p ∈ R d for p ∈ {1, 2, . . . , P } is the p-th column of σ α , by the usual arguments, 1 2 tr σ α σ α,T D 2 φ(x) = 1 2 P p=1 φ(x + kσ α p ) -2φ(x) + φ(x -kσ α p ) k 2 + O(k 2 ), (6)
b α Dφ(x) = φ(x + k 2 b α ) -φ(x) k 2 + O(k 2 ), (7)
for k > 0 and any smooth function φ. For brevity we write b α ≡ b α (t, x) and σ α ≡ σ α (t, x). When these approximations are used for points in N , the displaced points x + k 2 b α and x ± kσ α p do not generally coincide with nodes in N . Then, φ is approximated by its interpolant I ∆x φ. In the case of linear basis functions, the resulting scheme is referred to as the Linear Interpolation Semi-Lagrangian (LISL) scheme. We ignore for the time being the situation where x + k 2 b α or x ± kσ α p lies outside any of the mesh cells ("oversteps"), which of course will be the focus of the main body of the article.

The consistency error is easily seen to be (see [START_REF] Debrabant | Semi-Lagrangian schemes for linear and fully nonlinear diffusion equations[END_REF])

O k 2 + ∆x 2 k 2 .
The first term is the consistency error for the finite difference approximation of the first and second order derivatives in ( 6) and ( 7), whereas the second term corresponds to the linear interpolation error when replacing φ by its interpolant in ( 6) and [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]. The optimal choice k = √ ∆x makes the consistency error proportional to ∆x. Following the notation in [START_REF] Debrabant | Semi-Lagrangian schemes for linear and fully nonlinear diffusion equations[END_REF], the LISL approximations to (4) can be expressed as

L α ∆x [I ∆x φ](t, x) := M p=1 [I ∆x φ](t, x + y α,+ p (t, x)) -2φ(t, x) + [I ∆x φ](t, x + y α,- p (t, x)) 2∆x , (8)
for x ∈ N , and some M ≥ 1. The functions y α,± • (t, x) determine the stencil of the scheme at (t, x).

Different spatial schemes can be obtained depending on the values taken by M and y α,± p (t, x) in [START_REF] Debrabant | Semi-Lagrangian schemes for linear and fully nonlinear diffusion equations[END_REF]. In the following, we study specifically the approximation in [START_REF] Debrabant | Semi-Lagrangian schemes for linear and fully nonlinear diffusion equations[END_REF] with

y α,± p = ± √ ∆xσ α p , p ≤ P, y α,± P +1 = ∆xb α , M = P + 1.
Other schemes are defined similarly. The scheme above has more flexibility in defining consistent boundary modifications, as explained in [START_REF] Reisinger | Boundary treatment and multigrid preconditioning for semi-Lagrangian schemes applied to Hamilton-Jacobi-Bellman equations[END_REF]. Finally, a fully discrete scheme is obtained by combining (8) with a time stepping scheme. We introduce a time mesh T ∆t = {t n : 0 ≤ n ≤ N t } ⊆ [0, T ], for simplicity with t n = n∆t, i.e. with uniform step size ∆t > 0. We then define the standard θ-scheme by

u(t n , x) -u(t n-1 , x) ∆t -inf α∈A {θL α ∆x [I ∆x u](t n-1 , x) + (1 -θ)L α ∆x [I ∆x u](t n , x)} = 0, (9)
where we have set c = f = 0 in (1) for simplicity. In the tests we will focus on θ = 0 (explicit Euler) and θ = 1 (implicit Euler). Although the scheme can in principle be defined in this way for all x ∈ Ω, we restrict the scheme to x ∈ N .

Domain overstepping and stencil truncation

We now focus on the boundary of the domain and the definition of the scheme there. We take Ω ⊂ R d for d ≥ 1 a general domain with curved boundary, but for simplicity illustrate the method for C defined by a Cartesian mesh on R d with uniform mesh width ∆x. Then we choose C and N as explained in Section 1. See Figure 1.

a b Ω C C
Figure 1: An elliptical domain and a mesh made of square cells. In situation a, x + y α,± p (t, x) / ∈ Ω, while in situation b, x + y α,± p (t, x) ∈ Ω, but the cell it is contained in has vertices outside Ω. The modified stencil in Sections 2.2 -2.4 uses values from the boundary.

Overstepping

Now consider a mesh node x ∈ N in the vicinity of the boundary. In the two situations sketched in Fig. 1 the interpolation at the point x + y α,± p (t, x) fails for given t, α and p. We say in these cases that the stencil "oversteps". We now discuss several possibilities to deal with these situations. They are all based on a local modification of the step y α,± p (t, x) and of k, combined in some cases with a modification of the interpolation weights.

Where the stencil oversteps, we define ŷα,± p (t, x) = µ α,± p (t, x)y α,± p (t, x), where µ α,± p (t, x) = min µ ≥ 0 : x + µy α,± p (t, x) ∈ ∂Ω .

In case a in Fig. 1 this means µ < 1, while in case b we have µ > 1.

Remark 2.1. On rectangular domains, the elements of the Cartesian mesh cover exactly the domain and case b does not occur. Moreover, interior mesh points cannot be arbitrarily close to the boundary, but are always at least ∆x away. This can be enforced in the general case by removing the outermost layer of cells in C, such that again a distance of ∆x between non-boundary mesh points and the domain boundary is ensured. This allows the derivation of CFL conditions for the explicit schemes as given below (Proposition 2.2 for the scheme in Section 2.4 and similar for other schemes).

Constant extrapolation

When the stencil oversteps, constant extrapolation simply uses instead the value at the boundary in the direction of the stencil step. This can be written as

Lα ∆x [I ∆x φ](t, x) := M p=1 [I ∆x φ](t, x + ŷα,+ p (t, x)) -2φ(t, x) + [I ∆x φ](t, x + ŷα,- p (t, x)) 2∆x . (11)
Note that the scheme is generally not consistent up to the boundary. A proof of convergence is not available to the best of our knowledge.

Stencil cropping

The scheme in [START_REF] Feng | Convergent semi-lagrangian methods for the monge-ampère equation on unstructured grids[END_REF] shrinks the stencil so that it does not overstep. This corresponds to

Lα ∆x [I ∆x φ](t, x) := M p=1 [I ∆x φ](t, x + ỹα,+ p (t, x)) -2φ(t, x) + [I ∆x φ](t, x + ỹα,- p (t, x)) 2 k2 , (12)
where ỹα,±

p = ± kσ α p ∀p = 1 . . . P, ỹα,± P +1 = k2 b α and 0 < k ≡ k(t, x, α) = θ∆x for θ ∈ (0, 1] is the biggest step such that x + ỹα,± p ∈ Ω, ∀p = 1 . . . M .
Generally this means the scheme is not consistent up to the boundary in the sense of [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF], however, [START_REF] Feng | Convergent semi-lagrangian methods for the monge-ampère equation on unstructured grids[END_REF] can still prove a generalised consistency condition and convergence for viscosity solutions on convex domains.

Stencil truncation

In contrast to the above two schemes, the one in [START_REF] Reisinger | Boundary treatment and multigrid preconditioning for semi-Lagrangian schemes applied to Hamilton-Jacobi-Bellman equations[END_REF] is designed to be consistent in the whole of the domain. Hence, the objective is to find truncated or extended stencil vectors ŷα,± p (t, x) and corresponding finite difference weights

A α p ≡ A α p (t, x) and B α p ≡ B α p (t, x), such that x + ŷα,± p (t, x) ∈ ∂Ω and the truncated scheme Lα ∆x [I ∆x φ](t, x) := M p=1 A α p [I ∆x φ](t, x + ŷα,+ p (t, x)) -(A α p + B α p )φ(t, x) + B α p [I ∆x φ](t, x + ŷα,- p (t, x)) 2∆x ( 13
)
is a consistent approximation of (4) as ∆x → 0. If the stencil does not overstep, we have that ŷα,± p (t, x) = y α,± p (t, x) and

A α p = B α p = 1. For all p ∈ [[1, P + 1]], let µ α,± p ∈ [0, 1]
be as in 10, then define

A α P +1 = B α P +1 = 1 µ α,+ P +1 = 1 µ α,- P +1 (14) and, for p ∈ [[1, P ]], A α p = 2 (µ α,+ p ) 2 + µ α,+ p µ α,- p , B α p = 2 (µ α,- p ) 2 + µ α,- p µ α,+ p . ( 15
)
Proposition 2.1 (Consistency, see Corollary 2.3 in [START_REF] Reisinger | Boundary treatment and multigrid preconditioning for semi-Lagrangian schemes applied to Hamilton-Jacobi-Bellman equations[END_REF]). For the truncated scheme, ( 13), ( 14) and ( 15), the following holds: The local consistency error for points with truncation and p = P + 1 is O( √ ∆x) if only one side of the stencil oversteps, and O(1) if both sides overstep.

Remark 2.2. The scheme can be made consistent (of order O(∆x)) for the case where both sides of the stencil overstep by using the exact boundary value in the truncated scheme (13) instead of the interpolant. We therefore assume in the following that this is done. Proposition 2.2 (Monotonicity and stability, see Corollary 2.5 in [START_REF] Reisinger | Boundary treatment and multigrid preconditioning for semi-Lagrangian schemes applied to Hamilton-Jacobi-Bellman equations[END_REF]). In the case of overstepping and θ < 1 in ( 9), monotonicity requires that ∆t ≤ C 1 ∆x 3/2 if only one side of the diffusion stencils oversteps, or ∆t ≤ C 2 ∆x 2 if both sides overstep. However, if the stencil is not truncated, the positivity condition remains as in [START_REF] Debrabant | Semi-Lagrangian schemes for linear and fully nonlinear diffusion equations[END_REF], that is ∆t ≤ C 3 ∆x, where C 1 , C 2 , C 3 > 0 are sufficiently small constants depending on the coefficients σ α , b α and c α , but independent of ∆x and ∆t.

As the scheme is consistent up to the boundary in the classical sense, convergence follows directly from the framework in [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF].

Local mesh refinement

We now consider a refinement of the mesh in boundary layers of width O( √ ∆x) where the semi-Lagrangian scheme oversteps. The objective is to improve the local consistency error of the truncated stencil in Section 2.4 from O( √ ∆x) to O(∆x), and for all schemes to reduce the width of the region where the stencil oversteps from O( √ ∆x) to O(∆x). For this purpose, we combine a local refinement of the mesh with appropriate changes to the stencil step k in [START_REF] Camilli | An approximation scheme for the optimal control of diffusion processes[END_REF] in the region near the boundary.

General mesh construction

Let us consider a mesh defined by cells C and N x nodes N ⊂ Ω with refinement parameter ∆x > 0, as in Section 1. For simplicity, we assume that σ α (t, x) = σ(x).

Since it is primarily the overstepping of the diffusion stencil that reduces the local truncation error, we split N and C into three subsets. We will subsequently define new step sizes k i for nodes in N i , and a refinement with mesh size h i for the cells in C i .

Definition 3.1. Define C 1 = {C ∈ C : ∃x ∈ C, θ ∈ [0, 1], 1 ≤ p ≤ P : x + θ √ ∆xσ p (x) ∈ Ω or x -θ √ ∆xσ p (x) ∈ Ω} * C 2 = {C ∈ C\C 1 : ∃x ∈ C 1 , 1 ≤ p ≤ P : x + k 1 σ p (x) ∈ C or x -k 1 σ p (x) ∈ C} C 3 = C\(C 1 ∪ C 2 ).
Then define

N 1 = C 1 ∩ N the nodes of elements in C 1 , N 2 = (C 2 ∩ N )\N 1
, the nodes of elements in C 2 which are not already in N 1 , and

N 3 = N \(N 1 ∪ N 2 ). Therefore, C = C 1 ∪ C 2 ∪ C 3 with C i ∩ C j =
∅ for i = j and i, j ∈ {1, 2, 3}, and similarly for the nodes. This ensures that if k 1 , k 2 ≤ k, all nodes that overstep are contained in N 1 , and that no node in C 1 steps into C 3 after refinement.

We emphasise the need for the three region construction, with an overlapping layer C 2 of a fine mesh (as in C 1 ) and wide stencil (as in C 3 ). This prevents the situation where a narrow stencil steps into the coarse mesh, which would spoil consistency.

Next, we refine the mesh elements in C 1 and C 2 with mesh-refinement parameter proportional to h 1 = h 2 ∼ ∆x γ , where γ > 1 is a parameter to be determined so that the resulting local consistency error is at least O(∆x). After refinement, we remove the cells which are not fully in Ω to create a new set of cells C with vertices N . By refining C and then pruning the refined cells outside Ω, the space between C and ∂Ω gets refined, which would not happen if we simply refined C.

To determine γ, we procede in reverse order starting with C 3 . As the stencil may step into the finer regions C 1 and C 2 , the consistency error there is

O k 2 3 + h 2 1 k 2 3 + h 2 2 k 2 3 + h 2 3 k 2 3 = O k 2 3 + ∆x 2 k 2 3 + ∆x 2γ k 2 3 ,
where the first term corresponds to the consistency error of the finite difference approximation of the second order derivative and the last terms to the interpolation error in the original and the refined regions respectively. As γ ≥ 1, we do not modify the stencil step, i.e. k 3 ∼ O( √ ∆x). Similarly, the local consistency error for nodes N 2 after refinement is

O k 2 2 + h 2 1 k 2 2 + h 2 2 k 2 2 + h 2 3 k 2 2 = O k 2 2 + ∆x 2 k 2 2 + ∆x 2γ k 2 2 .
Choosing k 2 ∼ O( √ ∆x) the local consistency error is O(∆x), for all γ ≥ 1. Finally, the local consistency error for points in C 1 after refinement is

O k 1 + h 2 1 k 2 1 + h 2 2 k 2 1 = O k 1 + ∆x 2γ k 2 1 ,
where we have assumed that the exact boundary value is used in the case of stencil truncation (see Remark 2.2). Choosing γ = 3 2 and k 1 ∼ O(∆x) the truncation error is O(∆x).

Figure 2 shows a locally refined mesh and describes the effects of the refinement. Figure 2 gives a distorted view for illustration purposes, where in reality C 2 is substantially smaller than C 1 , and C 1 is substantially smaller than C 3 . Remark 3.1. Figure 2 shows that the local refinement leaves 'hanging nodes' at the interface between C 2 and C 3 . These nodes do not pose a problem for semi-Lagrangian discretizations. For the interpolation, the 'hanging nodes' are not used for stencil points with any neighbours belonging to C 3 . coarse black mesh for C 1 , fine black mesh for C 2 and fine white mesh for C 3 . Shown dashed is also the original coarse mesh. The stencil is shrunk for the nodes a and b from Figure 1 earlier, which both lie in C 3 , and does not overstep anymore. The stencil for the new node c oversteps in both ways that a and b did before refinement, but by a lesser distance. The stencil is not shrunk for nodes in C 1 and C 2 . Point d illustrates the situation where the larger stencil steps from region C 3 into the finer region C 2 , which can only improve the interpolation error. In e, the larger stencil steps from region C 2 into both the fine region C 1 and the coarse region C 3 . The latter highlights the importance of the three-region construction, which guarantees that no fine stencil steps into the coarse mesh. This would make the scheme inconsistent for those points.

Mesh construction in one dimension

In our numerical tests we focus on one-dimensional examples, i.e. the case Ω = (x min , x max ) ⊂ R. The construction of the refined mesh is simpler in this case and we report here for completeness its main steps as implemented in the code used in Section 4.

Let

∆ max = √ ∆x σ ∞ ,
where σ ∞ ≡ sup (t,x,α)∈[0,T ]×Ω×A |σ α (t, x)|, denote the maximum step of the SL scheme subject to the volatility σ. Defining a global maximum step is not strictly necessary and it is only done to make the a priori definition of the sets C i , i = 1, 2, 3 easier. The width of C 2 especially is not optimal here, but this does not affect the complexity significantly. Define

x i 1 = min {x i ∈ N : x min + ∆ max ≤ x i } , x i 2 = min {x i ∈ N : x i 1 + ∆ max ≤ x i } x i 3 = max {x i ∈ N : x max -∆ max ≥ x i } , x i 4 = max {x i ∈ N : x i 3 -∆ max ≥ x i } .
Then, the sets C i are defined as follows:

C 1 = [x min , x i 1 ] ∪ [x i 3 , x max ], C 2 = (x i 1 , x i 2 ] ∪ [x i 4 , x i 3 ), C 3 = (x i 2 , x i 4 ).
At this point a refinement of the mesh proportional to ∆x 3/2 is considered in C 1 ∪ C 2 (see Figure 3) defining the new mesh N . Given that the width of the stencil is of length O( √ ∆x), the cardinality of

x min x i 1 x i 2 x i 4 x i 3 x max N ∩ C 1 N ∩ C 2 N ∩ C 3 x min + ∆max x i 1 + ∆max x i 3 -∆max xmax -∆max x min x i 1 x i 2 x i 4 x i 3 x max N ∩ C 1 N ∩ C 2 N ∩ C 3
N 1 ∪ N 2 is |N 1 ∩ N 2 | ∼ O( √ N x )
. Moreover, after the refinement, the number of nodes in this region of width O( √ ∆x) is O(N x ).

Properties of the refined scheme

Let ∆t, ∆x > 0 be the time and space mesh refinement parameters and T ∆t ⊆ [0, T ] and

C ⊆ Ω ⊆ C ⊂ R d as above, with N t = |T ∆t | -1 and N x = |N | = O(∆x -d ).
The cardinality of the refined mesh N in d dimensions is still O(N x ) = O(∆x -d ). Moreover, the analysis in Section 3.1 gives the following result. Proposition 3.1 (Complexity and consistency). If we define sets of mesh cells according to Definition 3.1 and further refine the ones in C 1 ∪ C 2 with mesh refinement parameter O(∆x 3/2 ), the complexity of the method is O(N t N x ) = O(∆t -1 ∆x -d ). If for the nodes requiring truncation in N 1 we use k 1 ∼ O(∆x), then, globally the consistency error of this modified scheme becomes

|1 -2θ| 2 |φ tt | 0 ∆t + C ∆t 2 |φ ttt | 0 + ∆x(|D 2 φ| 0 + |D 3 φ| 0 + |D 4 φ| 0 ) . ( 16
)
As shown in Corollary 2.2, an undesirable side-effect of stencil truncation is the worsening of the CFL condition of the scheme. The local refinement of the grid results in a stricter CFL condition compared to the one in Corollary 2.2, as shown in the following. Proposition 3.2 (Monotonicity and stability). Additionally, provided that all the nodes are O(∆x 3/2 ) away from the boundary of the domain, a scheme with θ < 1 requires ∆t ∼ O(∆x 5/2 ) if only one side of the stencil oversteps or ∆t ∼ O(∆x 3 ) if both sides of the stencil overstep. Remark 3.2. Similar to Remark 2.1, ensuring that all nodes in the mesh are at least O(∆x 3/2 ) away from the boundary of the domain can be achieved by removing the outermost layer of the cells inside the domain after refinement.

Numerical tests

We consider the HJB equation in a bounded set Ω := (x min , x max ). Denoted by N x the number of mesh points, in one dimension, ∆x =

x max -x min N x -1 .

Let (x j ) be a uniform mesh on [x min , x max ] with x j = x min + j∆x, j = 1, . . . , N x and t n = n∆t, ∆t = T Nt . We are going to test both the explicit and the implicit scheme (θ = 0 and θ = 1, respectively, in ( 9)).

Test 1: Linear equation

We first test the scheme on a linear second order equation since, already in this simplified case, the main features of the scheme can be observed. In particular, we consider a classical Black-Scholes equation in Ω with a smooth initial datum: with

u t -1 2 (σx) 2 u xx -(bx)u x = f (t, x), t ∈ (0, T ), x ∈ Ω, (17)
u(t, x) = v(t, x), t ∈ (0, T ), x ∈ ∂Ω, (18)
u(0, x) = sin(πx), x ∈ Ω, (19)
f (t, x) = 1 2 (1 -t)(πσx) 2 -1 sin π x - t 2 -(1 -t) π 2 + πbx cos π x - t 2 .
The equation has the exact solution:

v(t, x) = (1 -t) sin π x - t 2 .
x min x max b σ T -1 1 2 1 0.5 for the truncated stencil scheme without mesh refinement,

N x N t = Nx-1 2 N t = (N x -1) 3/2 N t = (Nx-1) 5/2 20 L ∞ -error rate L ∞ -
N t ∼ N 5/2 x
for the scheme with mesh refinement) is confirmed by the results in Tables 2 to 5. The mesh refinement has little impact on the performance of the scheme with truncated stencil (last column of Tables 2 and3). Clear improvements can be observed if a constant extrapolation of the boundary conditions is performed outside the domain (Tables 4 and5). It is worthwhile to recall that in this last case the scheme is not consistent and the convergence is numerically achieved because consistency is lost only for points within a distance of √ ∆x of the boundary, and ∆x after mesh refinement. Similar results are obtained for the implicit scheme (Table 6). Here, no CFL condition is required and this allows us to take ∆t proportional to ∆x, i.e., N t proportional to N x .

Test 2: Controlled drift-diffusion equation

The second test we propose is a HJB equation with coefficients independent of (t, x):

u t + sup a 1 ,a 2 -1 2 a 2 u xx -2au x = 0, t ∈ (0, T ), x ∈ Ω, (20)
u(t, x) = ψ(t, x), t ∈ (0, T ), x ∈ ∂Ω, (21)
u(0, x) = ψ(0, x), x ∈ Ω, (22)
where ψ(0, •) is defined by

ψ(0, x) =    5x(1 + x) 4 if -1 < x ≤ 0 5x(1 -x) 4 if 0 ≤ x < 1 0 if |x| ≥ 1.
We use the parameters in Table 7. In order to compare our numerical results we use a reference numerical solution computed in a larger domain Ω 1 = (-5, 5) with zero boundary conditions, using a second order (in time and space) BDF scheme (see [START_REF] Bokanowski | High-order filtered schemes for time-dependent second order HJB equations[END_REF]) with N x = 5121 × 5 and N t = (N x -1)/10. The boundary value ψ(t, x) at x = ±1 is obtained using this reference numerical solution. The initial condition and the solution at terminal time are shown in Figure 5. We report here, in Table 8, only the results obtained by the implicit scheme. However, under the CFL condition N t ∼ N 5/2

x , similar results can be obtained using the explicit scheme. The truncated scheme converges with order one (first two columns in Table 8). The scheme with constant extrapolation of boundary conditions converges with asymptotic order 1/2 (third column in Table 8) which becomes order 1 applying the mesh refinement (fourth column in Table 8). We point out that in this example the main contribution to the error comes from the points where the solution changes concavity (Figure 6), which correspond to discontinuities in the second order derivative. This explains the almost absence of differences in Table 8 (columns 1,2,4) and the fact that the order 1/2 in the third column is recovered after a quite big number of mesh refinements. Figure 6 compares the error obtained using the constant extrapolation of boundary conditions (left) and the truncated stencil (right). One can notice that the use of the constant extrapolation creates some instability at the left hand boundary, that is where such a scheme strongly modifies the nature of the exact solution outside the computational domain.

Test 3: Nonsmooth initial data

Let us consider the following problem:

u t + sup σ min ,σmax -1 2 (σx) 2 u xx -bxu x + ru = 0, t ∈ (0, T ), x ∈ Ω, (23)
u(t, x) = 0, t ∈ (0, T ), x ∈ ∂Ω, (24)
u(0, x) = ψ(0, x), x ∈ Ω, (25)
where ψ(0, •) is defined by with K 1 , K 2 > 0. This type of equations typically arises in financial applications when pricing options under uncertain volatility model, see [START_REF] Lyons | Uncertain volatility and the risk-free synthesis of derivatives[END_REF]. The initial condition ψ(0, •) (Figure 7 (left)) has the shape of a butterfly payoff, for which a detailed study of finite difference numerical approximation has been proposed in [START_REF] Pooley | Numerical convergence properties of option pricing PDEs with uncertain volatility[END_REF].

ψ(0, x) = max -2K 1 |x -K 2 | + K 1 , 0
In our numerical tests we use the parameters in Table 9. As in the previous example, in order to compute errors and rate of convergence, a reference numerical solution computed using a second order BDF scheme with N x = 5120 and N t = (N x -1)/10. The initial condition and the solution at terminal time are shown in Figure 7. The convergence orders are as previously. Figure 8 compares the error obtained using the constant extrapolation of boundary conditions (left) and the truncated stencil (right). The instability at the boundary created by the constant extrapolation of the boundary conditions is even more evident in this example.

Last, we test the stencil cropping presented in [START_REF] Feng | Convergent semi-lagrangian methods for the monge-ampère equation on unstructured grids[END_REF] (see Section 2.3). In order to

Conclusions

In this paper we have presented and discussed a local mesh refinement for SL schemes approximating viscosity solutions to second order HJB equations in bounded domains with Dirichlet boundary conditions. In order to test our mesh refinement, we have considered different treatments for the 'overstepping' phenomena, which typically arise when this kind of wide stencil schemes is used. When a constant extrapolation of boundary conditions (Section 2.2) is applied, the local refinement of the mesh improves the observed order of convergence from order 1/2 to order 1. The scheme remains non-consistent with the differential operator in the neighborhood of the boundary and the mesh refinement has only the role of reducing the region of non-consistency. For the scheme with cropped stencil described in [START_REF] Feng | Convergent semi-lagrangian methods for the monge-ampère equation on unstructured grids[END_REF] (see also Section 2.3) the tests we performed do not show any benefits coming from the local mesh refinement since the scheme alone, even if not consistent up to the boundary, already shows a first order of convergence. The effects of the mesh refinement on the scheme with truncated stencil defined in [START_REF] Reisinger | Boundary treatment and multigrid preconditioning for semi-Lagrangian schemes applied to Hamilton-Jacobi-Bellman equations[END_REF] (also Section 2.4) consist in the improvement of the global truncation error from order 1/2 to order 1. Among other things, this permits to estimate the order of convergence of the scheme using the techniques in [START_REF] Barles | Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Barles | Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Debrabant | Semi-Lagrangian schemes for linear and fully nonlinear diffusion equations[END_REF]. However, from the numerical point of view, no substancial difference in the rate of convergence of the scheme is observable, due to the fact that the truncation error of order √ ∆x of the scheme without mesh refinement only occurs in a region of diameter √ ∆x.

In the numerical tests presented in Section 4 only the one-dimensional case was taken into account. Already in this simple setting the main feature of the scheme and the effects of the mesh refinement can be clearly observed. Numerical experiments in more dimensions and a deeper understanding of the numerical instabilities observed when a constant extrapolation of the boundary conditions is applied are the subject of future research.

3 Figure 2 :

 32 Figure 2: Locally refined mesh. The regions in Definition 3.1 are shown in different styles:coarse black mesh for C 1 , fine black mesh for C 2 and fine white mesh for C 3 . Shown dashed is also the original coarse mesh. The stencil is shrunk for the nodes a and b from Figure1earlier, which both lie in C 3 , and does not overstep anymore. The stencil for the new node c oversteps in both ways that a and b did before refinement, but by a lesser distance. The stencil is not shrunk for nodes in C 1 and C 2 . Point d illustrates the situation where the larger stencil steps from region C 3 into the finer region C 2 , which can only improve the interpolation error. In e, the larger stencil steps from region C 2 into both the fine region C 1 and the coarse region C 3 . The latter highlights the importance of the three-region construction, which guarantees that no fine stencil steps into the coarse mesh. This would make the scheme inconsistent for those points.

Figure 3 :

 3 Figure 3: Mesh refinement in one dimension. Definition of sets C i , i = 1, 2, 3 (top) and refined mesh (bottom).

Figure 4 :

 4 Figure 4: (Test 1) Plot of the value function at time t = 0 (left) and t = T (right).

x min x max a 1 a 2 TTable 7 :

 27 Parameters used in numerical experiments for Test 2.

Figure 5 :

 5 Figure 5: (Test 2) Plot of the value function at time t = 0 (left) and t = T (right).

Figure 6 :

 6 Figure 6: (Test 2) Exact solution (green), numerical solution (blue) and error (red) obtained with N x = 161 and N t = 16 using the constant extrapolation of boundary conditions (left) and the truncated stencil (right).

Figure 7 :

 7 Figure 7: (Test 3) Plot of the value function at time t = 0 (left) and t = T (right).

Figure 8 :

 8 Figure 8: (Test 3) Exact solution (green), numerical solution (blue) and error (red) obtained with N x = 321 and N t = 32 using the constant extrapolation of boundary conditions (left) and the truncated stencil (right).

Table 1 :

 1 Parameters used in numerical experiments for Test 1.

Table 2 :

 2 (Test 1) Explicit scheme. Truncated stencil without mesh refinement.For our numerical tests we used the parameters in Table1. The initial condition and the solution at terminal time are shown in Figure 4. The necessity of the CFL condition for No refinement With refinement No refinement With refinement L ∞ -error rate L ∞ -error rate L ∞ -error rate L ∞ -error rate

	error	rate L ∞ -error	rate

Table 8 :

 8 No refinement With refinement No refinement With refinement L ∞ -error rate L ∞ -error rate L ∞ -error rate L ∞ -error rate (Test 2) Implicit scheme.

			Truncated stencil	Constant extrapolation
	N x	N t
	21	2	2.61E-01	-2.63E-01	-2.61E-01	-2.63E-01	-
	41	4	1.75E-01 0.58 1.75E-01	0.58 1.75E-01 0.58 1.75E-01	0.58
	81	8	1.05E-01 0.73 1.05E-01	0.73 1.05E-01 0.74 1.05E-01	0.73
	161	16 5.62E-02 0.91 5.62E-02	0.91 5.61E-02 0.91 5.61E-02	0.91
	321	32 2.88E-02 0.96 2.88E-02	0.96 2.87E-02 0.96 2.88E-02	0.96
	641	64 1.41E-02 1.03 1.41E-02	1.03 1.40E-02 1.03 1.41E-02	1.03
	1281 128 6.50E-03 1.11 6.50E-03	1.11 1.08E-02 0.38 6.50E-03	1.11
	2561 256 3.45E-03 0.91 3.45E-03	0.91 8.58E-03 0.33 3.45E-03	0.91

avoid the case where the cropped stencil falls exactly on mesh points (this would make the scheme consistent since the interpolation would not be performed) we progressively divide the stencil by π until it fits into the domain Ω. The scheme shows a first order of convergence even without mesh refinement (Table 11). We observed the same also for the other examples in this section (not reproduced here). However, since consistency is not satisfied for those points where the cropping is performed, we cannot expect this to hold in general and the observed behavior in our tests might also be related to the fact that our solutions are almost linear close to the boundary, which makes the contribution to the error coming from the interpolation negligible.


Miguel Díaz-Rodríguez

Juan A Carretero

Ricardo Bautista-Quintero

Cinem ática y Din ámica Inversa de un Robot Paralelo 3-PRS

Keywords: cinemática inversa, dinámica inversa, robots paralelos, matriz Jacobiana

En este trabajo se obtiene el modelo cinem ático y din ámico inverso de un robot paralelo 3-PRS. Este robot presenta la característica de tener menos de 6 grados de libertad lo cual lo hace adecuado para aplicaciones de bajo costo. El robot presenta la particularidad de que cada pierna cuenta de dos eslabones unidos mediante pares prism ático-revoluta-esf érico. El trabajo se diferencia de trabajos anteriores por el hecho de que el an álisis de cada pierna ha permitido desarrollar la matriz Jacobiana de dimensiones homog éneas, que relaciona las coordenadas generalizadas del elemento terminal con la coordenadas de las juntas actuadas. El modelo din ámico inverso es obtenido al considerar que la plataforma se separa de las piernas en los pares esf éricos. Así, las coordenadas de juntas son utilizadas para modelar la din ámica de las piernas y las coordenadas del elemento terminal para modelar la din ámica de la plataforma. Luego, el modelo din ámico viene dado por la proyecci ón de la din ámica de las piernas y la plataforma sobre el espacio de juntas actuadas. El enfoque propuesto permite modelar indistintamente los casos en que el robot es actuado en el par prism ático o en el par de revoluta lo cual representa el otro de aporte del trabajo. Adem ás, el modelo presenta un n úmero de operaciones, sumas, restas, multiplicaciones y divisiones, comparable con las obtenidas en otros trabajos.

Introducci ón

El robot paralelo 3-PRS consiste de una base fija unida mediante tres cadenas cinemáticas abiertas a una platforma móvil. Cada cadena cinemática abierta presenta dos eslabones, donde el primer eslabón va unido a la base por medio de un par prismático y, al eslabón intermedio por medio de un par de revoluta. El eslabón intermedio va unido a la plataforma móvil mediante un par esférico. Por tanto, la configuración de cada cadena es PRS (prismático, revoluta y esférico). La estructura del robot puede verse en la Figura 1, donde se muestra un prototipo desarrollado previamente por uno de los autores de este trabajo (Valles y col., 2013).

El robot paralelo 3-PRS fue inicialmente propuesto para una aplicación telescópica (Carretero y col., 2000a) y luego ha sido propuesto tanto como componente de un centro de mecanizado (Fan y col., 2003), así como microrobot para aplicaciones médicas [START_REF] Merlet | Micro parallel robot MIPS for medical applications[END_REF]. Mas recientemente, este tipo de robot ha sido propuesto como dispositivo para rehabilitación de tobillo (Valles y col., 2015). El robot 3-PRS ha llamado la atención de los investigadores y es así como su cinemática y el análisis del espacio de trabajo del 3-PRS han sido ampliamente estudiadas, ver (Carretero y col. 2000a, Carretero y col. 2000b[START_REF] Tsai | Direct kinematic analysis of a 3-PRS parallel manipulator[END_REF], Li y col. 2007) por solo mencionar unos pocos. Por otra parte, la dinámica inversa ha sido abordada desde el punto de vista del principio dinámico. (Li y col., 2004) emplea las ecuaciones de Lagrange e incluyen las restricciones (debido a separar la plataforma móvil de las piernas del robot) mediante los multiplicadores de Lagrange. El inconveniente de este método es que el modelado dinámico resulta altamente complicado y adicionalmente computacionalmente complejo (alto número de operaciones, sumas restas, multiplicaciones y divisiones). (Li y col., 2004) utilizan el Principio de los Trabajos Virtuales pero en su aplicación simplifican la Inercia del eslabón intermedio y consideran su masa dividida en dos porciones, una ubicada en el primer eslabón y la otra unida a la plataforma móvil. (Tsai y col., 2010) resuelven el modelo dinamico junto con las fuerzas de restricción mediante una descomposición especial de las fuerzas de reacción que conectan la plataforma móvil al eslabón intermedio. Si bien las fuerzas de reacción son necesarias para el diseño estructural del robot, estas son innecesarias para el modelado dinámico inverso y además su calculo incrementa la complejidad del modelo. [START_REF] Staicu | Matrix modeling of inverse dynamics of spatial and planar parallel robots[END_REF] analiza y compara el consumo de potencia del robot 3-PRS cuando este es actuado bien en los pares prismático o en los pares de revoluta. El procedimiento empleado es recursivo similar al empleado por (Mata y col., 2008).

De los trabajos anteriores se observa que si bien se ha abordado el problema dinámico del robot 3-PRS, el aspecto del coste computacional no ha sido abordado. Este aspecto es importante en el diseño optimo ya que la evaluación de la función objetivo influye en el tiempo requerido por el algoritmo para llegar a la solución optima. Otra área de interés es en el control basado en el modelo dinámico, donde se requiere un modelo con un número de operaciones mínimas a fin de computarlo en linea, más ahora con el empleo cada vez mayor de plataforma de prototipado de bajo costo cuyo capacidad de procesamiento es limitado con respecto a una unidad convencional. Lo anterior a motivado el desarrollo del presente trabajo.

Este articulo propone un enfoque para resolver el problema cinemático y dinámico inverso de un robot paralelo 3-PRS. Para obtener el modelo, se parte de la solución general propuesta por (Khalil y col., 2007) y se particulariza para el caso del 3-PRS. A partir del estudio del movimiento de las piernas del robot, se ha obtenido una matrix Jacobiana de dimensiones homogéneas evitando así la inversión de una matrix con unidades de posición y angulo por una homogénea que relaciona las coordenadas de posición del elemento terminal con las de posición del par prismático o con la rotación de los pares de revoluta. Además, el modelado desarrollado considera indistintamente la posibilidad de que el 3-PRS sea actuado en los pares prismáticos o los pares de revoluta y presenta un número de operaciones, sumas/restas y multiplicaciones/divisiones, comparable o incluso menor con las obtenidas en otros trabajos.

La estructura del trabajo es la siguiente: en el apartado II se describe el robot 3-PRS. El apartado III presenta el desarrollo del modelo cinemático. El apartado IV detalla como se obtiene el modelo dinámico. El apartado V incluye un ejemplo de aplicación y finalmente, el apartado VI presenta las conclusiones. Para modelar cada una de las piernas se emplea la notación de Denavit-Hartenberg (D-H) modificada ? . La Tabla 1 presenta los parámetros D-H para cada pierna hasta el sistema coordenado {O i,2 }. Note que el primer subindice i indica la pierna respectiva y el segundo subindice indica el par cinemático numerados desde 1 la base hasta 3 la plataforma móvil.

Table 1. Parámetros D-H del Robot Paralelo 3-PRS i θ i,1 d i,1 a i,1 α i,1 θ i,2 d i,2 a i,2 α i,2 1 π/2 d 1 0 γ 1 θ 1 0 0 -π/2 2 π/2 d 2 0 γ 2 θ 2 0 0 -π/2 3 π/2 d 3 0 γ 3 θ 3 0 0 -π/2
Adicionalmente, para modelar las coordenadas del elemento terminal, se emplea el sistema de Angulos de Euler XYZ alabeo-cabeceo-guiñada (α, β , φ ) del sistema {p} con respecto al sistema global de coordenadas {O}. La matriz de rotación o R p viene dada por:

o R p =   c α c β c α s β s φ -s α c φ c α s β c φ + s α s φ s α c β s α s β s φ + c α c φ s α s β c φ -c α s φ -s β c β s φ c β c φ   (1) siendo c * = cos ( * ) y s * = sin ( * ).
En resumen las coordenadas empleadas para el modelado del robot son las: para la plataforma móvil, las coordenadas del elemento terminal x y para cada pierna del robot las coordenadas de junta q. Así:

x = x p y p z p φ β α T (2) q = q T 1 q T 2 q T 3 T (3) (4) donde, q 1 = d 1 θ 1 q 2 = d 2 θ 2 q 3 = d 3 θ 3 (5)
Las componentes de b i con respecto al sistema local {p} son:

p b 1 =   h 0 0   p b 2 =   -1 2 h √ 3 2 h 0   p b 3 =   -1 2 h - √ 3 2 h 0   . ( 6
)

3/10

Finalmente las componentes de a i con respecto al sistema local {O} son:

p a 1 =   g 0 0   p a 2 =   -1 2 g √ 3 2 g 0   p a 3 =   -1 2 g - √ 3 2 g 0   . (7)
3 Cinem ática

Problema de Posici ón

La ecuación de lazo vectorial para la pierna i viene dada por:

p + b i = a i + q i u i,1 + l i u i,2 (8)
En la ecuación anterior u i,1 es un vectores unitario que define la línea de acción del par prismático i, y u i,2 es también un vector unitario dirección va desde el centro del par de revoluta hasta el del par esférico. Además, l i define la longitud del eslabón intermedio o acoplador, mientras que q i define la magnitud de movimiento del par prismático. La ecuación (8) es función de: los parámetros cinemáticos, las coordenadas del elemento terminal y las coordenadas de junta. La cinemática inversa consiste en encontrar las coordenadas de junta en función de las coordenadas del elemento terminal. Los vectores u i,1 y u i,2 son función de d i y θ i respectivamente. Se puede eliminar la variable θ i de la ecuación Eq. ( 8) si se despeja el termino l i u 2 i hacia el lado derecho de la ecuación y se eleva al cuadrado ambos lados de la ecuación (Li y col., 2004). Así:

[p + b i -a i -q i u i,1 ] T [p + b i -a i -q i u i,1 ] = l 2 i (9)
Luego de varias manipulaciones algebraicas se obtiene la relación entre la coordenada asociada con el par prismático:

d i = c T i • u i,1 ± c T i • u i,1 -c T i • c i + l 2 i (10) siendo c i = p + b i -a i .
El ángulo θ i se determina despejando de la ecuación obtenida luego de sustituir d i en la ecuación Eq. ( 8):

θ i = f (d i , x, l i , h, g, γ i ) . ( 11
)
El robot paralelo estudiado presenta 3 grados de libertad, por lo que de las 6 coordenadas del elemento terminal, es necesario determinar la relación que hay entre 3 de ellas con el resto. Para ello se estudia el movimiento del centro de los pares esféricos (B i ), el cual tiene la característica de que su movimiento ocurre en el plano definido por el vector normal alineado con la dirección del eje del par de revoluta. En otras palabras, si se observa el movimiento de la pierna i respecto al sistema coordenado {O i,2 } se tiene que la componente de movimiento de la coordenada local z es igual a cero. Por tanto si determinamos la posición del punto B i en función del punto p y la representamos respecto al sistema {O i.2 } la tercera componente debe ser igual a cero. Esto es:

z T 1 1,2 R o p + 1,2 R o o R p p b 1 = 0 (12) z T 2 2,2 R o p + 2,2 R o o R p p b 2 = 0 (13) z T 3 3,2 R o p + 3,2 R o o R p p b 3 = 0 ( 14
)
siendo z i = 0 0 1 T , i,2 R o es la matriz de rotación, (usando parámetros de D-H) entre el sistema local {O i,2 } respecto del sistema global {O} para la pierna i.

El sistema de ecuaciones ( 12)-( 14) se puede resolver para x p , y p and α de tal manera que:

x p = -hs α c β y p = - 1 2 h c α c β -s α s β s φ -c α c φ (15) α = atan2 s β s φ , c φ + c β
Como puede observarse se ha seleccionado como coordenadas independientes la coordenada z p y los ángulos β y φ , siendo estos los grados de libertad del robot. A partir de estas tres coordenadas, las restantes coordenadas del robot pueden ser determinadas.

4/10

Velocidad

Para resolver el problema de velocidad, primero se determina la velocidad del B i respecto al sistema {O i,2 } se tiene:

i,2 v B i = i J q q i (16) siendo i J q =   -sin (θ i ) 0 -cos (θ i ) l i 0 0   (17)
La velocidad del elemento terminal puede ser escrita con respecto a la velocidad del punto B i :

i,2 v B i = i J v ˙ x p = i,2 R o -i,2 R o o R p p b i ẋp (18)
siendo ẋp = v p x v p y v p z ω x ω y ω z T , b representa la matriz antisimétrica que permite sustituir el producto vectorial

b i ×. Además: i,2 R p =   -c σ i s γ i +θ i -c σ i c γ i +θ i s σ i -s σ i s γ 1 +θ i -s σ i c γ 1 +θ i -c σ i c γ+θ i -s γ i +θ i 0   (19)
Las dos primeras columnas de las ecuaciones ( 17) y ( 18) relacionan las coordenadas del elemento terminal con las de junta. De esta forma:

J q q = J x ẋ (20) siendo, J q =   1 J q 0 0 0 2 J q 0 0 0 3 J q   (21) y, J x =      1,2 R o (1..2, :) -1,2 R o o R p p b 1 (1..2, :) 2,2 R o (1..2, :) -2,2 R o o R p p b 1 (1..2, :) 3,2 R o (1..2, :) -3,2 R o o R p p b 1 (1..2, :)      (22)
En la ecuación anterior R (1..2, :) indica que solo son empleadas las filas 1 y 2 de la matriz R y 0 es una matriz nula de dimensiones 2 × 2.

La ecuación (20) resuelve el problema de velocidad. Conocidas las coordenadas del elemento terminal, la ecuación permite determinar las velocidades de las juntas del robot. Sin embargo es necesario determinar la relación respecto a las coordenadas independientes, es decir, en función de los 3 grados de libertad. con este fin se tiene que:

qa = J -1 p ẋp (23) siendo J -1 p = J -1 q J x ([1 3
5] , :) si el robot es actuado en el par prismático PRS y J -1 p = J -1 q J x ([2 4 6] , :) si el robot es actuado en el par de revoluta PRS.

La matriz anterior no es cuadrada por lo que no puede invertirse. Para obtener una matriz cuadrada se analiza la tercera columna de la ecuación (18):

1,2 R o (3, :) v p -1,2 R o o R p p b 1 (3, :) ω p = 0 2,2 R o (3, :) v p -2,2 R o o R p p b 2 (3, :) ω p = 0 (24) 3,2 R o (3, :) v p -3,2 R o o R p p b 3 (3, :) ω p = 0 5/10
Desarrollando la ecuación (24) se obtiene:

J r 1 v p = J r 2 ω p (25)
La ecuación anterior es un sistema lineal que relaciona las velocidades tanto angulares como lineales del elemento terminal. Con el objeto de obtener una matriz Jacobiana homogénea, se estudió el conjunto de coordenadas independientes que pueden ser utilizadas para resolver el sistema lineal. Del estudio se tiene que a) Las tres traslaciones pueden ser empleadas (x, y, z), b) Dos rotaciones y una traslación (α, β , z) or c) dos traslaciones y una rotación (α, x, z). Si se toma la velocidad lineal del punto p se obtiene un Jacobiano cuyas dimensiones son homogéneas. Así:

ω p = J r v p (26) siendo J r = J -1 r 2 J r 1 .
Las velocidades de las componentes components v p x and v p x se obtienen derivando de las ecuaciones para x p y y p obtenidas en (15).

Las coordenadas del elemento terminal se obtienen a partir de las coordenadas independientes de la siguiente forma:

v = I 3×3 J r vp ( 27
)
Si se sustituye la ecuación ( 27) en la ecuación ( 23) se tiene:

qa = J c v p ( 28
)
siendo

J c = J -1 p I 3×3 J r . ( 29
)
La matriz Jacobiana que relaciona las coordenadas del elemento terminal con las de junta viene entonces dado por:

J k = J r J -1 c (30) con lo cual ẋp = J k qa . ( 31
)

Acceleraci ón

Similar a lo procedimiento utilizado para la velocidad, el problema de aceleración se resuelve relacionando la aceleración del punto B i respecto al elemento terminal y respecto a cada una de las piernas. Se tiene que:

qi = i J q a B i -i Jq q (32) siendo i Jq q = -c θ i θi di -s θ i θi di (33)
La aceleración a B i se relaciona mediante ẍp = a p ωp de la siguiente manera:

a B i = i,2 R o -i,2 R o o R p p b i ẍp + i,2 R o ω p × ( ω p × o R p p b i ) (34)
Finalmente, la aceleración del elemento terminal respecto a las coordenadas independientes viene dada por :

ωp = J r a p + Jr v p (35) 6/10
4 Modelo Din ámico Inverso

El modelo dinámico inverso se obtiene considerando el robot como dos sistemas. Uno referido a la plataforma móvil y el segundo referido a las piernas. Los sistemas se obtienen al separar el robot en las juntas esféricas. Se modela el sistema referido a las piernas utilizando las coordenadas de juntaq y el sistema de coordenadas del elemento terminal x se emplea para modelar la plataforma. De esta manera cada pierna puede ser vista como una cadena cinemática abierta o robot serial. La ventaja es que la dinámica inversa de cada pierna h i puede obtenerse mediante algoritmos recursivos eficientes (Provenzano y col., 2002). La dinámica de la plataforma móvil en función de las coordenadas del elemento terminal viene dada por:

f p = m p g -m p a p -I p ωp -ω p × (I p ω p ) (36)
siendo m p la masa de la plataforma y I p el tensor de inercia respecto al centro de masas de la plataforma móvil. Luego de obtener la dinámica de cada pierna y de la plataforma, el modelo dinámico inverso se obtiene proyectando la dinámica sobre el espacio de juntas activas (Khalil y col., 2007). De esta forma:

τ = J T k f p + G T 1 h + J T k f e (37)
donde G 1 representa la projección de la dinámica de las juntas pasivas de las piernas con respecto a las juntas activas. Para completar el modelo dinámico, en la ecuación (37), las dinámica asociada a las jutnas pasivas pueden proyectarse primero al espacio del elemento terminal y luego al espacio de juntas activas. Esto permite tener un modelo dinámico que considera las posibilidad de que el robot sea actuado en las juntas prismáticas o en las juntas de revoluta. Específicamente:

τ = h a i + J T k f p + G T 2 h p i (38)
siendo G 2 de 6 × 3 y se determina como sigue:

G 2 = J -1 q J x ([2 4 6] , :) PRS (39) G 2 = J -1 q J x ([1 3 5] , :) PRS (40)
Para resolver la ecuación (38) se requiere aproximadamente de 860 multiplicaciones/divisiones y 680 sumas/restas lo cual es relativamente bajo si se compara con un total de 1163 y 735 obtenido por (Díaz-Rodríguez y col., 2012) y las 1352 y 778 reportadas en (Tsai y col., 2010).

Ejemplo de Aplicaci ón

La tabla 2 muestra el valor de los parámetros empleados para resolver el problema dinámico inverso del robot 3-PRS cuando es actuado en el par P.

Table 2. Parámetros cinemáticos y dinámicos utilizados en las simulaciones

γ [rad] l i [m] h [m] g [m] π/2 1 0.5 1 m 1 [Kg] m 2 [Kg] m p [Kg] I zz 2 [Kg m 2 ] 0.1m p 0.1m p 5 1 12 m 2 l 2 m x 2 [Kg m] m y 2 [Kg m] I xx 3 [Kg m 2 ] I yy 3 [Kg m 2 ] l 1 2 m 2 0 m p h 2 4 m p h 2 4 I zz 3 [Kg m 2 ] I xy 3 [Kg m 2 ] I xz 3 [Kg m 2 ] I yz 3 [Kg m 2 ] m p h 2 2 0 0 0
Se evaluó un total de cuatro escenarios del comportamiento del robot que permitieron comparar los resultados del enfoque de modelado propuesto en este artículo, con los obtenidos utilizando el algoritmo empleado en (Mata y col., 2008). Para todas las simulaciones los resultados fueron similares dentro del rango de precisión (10 -5 ). Los escenarios estudiados se describen seguidamente: 1) el robot se mueve en vertical (z) siguiendo una trayectoria lineal, 2) la trayectoria vertical es armónica, 3) El ángulo de giro alrededor del eje x de la plataforma tiene un movimiento armónico y 4 ) El ángulo de giro alrededor del eje y de la plataforma tiene un movimiento armónico. Por limitaciones de espacio se presenta los resultados del segundo y tercer caso donde para el segundo caso la trayectoria del elemento terminal viene dada por:

z = 0.81 + 0.005 (1 -cos (π/3 t)) (41) φ = 0 (42) β = 0 (43)
siendo el tiempo de simulación igual a 5 segundos. La figura 3 muestra la potencia requerida en los actuadores para el caso segundo caso. Puede apreciarse que, debido a que la plataforma sube sin girar, es de esperar que todos sus actuadores requieran de la misma potencia. De igual forma la potencia debe tener un comportamiento similar al armónico.

Para el caso 3 el movimiento del elemento terminal viene dado por:

z = h (44) φ = π/36 (1 -cos (π/3 t)) (45)
β = 0 (46)
siendo el tiempo de simulación igual a 5 segundos. La figura 4 muestra la potencia requerida en los actuadores para el caso tercer caso. Puede apreciarse que, para lograr el movimiento armónico en la plataforma, dos de sus actuadores requieren potencia dentro del mismo rango pero en direcciones opuestas. Así mismo se observa que el tercer requiere menos del 10% del requerido por los otros dos actuadores.

Conclusiones

Se desarrolló el problema cinemático y dinámico inverso de un robot paralelo 3-PRS. Dicho robot cuenta con solo 3 grados de libertad, lo cual lo hace una alternativa de estudio para diseño de robot de bajo costo.

El trabajo propuso un enfoque para determinar la matrix Jacobiana que relaciona las coordenadas dependientes e independientes. La relación se obtuvo de analizar el movimiento permitido por cada una de las piernas del robot. Se determinó una matriz Jacobiana cuyas componentes son de dimensiones homogéneas lo cual resulta importante ya que el modelo dinámico involucra la inversa de dicha matriz. Otra de las ventaja del enfoque propuesto es que se puede analizar indistintamente los casos en que el robot es actuado en los pares prismáticos o en las de revoluta.

Se evalúo numéricamente cuatro escenarios de movimiento de plataforma móvil. Se obtuvó que la diferencia de los resultados comparados con la de otros autores fue igual dentro del margen de tolerancia establecido Presentando además la característica que, el modelo propuesto presenta un menor número de operaciones multiplicación/division y sumas/restas comparado con trabajos anteriores reportados.

El enfoque de modelado aquí propuesto puede ser extendido en futuros trabajos para estudiar otros robots de 3 grados de libertad tal como 3-RPS and 3-RRS.

Agradecimientos

Se agradece el apoyo financiero del Fondo Nacional de Ciencia y Tecnología (FONACIT-Venezuela), el Programa Plan III de Intercambio Científico de la Universidad de los Andes (Venezuela), al CONACYT beca 326912/381134 y el SNI-México.

Figure 1 .

 1 Figure 1. Prototipo Robot Paraelo 3-PRS.

Figure 2 .

 2 Figure 2. Diagrama Esquemático del Robot Paraelo 3-PRS.

Figure 3 .

 3 Figure 3. Potencia requerida en los actuadores para trajectoria caso 2.

Figure 4 .

 4 Figure 4. Potencia requerida en los actuadores para trajectoria caso 3.

El Robot Paralelo

3-PRSLa Figura 2 muestra el diagrama esquemático del robot paralelo 3-PRS. El sistema de referencia {O} representa el sistema global de coordenadas (x, y, z) y {p} define el sistema coordenado local unido a la plataforma móvil. Los pares esféricos, se localizados en la plataforma móvil, forman un triángulo equilátero circunscrito en un circulo con centro en p y radio h. Los ejes de los pares prismático interceptan a la plataforma base formando también un triángulo equilátero. La distancia desde O al punto de intercepción del eje del par i y el plano base es s.2/10

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