Monique Vianey-Liaud

Raef Minwer-Barakat

Laurent Marivaux

The rodents from Santpedor-2 (Ebro Basin, NE Spain) confirm the Oligocene age of the latest primates from the Paleogene of Europe §

Keywords:

Paleontologia Miquel Crusafont, ICP) and the

Introduction

The fossil site of Santpedor (Ebro Basin, NE Spain), originally named as ''Costa de la Vila'', was first mentioned in the 1950's, when [START_REF] Masachs | Sur l'a ˆge du gisement potassique de la Catalogne[END_REF] briefly described the presence of two perissodactyls and an artiodactyl: Paleotherium medium, Plagiolophus annectens, and an undetermined Anoplotheriidae (later referred to Anoplotherium commune by [START_REF] Golpe | Paleobiologia dels jaciments amb vertebrats al tra `nsit Eoce `-Oligoce `a la Catalunya central[END_REF]. These authors assigned the locality to the late Eocene. In the 1980's, the teams of the Institut de Paleontologia de Sabadell (now Institut Catala `de aff. aquatilis, Gliravus aff. priscus, Eucricetodon atavus, and Pseudoltinomys gaillardi. Based on this rodent assemblage, they assigned the locality to the early Oligocene, an age maintained in several later works [START_REF] Cuenca | Bio y cronoestratigrafı ´a con mamı ´feros en la Cuenca Terciaria del Ebro: ensayo de sı ´ntesis[END_REF][START_REF] Aguilar | Synthe `ses et tableaux de corre ´lations[END_REF][START_REF] Ko ¨hler | A finding of Oligocene primates on the European continent[END_REF][START_REF] Barbera | A complete terrestrial Oligocene magnetobiostratigraphy from the Ebro Basin Spain[END_REF][START_REF] Hooker | Discussion on the Eocene-Oligocene boundary in the UK[END_REF][START_REF] Hooker | The ''Grande Coupure'' in the Hampshire Basin UK: taxonomy and stratigraphy of the mammals on either side of this major Paleogene faunal turnover[END_REF][START_REF] Minwer-Barakat | A New Species of Pseudoloris (Omomyidae, Primates) from the Middle Eocene of Sant Jaume de Frontanya (Eastern Pyrenees Spain)[END_REF], 2012, 2013[START_REF] Minwer-Barakat | New material of Pseudoloris parvulus (Microchoerinae, Omomyidae, Primates) from the Late Eocene of Sossı ´s (northeastern Spain) and its implications for the evolution of Pseudoloris[END_REF][START_REF] Costa | The age of the ''Grande Coupure'' mammal turnover: new constraints from the Eocene-Oligocene record of the Eastern Ebro Basin (NE Spain)[END_REF][START_REF] Marigo | The primate fossil record in the Iberian Peninsula[END_REF][START_REF] Sanjuan | Calibration of Eocene-Oligocene charophyte biozones in the Eastern Ebro Basin (Catalonia Spain)[END_REF]. Even more, Agustı ét al. (1987) proposed a biozone based on the rodent assemblage from Santpedor-2. They defined the Theridomys aff. aquatilis zone, which would represent the earliest Oligocene mammal level (MP21), thereby being correlated with the locality of Hoogbutsel in Belgium. However, neither [START_REF] Agustı | Biostratigraphical characteristics of the Oligocene sequences of North-Eastern Spain (Ebro and Campins Basins)[END_REF] nor [START_REF] Arbiol | Sobre la edad oligoce ´nica inferior del yacimiento de Santpedor (Cuenca del Ebro, provincia de Barcelona)[END_REF] provided descriptions, measurements or illustrations of the rodent material from Santpedor-2, and their taxonomic ascriptions remained non verifiable as a result.

The relevance of this locality increased when Ko ¨hler and Moya `-Sola `(1999) reported the presence of scarce primate remains in Santpedor-2 and in the nearby and somewhat younger locality of Fonollosa-13. The fossil primates from Santpedor-2 consisted of two fragments of lower molars, whereas those from Fonollosa-13 documented an upper tooth row, the dental morphology of which allowed to undoubtedly assign them to the genus Pseudoloris. Interestingly, primates were so far supposed to have disappeared from Europe in relation to the drastic cooling recorded at the Eocene-Oligocene transition and the subsequent faunal turnover that dramatically affected continental vertebrate fauna, known as the ''Grande Coupure''. The finding of Pseudoloris (P. godinoti) from Santpedor-2 and Fonollosa-13 represented the unique record of Oligocene primates in the continent, a presence that was shortly after confirmed by the discovery of a single lower molar of the genus Microchoerus in the early Oligocene locality of Aguato ´n, also in Spain [START_REF] Pela ´ez-Campomanes | Mammalian faunas from the Paleogene of the Sierra Palomera (Teruel, Spain)[END_REF]. However, the age proposed for Santpedor was based only on the faunal list [START_REF] Arbiol | Sobre la edad oligoce ´nica inferior del yacimiento de Santpedor (Cuenca del Ebro, provincia de Barcelona)[END_REF]. C. Simplified column of the Santpedor section indicating the stratigraphic position of the sampled level (modified from [START_REF] Costa | The age of the ''Grande Coupure'' mammal turnover: new constraints from the Eocene-Oligocene record of the Eastern Ebro Basin (NE Spain)[END_REF].

provided by [START_REF] Agustı | Biostratigraphical characteristics of the Oligocene sequences of North-Eastern Spain (Ebro and Campins Basins)[END_REF] and [START_REF] Arbiol | Sobre la edad oligoce ´nica inferior del yacimiento de Santpedor (Cuenca del Ebro, provincia de Barcelona)[END_REF] and, despite the fact that the interest of the locality lies precisely on its Oligocene age, a detailed study of its rodents was never developed.

In this paper we present a thorough study of the rodents from Santpedor-2. The rodent assemblage from this site includes eight different forms of theridomyids, eomyids, cricetids and glirids, and is therefore more diverse than previously thought. The detailed study of these rodents supports the early Oligocene age of the locality.

Geological setting

The Ebro Basin is the southern foreland basin of the Pyrenean Range, which was formed in response to convergence and collision between the Iberian and European plates from the Late Cretaceous to the Miocene [START_REF] Puigdefa `bregas | Thrusting and Foreland Basin Evolution in the Southern Pyrenees[END_REF][START_REF] Busquets | New advances in Eocene Biostratigraphy (IGCP Project No. 393): a contribution from the Southeastern Pyrenean Foreland Basin[END_REF]. This basin is delimited by three mountain ranges formed during the Eocene-Oligocene compression: the Pyrenees to the North, the Iberian Range to the Southwest, and the Catalan Coastal Ranges to the Southeast. The marine connection of the Ebro Basin was interrupted during the Priabonian (the marine-continental transition has been precisely dated at 36.0 Ma by [START_REF] Costa | Closing and continentalization of the South Pyrenean foreland Basin (NE Spain): magnetochronological constraints[END_REF]. From the late Eocene to the late Miocene, the Ebro Basin displayed a closed drainage system, and its depositional framework consisted of distributive alluvial systems fed from the tectonically active surrounding ranges, laterally changing to shallow lacustrine zones [START_REF] Agustı | Biostratigraphical characteristics of the Oligocene sequences of North-Eastern Spain (Ebro and Campins Basins)[END_REF][START_REF] Barbera | A complete terrestrial Oligocene magnetobiostratigraphy from the Ebro Basin Spain[END_REF][START_REF] Sa ´ez | Variable style of transition between Palaeogene fluvial fan and lacustrine systems, southern Pyrenean foreland, NE Spain[END_REF]. The basin infill (up to 2000 m thick) consists therefore of alluvial, fluvial and lacustrine deposits, in which several vertebrate sites have been identified [START_REF] Agustı | Biostratigraphical characteristics of the Oligocene sequences of North-Eastern Spain (Ebro and Campins Basins)[END_REF][START_REF] Anado ´n | Gisements a verte ´bre ´s du pale ´oge `ne de la zone orientale du bassin de l'Ebre et leur apport a la stratigraphie[END_REF][START_REF] Cuenca | Bio y cronoestratigrafı ´a con mamı ´feros en la Cuenca Terciaria del Ebro: ensayo de sı ´ntesis[END_REF]. Nevertheless, mammal sites corresponding to the earliest phase of endorheic sedimentation are relatively scarce: the only two localities assigned to the late Eocene (Priabonian, MP19/20) are Sant Cugat de Gavadons [START_REF] Crusafont-Pairo | Sur quelques prosimiens de l'E ´oce `ne de la Zone Pre ´axiale Pyre ´naique et un essai provisoire de reclassification[END_REF][START_REF] Hooker | Refined correlation of the UK Late Eocene-Early Oligocene Solent Group and timing of its climate history[END_REF][START_REF] Minwer-Barakat | Redescription and designation of a neotype for Pseudoloris reguanti Crusafont-Pairo ´, 1967, an Eocene primate from the Iberian Peninsula[END_REF], 2016) and Rocafort de Queralt [START_REF] Anado ´n | Gisements a verte ´bre ´s du pale ´oge `ne de la zone orientale du bassin de l'Ebre et leur apport a la stratigraphie[END_REF], while the first mammal site that, according to the published data, postdates the ''Grande Coupure'' is Santpedor [START_REF] Ko ¨hler | A finding of Oligocene primates on the European continent[END_REF][START_REF] Barbera | A complete terrestrial Oligocene magnetobiostratigraphy from the Ebro Basin Spain[END_REF][START_REF] Costa | The age of the ''Grande Coupure'' mammal turnover: new constraints from the Eocene-Oligocene record of the Eastern Ebro Basin (NE Spain)[END_REF].

Material and methods

The described rodent specimens from Santpedor-2 consist of 33 isolated teeth, housed in the collections of the Institut Catala `de Paleontologia Miquel Crusafont (Sabadell, Barcelona, Spain). The nomenclature used in the descriptions (Fig. 2) follows Vianey-Liaud and Marivaux (2017) for theridomorphs, [START_REF] Dienemann | Die gattungen Eucricetodon und Pseudocricetodon (Rodentia, Mammalia) aus dem Oligoza ¨n Su ¨ddeustchlands. Bayerische Akadenie der Wissenschaften[END_REF] for cricetids, [START_REF] Vianey-Liaud | La radiation des Gliridae (Rodentia) a `l'Eoce `ne supe ´rieur en Europe Occidentale, et sa descendance Oligoce `ne[END_REF] for glirids, and [START_REF] Maridet | New data about the diversity of Early Oligocene eomyids (Mammalia Rodentia) in western Europe[END_REF] for eomyids. Measurements (Table 1) were taken using an optic calliper ''Nikon measuroscope 10'' connected to a monitor ''Nikon SC112''; measured parameters include the mesiodistal maximum length and the maximum width perpendicular to the length, taken in keeping the wear surface horizontal. Micrographs were obtained using the Environmental Scanning Electron Microscope (ESEM) at the Universitat de Barcelona.

Systematic paleontology

Class Mammalia Linnaeus, 1758Order Rodentia Bowdich, 1821Family Theridomyidae Alston, 1876Subfamily Oltinomyinae Hartenberger, 1971Genus Ectropomys Bosma et Schmidt-Kittler, 1972 Ectropomys exiguus [START_REF] Bosma | Ectropomys exiguus n. gen., n. sp., member of the Oltinomyinae n. subfam. (Theridomyidae Rodentia), from Paleogene deposits of the Isle of Wight (England) and southem Germany[END_REF] Fig. 3(A-C) Material: DP4: IPS 105269; p4: IPS 105286, m2: IPS 105287. Description: Upper teeth. The unique upper tooth is a DP4 (Fig. 3(A)), the lingual side of its crown being relatively high and narrow with respect to the buccal one; it is not anteriorly reduced. The postprotocrista makes the high endoloph, which does not join the hypocone, and then the syncline II communicates with the shallow and narrow sinus. The high anteroloph joins the paracone to the protocone, closing the syncline I (anteroflexus) buccally. The continuous protoloph joins the apex of the protocone. The mesoflexus (syncline II + syncline I) is closed buccally by the mesostyle and the extensions of the postparacrista mesially and of the premetacrista distally. The mesoloph is lower than the other lophs and breaks at mid-length. Its buccal part weakly attaches its lingual part, which fuses with the thicker lingual metaloph. The mesoloph separates the long buccolingually syncline II and the shorter syncline III. The buccal metaloph is transverse but slightly mesially oriented, connected to the lingual metaloph, and separated from the long posteroloph. As a result, the syncline IV is more developed than on the known populations of E. exiguus or E. monacensis. This syncline IV could be subdivided into a much worn DP4, since a very low bead is visible at its bottom both on the distal flank of the buccal metaloph and on the lingual flank of the posteroloph.

Lower teeth. The p4 from Santpedor-2 (Fig. 3(B)) is more worn than that of Rocafort de Queralt, the front part of the latter being incomplete. It shows a break from the entoconid to the sinusid. On this p4, the protoconid and the larger metaconid are linked by a complete transverse metalophulid I. The latter is notched mesially by a short antesinusid, limited mesially by a low anterolophid. A high postmetacristid joins the mesostylid. The mesostylid is connected to a mesolophid that reaches the mesoconid swelling at the distal end of a strong postprotocristid. The synclinid II is divided by a short mesiodistal cristid, extending from the mesolophid to the metaconid. The synclinid III is closed lingually. The short distal ectolophid is separated by a notch from the mesoconid swelling. The entolophid and posterolophid are high and complete, between the strong entoconid and hypoconid. The morphology of m2 (Fig. 3(C)) is reminiscent to that of the tooth from San Cugat de Gavadons, in which the small anteroconid is framed by the anterolophid and anterocingulid, which delimit narrow anteroflexid and antesinusid. The sinusid is slightly deeper transversely, and the mesolophid is complete, reaching the mesostylid-postmetacristid junction.

Remarks: The teeth are larger than those of Ectropomys exiguus from the Isle of Wight, Rocafort de Queralt and San Cugat de Gavadons, but they are clearly smaller than Oltinomys platyceps [START_REF] Bosma | Ectropomys exiguus n. gen., n. sp., member of the Oltinomyinae n. subfam. (Theridomyidae Rodentia), from Paleogene deposits of the Isle of Wight (England) and southem Germany[END_REF][START_REF] Vianey-Liaud | Pairomys et Ectropomys : la fin d'une ambiguı ¨te ´; mise au point sur les Oltinomyinae et les Remyinae (Rodentia, Theridomyinae)[END_REF]. The general features of these teeth are compatible with the species E. exiguus, whereas the few differences are more derived than in the material from the above mentioned localities (larger size, much more marked lophodonty, higher crown, deeper sinusid). These characteristics argue for a form younger than the late Eocene populations of E. exiguus. anterolophid; anlpd: anterolophulid; antsd: anterosinusid; buald: buccal anterolophid; eclpd: ectolophid; ecmslpd: ectomesolophid; ecstd: ectostylid; encd: entoconid; enlpd: entolophid; ensd: entosinusid; hyd: hypoconid; hyld: hypoconulid; lc: longitudinal crest; linald: lingual anterolophid; mecd: metaconid; melpd: metalophid; melpd I: metalophid I; melpd II: metalophulid II; mescd: mesoconid; mestd: mesostylid; metstd: metastylid; mslpd: mesolophid; papcrd: posterior arm of protoconid; prcd: protoconid; prhycd: prehypocristid; prsd: protosinusid; psthyd: posterior arm of hypoconid; pstlpd: posterolophid; pstmecd: postmetacristid; pstprcd: postprotocristid; sd: sinusid; syd I: synclinid I;. syd II: synclinid II; syd III: synclinid III; syd IV: synclinid IV. Abbreviations for upper molars: ac: anterocone; alp: anteroloph; alpl: anterolophule; asin: anterosinus; ast: anterostyle; bucalp: buccal anteroloph; bucg: buccal cingulum; endlp: endoloph; ensin: entosinus; enst: entostyle; hy: hypocone; linalp: lingual anteroloph; lincg: lingual cingulum; me: metacone; mesc: mesocone; mesin: mesosinus; mesl: mesoloph; mest: mesostyle; metcl: metaconule; metlp: metaloph; metlpl: metalophule; metst: metastyle; mu: mure; pa: paracone; pacl: paraconule; past: parastyle; prc: protocone; prhyc: prehypocrista; prl: protoloph; prlp1: protolophule 1; prlp2: protolophule 2; prmec: premetacrista; prprc: preprotocrista; prsin: protosinus; prst: protostyle; psc: posterocone; pspac: postparacrista; pstl: posteroloph; pstsin: posterosinus; sy I: syncline I;. sy II: syncline II; sy III: syncline III; sy IV: syncline IV.

Subfamily

P

The protocone is not strongly pinched, and rather appears only slightly stretched obliquely (mesiobuccal to distolingual), giving the same orientation to the sinus. The syncline I and syncline II are equally long and shorter than syncline III. The enamel of the distal flank of the protocone is only a little thinner than on the mesial flank; the enamel of the hypocone and of its arms is particularly thick, and as thick as the distal border of the crown.

Lower molar. Only one m3 can be referred to this species (Fig. 3

(F))

. There is no synclinid I; the metalophid I is mesial and the synclinid II is not compressed. The postmetacristid is high and joins the mesostylid at the lingual extremity of the transverse mesolophid. The postprotocristid is not markedly oblique and it joins the (pseudo)mesoconid at the mesial extremity of the short and slightly oblique buccomesial to distolingual short ectolophid. The synclinid II is nearly as long as the synclinid IV, but wider. Synclinid III and synclinid IV are equal in width.

Remarks: The characters of the upper molars described above are found on the M2 of the type population from Hoogbutsel (Belgium, MP21;[START_REF] Vianey-Liaud | L'e ´volution du genre Theridomys a `l'Oligoce `ne moyen. Inte ´re ˆt biostratigraphique[END_REF][START_REF] Vianey-Liaud | Paralle ´lisme chez les Theridomyinae (rodentia) de l'Oligoce `ne : e ´tude de deux nouvelles espe `ces des genres Theridomys et Blainvillimys[END_REF]. These teeth differ from those of Blainvillimys langei in their more regular enamel thickness, their lower crown, less pinched protocone and less oblique sinus. The features of the m3 described above are very similar to those of the m3 of P. margaritae lacking synclinid I from Hoogbutsel [START_REF] Vianey-Liaud | L'e ´volution du genre Theridomys a `l'Oligoce `ne moyen. Inte ´re ˆt biostratigraphique[END_REF]. The m3 from Santpedor is lower crowned, with a more regular enamel thickness and less oblique posprotocristid than the m3 of B. langei. The teeth from Santpedor differ from Theridomys aquatilis in their smaller size.

?Issiodoromyinae Tullberg, 1899 ?Issiodoromyinae nov. gen., nov. sp. Description: Upper teeth. The crowns of the P4 and M1-2, and probably of the worn M3, show a constriction of the outline at the levels of the termination of the anterior arm of the protocone (at the junction with the anteroloph) and of the posterior arm of the hypocone (at its junction with the posteroloph). These shrinkings, well distinct on the less worn teeth only, are never observed on the M1-2 of Paratheridomys margaritae, Theridomys aquatilis, and Blainvillimys langei, as well as on M1-2 of any species of Pseudoltinomys. On the P4 and M1-2, the paracone and metacone are markedly bulged and the protoloph and metaloph are thick. The mesoloph is thinner and lower than these lophs. On P4 (Fig. 3(G)), the anteroloph is short but the anterosyncline (syncline I) is longer than in the late Eocene and Oligocene species of Pseudoltinomys. The posteroloph is short, and the posterosyncline is particularly reduced, even on the weakly worn M1-2 IPS 105275 (Fig. 3(H)). The sinus is shorter buccolingually than the opposite syncline II, or has similar length in the most worn teeth. Contra what is seen on Pseudoltinomys, the sinus and the syncline II are not communicating: they are separated by a very short mure as seen on the weakly worn M1-2 (Fig. 3(H)).

Lower teeth. The dp4 (Fig. 3(M)) is slightly shorter than p4 (Fig. 3(N)), and the latter is similar in length then that of the m2 and m3. On dp4, the weak protoconid and the following postprotocristid are lowered by wear, and then the metaconid and entoconid remain higher. There is a very low mesiodistal ridge descending from the metaconid to the middle of the mesolophid. The latter is complete, and joins the postmetacristid at the mesostylid level. The lingual opening of the synclinid III is narrow. The sinusid is buccolingually short. The long and curved posterolophid is thickened at the hypoconulid level, lingually to the oblique posthypocristid. On the unworn p4, the metaconid connects to the protoconid by a high mesial metalophid. There is no antesinusid. The mesolophid is slightly lower than the metaand ento-lophids. The mesolophid is broken before joining the mesostylid level by a short and narrow notch. The synclinid II displays a short and weak mesiodistal ridge. The ectolophid is short, showing a very shallow and narrow break in its middle. The synclinid III is lingually open. The sinusid is buccolingually shorter than the synclinid III. The entolophid is as high as the other lophids, and the posterolophid is long, without distinct swelling at the hypoconulid level.

The m2 (Fig. 3(O)) is worn and the enamel shows a similar thickness on the mesial, buccal and distal surfaces of the crown. There is an indication of the presence of an antesinusid as an angle on the mesial face of the metalophulid I. The synclinid II is divided by a mesiodistal ridge. The sinusid is oblique and directed to the buccal end of the synclinid IV. The synclinid II is closed lingually, the synclinid III open, and the synclinid IV shallowly and narrowly open. The m3 IPS 105283 (Fig. 3(P)) displays the same features as those of the m2, and, as it is more worn, the synclinid III and synclinid IV are weakly closed lingually. A very narrow break is observed between the oblique sinusid and the synclinid IV. As for the m2, an angle of enamel on the metalophulid I indicates the presence of a possible antesinusid. It is well marked on the m3 IPS 105285 (Fig. 3(Q)).

Remarks: This material documents the best represented species of theridomyid from Santpedor-2. It was previously referred to Pseudoltinomys gaillardi (see Section 1). Nevertheless, these teeth show strong differences with this species, and moreover, with the genus Pseudoltinomys, like the absence of break of the longitudinal crest on upper teeth, the reduced antesinusid or the relatively larger p4. It would be necessary to get more material to define a new species, as it appears to be, and probably also a new genus. The teeth have been directly compared with several theridomyid species from the late Eocene and early Description and remarks: The size of these teeth is larger than that of Eomys antiquus, close to that of E. molassicus [START_REF] Engesser | New Eomyidae, Dipodidae and Cricetidae (Rodentia Mammalia) of the Lower Freshwater Molasse of Switzerland and Savoy[END_REF], and slightly smaller than that of E. zitteli [START_REF] Comte | Eomyidae (Rodentia) de l'Oligoce `ne d'Europe Occidentale[END_REF]). However, some features of these two teeth differ from the latter two species. The p4 of Eomys sp. (Fig. 4(A)) is less elongated, and its metaconid and protoconid are more robust than on E. zitteli. Contrary to E. antiquus, E. molassicus and E. zitteli, in the teeth from Santpedor the anteroconid/anterolophid is absent: there is only a high mesial connection between the protoconid and the metaconid, which is the metalophulid I. These cuspids are closer to each other than on E. molassicus, on which the 'trigonid' appears less short buccolingually than on the species from Santpedor. On p4 and m2, the mesolophid is moderately long as observed on E. zitteli or on the Eomys sp. from Mo ¨hren 20 [START_REF] Maridet | New data about the diversity of Early Oligocene eomyids (Mammalia Rodentia) in western Europe[END_REF]. It is longer than on the type of E. zitteli, but shorter and weaker than on E. molassicus. On the latter (type from Oensingen), it makes an angle, being firstly directed posteriorly before being transverse, whereas it is transverse since the mesoconid on Eomys sp. from Santpedor. On the fragment of m2 IPS 105262 (Fig. 4(B)), the attachment of the entolophid on the posterolophid is more buccal than on E. molassicus, and thus the posterosynclinid and the posterolophid are longer mesiodistally. Therefore, the species from Santpedor displays original features.

Family Cricetidae Rochebrune, 1883 Remarks: When studying the rare cricetid teeth from Santpedor-2, we were confronted with the absence or incompleteness of diagnosis for several Oligocene species or genera. The history of these European Oligocene cricetids, resulting from several phases of immigration and local developments in Western Europe, is complex. An exhaustive review with direct comparison of populations from numerous localities in this geographical area would be essential to achieve consistent results. It would be necessary to take into account all their features, dental (morphology and ultrastructure) and cranial.

Subfamily Eucricetodontinae Mein et Freudenthal, 1971Genus Eucricetodon Thaler, 1966 Eucricetodon atavus Misonne, 1957 Fig. 4(C-E) Material: M1: IPS 105268; M2: IPS 105266; M3: IPS 105264. Description: M1. The enamel of the outskirts of the crown is wrinkled. The buccal side is convex. There is only a single anterocone, from which a weak anterocone spur slopes and joins the long protocone spur. Lingually, the anteroloph slopes to the level of the protostyle, from which it is separated by a notch. Buccally, the cingulum is longer and stronger than the anteroloph. The protolophule joins the short distal arm of the protocone, and the metalophule reaches the short mesial arm of the hypocone. The entoloph is long and angles at the level of the mesocone. The mesoloph is short and the mesostyle is reduced to short cingular elements. Lingually, the entostyle is also reduced.

M2. The buccal anteroloph is well developed, reaching the buccal corner of the tooth. The lingual anteroloph is thinner, lower and long. The protoloph is oblique forwardly and joins the anterolophule. The posterior arm of the protocone is present. The mesoloph is short. The metalophule is directed anteriorly. A thin entostyle underlines the buccal border of the mesosinus. It differs from the M2 of cf. Heterocricetodon from Santpedor-2 in its smaller size, lophs and mesosinus less transverse and in the different wear pattern (horizontal only on the lingual half of Eucricetodon, the paracone and metacone being bulged and protruding, whereas the buccal cusps of Heterocricetodon are also flattened).

M3. This tiny tooth is too much eroded to depict its features.

Remarks: The small species of Eucricetodon, including atavus, have been referred to the genus Atavocricetodon by [START_REF] Freudenthal | The Early Oligocene rodent fauna of Olalla 4A (Teruel Spain)[END_REF], but this author did not give any differential diagnosis with respect to the genus Eucricetodon. A differential diagnosis is also missing in the redescription of the species atavus made by [START_REF] Freudenthal | Redescription of Eucricetodon atavus (Misonne, 1957) (Rodentia) from Hoogbutsel (Belgium)[END_REF]. The validity of Atavocricetodon was then discussed and rejected by other authors (e.g., [START_REF] De Bruijn | A rodent assemblage from the Eo/ Oligocene boundary interval near Su ¨ngu ¨lu ¨, Lesser Caucasus Turkey[END_REF][START_REF] Gomes Rodrigues | On the status of early Eucricetodontinae (Muroidea Rodentia) with a special focus on the Atavocricetodon vs. Eucricetodon issue: morphometrical and microstructural aspects[END_REF], who ''found no obvious reason, even pragmatical, to keep Atavocricetodon as a genus or even a subgenus''. This is the reason why we decide here to use Eucricetodon rather than Atavocricetodon. As this cricetid is poorly represented in the locality, we cannot contribute more to this discussion. Its size and morphology fit those of the genus Eucricetodon: bunodont teeth, larger than those of Pseudocricetodon and with simpler lophs and ridges; anterocone of the M1 generally simple, sometimes tending to be slightly duplicated, convex buccal contour of the M1 and posterior lobe of the M3 markedly reduced [START_REF] Vianey-Liaud | L'e ´volution du genre Theridomys a `l'Oligoce `ne moyen. Inte ´re ˆt biostratigraphique[END_REF]. The size and morphology of the M1 and M2 match those of the lower Oligocene populations of E. atavus (e.g., Hoogbutsel, Aubrelong 1, Ravet, Montalba ´n, Me `ge, Pech Crabit, Itardies; [START_REF] Vianey-Liaud | L'e ´volution du genre Theridomys a `l'Oligoce `ne moyen. Inte ´re ˆt biostratigraphique[END_REF][START_REF] Vianey-Liaud | L'anatomie des genres Eucricetodon et Pseudocricetodon (Cricetidae, Rodentia Mammalia)[END_REF][START_REF] Freudenthal | Redescription of Eucricetodon atavus (Misonne, 1957) (Rodentia) from Hoogbutsel (Belgium)[END_REF], better than to 'Atavocricetodon' aff. nanus from Valbro (Peigne ´et al., 2014: fig. 16). The more distinctive difference in size is between the lower m1 of these two species, but there is no available m1 in Santpedor-2. If considering the type population of nanus from Valdecollares (Pela ´ez-Campomanes, 1995), the size of the M1 and M2 from Santpedor-2 is clearly larger. Eucricetodon atavus is recorded from MP21 to MP23 [START_REF] Vianey-Liaud | L'e ´volution du genre Theridomys a `l'Oligoce `ne moyen. Inte ´re ˆt biostratigraphique[END_REF][START_REF] Vianey-Liaud | Diversite ´, Datation et Pale ´oenvironnement de la faune de mammife `res oligoce `ne de Cavale ´(Quercy, SW France) : contribution de l'analyse morphome ´trique des Theridomyinae (Mammalia Rodentia)[END_REF].

Subfamily Pseudocricetodontinae [START_REF] Engesser | New Eomyidae, Dipodidae and Cricetidae (Rodentia Mammalia) of the Lower Freshwater Molasse of Switzerland and Savoy[END_REF]Genus Heterocricetodon Schaub, 1925 cf. Heterocricetodon sp. Fig. 4(F-L Description: We refer the teeth from Santpedor to a small primitive species of Heterocricetodon better than to Pseudocricetodon, on the basis of their rather lophodont dental pattern and the presence of a long mesoloph on one of the M2 (Fig. 4(G)) and on the M3 (Fig. 4(I)). Even if this latter feature is variably found in Pseudocricetodon incertus, it is observed in the material from Santpedor together with other features found in Heterocricetodon: the long lingual anteroloph of M2, the high lophs on the unworn M3; the flat wear surface of the M2 from the lingual edge of the crown to the 2/3 of the occlusal surface, with moderately prominent buccal upper main cusps; the long postmetacristid on m1 (Fig. 4(J)), the flat wear surface of the m3, with its long posterior arm of the protoconid, and the reduced width of its posterior lobe (Fig. 4(L)). The anteroconid and metaconid of m1 are strongly worn, and therefore the lophids are difficult to distinguish, except the occurrence of a long postmetacristid and a short thick posterior arm of the protoconid, weakly separated from the lingual metalophulid; there are two short mesolophids, the more mesial being the longest. There is no posterior arm of the hypoconid. The fragment of m2 shows a long postmetacristid ending in a small swelling as a metastylid; its posterior arm of the protoconid is long and oblique towards the base, and ends with a thickening against the middle of the buccal flank of the metaconid; the mesolophid has same length than the posterior arm of the protoconid. Flat and worn on its buccal two-third, its lingual cuspids are not worn. The enamel of the surroundings of the crown is wrinkled.

Remarks: The genus Heterocricetodon was so far unknown in the lower Oligocene. Until now, the earliest species referred to this genus were H. landroveri [START_REF] Daams | Upper Oligocene micromammals from Pareja, Loranca Basin, prov. of Guadalajara, Spain[END_REF] from Pareja (MP 26, early Chattian; [START_REF] Aguilar | Synthe `ses et tableaux de corre ´lations[END_REF] and H. hausi from Bumbach (MP24-25?, earliest Chattian; [START_REF] Engesser | New Eomyidae, Dipodidae and Cricetidae (Rodentia Mammalia) of the Lower Freshwater Molasse of Switzerland and Savoy[END_REF].

As noticed by Engesser (1987: p. 990) for the species hausi, it is difficult to refer isolated teeth to Heterocricetodon, and especially to differentiate them from the genus Pseudocricetodon until the discovery of more abundant material. It is even much more difficult to carry out such a distinction in an older Rupelian locality, where very few teeth are documented. Daams et al. (1989: pp. 47-48) propose to include the species incertus within the genus Heterocricetodon even if cranial features ''prove the value of this genus''.

The teeth from Santpedor are larger than those of 'Pseudo-Pseudocricetodon' incertus from some lower Chattian localities (Vianey-Liaud et al., 2014: table 5, p. 594) but smaller than H. hausi from Bumbach, and similar in size to H. landroveri (m1 smaller than the smallest m1 from Pareja, m3 like the largest). The species from Santpedor-2 differs from H. landroveri in the more reduced posterior lobe of m3 and the mesoconid more distinct on m1. Its morphology is closer to H. hausi, but the sinus seems less deep buccolingually. As the sample is scarce, we cannot define the morphological and size variability, and prefer to keep this Heterocricetodon sp. in open nomenclature. This species thus represents the earliest record of the genus documented so far.

Genus Pseudocricetodon Thaler, 1969 cf. Pseudocricetodon sp. Fig. 4(M) Material: An incomplete m2: IPS 105274. Description and remarks: This fragment of tooth differs from that ascribed to Heterocricetodon in its slightly smaller size, the enamel smooth, and the more bulbous and rounded main cuspids, as well as in the different arrangement of the talonid basin area. The postmetacristid is shorter and the metastylid less distinct, by contrast there is a mesostylid blocking the opening of the entosinusid. There is no mesolophid distinct from the posterior arm of the protoconid. The latter is widely oblique and is connected to the mesial extremity of the ectolophid: at the junction it curves buccolingually. If this interpretation is right, the mesolophid is absent, because there is no distinct mesoconid, and no ectomesolophid, but only a faint vertical swelling on the buccal flank of the short ectolophid.

Family Gliridae Thomas, 1897 Subfamily Glamyiinae [START_REF] Vianey-Liaud | La radiation des Gliridae (Rodentia) a `l'Eoce `ne supe ´rieur en Europe Occidentale, et sa descendance Oligoce `ne[END_REF]Genus Glamys Vianey-Liaud, 1989 Glamys devoogdi [START_REF] Bosma | Eocene and Oligocene Gliridae (Rodentia, Mammalia) from the Isle of Wight, England. Part l. The Gliravus priscus -Gliravus fordi lineage[END_REF] Fig. 4(N-P) Material: p4: IPS 105272; m2: IPS 105270; m2 incomplete: IPS 105271.

Description: The wear surface of p4 (Fig. 4(P)) is nearly flat and the crown relatively low. The metaconid and protoconid are linked by two short and thick lophids: an anterolophid and a metalophulid II. The anterior arm of the hypoconid is faintly developed; therefore, the sinusid is confluent with the posterosynclinid. The posterolophid is long and strong, ending against the bulged entoconid. On m2, the anterolophid is long, separates from the protoconid and joins the metaconid; the postmetacristid is present and slopes gently to the mesoflexid lingual opening; the metalophid is transverse and thick; the mesoconid is present, and the mesolophid is long, connected to the entoconid by a low entolophid. One extra-ridge (Fig. 4(N)) and granules (Fig. 4(O)) are present in the mesoflexid and posteroflexid.

Remarks: The size of the p4 and of the complete m2 fits the size range of Glamys devoogdi from Hamstead Beds and Hoogbutsel [START_REF] Bosma | Eocene and Oligocene Gliridae (Rodentia, Mammalia) from the Isle of Wight, England. Part l. The Gliravus priscus -Gliravus fordi lineage[END_REF][START_REF] Vianey-Liaud | La radiation des Gliridae (Rodentia) a `l'Eoce `ne supe ´rieur en Europe Occidentale, et sa descendance Oligoce `ne[END_REF]) and the morphology is also close to that of this species. On the p4 figured in Bosma and de Bruijn (1979: pl. 2, fig. 5), the mesial connection (anterolophid) is low and does not reach the top of the metaconid and protoconid, whereas this connection is higher and complete on the p4 from Santpedor. The paths of the other extra-ridges and granules, into the sinusid and the talonid, occupy similar positions but their connections vary. The species is recorded from the late Priabonian (MP19/20) to the early Rupelian (MP21).

Discussion

The Eocene-Oligocene transition was characterized by a drastic decrease in global temperatures, which involved a significant change in the faunas of both marine and terrestrial realms, known as the ''Grande Coupure'' (e.g., [START_REF] Stehlin | Remarques sur les faunules de Mammife `res des couches E ´oce `nes et Oligoce `nes du Bassin de Paris[END_REF][START_REF] Prothero | The late Eocene-Oligocene extinctions[END_REF][START_REF] Zachos | Trends, rhythms and aberrations in global climate 65 Ma to present[END_REF][START_REF] Hooker | Eocene-Oligocene mammalian faunal turnover in the Hampshire Basin, UK: calibration to the global time scale and the major cooling event[END_REF][START_REF] Hooker | The ''Grande Coupure'' in the Hampshire Basin UK: taxonomy and stratigraphy of the mammals on either side of this major Paleogene faunal turnover[END_REF][START_REF] Hren | Terrestrial cooling in Northern Europe during the Eocene-Oligocene transition[END_REF]. In the European continent this faunal turnover was marked, among other changes, by the arrival of some immigrant groups, such as cricetid rodents [START_REF] Vianey-Liaud | Evolution des rongeurs a `l'Oligoce `ne en Europe Occidentale[END_REF][START_REF] Hooker | Eocene-Oligocene mammalian faunal turnover in the Hampshire Basin, UK: calibration to the global time scale and the major cooling event[END_REF][START_REF] Maridet | A new cricetid rodent from the early Oligocene of Yunnan, China, and its evolutionary implications for early Eurasian cricetids[END_REF][START_REF] Freudenthal | A review of Oligocene and early Miocene European Cricetidae (Mammalia)[END_REF], and by the nearly complete disappearance of primates. Until the end of the XX th century, it was generally accepted that microchoerine primates became extinct at the Eocene-Oligocene transition, but the identification of scarce remains of Pseudoloris in Fonollosa and Santpedor (Ko ¨hler and Moya `-Sola `, 1999) and of Microchoerus in Aguato ´n [START_REF] Pela ´ez-Campomanes | Mammalian faunas from the Paleogene of the Sierra Palomera (Teruel, Spain)[END_REF] testified to the persistence of this group in the Iberian Peninsula during the earliest Oligocene. For this reason, the detailed study of the fossil sites documenting this time window in Spain is particularly relevant, in order to ensure precise dating of such localities, and to analyze the mammal assemblages that coexisted with these last surviving Paleogene primates in Europe.

The reference faunas of the latest Priabonian (MP20, St-Capraise-d'Eymet) and earliest Rupelian (MP21, Soumailles) MPlevels are poor, and have been chosen mainly because they have provided both small and large mammals. However, the evolutionary stages of the documented lineages are not very discriminating, either for the MP20 or for the MP21 [START_REF] Aguilar | Synthe `ses et tableaux de corre ´lations[END_REF]. The characterization of the MP20, which is mainly based on the terminal stage of Palaeotherium curtum (sub-species frohnstettense), is not evident (e.g., [START_REF] Hooker | The ''Grande Coupure'' in the Hampshire Basin UK: taxonomy and stratigraphy of the mammals on either side of this major Paleogene faunal turnover[END_REF]. For the rodents, 'Theridomys' bonduelli could be a good marker, but this species is only known from the Paris and Hampshire basins. This has a definite impact on the precise definition and correlations of faunas around the ''Grande Coupure''.

The ''Grande Coupure'' would mark the base of the Rupelian, currently dated at 33.9 Ma [START_REF] Vandenberghe | The Paleogene Period[END_REF]. Several correlation attempts with respect to mammalian lineages have been made, using magnetostratigraphy and geochemistry (Solent Group, Whitecliff Bay, Isle of Wight, England; [START_REF] Gale | Correlation of Eocene-Oligocene marine and continental records: orbital cyclicity, magnetostratigraphy and sequence stratigraphy of the Solent Group, Isle of Wight UK[END_REF][START_REF] Hooker | Refined correlation of the UK Late Eocene-Early Oligocene Solent Group and timing of its climate history[END_REF] or only magnetostratigraphy in the case of the Santpedor section [START_REF] Barbera | A complete terrestrial Oligocene magnetobiostratigraphy from the Ebro Basin Spain[END_REF][START_REF] Costa | The age of the ''Grande Coupure'' mammal turnover: new constraints from the Eocene-Oligocene record of the Eastern Ebro Basin (NE Spain)[END_REF]. According to the latter authors, the magnetostratigraphic location of the MP21 level, possibly represented at Santpedor at the upper part of the 13r chron, would date this MP around 33.4 Ma, but this datum needs to be confirmed. This locality contains both immigrant rodents, such as the Cricetidae, as well as native species of Gliridae and Theridomyidae. The scarcity of rodent material in the MP20 and MP21 reference levels, combined with the geographical differentiation observed between theridomyoids at the beginning of the Oligocene, does not facilitate precise correlations. Nonetheless, it is easy to identify the arrival of immigrant taxa and also to distinguish the evolutionary stages of certain lineages when they have represented by sufficient teeth.

There are currently few Rupelian and earliest Chattian MP, four in number (MP21 to MP24), over a total stage duration of 6 myr. However, based on the evolutionary degrees of several Theridomyoidea lineages, at least one intermediate level MP23/24 can be inserted even if it has not yet been formally defined (Fig. 5). It would correspond to the evolutionary level of the fauna from St-Martin-de-Castillon (Vaucluse), in particular to the species Protechimys truci [START_REF] Hugueney | Theridomys truci de l'Oligoce `ne de Saint-Martin-de-Castillon (Vaucluse, France), nouvelle espe `ce du genre Theridomys (Rodentia, Mammalia) et sa relation avec la ligne ´e de Theridomys lembronicus[END_REF][START_REF] Vianey-Liaud | La radiation des Theridomyidae (Rodentia) a `l'Oligoce `ne infe ´rieur : modalite ´s et implications biochronologiques[END_REF][START_REF] Vianey-Liaud | Autopsie d'une radiation adaptative : Phyloge ´nie des Theridomorpha, rongeurs ende ´miques du Pale ´oge `ne d'Europehistoire, dynamique e ´volutive et inte ´re ˆt biochronologique[END_REF]. The standard locality of MP24 (Heimersheim) is located at the base of the Cyrenen Mergel, and correlated with the Kasseler Meeressand, which provided nannoplankton from the NP24 area (Bahlo, 1975: p. 12). Thus, the age of MP24 is close to the Rupelian-Chattian boundary (28.3 Ma;[START_REF] Vandenberghe | The Paleogene Period[END_REF] within the lowermost Chattian.

The study of the rodents from Santpedor-2 reveals a higher diversity than previously thought. Even if the number of rodent teeth is small, they are characteristic enough either to be identified as already known species, or at least compared to morphologically close species. Among them, one species is known only from late Please cite this article in press as: Vianey-Liaud, M., et al., The rodents from Santpedor-2 (Ebro Basin, NE Spain) confirm the Oligocene age of the latest primates from the Paleogene of Europe. Geobios (2019), https://doi.org/10.1016/j.geobios.2019.06.002 Eocene localities (MP18 to MP20): the Theridomyidae Ectropomys exiguus [START_REF] Vianey-Liaud | Pairomys et Ectropomys : la fin d'une ambiguı ¨te ´; mise au point sur les Oltinomyinae et les Remyinae (Rodentia, Theridomyinae)[END_REF]. The glirid Glamys devoogdi is known from late Priabonian to early Rupelian localities (MP19 to MP21; Vianey-Liaud, 1994: table 2). The other identified rodents belong exclusively to Oligocene taxa (Fig. 5). One Theridomyidae (Paratheridomys margaritae) is known from the standard locality of MP21 (Hoogbutsel, Belgium;[START_REF] Vianey-Liaud | L'e ´volution du genre Theridomys a `l'Oligoce `ne moyen. Inte ´re ˆt biostratigraphique[END_REF], which is the type locality of the species, and from La Plante 2 (Quercy, France, MP22; [START_REF] Vianey-Liaud | Paralle ´lisme chez les Theridomyinae (rodentia) de l'Oligoce `ne : e ´tude de deux nouvelles espe `ces des genres Theridomys et Blainvillimys[END_REF]. The other Theridomyidae (?Issiodoromyinae) is probably a new species, and perhaps a new genus; its features (e.g., size of premolars) better indicate Oligocene rather than late Eocene evolutionary grades. Two immigrant families of the ''Grande Coupure'' are recognized: the Eomyidae and the Cricetidae, among which one species (Eucricetodon atavus) and one genus (Pseudocricetodon) are recorded since MP21. The other genera are represented by original species, a Heterocricetodon more primitive, smaller, and plausibly older than the earliest species known until now, and an Eomys larger than the MP21 species E. antiquus, but with features different from all known Eomys species.

This rodent assemblage is most probably early Oligocene in age, with a certain diversity of well-characterized immigrant families, and Oligocene evolutionary grades of autochthonous lineages (i.e., P. margaritae and G. devoogdi). One rodent species (Ectropomys exiguus) represents an Eocene relict, as is the case of Pseudoloris godinoti, the only primate recorded in this locality. As there are only 33 teeth for eight rodent species found so far, some of them remain in open nomenclature, and therefore we cannot precise if the assemblage is close to MP21 or to MP22. Further field campaigns in this fossil site, leading to the recovery of more abundant material, will probably allow for more precise taxonomic determinations and a more accurate dating of the locality. [START_REF] Vianey-Liaud | La radiation des Gliridae (Rodentia) a `l'Eoce `ne supe ´rieur en Europe Occidentale, et sa descendance Oligoce `ne[END_REF][START_REF] Freudenthal | The Early Oligocene rodent fauna of Olalla 4A (Teruel Spain)[END_REF][START_REF] Hooker | Eocene-Oligocene mammalian faunal turnover in the Hampshire Basin, UK: calibration to the global time scale and the major cooling event[END_REF][START_REF] Vianey-Liaud | Diversite ´, Datation et Pale ´oenvironnement de la faune de mammife `res oligoce `ne de Cavale ´(Quercy, SW France) : contribution de l'analyse morphome ´trique des Theridomyinae (Mammalia Rodentia)[END_REF], [START_REF] Peigne | Valbro : un nouveau site a verte ´bre ´s de l'Oligoce `ne infe ´rieur (MP22) de France (Quercy). I -Contexte ge ´ologique ; Mammalia : Rodentia, Hyaenodontida Carnivora[END_REF]Vianey-Liaud and[START_REF] Vianey-Liaud | Autopsie d'une radiation adaptative : Phyloge ´nie des Theridomorpha, rongeurs ende ´miques du Pale ´oge `ne d'Europehistoire, dynamique e ´volutive et inte ´re ˆt biochronologique[END_REF]. Dotted lines indicate the extended ranges of some taxa after the identification of the rodents from Santpedor-2 (MP21 or MP22). The MP reference levels are correlated to the GTS scale [START_REF] Vandenberghe | The Paleogene Period[END_REF], with position of these levels discussed in Vianey-Liaud and Marivaux (2017). The Theridomorpha from Gousnat (MP18a) represent evolutionary grades more primitive than those from La De ´bruge (MP18b) (Vianey-Liaud and Marivaux, 2017).

Conclusions

The rodent fossil material from Santpedor-2 (Ebro Basin, NE Spain) was described in detail for the first time. The assemblage includes eight different rodent taxa: Ectropomys exiguus, Paratheridomys margaritae, ?Issiodoromyinae nov. gen., nov. sp., Eomys sp., Eucricetodon atavus, cf. Heterocricetodon sp., cf. Pseudocricetodon sp., and Glamys devoogdi; it is notably different from that reported in the preliminary faunal lists provided by [START_REF] Agustı | Biostratigraphical characteristics of the Oligocene sequences of North-Eastern Spain (Ebro and Campins Basins)[END_REF] and [START_REF] Arbiol | Sobre la edad oligoce ´nica inferior del yacimiento de Santpedor (Cuenca del Ebro, provincia de Barcelona)[END_REF]. Although the locality has yielded one species that can be considered as an Eocene relict (Ectropomys exiguus), the presence of two rodent families that arrived in Europe as part of the ''Grande Coupure'' event (Eomyidae and Cricetidae), as well as the evolutionary grade of species of Gliridae and Theridomyidae, might indicate an early Oligocene age for this assemblage. Nevertheless, the scarcity of the material prevents a precise taxonomic determination of some taxa and a conclusive assignment to one of the earliest Oligocene standard levels (MP21 or MP22). In any case, this study confirms the early Oligocene age of Santpedor-2, one of the few European localities that documents the persistence of omomyid primates after the ''Grande Coupure''.

Fig. 1 .

 1 Fig. 1. Geographic and geological setting of the Santpedor-2 fossil locality. A. Situation of the studied section in the Iberian Peninsula. B. Detailed map showing the position of Santpedor-2 (modified from Arbiol and Sa ´ez, 1988). C. Simplified column of the Santpedor section indicating the stratigraphic position of the sampled level (modified from Costa et al., 2011).

  Theridomyinae Alston, 1876 Genus Paratheridomys Vianey-Liaud et Marivaux, 2017 Paratheridomys margaritae (Vianey-Liaud, 1972) Fig. 3(D-F) Material: M2: IPS 18941; M3: IPS 105257; m3: IPS 105284. Description: Upper molars. The mesial and distal flanks of the crown are relatively flat, vertical and parallel on M2 (Fig.3(D)). This tooth is worn, and the wear reduces the posterosyncline (syncline IV) to an islet. The paracone, mesostyle and metacone are slightly bulged.

Fig. 2 .

 2 Fig. 2. Dental nomenclature employed for the description of the teeth of Theridomyidae (A, B), Cricetidae (C-F), Eomyidae (G) and Gliridae (H). A, C, G and H represent m1; B and E represent M1; D corresponds to a m2, and F to a M2. A and B are modified from Bonilla-Salomo ´n et al. (2016); G is modified from Daxner-Ho ¨ck and Ho ¨ck (2009); the rest of figures correspond to original drawings. Abbreviations for lower molars: aaprcd: anterior arm of protoconid; abcld: anterobuccal cingulid; acd: anteroconid, anld: anterolophid; anlpd: anterolophulid; antsd: anterosinusid; buald: buccal anterolophid; eclpd: ectolophid; ecmslpd: ectomesolophid; ecstd: ectostylid; encd: entoconid; enlpd: entolophid; ensd: entosinusid; hyd: hypoconid; hyld: hypoconulid; lc: longitudinal crest; linald: lingual anterolophid; mecd: metaconid; melpd: metalophid; melpd I: metalophid I; melpd II: metalophulid II; mescd: mesoconid; mestd: mesostylid; metstd: metastylid; mslpd: mesolophid; papcrd: posterior arm of protoconid; prcd: protoconid; prhycd: prehypocristid; prsd: protosinusid; psthyd: posterior arm of hypoconid; pstlpd: posterolophid; pstmecd: postmetacristid; pstprcd: postprotocristid; sd: sinusid; syd I: synclinid I;. syd II: synclinid II; syd III: synclinid III; syd IV: synclinid IV. Abbreviations for upper molars: ac: anterocone; alp: anteroloph; alpl: anterolophule; asin: anterosinus; ast: anterostyle; bucalp: buccal anteroloph; bucg: buccal cingulum; endlp: endoloph; ensin: entosinus; enst: entostyle; hy: hypocone; linalp: lingual anteroloph; lincg: lingual cingulum; me: metacone; mesc: mesocone; mesin: mesosinus; mesl: mesoloph; mest: mesostyle; metcl: metaconule; metlp: metaloph; metlpl: metalophule; metst: metastyle; mu: mure; pa: paracone; pacl: paraconule; past: parastyle; prc: protocone; prhyc: prehypocrista; prl: protoloph; prlp1: protolophule 1; prlp2: protolophule 2; prmec: premetacrista; prprc: preprotocrista; prsin: protosinus; prst: protostyle; psc: posterocone; pspac: postparacrista; pstl: posteroloph; pstsin: posterosinus; sy I: syncline I;. sy II: syncline II; sy III: syncline III; sy IV: syncline IV.

  Fig. 3(G-Q) Material: P4: IPS 105276; M1-2: IPS 105275, 105277, 105278; M3: IPS 105279, 105280; dp4: IPS 18940; p4: IPS 105281; m2: IPS 105282; m3: IPS 105283, 105285.

Fig. 3 .

 3 Fig. 3. ESEM pictures of the teeth of Theridomyidae from Santpedor-2. A-C. Ectropomys exiguus Bosma et Schmidt-Kittler, 1972. A: right DP4 (IPS 105269); B: right p4 (IPS 105286); C: left m2 (IPS 105287). D-F. Paratheridomys margaritae (Vianey-Liaud, 1972). D: right M2 (IPS 18941); E: right M3 (IPS 105257); F: right m3 (IPS 105284). G-Q. ?Issiodoromyinae nov. gen., nov. sp. G: left P4 (IPS 105276); H: right M1-2 (IPS 105275); I: right M1-2 (IPS 105277); J: right M1-2 (IPS 105278); K: left M3 (IPS 105279); L: left M3 (IPS 105280); M: right dp4 (IPS 18940); N: right p4 (IPS 105281); O: right m2 (IPS 105282); P: left m3 (IPS 105283); Q: right m3 (IPS 105285). Scale bar: 3 mm.

Fig. 4 .

 4 Fig. 4. ESEM pictures of the teeth of Eomyidae, Cricetidae and Gliridae from Santpedor-2. A, B. Eomys sp. A: right p4 (IPS 105265); B: right m1-2 (IPS 105262). C-E. Eucricetodon atavus Misonne, 1957. C: right M1 (IPS 105268); D: right M2 (IPS 105266); E: right M3 (IPS 105264). F-L. cf. Heterocricetodon sp. F: left M1 (IPS 105273); G: left M2 (IPS 105258); H: right M2 (IPS 105260); I: left M3 (IPS 105267); J: right m1 (IPS 105259); K: left m2 (IPS 105261); L: right m3 (IPS 105263). M. cf. Pseudocricetodon sp., left m2 (IPS 105274). N-P. Glamys devoogdi (Bosma et de Bruijn, 1979). N: left m2 (IPS 105270); O: left m2 (IPS 105271); P: right p4 (IPS 105272). Scale bar: 1 mm.

  ) Material: M1 (distal part): IPS 105273; M2: IPS 105258, 105260; M3: IPS 105267; m1: IPS 105259; m2: IPS 105261; m3: IPS 105263.

Fig. 5 .

 5 Fig.5. Known stratigraphical extension of the rodents identified in Santpedor-2, after Vianey-Liaud (1994),[START_REF] Freudenthal | The Early Oligocene rodent fauna of Olalla 4A (Teruel Spain)[END_REF],Hooker et al. (2004), Vianey-Liaud and[START_REF] Vianey-Liaud | Diversite ´, Datation et Pale ´oenvironnement de la faune de mammife `res oligoce `ne de Cavale ´(Quercy, SW France) : contribution de l'analyse morphome ´trique des Theridomyinae (Mammalia Rodentia)[END_REF],[START_REF] Peigne | Valbro : un nouveau site a verte ´bre ´s de l'Oligoce `ne infe ´rieur (MP22) de France (Quercy). I -Contexte ge ´ologique ; Mammalia : Rodentia, Hyaenodontida Carnivora[END_REF] Vianey-Liaud and[START_REF] Vianey-Liaud | Autopsie d'une radiation adaptative : Phyloge ´nie des Theridomorpha, rongeurs ende ´miques du Pale ´oge `ne d'Europehistoire, dynamique e ´volutive et inte ´re ˆt biochronologique[END_REF]. Dotted lines indicate the extended ranges of some taxa after the identification of the rodents from Santpedor-2 (MP21 or MP22). The MP reference levels are correlated to the GTS scale[START_REF] Vandenberghe | The Paleogene Period[END_REF], with position of these levels discussed in Vianey-Liaud and Marivaux (2017). The Theridomorpha from Gousnat (MP18a) represent evolutionary grades more primitive than those from La De ´bruge (MP18b) (Vianey-Liaud and Marivaux, 2017).

Table 1

 1 Measurements (in mm) of the rodent teeth from Santpedor-2.

	Taxon	Catalogue number Tooth Length Width
	Ectropomys exiguus	IPS 105269	DP4	1.66	1.86
		IPS 105287	m2	1.91	1.80
		IPS 105286	p4	1.75	1.44
	Paratheridomys margaritae	IPS 18941	M2	1.94	2.63
		IPS 105257	M3	1.96	2.23
		IPS 105284	m3	-	1.72
	Issiodoromyinae nov. gen., nov sp. IPS 105276	P4	2.23	2.69
		IPS 105275	M1-2 2.16	2.72
		IPS 105277	M1-2 2.03	2.75
		IPS 105278	M1-2 -	-
		IPS 105279	M3	2.20	2.69
		IPS 105280	M3	2.03	2.42
		IPS 18940	dp4	2.14	1.41
		IPS 105281	p4	2.35	1.97
		IPS 105282	m2	2.39	2.07
		IPS 105283	m3	2.30	1.93
		IPS 105285	m3	-	-
	Eomys sp.	IPS 105265	p4	1.08	1.03
		IPS 105262	m1-2 -	1.09
	Eucricetodon atavus	IPS 105268	M1	1.92	1.23
		IPS 105266	M2	1.32	-
		IPS 105264	M3	-	-
	cf. Heterocricetodon sp.	IPS 105273	M1	-	-
		IPS 105258	M2	1.50	1.48
		IPS 105260	M2	1.51	-
		IPS 105267	M3	1.18	1.24
		IPS 105259	m1	1.72	1.23
		IPS 105261	m2	-	-
		IPS 105263	m3	1.58	1.33
	cf. Pseudocricetodon sp.	IPS 105274	m2	-	1.19
	Glamys devoogdi	IPS 105272	p4	0.93	0.86
		IPS 105270	m2	1.01	1.11
		IPS 105271	m2	-	-

Acknowledgements

We are truly grateful to the editor G. Escarguel and three anonymous reviewers for their constructive comments that greatly improved the manuscript. This work has been supported by the CERCA Programme/Generalitat de Catalunya, the project CGL2017-82654-P, MINECO/FEDER, UE (Ministerio de Economı´a, Industria y Competitividad, Spanish Government and European Union), and the research groups 2017 SGR 86 GRC (Generalitat de Catalunya) and RNM190 (Junta de Andalucı´a). R. M.-B. is funded by the Plan Propio de Investigacio ´n, Vicerrectorado de Investigacio ´n y Transferencia, Universidad de Granada. This work has made use of the collections of the Institut des Sciences de l'E ´volution de Montpellier (ISE-M, UMR 5554). Publ. ISEM n o 2019-125.


P Long

W Khalil

S Caro

Control of a Lower Mobility Dual Arm System

Keywords: Co-operative control, Robot control, Robotic manipulators

This paper studies the kinematic modeling and control of two cooperative manipulators. The cooperative system is composed of the two arms of the humanoid Nao robot of Aldebaran. The serial structure of each arm has five degrees of freedom, in the closed formulation, when transporting a common object, it has 4-DOF. The kinematic and dynamics representing the closed chain system is studied. A control scheme based on the cooperative task space with a minimum representation of the task is implemented. Furthermore by modeling the object grasp as a passive joint, all 6-DOF of the object can be controlled.

INTRODUCTION

The capability of dual arm system when processing parts reduces the need for custom fixtures and permits the use of a simpler end effector. The system can then execute sophisticated tasks that may be difficult for a single arm system. For example, rather than using a large serial robot a cooperative system distributes a heavy load among several smaller robots. Similarly if the object is of an unwieldy, non-rigid or awkward composition, the single arm robot may struggle to manipulate it.

By using a cooperative system, both the location and the internal forces of the object can be controlled. The two principal approaches to control such a system are hybrid position/force control and impedance control. An example of a position/force scheme for cooperative systems is found in [START_REF] Uchiyama | A symmetric hybrid position/force control scheme for the coordination of two robots[END_REF]. The control variables are split into force and position controlled variables. The final control input is the sum of these variables after converting them to joint torque. Decentralized impedance control schemes, proposed by [START_REF] Bonitz | Internal force-based impedance control for cooperating manipulators[END_REF] and [START_REF] Sadati | Decentralized impedance control of nonredundant multi-manipulator systems[END_REF], control the internal forces for multi arm systems. An impedance relationship is enforced between each end effector velocity and the internal force at the grasp location. In [START_REF] Caccavale | Six-DOF impedance control of dual-arm cooperative manipulators[END_REF] impedance control schemes for cooperative manipulators are extended to manage the effect of the environmental forces on the object.

Another approach is to formulate kinematic relations that create a task space describing the multi-arm system while grasping an object. The main methods are known as Symmetric formulation [START_REF] Uchiyama | A symmetric hybrid position/force control scheme for the coordination of two robots[END_REF], and cooperative task space [START_REF] Chiacchio | Direct and inverse kinematics for coordinated motion tasks of a two-manipulator system[END_REF], [START_REF] Caccavale | Task-space regulation of cooperative manipulators[END_REF]. The cooperative task space is a control scheme used to manage both the external motion of the object and the forces experienced by the object via an inverse kinematic controller. Furthermore it is demonstrated that an undesired motion in the object grasp can be represented as an extra joint in the kinematic chain, which increases the degree of redundancy or in this case increases the mobility of the object. On the other hand, the system can be viewed as a redundantly actuated parallel manipulator. Kinematic constraint equations are derived that establish a relationship between the chosen independent and dependent joint variables [START_REF] Yeo | A single closed-loop kinematic chain approach for a hybrid control of two cooperating arms with a passive joint: an application to sawing task[END_REF], [START_REF] Cheng | Dynamics and control of redundantly actuated parallel manipulators[END_REF]. The dependent joint variables adopt values that ensure loop closure at each instant, the issue of inadmissible actuation schemes is discussed in Özkan and Özgören (2001).

Most of the preceding work has been carried out with dual arm systems, where both arms are either of planar or 6-DOF spatial structure. On the other hand the study of lower mobility cooperative manipulators has been limited. In [START_REF] Zielinski | Control of two 5 dof robots manipulating a rigid object[END_REF] admissible path planning for two 5-DOF robots is explored. In [START_REF] Bicchi | On the mobility and manipulability of general multiple limb robots[END_REF] a generalized method based on the Jacobian matrix of each arm and their constraint relations with the object, permits the calculation of the mobility, possible first order differential motions, and manipulability velocity ellipsoids of general multiple limb robots. In [START_REF] Yeo | A single closed-loop kinematic chain approach for a hybrid control of two cooperating arms with a passive joint: an application to sawing task[END_REF] the cooperation between a 5-DOF and 4-DOF robot is used in conjunction with a passive joint in order to execute a 4-DOF position/force task. By adding a passive joint to the closed chain, the mobility is increased.

The main contribution of this paper is the formulation of two control approaches for lower mobility cooperative manipulators. Firstly by using the cooperative task space in conjunction with a minimum description of task [START_REF] Dombre | Analyse des performances des robotsmanipulateurs flexibles et redondants: contribution a leur modelisation et a leur commande[END_REF], the feasible directions of the object are controlled. The liberated DOF are then used to optimize extra criterion while satisfying the closed loop kinematic constraint equations. Secondly, by modeling the object grasp of each manipulator as a passive joint, the mobility of the object can be increased to 6-DOF while ensuring the loop closure. The cooperative control theory is applied to the two arms of Aldebaran NAO T14 humanoid robot, in simulation, and where possible experimentally. In this case, the object has 4 DOF when firmly grasped by the manipulators. The system is modeled as a closed chain mechanism in Sections 2, 3, and 4, the control system is illustrated in Section 5. The NAO T14 robot illustrated in Fig. 1 is the experimental platform used. The Modified Denavit-Hartenberg (MDH) notation from [START_REF] Khalil | A new geometric notation for open and closed-loop robots[END_REF] is used to describe the kinematics of system as given in Table 1.

DESCRIPTION OF THE SYSTEM

The right arm consists of joints 1-5 and the left arm of joints 6-10. Once the object is grasped a closed loop is formed from the two arms, the object, and the common robot torso. As illustrated in Fig. 2 link 5 of the closed chain is composed of link 5 and link 10 of the open chain and the object. Frame 10 is thus fixed on link 5. We introduce frame 11, that is equivalent to frame 10, but its antecedent is frame 5. The system has in this case only nine bodies. Hence joint 10 is denoted as the cut joint. The table parameters are explained as:

• j is the joint number, a(j) is its antecedent joint • σ j is the joint type: revolute (σ j =0), prismatic (σ j =1) or fixed (σ j =2) • z j is the jth joint axis • u j is the common normal between z a(j) and z j • x j is perpendicular to z j and one of the succeeding axes, z k such that a(k) = j • γ j is the angle between x a(j) and u j about z a(j)

• b j is the distance between x a(j) and u j along z a(j) • d j is the distance between z a(j) and z j along u j • α j is the angle between z a(j) and z j about u j • θ j is the angle between u j and x j about z j • r j is the distance between u j and x j along z j • γ j = b j = 0 when x a(j) is perpendicular to z j Table 1. MDH Parameters of the closed loop

j a(j) σ(j) γ(j) b d α θ r 1 0 0 0 b 1 0 -π 2 θ 1 -r 1 2 1 0 0 0 0 π 2 θ 2 0 3 2 0 0 0 -d 3 π 2 θ 3 r 3 4 3 0 0 0 0 -π 2 θ 4 0 5 4 0 0 0 0 π 2 θ 5 r 5 6 0 0 0 b 1 0 -π 2 θ 6 r 1 7 6 0 0 0 0 π 2 θ 7 0 8 7 0 0 0 d 3 π 2 θ 8 r 3 9 8 0 0 0 0 -π 2 θ 9 0 10 9 0 0 0 0 π 2 θ 10 r 5 11 5 2 γ 11 b 11 d 11 α 11 θ 11 r 11
The parameters of frame 11 in Table 1 are defined by the grasp. Therefore, when the robot grasps the object, the transformation matrices of each serial arm are calculated.

The parameters defining frame 11 with respect to frame 5 are calculated such that they yield5 T 11 .

trans x (d 11 ) • rot z (θ 11 ) • trans z (r 11 ) (2)
where rot i (θ) indicates a rotation of θ radians about the ith axis and trans i (l) a translation of l meters along the ith axis.

KINEMATIC MODELING

Kinematic Constraint Equations

The location, velocity, and acceleration of the frame at the cut joint must be equivalent when calculated via either chain. This ensures a constant object grasp throughout the trajectory. In the closed loop formulation, some joints may be actuated and some others could be passive. Let q a contain the joint variables that are actuated, q p contain the passive joint variables and q c contain the passive joint variable where the chain is considered to be cut. The passive and cut joint variables can be obtained in terms of the active joint variables using the following geometric constraint equations:

0 T 1 1 T 2 2 T 3 3 T 4 4 T 5 5 T 11 = 0 T 6 6 T 7 7 T 8 8 T 9 9 T 10 (3)
The kinematic constraints are given by:

v 11 ω 11 = 0 J 11      q1 q2 q3 q4 q5      = 0 J 11 qr (4) v 10 ω 10 = 0 J 10      q6 q7 q8 q9 q10      = 0 J 10 ql (5)
where qr and ql contain the joint velocities of the right arm and the left arm, respectively. 0 v j is the linear velocity and 0 ω j the angular velocity of frame j in frame 0, 0 J j is the kinematic Jacobian matrix of frame j w.r.t. frame 0. As frames 10 and 11 are coincident:

v 11 ω 11 = v 10 ω 10 (6)
by substituting ( 4) and ( 5) into ( 6)

0 J 11 -0 J 10 qr ql = 0 (7)
By rearranging the rows and columns of ( 7), a relationship is obtained between the passive joint velocities and the actuated joint velocities

G a G p 0 G ac G pc G c qa qp qc = 0 (8)
that can be rewritten as J c q = 0 (9) From the first row of (8), we obtain:

qp = G qa (10) where G = -G -1 p G a (11)
From the second row of (8), we obtain:

qc = -G -1 c (G ac + G pc G) qa (12
) Upon differentiation of (8) with respect to time the acceleration constraints equation is expressed as:

G a G p 0 G ac G pc G c qa qp qc + Jc q = 0 (13)

Mobility Analysis

The mobility of a closed loop mechanism can be calculated exactly by using the rank of the closed loop kinematic constraint equations (9). The rank of (J c ) is obtained numerically using random configurations.

J c = 0 J 11 -0 J 10 (14)
The DOF can be found as the number of independent joints l before loop closure, minus those that lose their independence after the loop is closed (the degree of the constraint equation c):

n = l -c = l -rank(J c ) (15) n = 10 -6 = 4
(16)

Closed Loop Jacobian Matrix

A new Jacobian matrix is derived that relates the actuated joint velocities to the task space velocities. The velocity is equal when calculated via either chain, from (4) and ( 5), and also via the actuators:

v ω = 0 J 11 qr = 0 J 10 ql = J a qa (17)
When a closed chain is formed, the object has 4-DOF. It can thus be fully controlled by four actuated joints, a feasible actuation scheme is chosen as q a = [q 1 q 2 q 3 q 7 ] T .

The passive joint vector is q p = [q 4 q 5 q 6 q 8 q 9 q 10 ] T . Therefore using ( 10) and ( 12):

qr =      1 0 0 0 0 1 0 0 0 0 1 0 G(1, :) G(2, :)      qa ( 18
)
where G(i, :) indicates the i th row of the matrix G.

The Jacobian matrix J a can be calculated by using the Jacobian matrix from either arm. For the right arm it is obtained by combining ( 17) and ( 18):

J a = 0 J 11      1 0 0 0 0 1 0 0 0 0 1 0 G(1, :) G(2, :)      (19)
(20)

DYNAMIC MODELING OF CLOSED LOOP

Calculation of the Inverse Dynamic Model

The inverse dynamic model (IDM) calculates the active motor torques τ in terms of q, q, and q of the joints.

Let q tr = q T a q T p T denote the joint variables of the tree structure obtained when considering q 10 as virtually cut. In order to find the Closed Loop Inverse Dynamic Model (CLIDM), first the IDM for the tree structure is found and then converted to CLIDM by using the following relation given in [START_REF] Khalil | Modeling, identification & control of robots[END_REF]:

τ = ∂q tr ∂q a T Γ tr = Γ a + ∂ qp ∂ qa T Γ p (21)
where Γ tr denotes the joint torques of the tree structure.

It can be expressed as:

Γ tr = Γ a Γ p = A tr (q tr ) qa qp + H tr (q tr , qtr ) (22)
Γ a , Γ p represent the torque on the actuated and passive joints of the tree structure, respectively. A tr and H tr are tree structure inertia matrix and the tree structure matrix of Coriolis, Centrifugal, and Gravity forces respectively. Using ( 10) and ( 21), we obtain the dynamic model:

τ = I N G T Γ a Γ p (23)
G is defined by ( 11) and I N is the identity matrix of dimension N , where N is equal to the DOF of the system. Substituting the general expression for the tree dynamic model given by ( 22) into ( 21), the closed loop dynamic model is obtained as:

τ = I N G T A tr qa qp + I N G T H tr (24)

Dynamic Model with Redundant Actuators

When the number of actuated joints is greater than four, the inverse dynamic model will be given by:

τ ra = ∂q a ∂q ra T τ (25)
where τ ra is the torque vectors composed of all actuated joints. q ra is defined as:

where q r is the vector of redundant actuated joints. Thus, τ ra can be obtained as:

q ra = q a q r ( 26
)
τ ra = I N ∂q r ∂q a + τ (27)

Calculation of the Direct Dynamic Model

The direct dynamic model (DDM) calculates the independant joint accelerations qa from the motor torques τ . It can be obtained after expressing qp in terms of qa as shown in ( 13), then substituting the result into (24).

CONTROL AND SIMULATION

Kinematic controller

A kinematic controller obtains the joint velocities that realize a desired task frame velocity. One implementation uses ( 17) to obtain the actuated joint velocities for a certain task velocity. Then from (8) the remaining velocities, which satisfy the closed loop constraints, can be resolved. Instead of this two step process, by using the cooperative task space as defined by [START_REF] Chiacchio | Direct and inverse kinematics for coordinated motion tasks of a two-manipulator system[END_REF], the task velocities and constraints can be embedded into one equation. This compact representation is applied for the first of our control schemes. Twelve variables are fully defined, six controlling the object location in space and a further six controlling the relative location between the end effectors, denoted here as the task and relative variables respectively. The direct kinematic model that relates the joint velocities to the end effector velocities is recalled here:

V tsk V rel = J tsk J rel qr ql = J ext qr ql (28)
where

V tsk = v tsk ω tsk , V rel = v rel ω rel (29)
J tsk = 1 2 0 J 11 0 J 10 J rel = -0 J 11 0 J 10 (30)
J ext = J tsk J rel (31)

Kinematic Control with Minimum representation

Minimum representation is used when the location of the end effector does not need to be not fully defined, see [START_REF] Dombre | Analyse des performances des robotsmanipulateurs flexibles et redondants: contribution a leur modelisation et a leur commande[END_REF]. The degree of redundancy is exploited to improve the performance of the robot with regard to some optimization criteria. Since the mobility of the system is 4, the maximum dimension of the task coordinates is 4, dim(V tsk ) ≤ 4. Therefore the task is defined as transporting a common object from one position to another meaning some variables do not need to be assigned. This leads to a minimum representation of the task variables denoted as ḣ while fully controlling all the relative variables. The minimum representation is implemented by pre-multiplying the Jacobian matrix of equation ( 31) by a selection matrix H:

ḣ V rel = H J ext qr ql (32)
The inverse kinematic control law with optimization function Z is defined as:

qr ql = (H J ext ) + ḣ V rel + PZ (33)
with V rel = 0 ensures the validation of the loop constraint. P is the projection into the null space of HJ ext defined as:

P = I n -(H J ext ) + (H J ext ) (34)
The minimum representation of task is implemented with different degrees of redundancy, by selecting an appropriate H matrix.

Point on plane A point on the object is moved to a plane by a series of elementary motions denoted as ḣ along the normal to that plane. The task is defined by one direction whereas the relative variables contain six coordinates. Thus in total seven variables are controlled, maximizing the degrees of redundancy, while maintaining loop closure. The normal to the plane is used to project the velocity of the object to that directing towards the plane. The motion is uncontrolled in five directions but the normal projection guarantees that the point on the object will reach the plane. The orientation of the plane can be changed through the trajectory. The total kinematic equation is defined as: ḣ

V rel = H 1 J ext qr ql (35)
H 1 = u T 0 1×9 0 6 I 6 (36)
u is the unit vector along the normal to the desired plane, for example to attain a horizontal plane:

u T = [0 0 1 ] ( 37
)
Point on Point A point on the object is moved to a point in space by a series of elementary motions with an arbitrary orientation. Three degrees of freedom of the object are controlled. Equation ( 32) is modified as follows: The redundancy created by the minimum representation is employed to maximize the distance from the joint limits, utilizing the workspace as much as possible. The figures from Fig. 4 to 8 compare the joint values of the right arm, during a point to point trajectory. Joint limit motion is the solution when the joint limit optimization criterion is applied. Pseudo is the solution using the pseudoinverse solution. The system described in table 1 is modified to include two additional passive revolute joints, the new M-DH parameters are given in table 2. These joints represent the object grasp conditions as shown in Fig. 9 where the passive joint is denoted as q 11 (q 12 of the second chain). It is assumed that the object grasp constrains five degrees of freedom but allows one degree of rotation about an axis that passes through the two contact points. By the addition of these passive joints the object gains 2-DOF. Thus q a becomes a vector of six components. In contrast to Section 5.2, arbitrary forces can no longer be applied to the object since the chain contains passive joints. The parameters of frame 13 are defined by the grasp. The procedure defined in section 2 is carried out to find these parameters θ 12 and θ 11 represent the passive rotation of the aforementioned grasp condition.

v tsk V rel = H 2 J ext qr ql (38)
H 2 = I 3 0 3×9 0 6 I 6 (39)
Control By modeling the robot as a closed loop kinematic chain, the entire system can be controlled as one This work has presented a study of the two arms of the NAO robot when engaged in a cooperative task. By using cooperative task space formulation, an inverse kinematic controller with minimum representation of task is implemented. The minimum representation allows loop closure and also the creation a degree of redundancy. This redundancy is then employed to avoid the violation of joint limits. Secondly a task space PID is used to demonstrate the feasibility of the virtual passive joint. By including this passive joint two of the redundant actuators can be used to control the object directions in space.

Fig. 1 .

 1 Fig. 1. NAO T14 (Courtesy of Aldeberan-Robotics) with schematic representation of its arm

  Fig. 3. Cooperative Control with Minimum Representation

Fig. 4 .

 4 Fig. 4. Effect of joint limit optimization on joint q1

Fig. 6 .

 6 Fig. 6. Effect of joint limit optimization on joint q3

Fig

  Fig. 9. Nao with passive jointTable 2. MDH Parameters with passive graspj a(j) σ(j) γ(j) b d α θ r 1 0 0 0 b 1 0 -π 2 θ 1 -r 1 2 1 0 0 0 0 π 2 θ 2 0 3 2 0 0 0 -d 3 π 2 θ 3 r 3 4 3 0 0 0 0 -π 2 θ 4 0 5 4 0 0 0 0 π 2 θ 5 r 5 6 0 0 0 b 1 0 -π 2 θ 6 r 1 7 6 0 0 0 0 π 2 θ 7 0 8 7 0 0 0 d 3 π 2 θ 8 r 3 9 8 0 0 0 0 -π 2 θ 9 0 10 9 0 0 0 0 π 2 θ 10 r5 11 5 0 0 0 d 11 0 θ 11 r 11 12 6 0 0 0 d 12 0 θ 12 r 12 13 11 2 γ 13 b 13 d 13 α 13 θ 13 r 13dynamic entity. Fig.10illustrates a PID controller in task space, where K represents appropriate gain matrices.

T 11 = 0 T 5 -1 0 T 10(1) i T j is the 4 × 4 transformation matrix from frame i to frame j. The six parameters defining frame 11 can be found by solving the following equation: 5 T 11 = rot z (γ 11 ) • trans z (b 11 ) • rot x (α 11 ) •