Design of a Spherical Wrist with Parallel Architecture: Application to Vertebrae of an Eel Robot

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1 Chablat D. et Wenge P., Desgn of a Sphecal Wst wth Paallel Achtectue: Applcaton to Vetebae of an Eel obot, Poc. IEEE Int. Conf. ob. utoaton, Bacelone, -6 Avl 5. Desgn of a Sphecal Wst wth Paallel Achtectue: Applcaton to Vetebae of an Eel obot Daen Chablat and Phlppe Wenge Insttut de echeche en Councatons et Cbenétque de Nantes, ue de la Noë, BP 9, 44 Nantes Cedex Fance Daen.Chablat@ccn.ec-nantes.f Abstact - he desgn of a sphecal wst wth paallel achtectue s the object of ths atcle. hs stud s pat of a lage poject, whch as to desgn and to buld an eel obot fo nspecton of esed ppng. he kneatc analss of the echans s pesented fst to chaactee the sngula confguatons as well as the sotopc confguatons. We add the desgn constants elated to the applcaton, such as () the copactness of the echans, () the set of the eleents n ode to ensue statc and dnac balance and () the possblt of the echans to fll the ellptc fo of the ell sectons. Kewods - Sphecal wst, paallel obots, sotopc desgn. Caangdae as jacks, hose ackeel o popano [8]) d on oscllatons of the bod and () the angullfo swng (of snake tpe, eel, lape, etc.) d on undulatons of the bod. An angullfo swe popels tself fowad b popagatng waves of cuvatue backwad along ts bod []. I. INDUCIN ve llons of eas, fsh have evolved swng capact fa supeo n an was to what has been b nautcal scence and technolog. he use the stealned bodes to explot flud-echancal pncples. hs wa, the can acheve extaodna populson effcences, acceleaton and aneuveablt not feasble b the best naval achtects []. Paallel kneatc achtectues ae coonl claed to offe seveal advantages ove the seal countepats, lke hgh stuctual gdt, hgh dnac capactes and hgh accuac []. hus, the ae nteestng fo applcatons whee these popetes ae needed, such as flght sulatos [] and hghspeed achnes. ecentl, new applcatons have used such echanss to buld huanod obots [4], o snake obots [5]. he pupose of ths atcle s to desgn vetebae of an eel obot b usng the advantages of the paallel achtectues whle appoachng eel opholog. he next secton pesents the objectves of the boetc as well as sutable sphecal achtectues. he desgn paaetes and the kneatcs of the echans to be opted ae epoted n Secton. Fg. : Change n bod shape n swng and a subdvson of ts bod o ca out angullfo swng, the bod of the eel s ade of a successon of vetebae whose undulaton poduces oton, as depcted n Fg.. In natue, thee s onl one degee of feedo between each veteba because the oton contol of the vetebae s coupled wth the oton of the dosal and vental fn. hese two fns beng not easl epoducble, we wll gve to each veteba, oe oblt to account pobles of ollng, fo exaple. he assebl of these vetebae, coupled to a head havng two fns ust allow the epoducton of the eel swng. Fo the obsevaton of Euopean eel, Angulla angulla, we have data concenng hs kneatc swng such as wave speed, ccle fequenc, apltude o local bendng [9]. he aw s gven fo fowad and backwad swng on total bod length, as depcted n Fg.. he othe angles ae obtaned usng Nave-Stokes equatons on chaactestc tajectoes []. Fo ou pototpe, we took as constants of desgn, ± degees n aw fo fowad swng, ±5 degees n ptchng fo dvng and ±4 degees n ollng to copensate fo toson n dvng. II. PELIMINAIES A. Boetc obotcs he object of the boetc obotcs s to c lfe, to tate bologcal sstes o to conceve new technologes dawn fo the lesson of the stud [6]. Fo the last twent eas o so, an eseaches have been ade n the undewate feld n Aeca and n Japan [7]. Aong those, a good nube attepted to epoduce fsh. In ths context, two odes of locooton anl attact the attenton of eseaches () the caangd swng (fal Ptchng Yaw ollng Fg. : ollng, ptchng and aw angles of vetebae he objectve of ou stud s to buld an eel obot wth vetebae and an oveall length of 5 (wth the head and

2 Chablat D. et Wenge P., Desgn of a Sphecal Wst wth Paallel Achtectue: Applcaton to Vetebae of an Eel obot, Poc. IEEE Int. Conf. ob. utoaton, Bacelone, -6 Avl 5. the tal ncluded). Each veteba wll have an ellptc secton of 5 and focal dstance espectvel and wll be a thck. B. Mechancal achtectues he desgn of the vetebae of an eel s equvalent to the desgn of sphecal wsts. Indeed, an eel beng able to be copaable wth a bea, two theoes can goven ts oves, () the theo of essne [], whch poses a 6 DF kneatc echans (ealable b the stackng of Gough-Steewat's platfos) and () the theo of Kchoff of the nextensble beas, whch poses the kneatcs of the ball jont tpe []. Sphecal wsts wth seal achtectue ase seveal pobles, whch lead us to stud paallel achtectues. he fst poble s elated to the copactness of the wst because the dstance between the successve vetebae ust be ned n ode to ceate a contnuous defoaton of the eel bod. he second poble s elated to the sngulat of seal wst (fst and last axs algned) and, f we use such achtectue, onl the second evolute jont s anl used to poduce the oscllaton of the bod, whch elds pobles of denson. Convesel, sphecal wsts wth paallel achtectue ae nueous [-]. If we want to c the dsplaceents ceated b the uscles, the coespondng jont s a psatc actuato. Most exstng actuatos ae d on the use of a ota oto, a educe and a ball scew. Such devces ae used, fo exaple, n the flght sulatos usng Gough-Stewat's platfos. In the next secton, we wll pesent an achtectue that uses evoluton jonts to poduce equvalent otons. III. KINEMAIC SUDY F HE SELECED ACHIECUE A. Descpton Sphecal paallel echanss can be classfed nto two an goups, setcal o asetcal echanss, whch can be oveconstant o non-oveconstant [4]. Fo the fst goup, we have the agle ee [5] whch uses evolute jonts fxed on the (Fg. ). It s fstl developed fo the apd oentaton of a caea but t s also used fo cang a tool [6]. Fg. : he agle ee [5] Fo exaple, the natue caea attached to the endeffecto can be ponted n a cone of vson of 4 wth ± n toson [7]. Such popetes ae not asked fo ou pototpe because onl the aw angle ust be hghe and t s dffcult to place the actuated jonts on an ellptc bass. hus, we wll stud a sphecal wst, whch can poduce hgh aw and whee engne toques can be added b usng the pncple of the dffeental echans. he selected achtectue s a non-oveconstaned asetcal achtectue that s epoted n [4] as an (, 6, 6) achtectue. he and the oble platfo ae connected b thee kneatc chans, as depcted n Fg. 4. C B x θ C A θ A θ x B Fg. 4: Stuctue of the studed sphecal wst hs achtectue esults fo the eseach aound the Le Goup of Eucldan dsplaceents []. hee ae () two kneatc chans, noted legs and, to poduce a geneal gd bod dsplaceent fo the subgoup {D} (6 DF) and () a kneatc chan, noted leg, fo the sphecal subgoup {S} and ade b thee coaxal evolute jonts ( DF). hee s onl one actuated jont on each leg ( θ, θ, θ ). If the ealaton of leg s eas (thee coaxal evolute jont), t s dffcult to enueate all the legs wth 6-DF. he ost cuent geneato of {D} s of the UPS tpe (Gough- Stewat's platfo, wth P psatc actuated jont, U fo unvesal jont and S fo sphecal jont), whch has the dsadvantage of usng a psatc actuato that s not fxed on the bass. In the lteatue, an equvalent echans exsts but the geneato of {D} s of PUS tpe (wth P psatc actuated jont). Fo legs and, the psatc actuated jonts ae n paallel to the vetebal colun whch s haful fo the copactness of the echans. he oentaton can be changed but the effcenc deceases consdeabl. Fo leg, the fst evolute jont (located on the ) s actuated. hus, we have changed the tpe of legs and, b a US tpe (wth evolute actuated jont) as depcted n Fg. 4. n the next secton, we wll justf the placeents and the densons b the stud of the Jacoban atx. B. Kneatcs A fxed efeence fae, noted fxed ( x,,, ) s located on the and s oented n such a wa that () plane x s planed b ponts C and, () the -axs s vetcal, () x-axs s dected fo A to A. he coodnates of ponts A n ae wtten as fxed [ A ] = [ a b c] and [ A ] [ ] a b c fxed fxed = () he lengths a, b, and c wll be chosen b the stud of the Jacoban atx n the next subsecton.

3 Chablat D. et Wenge P., Desgn of a Sphecal Wst wth Paallel Achtectue: Applcaton to Vetebae of an Eel obot, Poc. IEEE Int. Conf. ob. utoaton, Bacelone, -6 Avl 5. he oble platfo wll be otatng aound pont that s the ogn of the oble fae, noted oble. he oentaton of oble ( x,,, ) s defned so that () plane x s the plane defned b ponts C and, () x -axs s dected fo to C and () -axs s dected fo to C. Let θ be the vecto of jont coodnates assocated wth the actuated evolute jonts. he oentaton of the oble platfo wth espect to fxed fae s defned b the "ollng Ptchng Yaw" paaetes (PY) whee the fst paaete s the oentaton angle θ of the fst evolute jont of leg ). fxed θ= [ θ θ θ ] (, θ ) ( ', φ) ( '', ψ) oble = x he angles θ, φ and ψ ae assocated wth the followng cascaded otatons () a otaton of angle θ aound -axs, () a otaton of angle φ aound the '-axs (obtaned fo the pevous otaton and whose axs s the axs of the second evolute jont of leg ), () a otaton of ψ aound the x''-axs (obtaned fo the second otaton and whose axs s the axs of the thd evolute jont of leg ). C. Jacoban atces o chaactee the sngula confguatons, we wll use an nvaant fo, whch allows ou esults to be applcable to an achtectue studed hee. hus, thee s no poble of sngulat of tansfoaton n the otaton atx between fxed and oble. c b to have We wte the Chasles's elaton on ( ) ( ) = ( ) + ( ) ( ) c b c o o a b a () In ths equaton, all the vectos ae expessed n fxed. o splf calculatons, we set ( ), ( ), ( ) et ( ) = c b p = c o b = o a l = b a B dffeentatng Eq. () wth espect to te, we obtan, & = p& & l () wth fxed [ ] = [ ] p p (4) oble fxed oble Dffeentatng wth espect to te, we fnd fxed [ p ] = Q& [ p ] & oble (5) fxed oble snce vecto [ p ] s a constant vecto when expessed oble n fae oble. Moeove, the te devaton of the otaton atx can be wtten as Q& = Ω Q (6) whee Ω s the angula veloct tenso. Fnall, fo Eqs. () and (6), we get p& = Ω p = ω p whee denotes the coss poduct of the two vectos and ω s the angula veloct vecto. We note and, the unt vectos passng though the axs of the fst evolute jont of legs and, espectvel. Moeove, we can wte vecto l& as functon of angula veloctes & θ and & θ & l = l (& θ. ) and & l = l (& θ. ) hus, Eq. () can be wtten n the fo & = ω p l (& θ. ) We ultpl the pecedng equaton b because. & =. hus, we have.( ω p) =.( l ( & θ. )) ( p ). ω = ( l ).(& θ. ) hese two equatons can be cast n vecto fo wth Aω + Bq& = (7) ( p ) ( ) A = p (8) ( l ). B = ( l ). (9) and q & = & θ & θ & θ hen, when B s not sngula, the nvese Jacoban atx s wtten, ( ) p ( l ). ( ) J = p ( l ). D. Sngula confguatons he paallel sngulates occu when the detenant of the atx A vanshes,.e. when det( A ) =. In such confguatons, t s possble to ove locall the oble platfo wheeas the actuated jonts ae locked. hese sngulates ae patculal undesable because the stuctue cannot esst an foce o toque. Fo Eq. (7), we have ( p ) ( p ) o ( p ) = o ( p ) = It s equvalent to have B, B and coplana o

4 Chablat D. et Wenge P., Desgn of a Sphecal Wst wth Paallel Achtectue: Applcaton to Vetebae of an Eel obot, Poc. IEEE Int. Conf. ob. utoaton, Bacelone, -6 Avl 5. to have ( B, ) o ( B, ) algned, as depcted n Fg. 5. B A l p p l B A l C A C p p C C A densons of each veteba to elnate the sngula confguatons fo the wokspace and to axe the kneatc popetes aound ts sotopc confguatons. he atx A s sotopc when p and p and ( p ) ( p ) and = p = = p = Moeove, A s equal to the dentt atx, as depcted n Fg. (7). he coss poduct ( p ) and ( p ) and the - axs fo an othogonal fae. l Fg. 5: Paallel sngulat when B, B and ae coplana and B and ae algned C p p C Seal sngulates occu when the detenant of the atx B vanshes,.e. when det( B ) =. At a seal sngulat, an oentaton exsts along whch an angula veloct cannot be poduced. Fo Eq. (8), we have ( ). l = o ( ). = l ( l ). = o l o ( ). = It s equvalent to have () l and algned, o () l and algned, o () and algned, o (v) and algned, as depcted n Fg.6. B B Fg. 7: Isotopc confguaton of atx A he atx B s sotopc and equal to the dentt atx when l and l and l and l and = l = he esult of the sotopc constants on A and B ae shown n Fg. 4 as an exaple. In fact, thee s an nfnt of soluton because no constant gves us the oentaton of copaed to. Fgue 4 shows and paallel but the can be dffeent. p p C C B B l l C p p A C l A F. Boetc constants and odel splfcatons Fo the pecedng esult, we wll pesent thee sutable solutons of ou pototpe. he fst soluton s the echans depcted n Fg. 4 that we could call "paallel axes". Equaton gves the locaton of ponts A n fo a unt echans, A A B B Fg. 6: Seal sngulat when l and ae algned and l and ae algned E. Condton nube and sotopc confguatons he Jacoban atx s sad to be sotopc when ts condton nube attans ts nu value of one [8]. he condton nube of the Jacoban atx s an nteestng pefoance ndex, whch chaacteses the dstoton of a unt ball unde the tansfoaton epesented b the Jacoban atx. he Jacoban atx of a anpulato s used to elate () the jont ates and the Catesan veloctes, () the statc load on the output lnk and the jont toques o foces. hus, the condton nube of the Jacoban atx can be used to easue the unfot of the dstbuton of the tool veloctes and foces n the Catesan wokspace. he a of ths secton s to defne the placeent and a =, b =, c = If ths soluton adts an sotopc confguaton, the behavou n fowad swng leads to use legs and sultaneous. When we appl as nput veloct θ & = [ ], the angula veloct obtaned s ω= [ ]. hs eans that we aplf the otatonal oton just afte havng used a educton gea on the ota oto to ncease the avalable toque. hus, the length of the otos s constaned b the shape of the coss secton of the eel, as depcted n Fg. 9. he second soluton, called "othogonal axes", s to place and othogonall as depcted n Fg. 8. he locaton of ponts A n concdes wth pont.

5 Chablat D. et Wenge P., Desgn of a Sphecal Wst wth Paallel Achtectue: Applcaton to Vetebae of an Eel obot, Poc. IEEE Int. Conf. ob. utoaton, Bacelone, -6 Avl 5. C x θ C Concenng the ntegaton nto the coss-secton of the eel, the placeent s less constaned, as shown n Fg.. A A Motos B C B C B θ θ x A A Fg. 8: Sphecal wst wth othogonal actuatos In ths case, the dect and nvese kneatc odels ae sple but t s oe dffcult to place the otos of legs and, as shown n Fg. 9. Moeove, thee also exsts an angula aplfcaton facto n the fowad swng. Motos A Motos A A A B C B C B B C B C Fg. 9: Placeent of the otos and the legs fo the "paallel axes" and the "othogonal axes" he last soluton has paallel actuatos and the axes ntesect the -axs, as depcted n Fg.. When the eel obot s swng, the angula veloct of the actuated jonts of legs and s equal to aw veloct. C B A θ x x θ C B A Fg. : Sphecal wst wth paallel actuatos hs eans that fo the fowad o backwad swng, the kneatc odels ae sple and the toque needed fo the oton s dstbuted. Howeve, onl A can be sotopc because we have ( l ). = fo =, Equaton gves the locaton of ponts A n fo a unt echans, a =, b =, c = θ Fg. : Placeent of the otos and the legs fo sphecal wst wth paallel actuatos G. Dect and nvese kneatc odels he dect kneatc odel can be wtten when we know the poston of B and C. hus, we have, C S B = + and n oble, o n fxed C S B = + C = [ ] C = [ ] CC φ C = SCφ S φ wth C = cos( θ ), S sn ( θ ) S φ CSS φ ψ SC ψ C = SSφSψ + CCψ CS φ ψ = fo =,, φ = cos( φ), = sn( φ) ψ = cos( ψ ) and S ψ = sn( ψ ). We add the constant that BC = CC φ + SC φ C + Sφ + S = () CSS φ ψ SC ψ + + SSS φ ψ + CC ψ C + S C S = φ ψ () o solve the dect kneatc, we know θ = [ θ θ θ ] and we use the followng substtutons Q Q sn( φ ) = cos( φ) = + Q + Q hus, we can eak that Eq. depends onl on φ and s a quadatc equaton of Q ( SQ QC QCS Q + CS C S) ( Q ) = ne soluton s Q =,.e. φ = π /+ kπ that does not depend on the actuated jonts. Fgue depcts the fou dect kneatc soluton fo θ =., θ =., θ = π / 4. Solutons and ae found when Q = and can be easl solated. Fo solutons (c) and (d), onl the second one s sutable that

6 Chablat D. et Wenge P., Desgn of a Sphecal Wst wth Paallel Achtectue: Applcaton to Vetebae of an Eel obot, Poc. IEEE Int. Conf. ob. utoaton, Bacelone, -6 Avl 5. can be solated b the dot poduct of b p. A A C C A B A C B C B A B A (c) (d) C A B A C B C Fg. : he fou dect kneatc solutons fo θ =., θ =., θ = π / 4 o solve the nvese kneatc, we use two substtutons, = tan( θ / ) and S = tan( θ / ) that pet us to have two quadatc and ndependent equatons as functon of and S espectvel. Fgue shows the fou nvese kneatc solutons fo θ = π /4, φ = π /, ψ = π / that we can solate b calculatng l. and l. fo legs and, espectvel. B A A B A B A B C C C C B A A A A B C C Fg. : he fou nvese kneatc solutons fo θ = π /4, φ = π /, ψ = π / o conclude, we have fou solutons fo the dect kneatc and fou solutons fo the nvese kneatc (two fo legs and, espectvel). B C B C B C B IV. CNCLUSINS he desgn of sphecal wsts takng nto account the constants on the boetc of the eel was ade n ths pape. A new achtectue s nvestgated and sotopc constants ae appled to poduce thee sutable solutons. he setcal constants lead us to choose the one whee the placeent of the actuated jonts s optal because the ae located on a edan plane whee the focal dstance s axa. ACKNWLEDGMENS hs eseach was patall suppoted b the CNS ("Angulle" Poject). EFEENCES [] antafllou, M. S. and antafllou G. S., An Effcent Swng Machne, Scentf Aecan, pp. 65-7, Mach 995 [ eb,. and Zn,., Slat laws of seal and paallel anpulatos fo achne tools, Poc. Int. Sena on Ipovng Machne ool Pefoances, pp. 5-, Vol., 998. [] Melet J-P, Paallel obots, Kluwe Acadec Publshes,. [4] Lenacc J., Stansc, M. M. and Paent-Castell, Kneatc Desgn of a Huanod obotc Shoulde Coplex, poceedngs of IEEE Intenatonal Confeence on obotcs utoaton, Apl 4-8,. [5] Lee K-M. et Ajunan S,. A thee-degees-of feedo cooton npaallel actuated anpulato, IEEE ans. on obotcs utoaton, ctobe 99. [6] Cha J.G., Bale S.A.lak J.E., Full.J.utkosk M.., Fast and obust: Hexapedal obots va Shape Deposton Manufactung, he Int. Jounal of obotcs eseach, vol., no., pp (4), ctobe. [7] Hose, S., Bologcall nsped obots: Snake-lke loconotos and anpulatos, xfod Unv. Pess, xfod, 99. [8] McIsaac, K.A., stowsk, J.P., A geoetc appoach to angullfo locooton odellng of an undewate eel obot, IEEE Int. Conf. obot. uto., ICA 999, pp [9] D'Août K., Ats P., A Kneatc Copaason of Fowad and Backwad Swng n the Eeel Angulla Angulla, Jounal of Expeental Bolog, pp. 5-5, 999. [] Calng J., Wllas. L., Bowtell G., Self-Popelled Angullfo Swng: Sultaneous Soluton of the wo-densonal Nave-Stokes Equatons and Newton's Laws of Moton, 998. [] So J.C. and Vu-Quoc L., n the dnacs n space of ods undegong lage otons - A geoetcall exact appoach, Cop. Meth. Appl. Mech. Eng., 66, 988, pp [] Boe, F., Pault, D., Fnte eleent of slende beas n fnte tansfoatons: a geoetcall exact appoach, Intenatonal Jounal fo Nuecal Methods n Engneeng, 4: 59, pp [] Hevé J. M., Analse stuctuale des écanses pa goupe des déplaceents, Mech. Mach. heo, Vol., No.4, pp.47-45, 978. [4] Kaoua, M., Concepton stuctuale de ecanses paalleles spheques, hèse de doctoat, I -6, École Centale de Pas,. [5] Gosseln C., Hael J.F., he agle ee: a hgh pefoance thee-degeeof-feedo caea-oentng devce, IEEE Int. confeence on obotcs utoaton, pp San Dego, 8- Ma 994. [6] Bdault, F., eng. P. ngeles, J., Stuctual optaton of a sphecal paallel anpulato usng a two-level appoach, Poc. ASME Desgn Engneeng echncal Confeences, Pttsbugh, PA, Sept. 9- D-M DEC /DAC-,. [7] Gosseln C., St-Pee E. and Gagné M., n the developent of the agle ee: echancal desgn, contol ssues and expeentaton, IEEE obotcs utoaton Socet Magane, Vol., No. 4, pp. 9-7, 996. [8] Golub, G. H. and Van Loan. F., Matx Coputatons, he John Hopkns Unvest Pess, Baltoe, 989.