Kinematic Analysis of the vertebra of an eel like robot

Transcription

1 Kneatc Analss of the vertebra of an eel lke robot Daen Chablat o cte ths verson: Daen Chablat. Kneatc Analss of the vertebra of an eel lke robot. 3nd Annual Mechanss and obotcs Conference (M), Aug 8, New-York, Unted States. pp.-, 8. <hal-8459> HAL Id: hal Subtted on 3 Jun 8 HAL s a ult-dscplnar open access archve for the depost and dssenaton of scentfc research docuents, whether the are publshed or not. he docuents a coe fro teachng and research nsttutons n France or abroad, or fro publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de docuents scentfques de nveau recherche, publés ou non, éanant des établsseents d ensegneent et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Proceedngs of the ASME 8 Internatonal Desgn Engneerng echncal Conferences & Coputers and Inforaton n Engneerng Conference IDEC/CIE 8 August 3-6, 8, New York, USA DEC KINEMAIC ANALYSIS F HE VEEBA F AN EEL LIKE B Daen Chablat Insttut de echerche en Councatons et Cbernétque de Nantes, rue de la Noë, 443 Nantes, France Daen.Chablat@rccn.ec-nantes.fr ABSAC he kneatc analss of a sphercal wrst wth parallel archtecture s the object of ths artcle. hs stud s part of a larger French project, whch as to desgn and to buld an eel lke robot to tate the eel swng. o pleent drect and nverse kneatcs on the control law of the prototpe, we need to evaluate the workspace wthout an collsons between the dfferent bodes. he tlt and torson paraeters are used to represent the workspace. INDUCIN Parallel kneatc archtectures are coonl claed to offer several advantages over ther seral counterparts, lke hgh structural rgdt, hgh dnac capactes and hgh accurac []. hus, the are nterestng for applcatons where these propertes are needed, such as flght sulators [] and hghspeed achnes. ecentl, new applcatons have used such echanss to buld huanod robots [3] or snake robots [4]. ver llons of ears, fsh have evolved swng capact far superor n an was to what has been b nautcal scence and technolog. he use ther strealned bodes to eplot flud-echancal prncples. hs wa, the can acheve etraordnar propulson effcences, acceleraton and aneuverablt not feasble b the best naval archtects [5]. In [6], we have ntroduced a new archtecture of sphercal wrst able to reproduce the vertebra of an eel. he purpose of ths artcle s to show the kneatc equatons and to stud ts workspace takng nto account echancal constrants to avod nternal collsons. he Euler angles are classcall used to copute the workspace of sphercal wrsts but the do not pert to vsuale the setrcal propertes. ecentl n [7], the lt-and-orson was ntroduced to represent the workspace of the agle ee. In ths paper, we wll present the kneatc equatons of the sphercal wrst and an algorth to copute the workspace and the jont space. PELIMINAIES Boetc robotcs he a of our project s to tate the eel swng and ts bologcal sstes and to conceve new technologes drawn fro the lesson of ther stud [8]. Man researches have been ade n the underwater feld n Aerca and n Japan [9]. In ths contet, two odes of locooton anl attract the attenton of researchers, () the carangd swng (fal Carangdae as the one of a jack, a horse ackerel or a popano []) based on oscllatons of the bod and () the angullfor swng (of snake tpe, eel, lapre, etc.) based on undulatons of the bod. An angullfor swer propels tself forward b propagatng waves of curvature backward along ts bod [5]. Fgure : Change n bod shape n swng and a subdvson of ts bod Coprght #### b ASME

3 o carr out angullfor swng, the bod of the eel s ade of a successon of vertebrae whose undulaton produces oton, as depcted n Fg.. In nature, there s onl one degree of freedo between each vertebra because the oton control of the vertebrae s coupled wth the oton of the dorsal and ventral fn. hese two fns beng not easl reproducble, we wll gve to each vertebra ore oblt to account probles of rollng, for eaple. he assebl of these vertebrae, coupled to a head havng two fns, ust allow the reproducton of the eel swng. Fro the observaton of European eel, Angulla angulla, we have data concernng ts kneatc swng such as wave speed, ccle frequenc, apltude or local bendng []. he aw s gven for forward and backward swng on total bod length, as depcted n Fg.. he other angles are obtaned usng Naver-Stokes equatons on characterstc trajectores []. For our prototpe, we took as constrants of desgn, ±3 degrees n aw for forward swng, ±5 degrees n ptchng for dvng and ±4 degrees n rollng to copensate for torson n dvng. ϕ a Fgure 3: he successve rotatons of the & angles: (a) tlt, (b) torson. For a gven torson angle, the workspace can be represented n a polar coordnate sste as s shown n Fg ϕ ψ φ a Ptchng Yaw ollng Fgure 4: Projected orentaton workspace n a polar coordnate sste (φ, ) for a gven torson ψ Fgure : ollng, ptchng and aw angles of vertebrae rentaton representaton he tlt-and-torson (&) angles are defned n [7] as a cobnaton of a tlt and a torson. hese angles are defned n two stages. In the frst one, llustrated n Fg. 3(a), the bod frae s tlted about a horontal as, a, at an angle, referred to as the tlt. he as a s defned b an angle ϕ, called the auth, whch s the angle between the projecton of the bod ' as onto the fed plane and the fed -as. In the second stage, llustrated n Fg. 3(b), the bod frae s rotated about the bod ' as at an angle ψ, called the torson. = ( ϕ) ( ) ( ψ φ) KINEMAIC SUDY F HE SELECED ACHIECUE Descrpton he archtecture chosen n [6] s a non-overconstraned asetrcal archtecture that s reported n [6] as an (3, 6, 6) archtecture. he base and the oble platfor are connected b three kneatc chans, as depcted n Fg. 5. hs archtecture results fro the research around the Le Group of Eucldan dsplaceents [5]. here are () two kneatc chans, noted legs and, to produce a general rgd bod dsplaceent fro the subgroup {D} (6 DF) and () a kneatc chan, noted leg, fro the sphercal subgroup {S} and ade b three coaal revolute jonts (3 DF). here s onl one actuated jont on each leg (,, ) 3. Coprght #### b ASME

4 3 A For the prototpe, we have replaced the jonts located n,, and b sphercal jonts. he otor assocated wth the jont 3 s not coaal but parallel and two gears transt the oton. Kneatcs A fed reference frae, noted fed (,,, ) s located on the base and s orented n such a wa that () plane s defned b ponts C and, () the -as s vertcal, () -as s drected fro A to A. he coordnates of ponts A and A n are wrtten as fed Fgure 5: Structure of the studed sphercal wrst If the realaton of leg s eas (three coaal revolute jonts), t s dffcult to enuerate all the legs wth 6-DF. he ost current generator of {D} s of the UPS tpe (Gough-Stewart's platfor, wth P prsatc actuated jont, U for unversal jont and S for sphercal jont), whch has the dsadvantage of usng a prsatc actuator that s not fed on the bass. In lterature, an equvalent echans ests but the generator of {D} s of PUS tpe (wth P prsatc actuated jont). For legs and, the prsatc actuated jonts are n parallel to the vertebral colun whch s harful for the copactness of the echans. he orentaton can be changed but the effcenc decreases consderabl. For leg, the frst revolute jont (located on the base) s actuated. hus, we have changed the tpe of legs and, b a US tpe (wth revolute actuated jont) as depcted n Fg. 5. In our project, we have bult an eel robot wth vertebrae and an overall length of 5, as shown n Fg. 6 (wth the head and the tal ncluded). Each vertebra wll have an ellptc secton of 5 and focal dstance respectvel and wll be a thck. he workspace analss s needed to avod collson and to check the jont lts. Fgure 6: wo vertebrae of the prototpe under constructon at ICCN [ A ] = [ a b c] and [ A ] [ ] a b c fed = () he lengths a, b, and c wll be chosen b the stud of the Jacoban atr n the net subsecton. he oble platfor wll be rotatng around pont that s the orgn of the oble frae, noted oble. he orentaton of oble (,,, ) s defned so that () plane s the plane defned b ponts C and, () -as s drected fro to C and () -as s drected fro to C. Let be the vector of jont coordnates assocated wth the actuated revolute jonts. he orentaton of the oble platfor wth respect to fed frae base s defned b the "ollng Ptchng Yaw" paraeters (PY) where the frst paraeter s the orentaton angle 3 of the frst revolute jont of leg ). fed = [ ] 3 fed (, ) ( ', φ) ( '', ψ) oble = 3 he angles 3, φ and ψ are assocated wth the followng cascaded rotatons () a rotaton of angle 3 around -as, () a rotaton of angle φ around the '-as (obtaned fro the prevous rotaton and whose as s the as of the second revolute jont of leg ), () a rotaton of ψ around the ''-as (obtaned fro the second rotaton and whose as s the as of the thrd revolute jont of leg ). Jacoban atrces o charactere the sngular confguratons, we wll use an nvarant for, whch allows our results to be applcable to an archtecture studed here. hus, there s no proble of sngulart of transforaton n the rotaton atr between fed and oble. 3 Coprght #### b ASME

5 We wrte the Chasles's relaton on ( ) c b to have ( ) = ( ) + ( ) ( ) c b c o o a b a () r r.( ω p ) = r.( l ( &. )) In ths equaton, all the vectors are epressed n splf calculatons, we set fed ( ), ( ), ( ) and ( ) r = c b p = c o b = o a l = b a B dfferentatng Eq. () wth respect to te, we obtan, wth. o r& = p& & l (3) fed [ ] = [ ] p p (4) oble fed oble Dfferentatng wth respect to te, we fnd fed [ p ] = Q& [ p ] & (5) oble fed oble snce vector [ p ] oble s a constant vector when epressed n frae oble. Moreover, the te dervaton of the rotaton atr can be wrtten as Q& = Ω Q (6) where Ω s the angular veloct tensor. Fnall, fro Eqs. () and (6), we get p& =Ω p = ω p where denotes the cross product of the two vectors and ω s the angular veloct vector. We note and, the unt vectors passng through the as of the frst revolute jont of legs and, respectvel. Moreover, we can wrte vector l& as functon of angular veloctes & and & & (&. ) & l = l (. ) and & l = l hus, Eq. (3) can be wrtten n the for r& = ω p l (&. ) We ultpl the precedng equaton b r because r. r& =. hus, we have ( p r). ω = ( l r).(&. ) hese two equatons can be cast n vector for wth Aω + Bq& = (7) ( p r) ( ) A = p r (8) ( l r). B = ( l r). (9) and & 3 q = & & & hen, when B s not sngular, the nverse Jacoban atr s wrtten, Sngular confguratons ( ) p r ( l r). ( ) J = p r ( l r). he parallel sngulartes occur when the deternant of the atr A vanshes,.e. when det( A ) =. In such confguratons, t s possble to ove locall the oble platfor whereas the actuated jonts are locked. hese sngulartes are partcularl undesrable because the structure cannot resst an force or torque. Fro Eq. (7), we have ( ) ( ) p r p r or ( p r ) = or ( p r ) = 4 Coprght #### b ASME

6 It s equvalent to have B, B and coplanar or to have ( B, ) or ( B, ) algned, as depcted n Fg. 7. B A l p p l r l r p p r r l A (b) (a) Condton nuber and sotropc confguratons he Jacoban atr s sad to be sotropc when ts condton nuber attans ts nu value of one []. he condton nuber of the Jacoban atr s an nterestng perforance nde, whch characteres the dstorton of a unt ball under the transforaton represented b the Jacoban atr. he atr A s sotropc when p r and p r and ( p r ) ( p r ) and r = p = r = p = he atr B s sotropc and equal to the dentt atr when l r and l r and l and l and r = l =. Fgure 7: Parallel sngulart when (a) B, B and are coplanar and (b) B and are algned Seral sngulartes occur when the deternant of the atr B vanshes,.e. when det( B ) =. At a seral sngulart, an orentaton ests along whch an angular veloct cannot be produced. Fro Eq. (8), we have ( ). l r = or ( ). ( l r ). = or r l = or r l ( ). = It s equvalent to have () l and r algned, or () l and r algned, or () r and algned, or (v) r and algned, as depcted n Fg.8. r p p r p p Fro these condtons, n [6], we have solated three cases. he frst soluton s the echans depcted n Fg. 5 that we could call "parallel aes". Equaton gves the locaton of ponts A and A n base for a unt echans, a =, b =, c = If ths soluton adts an sotropc confguraton, the behavor n forward swng leads to use legs and sultaneousl. When we appl as nput veloct & = [ ], the angular veloct obtaned s ω= [ ]. hs eans that we aplf the rotatonal oton just after havng used a reducton gear on the rotar otor to ncrease the avalable torque. hus, the length of the otors s constraned b the shape of the cross secton of the eel, as depcted n Fg. (a). he second soluton, called "orthogonal aes", s to place and orthogonall as depcted n Fg. 9. he locaton of ponts A and A n base concdes wth pont. l l (a) r l r A A (b) Fgure 8: Seral sngulart when (a) l and r are algned and (b) l and are algned 5 Coprght #### b ASME

7 3 oton s dstrbuted. However, onl A can be sotropc because we have ( l ). r = for =, Equaton gves the locaton of ponts A and A n a unt echans, base for Fgure 9: Sphercal wrst wth orthogonal actuators In ths case, the drect and nverse kneatc odels are spler but t s ore dffcult to place the otors of legs and, as shown n Fg. (b). Moreover, there also ests an angular aplfcaton factor n the forward swng. A a =, b =, c = Concernng the ntegraton nto the cross-secton of the eel, the placeent s less constraned, as shown n Fg.. A Motors Motors A Motors A Fgure : Placeent of the otors and the legs for sphercal wrst wth parallel actuators (a) Fgure : Placeent of the otors and the legs for (a) the "parallel aes" and (b) the "orthogonal aes" (b) Drect and nverse kneatc odels he drect kneatc odel can be wrtten when we know the poston of B and C. hus, we have, he last soluton has parallel actuators and ther aes ntersect the -as, as depcted n Fg.. When the eel robot s swng, the angular veloct of the actuated jonts of legs and s equal to aw veloct. 3 Fgure : Sphercal wrst wth parallel actuators hs eans that for the forward or backward swng, the kneatc odels are sple and the torque needed for the A C S B = + and n or n oble, fed C S B = + C = [ ] C = [ ] CC 3 φ C = S3Cφ S φ wth C = cos( ), S sn ( ) CSS 3 φ ψ SC 3 ψ C = S3SφSψ + C3Cψ CS φ ψ = for =,,3 φ = cos( φ), S φ = sn( φ) ψ = cos( ψ ) and S ψ = sn( ψ ). We add the constrant that BC = 6 Coprght #### b ASME

8 CC 3 φ + SC 3 φ C + Sφ + S = () A A CSS 3 φ ψ SC 3 ψ + + SSS 3 φ ψ + CC 3 ψ C + S C S = φ ψ 3 o solve the drect kneatc, we know = [ ] we use the followng substtutons () and (a) A (b) A Q Q sn( φ ) = cos( φ) = + Q + Q hus, we can reark that Eq. depends onl on φ and s a quadratc equaton of Q ( SQ QC3 QCS 3 Q + CS 3 C3 S) ( Q ) = ne soluton s Q =,.e. φ = π /+ kπ that does not depend on the actuated jonts. Fgure 3 depcts the four drect kneatc solutons for =., =., 3 = π / 4. Solutons (a) and (b) are found when Q = and can be easl solated. Fro solutons (c) and (d), onl the second one s sutable, t can be solated b the dot product of r b p. (c) (d) Fgure 3: he four drect kneatc solutons for =., =., = π / 4 3 o solve the nverse kneatc, we use two substtutons, = tan( / ) and S = tan( / ) that allow us to have two quadratc and ndependent equatons as functon of and S respectvel. Fgure 4 shows the four nverse kneatc solutons for 3 = π /4, φ = π /, ψ = π / that we can solate b calculatng l. r and l. r for legs and, respectvel. A A A A Fgure 4: he four nverse kneatc solutons for 3 = π /4, φ = π /, ψ = π / 7 Coprght #### b ASME

9 o conclude, we have four solutons for the drect kneatc and four solutons for the nverse kneatc (two for legs and, respectvel). Mechancal constrants o eplan the echancal constrants, we use a splfed odel depcted on Fg.5 but the nuercal constrants coe fro the real prototpe shown n Fg. 6. l a C X Y Z Fgure 6: Jont lts on unversal jonts L A C he ntersecton between segents B Fgure 5: Splfed odel of a vertebra For a parallel anpulator, the ost coon constrants are: he jont constrants that take nto account the varaton of the length of the legs that contan sldes, whch s not our case. he constrants on the jont lts, whch represent the constrants on the sphercal jonts. he ntersecton between segents, whch wll be the ntersecton of ( ) and ( ) or (B C ) and the leg (the latter s possble because of the echans archtecture). he ntersecton segents wth obstacles, whch wll be defned for our sste b the ntersecton of rods wth the low base. We wll stud the three constrants that est for the echans later on. he jont lts on unversal jonts ur sphercal jonts are lted b a 'la' so C B wll be requred wthn the cones (Fg. 6): he cone center B and a half angle 'La' and havng ts as of setr perpendcular to the rod A B. he cone center C and a half angle 'La' and havng ts as of setr perpendcular to the oble platfor. hese lts have the ost nfluence on the se of the workspace. L B A o avod these ntersectons, we have lted the nu dstance between segents. he ethodolog for fndng the nu dstance s as follows: We take a pont M, whch belongs to. ake M projecton of M on ( ), we easure the dstances M and M. If these two dstances are less than the length of ( ), we defne the dstance between the two segents as dstance M M. We are changng the poston of the M throughout the segent and the nu dstance between segents wll be the nu length M M. hs ethod requres a large coputaton te and there s lttle rsk to eet ths confguraton when the other constrants are valdated. he ntersecton between segents and the base he onl wa to have nternal collsons between a segent and a fed part s to have an ntersecton between segents (A B ) and the base. o elnate the, we lted the angle (=, ) as sn( ) L > l d, ld beng the dstance between A and the base. Fgure 7: he ntersecton between segents (A B ) and the base 8 Coprght #### b ASME

10 he algorth used to deterne the workspace After defnng all the eleents necessar for the deternaton of the workspace, we wll descrbe the algorth used to accoplsh ths work. hs algorth conssts n the followng steps. he upper part of the workspace: Intale two atrces Wϕ u and W, densons u (n ψ /+) n where ϕ n pss + s the odd nuber of equdstant plans ψ wth ψ = [-8 8 ] on whch the workspace wll be calculated, and n ϕ s the nuber of ponts that ust be calculated over each ψ. hese atrces record respectvel values φ and for each pont defned n the upper part of the border of the workspace. Let ψ =, we assue ( ϕc C ) = ( ) s the center of the horontal cross-secton of the workspace for ψ=. For the current ψ, buld a polar coordnate sste ( ϕc C ). Startng wth n ϕ equall spaced angles, and ncreasng the polar radus to solve the nverse kneatc proble, and test the valdt of solutons b checkng the constrants untl a pont appears where at least one constrant s broken. he last vald value (φ, ) s saved n Wϕ u and W as the border of the workspace. u Calculate the geoetrc center ( ϕc C ) of the workspace cross-secton to use t as a center for the net cross-secton. Increase ψ so that ψ = ψ +36 / n ψ whle ψ <36 or the last horontal cross-secton of the workspace s a sngle pont. he lower part of the workspace: Intale two atrces Wϕ l and W, densons l (n ψ /+) n. ϕ Let ψ =. Assue ( ϕc C ) = ( ) s the center of the horontal cross-secton of the workspace for ψ=. Proceed n the sae anner as n the upper part to check the collson. Calculate the geoetrc center ( ϕc C ) of the crosssecton of the workng space that wll serve as the geoetrc centre for the net secton. ake ψ = ψ -36 / n ψ. epeat untl ψ s saller than -8 or the last horontal cross-secton of the workspace s a sngle pont. reatent of values to plot the workspace: Let Wϕ =[ W W ] ϕu ϕ l and W = W W u, we buld the workspace n a polar coordnate sste as ψ ,j [ ] = W[,j] cos( Wϕ[,j]),j [ ] = W[,j] sn( Wϕ[,j]),j [ ] = ψ - (- )(36 /n ) a Fgure 8: Workspace wthout an collsons he cross-secton of the workspace perts to check ts setr for opposton value of ψ as s shown n Fg. 9 for ψ = 8 and ψ = 8. he se of the cross-secton decreases f we chose ψ < 8 or ψ > 8. ψ= Fgure 9: Cross-secton of the workspace for ψ = 8 and ψ = ψ ψ=8 For ψ =, we have the aal se of the cross-secton as shown n Fg.. φ 9 Coprght #### b ASME

11 φ CNCLUSINS In ths paper, we have presented the kneatc equatons as well as the sngular confguratons of a sphercal wrst able to be a part of an eel-lke robot. An algorth s wrtten to copute the workspace wthout an nternal collsons. he lt-and-orson paraeters are used to represent the workspace. hanks to these paraeters, we are able to see ts setrcal propert conversel to Euler paraeters. 4 7 Fgure : Cross-secton of the workspace for ψ = he torson of the oble plat-for s [ 8 8 ] ψ = and the range values of the actuated jonts are: 3 [ 7 38 ] [ 7 38 ] [ ] = = = Wth cad software, we can buld a surface passng through the border of the jont space b flterng the data obtaned wth the nverse kneatcs proble (Fg. ). We can easl appl an offset on ths surface to defne a securt dstance. 3 Fgure : he jont space odellng wth a CAD software 3 33 ACKNWLEDGMENS hs research was partall supported b the CNS ("Angulle" Project) and AN AAM. he author wshes to thank the techncal staff of the laborator to have realed the prototpe, Mchaël Canu, Gaël Branchu, Fabrce Brau and Paul Maulna as well as aster student, Gerges Fadel. EFEENCES [ reb,. and Zrn,., Slart laws of seral and parallel anpulators for achne tools, Proc. Int. Senar on Iprovng Machne ool Perforances, pp. 5-3, Vol., 998. [] Merlet J-P, Parallel obots, Kluwer Acadec Publshers,. [3] Lenarcc J., Stansc, M. M. and Parent-Castell, Kneatc Desgn of a Huanod obotc Shoulder Cople, proceedngs of IEEE Internatonal Conference on obotcs and Autoaton, Aprl 4-8,. [4] Lee K-M. and Arjunan S,. A three-degrees-of freedo crooton nparallel actuated anpulator, IEEE rans. on obotcs and Autoaton, ctobre 99. [5] rantafllou, M. S. and rantafllou G. S., An Effcent Swng Machne, Scentf Aercan, pp. 65-7, March 995. [6] Chablat D. and Wenger P., Desgn of a Sphercal Wrst wth Parallel Archtecture: Applcaton to Vertebrae of an Eel obot, Proc. IEEE Int. Conf. ob. and Autoaton, Barcelona, 3-6 Aprl 5. [7] Bonev, I.A., and u, J., rentaton workspace analss of 6-DF parallel anpulators, Proceedngs of the ASME 999 Desgn Engneerng echncal Conferences, Las Vegas, NV, 999. [8] Cha J.G., Bale S.A.lark J.E., Full.J.utkosk M.., Fast and obust: Heapedal obots va Shape Deposton Manufacturng, he Int. Journal of obotcs esearch, vol., no., pp (4), ctober. [9] Hrose, S., Bologcall nspred robots: Snake-lke locornotors and anpulators, ford Unv. Press, ford, 993. [] McIsaac, K.A., strowsk, J.P., A geoetrc approach to angullfor locooton odellng of an underwater eel robot, IEEE Int. Conf. obot. and Auto., IC999, pp [] D'Août K., Arts P., A Kneatc Coparason of Forward and Backward Swng n the Eeel Angulla Angulla, Journal of Eperental Bolog, pp. 5-5, 999. Coprght #### b ASME

12 [] Carlng J., Wllas. L., Bowtell G., Self-Propelled Angullfor Swng: Sultaneous Soluton of the wo-densonal Naver-Stokes Equatons and Newton's Laws of Moton, 998. [3] So J.C. and Vu-Quoc L., n the dnacs n space of rods undergong large otons - A geoetrcall eact approach, Cop. Meth. Appl. Mech. Eng., 66, 988, pp [4] Boer, F., Prault, D., Fnte eleent of slender beas n fnte transforatons: a geoetrcall eact approach, Internatonal Journal for Nuercal Methods n Engneerng, 4: 59, pp [5] Hervé J. M., Analse structurale des écanses par groupe des déplaceents, Mech. Mach. heor, Vol.3, No.4, pp , 978. [6] Karoua, M., Concepton structurale de ecanses paralleles spherques, hèse de doctorat, I 3-6, École Centrale de Pars, 3. [7] Gosseln C., Hael J.F., he agle ee: a hgh perforance three-degreeof-freedo caera-orentng devce, IEEE Int. conference on obotcs and Autoaton, pp San Dego, 8-3 Ma 994. [8] Bdault, F., eng. P. and Angeles, J., Structural optaton of a sphercal parallel anpulator usng a two-level approach, Proc. ASME Desgn Engneerng echncal Conferences, Pttsburgh, PA, Sept. 9- D-M DE/DAC-3,. [9] Gosseln C., St-Perre E. and Gagné M., n the developent of the agle ee: echancal desgn, control ssues and eperentaton, IEEE obotcs and Autoaton Socet Magane, Vol. 3, No. 4, pp. 9-37, 996. [] Golub, G. H. and Van Loan. F., Matr Coputatons, he John Hopkns Unverst Press, Baltore, 989. Coprght #### b ASME