KINEMATIC AND SINGULARITY ANALYSIS OF THE HYDRAULIC SHOULDER A 3-DOF Redundant Parallel Manipulator

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1 KINEMIC ND SINGULRIY NLYSIS OF HE HYDRULIC SHOULDER -DOF Redundant Parae Manpuator H. Sadjadan and H. D. aghrad dvanced Rootcs and utomated Sstems (RS, Eectrca Engneerng Department, K.N. oos Unverst of echnoog P.O. Bo 6-, ehran, Iran Ema: and Kewords: stract: Parae Manpuator, Knematc Modeng, Forward Knematcs, Jacoan, Snguart nass. In ths paper, knematc modeng and snguart anass of a three DOF redundant parae manpuator has een eaorated n deta. It s known, that on the contrar to seres manpuators, the forward knematc map of parae manpuators nvoves hgh couped nonnear equatons, whose cosed-form souton dervaton s a rea chaenge. hs ssue s of great mportance notng that the forward knematcs souton s a ke eement n cosed oop poston contro of parae manpuators. Usng the nove dea of knematc chans recent deveoped for parae manpuators, oth nverse and forward knematcs of our parae manpuator are fu deveoped, and a cosed-form souton for the forward knematc map of the parae manpuator s derved. he cosed form souton s aso otaned n deta for the Jacoan of the mechansm and snguart anass of the manpuator s performed ased on the computed Jacoan. INRODUCION Over the ast two decades, parae manpuators have een among the most consderae research topcs n the fed of rootcs. parae manpuator tpca conssts of a movng patform that s connected to a fed ase severa egs. he numer of egs s at east equa to the numer of degrees of freedom (DOF of the movng patform so that each eg s drven no more than one actuator, and a actuators can e mounted on or near the fed ase. hese roots are now used n rea-fe appcatons such as force sensng roots, fne postonng devces, and medca appcatons (Meret,. In the terature, most 6 DOF parae mechansms ased on the Stewart-Gough patform are anaed (Josh et a.,. However, parae manpuators wth DOF have een aso mpemented for appcatons where 6 DOF are not requred, such as hgh-speed machne toos. Recent, DOF parae manpuators wth more than three egs have een nvestgated, n whch the addtona egs separate the functon of actuaton from that of constrants at the cost of ncreased mechanca compet (Josh et a.,. Compete knematc modeng and Jacoan anass of such mechansms have not receved much attenton so far and s st regarded as an nterestng proem n parae rootcs research. It s known that unke sera manpuators, nverse poston knematcs for parae roots s usua smpe and straght-forward. In most cases jont varaes ma e computed ndependent usng the gven pose of the movng patform. he souton to ths proem s n most cases unque determned. But forward knematcs of parae manpuators s genera ver compcated. Its souton usua nvoves sstems of nonnear equatons whch are hgh couped and n genera have no cosed form and unque souton. Dfferent approaches are provded n terature to sove ths proem ether n genera or n speca cases. here are aso severa cases n whch the souton to ths proem s otaned for a speca or nove archtecture (Baron et a.,, Meret, 96, Song et a.,, Bonev et a.,. wo such speca DOF constraned mechansms have een studed n (Scano, 99, Fattah et a.,, where knematcs, Jacoan and dnamcs have een consdered for

2 ICINCO - ROBOICS ND UOMION such manpuators. Josh and sa (Josh et a., performed a detaed comparson etween a -UPU and the so caed rcept manpuator regardng the knematc, workspace and stffness propertes of the mechansms. In genera, dfferent soutons to the forward knematcs proem of parae manpuators can e found usng numerca or anatca approaches, or cosed form souton for speca archtectures (Ddrt et a., 98, Dasgupta et a.,. In ths paper, compete knematc modeng has een performed and a cosed-form forward knematcs souton s otaned for a three DOF actuator redundant hdrauc parae manpuator. he mechansm s desgned Haward (Haward, 94, orrowng desgn deas from oogca manpuators partcuar the oogca shouder. he nterestng features of ths mechansm and ts smart to human shouder have made ts desgn unque, whch can serve as a ass for a good epermenta setup for parae root research. In a former stud the authors, dfferent numerca approaches have een used to sove the forward knematc map of ths manpuator (Sadjadan et a., 4. he numerca approaches are an aternatve to estmate the forward knematc souton, n case such soutons cannot e otaned n cosed form. In ths paper, however, the nove dea of knematc chans deveoped for parae manpuators structures (Scano, 99, Fattah et a.,, s apped for our manpuator, and t s oserved that n a sstematc manner the cosed form souton for ths manpuator can aso e otaned n deta. he paper s organed as foowng: Secton contans the mechansm descrpton. Knematc modeng of the manpuator s dscussed n secton, where nverse and forward knematcs s studed and the need for approprate method to sove the forward knematcs s justfed. In secton 4, he Jacoan matr of the manpuator s derved through a compete veoct anass of the mechansm, and fna, n secton, snguart anass s performed usng the confguratondependent Jacoan. MECHNISM ESCRIPION schematc of the mechansm, whch s current under epermenta studes n RS Rootcs La, s shown n Fg.. he moe patform s constraned to spherca motons. Four hgh performance hdrauc pston actuators are used to gve three degrees of freedom n the moe patform. Each actuator ncudes a poston sensor of LVD tpe and an emedded Ha Effect force sensor. he four ms share an dentca knematc structure. passve eg connects the fed ase to the movng patform a spherca jont, whch suppresses the pure transatons of the movng patform. Smpe eements ke spherca and unversa jonts are used n the structure. compete anass of such a carefu desgn w provde us wth requred characterstcs regardng the structure tsef, ts performance, and the contro agorthms. From the structura pont of vew, the shouder mechansm whch, from now on, we ca t "the Hdrauc Shouder" fas nto an mportant cass of rootc mechansms caed parae roots. In these roots, the end effector s connected to the ase through severa cosed knematc chans. he motvaton ehnd usng these tpes of root manpuators was to compensate for the shortcomngs of the conventona sera manpuators such as ow precson, stffness and oad carrng capat. However, the have ther own dsadvantages, whch are man smaer workspace and man snguar confguratons. he hdrauc shouder, havng a parae structure, has the genera features of these structures. It can e consdered as a shouder for a ght weghed seven DOF rootc arm, whch can carr oads severa tmes, ts own weght. Smpe eements, used n ths desgn, add to ts ghtness and smpct. he workspace of such a mechansm can e consdered as part of a spherca surface. he orentaton anges are mted to var etween -π/6 and π/6. No sensors are avaae for measurng the orentaton anges of the movng patform whch justfes the mportance of the forward knematc map as a ke eement n feedack poston contro of the shouder wth the LVD poston sensors used as the output of such a contro scheme. KINEMICS Fg. depcts a geometrc mode for the hdrauc shouder manpuator whch w e used for ts knematcs dervaton. he parameters used n knematcs can e defned as: = C, p = CP d = PP α : he ange etween C 4 and C : Center of the reference frame P : Center of the movng pate : ctuator engths =,,, 4 P : : Movng endponts of the actuators Fed endponts of the actuators 4 k = PP 4 6

3 KINEMIC ND SINGULRIY NLYSIS OF HE HYDRULIC SHOULDER: -DOF Redundant Parae Manpuator Z 4 {B} Y 4 d P k P P X 4 p Z Fgure : he hdrauc shouder n movement wo coordnate frames are defned for the purpose of anass. he ase coordnate frame {}: s attached to the fed ase at pont C (rotaton center wth ts -as perpendcuar to the pane defned the actuator ase ponts 4 and an -as parae to the sector of ange C 4. he second coordnate frame {B}: 44 4 s attached to the center of the movng patform P wth ts -as perpendcuar to the ne defned the actuators movng end ponts (P P aong the passve eg. Note that we have assumed that the actuator fed endponts e on the same pane as the rotaton center C. he poston of the movng patform center P s defned : p = [ p, p, p ] ( so, a rotaton matr s used to defne the orentaton of the movng patform wth respect to the ase frame: = R B = R (θ R (θ R (θ + + whereθ, θ, θ are the orentaton anges of the movng patform denotng rotatons of the movng frame aout the fed,, and aes respectve. so and s θ denote cos(θ and sn(θ respectve. Wth the aove defntons, the 4 4 transformaton matr B s eas found to e: p B = ( Hence, the poston and orentaton of the movng patform are compete defned s varaes, from whch, on three orentaton ange, θ, θ are ndependent specfed as the task space varaes of the hdrauc shouder. ( {} X C Fgure : geometrc mode for the hdrauc shouder manpuator. Inverse Knematcs In modeng the nverse knematcs of the hdrauc shouder we must determne actuator engths ( as the actuator space varaes gven the task space varae, θ, θ as the orentaton anges of the movng patform. Frst, note that the passve eg connectng the center of the rotaton to the movng patform can e vewed as a -DOF open-oop chan defnng three jont varaes θ, θ, andθ as the jont space varaes of the hdrauc shouder. Hence, appng the Denavt-Hartenerg (D-H conventon, the transformaton B can aso e wrtten as: B = ( θ. ( θ. ( θ. B (4 he D-H transformaton matrces j are computed usng the coordnate sstems for the passve eg n Fg., accordng to the D-H conventon. s shown n Fg., the as of frame {} ponts aong the frst jont as of the passve eg; the frst nk frame s attached to the frst movng nk wth ts as pontng aong the second jont as of the passve eg; the second nk frame s attached to the second movng nk wth ts as pontng aong the thrd jont as of the passve eg; and the thrd nk frame s attached to the movng patform n accordance wth the D-H conventon. Usng the aove frames, the D-H parameters of the passve eg are found as n ae (. ae : D-H Parameters for the passve supportng eg α a d θ 9 θ 9 θ 9 θ B p α Y 4 4 7

4 ICINCO - ROBOICS ND UOMION {B} P P X 4 Y X Z Z 4 {} Y Y Z X α X C X 4 Z Y P Fgure : D-H Frame attachments for the passve supportng eg Z 4 Usng the D-H parameters n ae (, the D-H transformaton matrces n (4 can e found as: =, = ( =, B = p Susttutng ( nto (4 eds: p B = (6 Where: + + R B = (7 and: [ ] p p p p = (8 For the nverse knematcs, the three ndependent orentaton anges n ( are gven. Hence, equatng ( to (7 eds: θ = cos ( R p Y 4 = cos ( B(, Once θ s known, we can sove forθ and θ as: (, (, θ = tan (, ( nd (, (, θ = tan (, ( Provded that s θ.havng the jont space varaes θ, θ, and θ n hand, we can eas sove for the poston of the movng patform usng (8. d k (9 Now, n order to fnd the actuator engths, we wrte a knematc vector-oop equaton for each actuated eg as: B L =. s = p+ p a ( where s the ength of the th actuated eg and s s a unt vector pontng aong the drecton of the th actuated eg. so, p s the poston vector of the movng patform and R B s ts rotaton matr. Vectors a and B p denote the fed end ponts of the actuators ( n the ase frame and the movng end ponts of the actuators respectve, wrtten as: a ( snα cosα = =, and ( snα cosα ( snα cosα ( snα cosα a = =, a = =, a 4 4 = =, ( d k ( =, B p ( B p = d k, (4 Hence, the actuator engths can e eas computed dot-mutpng ( wth tsef to ed: B B L. L = = [ p+ p a ] [ p+ p a ] ( Wrtng ( four tmes wth the correspondng parameters gven n (7,(8,( and (4, and smpfng the resuts eds: = ( + k (6-a k + k + k + k4 c θ c θ = ( + k (6- k + k k k4 c θ = ( + k (6-c k k k + k4 c θ = ( + k (6-d 4 k k + k k4 c θ where: k = + d + ( p k k = ( p k cos( α k = ( p k sn( α (7 k 4 = d sn( α k = d cos( α Fna, the actuator engths are gven the square roots of (6, edng actuator space varaes as the unknowns of the nverse knematcs proem.. Forward Knematcs Forward knematcs s undouted a asc eement n modeng and contro of the manpuator. In forward knematc anass of the hdrauc shouder, we sha fnd a the posse orentatons of the movng patform for a gven set of actuated eg engths. Equaton (6 can aso e used for the 8

5 KINEMIC ND SINGULRIY NLYSIS OF HE HYDRULIC SHOULDER: -DOF Redundant Parae Manpuator forward knematcs of the hdrauc shouder ut wth the actuator engths as the nput varaes. In fact, we have four nonnear equatons to sove for three unknowns. Frst, we tr to epress the movng patform poston and orentaton n terms of the jont varaes θ, θ, and θ usng (7-(8. s t s ovous from (6, the on unknowns are the jont varaes θ, θ, andθ, snce actuator engths are gven and a other parameters are determned the geometr of the manpuator. Hence, we must sove the equatons for s unknowns from whch on three are ndependent. Summng (6-a and (6- we get: + = k + k + k (8 Smar addng (6-c and (6-d eds: + 4 = k k + k (9 Sutractng (9 from (8, we can sove for c θ as: + 4 = ( 4k Susttutng ( nto the trgonometrc dentt s θ + =, we get: = ± ( Havng and c θ n hand, we can sove for s θ from (8 as: + k k = k ( Smar: θ o sove for the remanng unknowns, and s θ, we sum (6- and (6-c to get: + = k k + k (4 Havng computed s θ and s θ, we otan: k + k = ( k nd fna: = ± (6 Hence, the jont space varaes are gven : θ = a tan (, θ = a tan (, (7 θ = a tan (, so, the movng patform poston p and orentaton R B are found usng (7-(8. he fna step s to sove for the orentaton anges θ, θ and θ usng ( whch competes the souton process to the forward knematcs of the hdrauc shouder. It shoud e noted that there are some addtona erroneous soutons to the forward knematcs as stated aove due to severa square roots nvoved n the process. hese soutons must e dentfed and omtted. nother mportant assumpton made n our souton procedure was that a four actuator fed = ± s ( endponts are copanar, just as the actuator movng endponts. 4 JCOBIN NLYSIS he Jacoan matr of a -DOF parae manpuator reates the task space near or anguar veoct to the vector of actuated jont rates n a wa that t corresponds to the nverse Jacoan of a sera manpuator. In ths secton, we derve the Jacoan for the Hdrauc shouder as a ke eement n snguart anass and poston contro of ths manpuator. For the Jacoan anass of the Hdrauc shouder, we must fnd a reatonshp etween the anguar veoct of the movng patform, ω, and the vector of eg rates as the actuator space varaes, = [ ] 4, so that: = Jω (8 From the aove defnton, t s eas oserved that the Jacoan for the Hdrauc shouder w e a 4 rectanguar matr as epected, regardng the mechansm as an actuator redundant manpuator. Usng the same dea of mappng etween actuator, jont and task space, we fnd that the Jacoan depends on the actuated egs as we as the passve supportng eg. herefore, we frst derve a 4 6 Jacoan, J, reatng the s-dmensona veoct of the movng patform, v, to the vector of actuated eg rates,. hen, we fnd the 6 Jacoan of the passve supportng eg, J p he Jacoan of the Hdrauc shouder w e fna derved as: J = J J p (9 4. Jacoan of the actuated egs he Jacoan of the actuated egs, J, reates the sdmensona veoct of the movng patform, v, to the vector of actuated eg rates,, such that: = Jv ( We can wrte the s-dmensona movng patform veoct as: p v = = [ p ] p p ω ω ω ( ω where p s the veoct of the movng patform center and ω s the anguar veoct of the movng patform. Dfferentatng the knematc vector-oop equaton ( wth respect to tme we get: B s + (ω s = p + ω p ( where ω s the anguar veoct of the th eg wrtten n the ase frame. Dot mutpng ( s we have: B = s p +( R p s ω ( B 9

6 ICINCO - ROBOICS ND UOMION wrtng the aove equaton four tmes for each actuated eg and comparng the resut to ( gves the actuated egs Jacoan as: B s ( p s B s ( p s J = (4 B s ( R B p s B s4 ( p s Jacoan of the passve eg In order to fnd the manpuator Jacoan, we need to fnd a reatonshp etween the s-dmensona veoct vector of the movng patform, v, and the anguar veoct of the movng patform, ω. Frst, dfferentatng (8 wth respect to tme we get: p p p = θ θ ps ( p p where θ = [ θ ] θ θ s the vector of passve jont rates. he anguar veoct of the movng patform can aso e epressed as: ω = R B (6 Susttutng from (7 and computng (6, we have: ω = θ (7 Sovng (7 for θ and susttutng t n ( eds: p p p = p p ω (8 p p Compementng the aove equaton wth the dentt map ω = Iω, we fna otan: p v = = J pω ω (9 where: p p p p p p J p = (4 6 Havng J and J p n hand, the Hdrauc shouder Jacoan, J 4 w e eas found usng (9. SINGULRIY NLYSIS s shown n the prevous secton, the near veoctes of the actuators are reated to the anguar veoct of the movng patform ω (8, n whch J was the 4 Jacoan matr of the hdrauc shouder. Snguartes w occur f the Jacoan rank s ower than three, the numer of DOF of the movng patform, or equvaent f: det( J J = (4 Such a case occurs on f the determnants of a mnors of J are dentca ero. hese square mnors correspond to the Jacoan matrces of the hdrauc shouder wth one of the actuatng egs removed. herefore, the redundant manpuator w e n a snguar confguraton on f a the nonredundant structures resuted suppressng one of the actuatng egs are n a snguar confguraton. Such a case w not occur n the workspace of the hdrauc shouder owng to the specfc and carefu desgn of the mechansm (Haward, 94. In fact, one of the remarkae features of addng the fourth actuator s the emnaton of the oc of snguartes. Fg.(, shows the determnants of the four mnor Jacoan matrces, computed n the workspace of the manpuator where DM-DM4 denote the nonredundant mnor determnants. hs can e of nterest n cases where one of the actuators mafunctons. It can e seen that the posst of gettng nto a snguar confguraton s ncreased when ether of the redundant actuators are removed. Fgure 4: Mnor Determnants for the non-redundant structures 6 CONCLUSIONS In ths paper, knematc modeng and snguart anass of a -DOF actuator redundant parae

7 KINEMIC ND SINGULRIY NLYSIS OF HE HYDRULIC SHOULDER: -DOF Redundant Parae Manpuator manpuator has een studed n deta. he cosed form souton to the forward knematc s otaned usng a vector approach consderng the ndvdua knematc chans nherent n such parae mechansms. It s proposed to consder sutae mappng etween actuator, jont and task spaces n oth knematc and Jacoan modeng of the manpuator. he proposed method paves the wa for the feedack poston contro of the manpuator, usng a cosed-form souton to the forward knematcs and eavng out the appromaton errors nherent n numerca dentfcaton methods. It s aso shown that the forward knematcs map provdes us wth some etra soutons whch shoud e regarded proper. Snguart anass was aso performed usng the anatc Jacoan otaned for the mechansm. he manpuator workspace was shown to e free of snguartes due to the redundanc n actuaton. Future work w consder Dnamc anass of the hdrauc shouder manpuator. REFERENCES J.P. Meret, St a ong wa to go on the road for parae mechansms, SME DEC Conference, Montrea, Canada, J.P. Meret, Parae Roots: Open proems, In 9th Int'. Smp. of Rootcs Research, Snowrd, 9- Octoer 999. S. Josh and L.W. sa, he knematcs of a cass of - DOF, 4-Legged parae manpuators, rans. Of the SME, vo., March, -6. S. Josh and L.W. sa, comparson stud of two -DOF parae manpuators: One wth three and the other wth four supportng egs, IEEE rans. On Rootcs & utomaton, vo. 9, No., pr, -9. L.Baron and J. ngees, he knematc decoupng of parae manpuators usng jont-sensor data, IEEE rans. On Rootcs & utomaton, vo. 6, No. 6, Dec, J.P.Meret, Cosed-form resouton of the drect knematcs of parae manpuators usng etra sensors data, IEEE Int. Conf. On Rootcs & utomaton, 99, - 4. J.P.Meret, Drect knematcs of panar parae manpuators, Int. Conf. On Rootcs & utomaton, Mnnesota, pr 996, S.K.Song and D.S.Kwon, Effcent formuaton approach for the forward knematcs of the -6 Stewart-Gough patform, Int. Conf. on Integent Roots and Sstems, Hawa, US, Oct., Bonev, I.., Ru, J., Km, S.-G., and Lee, S.-K., cosed-form souton to the drect knematcs of near genera parae manpuators wth optma ocated three near etra sensors, IEEE rans. On Rootcs & utomaton, vo. 7, No., pr, B. Scano, he trcept root: nverse knematcs, manpuat anass and cosed-oop drect knematcs agorthm, Rootca, vo. 7, pp.47-44, June Fattah and G. Kasae, Knematcs and dnamcs of a parae manpuator wth a new archtecture, Rootca, vo. 8, pp. -4, Sept.. O.Ddrt, M.Pettot and E.Water, Guaranteed souton of drect knematc proems for genera confguratons of parae manpuators, IEEE rans. On Rootcs & utomaton, pr 998, B. Dasgupta,.S. Mruthunjaa, he Stewart patform manpuator: a revew, Esever Scence, Mechansm & Machne theor,,-4. Haward, V. and Kurt, R.: Modeng of a parae wrst mechansm wth actuator redundanc, Int'. J. Laorator Rootcs and utomaton, VCH Pushers, Vo. 4, No..99, Haward, V.: Desgn of a hdrauc root shouder ased on a comnatora mechansm Epermenta Rootcs III: he rd Int' Smposum, Japan Oct Lecture Notes n Contro & Informaton Scences, Sprnger-Verag, 97-. Haward, V.: Borrowng some desgn deas from oogca manpuators to desgn an artfca one n Roots and Boogca Sstem, NO Seres, Sprnger-Verag, 99, -48. H. Sadjadan and H.D. aghrad, Numerca Methods for Computng the Forward Knematcs of a Redundant Parae Manpuator, IEEE Conf. on Mechatroncs and Rootcs, pp 7-6, Sept. 4, achen, German.