Knematc evaluaton and fowad knematc poblem fo Stewat platfom based manpulatos Domagoj Jakobovc Facult of Electcal Engneeng and Computng Unska 3, Zageb Coata domagoj.jakobovc@fe.h Abstact In ths pape two methods ae pesented. he fst s a knematc evaluaton method fo two dffeent heapod stuctues: standad Stewat platfom manpulato wth etensble stuts and second stuctue wth fed stut lengths but sldng gudewas on fed platfom. he second method addesses the fowad knematc poblem whee dffeent mathematcal epesentatons ae combned wth vaous optmaton algothms to fnd a sutable combnaton that ma be utled n eal-tme envonment. Addtonall, we note the estence of equvalent tajectoes of the moble platfom and suggest an adaptaton to the solvng method that, havng satsfed cetan assumptons, s able to successfull solve the fowad knematc poblem n eal-tme condtons wth ve hgh pecson. I. INRODUCION Paallel knematc manpulatos (PK) have been edscoveed n the last decade as mcopocesso s powe fnall satsfes computng foce equed fo the contol. Its geat paload capact, stffness and accuac chaactestc as esult of the paallel stuctue make them supeo to seal manpulatos n man felds. One of the most accepted PK s Stewat platfom based manpulato, also known as heapod o Gough platfom. Heapod, ognall, conssts of two platfoms, one fed on the floo o celng and one moble, connected togethe va s etensble stuts b sphecal o othe tpes of jonts. hat constucton gves moble platfom -DOF (degee of feedom). Heapod movement and contol s acheved onl though stut lengths changes. One vaaton to ths stuctue, also obseved hee, s when stuts ae fed n length but one of the ends s placed on gudewa. Contol s then obtaned onl b movng those jonts on gudewas. Although n ths model the foces actng on stuts aen t just along the as of the stuts, lke wth the ognal Stewat s desgn, pactcall attanable sldng chaactestcs of gudewas make t ve consdeable stuctue fo manpulatos. One of the qualtes we want fom a manpulato s ts good knematc chaactestcs. hose chaactestcs have dect mpact on manpulablt and wokng speed of a manpulato. One pat of ths pape pesents a method fo calculatng seveal knematc paametes. he method can be used to optme heapod stuctue fo bette knematc chaactes tcs o combned wth othe methods wee knematc can be just one measue n optmaton pocess. Second pat of ths pape deals wth fowad knematcs. he fowad knematc [7] of a paallel manpulato s the poblem of fndng the poston and oentaton of the moble platfom when the stut lengths ae known. hs poblem has no known closed fom soluton fo the most geneal - fom of heapod manpulato (wth s jonts on the base and s on the moble platfom). In ths wok seveal mathematcal epesentatons of the fowad Leonado Jelenkovc Facult of Electcal Engneeng and Computng Unska 3, Zageb Coata leonado.jelenkovc@fe.h knematc poblem, n the fom of optmaton functons, ae combned wth vaous optmaton algothms and adaptaton methods n ode to fnd an effcent pocedue that would allow fo pecse fowad knematc solvng n eal-tme condtons. II. KINEAIC EQUAIONS Standad Stewat Platfom based manpulato as shown n Fg. can be defned n man was but most common set of paametes ae: mnmal and mamal stuts length (l mn, l ma ad of fed and moble platfoms (, jont placement defned wth angle between closest jonts fo both platfoms (α, β) and jont movng aea (assumng cone wth angle γ). b O q w t A p O B Fg.. Stewat Platfom manpulato Fo knematc evaluaton we need elaton between actuatos speed and end effecto speed. Obsevng one vecto chan though th stut (Fg. the followng equaton can be deducted: v b + q w t + p () Snce w b s unt vecto and devaton of eq. t () elds eq. ( whee v andω ae lnea and angula end effecto veloctes. q& w + q ω w v + ω p () Eq. () can be easl tansfomed n fom of eq. (3) and then fnall n mat fom as on eq. (4). We have a knematc equaton, whee elaton between end effecto veloct and actuato veloct (stut lengths changes) s gven.
w q & w q& v w + ω (3) ( p ) w v q & J & ω he second obseved heapod model, shown n Fg., dffes fom standad Stewat manpulato at base platfom and stuts. Stut lengths ae constant and same fo all stuts but the jonts on one sde ae placed on sldng gudewas whee actuatos ae placed. Paametes whch descbe ths model dffe onl fo base platfom: B k, and B p, (4) defne th gude wa and t as value between [, ] dentf actual jont poston. If we obseve models lke on Fg., those vectos can be defned usng fou paametes: d as dstance between close paallel gude was, and as ad of ccles whee gude was ends ae placed wth heght dffeence h, as shown on Fg. 3. β O O m A B k, Fg.. Heapod wth fed stut lengths l Ltool b B B p, s s calculated fom quadatc equaton and theefoe can gve two possble jont poston on same gude wa. hs poblem must be solved n contol pocedues. Fom Fg., fo one vecto chan though th stut, the followng equaton can be deducted: b s l + q w t + p + () Devaton of eq. () elds eq. (7 and wth lttle moe mathematcal opeatons we get knematc equaton (8) ve smla to fst heapod model. s & l + q ω w v + ω p (7) w l L s& w O L w l s& w L s& K & s & L K & J & v ω (8) End effecto (tool) s placed on moble platfom wth heght l tool. heefoe, ogn of local coodnate sstem of moble platfom s put n that pont. Usng nvese knematc fo an pont and tool oentaton, t s possble to compute moble platfom poston and oentaton. Stuts lengths fo fst, o jont postons fo second model can then be calculated. If stut lengths ae wthn gven anges, o jonts can be placed on gudewas fo nd model, and othe constants ae fulflled, as jont angle constant and no collson between stuts, than heapod s capable of puttng ts end effecto n gven pont wth gven oentaton. In ths wa aea eachable wth gven oentaton the wokng aea, can be found b fndng all ponts whch satsfes all constants. Assumng that manpulato s used fo machnng fee suface peces, wokng aea can be bette defned as aea wee manpulato can wok fo not just one but an equed oentaton. Requed oentatons whch gve optmal suface chaactestcs can usuall be defned wth vectos wthn a cone wth defned angle as n Fg. 4. Wokng aea calculated usng ths defnton gves supeo vsual and numec descpton of manpulato. P d h Fg. 3. Paametes that defne heapod stuctue Invese knematc fo ths model s slghtl moe comple than standad heapod and can be computed usng equatons: A B t + p, l B + s p, d ( A, B ( B B ) k, p, (5) ϑ L s ϑma Fg. 4. Oentatons used n calculatons
When dealng wth -DOF heapod manpulatos, whch on ts end effecto have tool on spndle, nvese knematc can t geneall gve unque esult. hs gves feedom to apo choose otaton angle of movng platfom as the th DOF. Fo smplct, no otaton angle was used wheneve such oentaton was feasble. III. KINEAIC PARAEERS As equatons (4) and (8) show, elaton between end effecto veloctes and stut changes s gven b a mat commonl called jacoban. Knematc chaactestcs must theefoe be etacted fom that mat. Commonl used values fo knematc evaluaton of manpulato ae sngula values of jacoban [], []. Sngula values of mat A ae calculated b fomula shown n eq. (9 whee? s -th egenvalue of A. ( A ) A, {,..} σ (9) λ Geometcal meanng of sngula values can be vewed tough equatons () and (). A ma ma () A σ σ mn A σ ma, () If s unt vecto then A s a hpeellpsod wth sngula values as as length values. If A s jacoban mat and s veloct, hpeellpsod epesent ablt to geneate end effecto s veloctes n gven dectons. Hpeellpsod volume s popotonal to detemnant of a jacoban. Rato between mamum and mnmum sngula value s measue fo homogenet. he sma lle that ato s, the ablt of geneatng speed s less dependent of decton. Pactcall, the geate the sngula value s, the geate stut change s needed fo achevng same end effecto movement fo that patcula decton, and vce vesa, the smalle sngula value means less actuato actvt s needed fo end effecto movement. Ve small sngula values can cause bg poblems. If ve small actuato movement s enough to move end effecto, than even eos as a esult of mpefect mateal, tempeatue and pessue dlataton, can have bg nfluence on end effecto path. In othe wods, end effecto can t be contolled well enough. hee paametes based on sngula values ae usuall called fo knematc evaluaton:. condton numbe: κσ ma /σ mn bette smalle values. mnmal sngula value: σ mn bette lage values 3. manpulablt: det(j)? σ bette lage values. he method we popose to evaluate manpulato fom a knematc aspect s to calculate those thee paametes tough whole wokspace of the manpulato o just on some pat of t. Fo eve pont whee calculatons ae to be pefomed, those thee paametes ae calculated not onl fo one end effecto oentaton but fo all oentatons as shown on Fg. 4. he value fo patcula knematc paamete s then calculated as aveage value. It s sometmes appopate to calculate those values on ente volume o just n coss-secton wth vetcal o hoontal plane to fnd a spot whee knematc chaactestcs ae bette. One such eample s on Fg. 5. Fg. 5 Condtonal numbe (left) and mnmal sngula numbe (ght) wo heapod models, fst as on Fg. and second as on Fg., wth paametes shown n ABLE I ae evaluated. ABLE I Heapod model paametes st model nd model paameta value paamete value l ma 85 l 7 l mn 45 75 5 5 h l alat d 3.5 a b 3 ß ß φ ma 45 l alat φ ma 45 Wokng Aea Volume 99 77 Knematc paametes ae calculated ove wokng aea and aveage values ae pesented n ABLE II. ABLE II Aveage knematc paamete values st model nd model κ.835.55 σ mn..5 det(j) 5.84 5.77 Fo compason onl, models poposed n [], [3] and [4] ae evaluated also and pesented n [5]. Some of models have bette paametes than st model shown n ABLE II, but wth at least halved wokng aea volume. IV. HE FORWARD KINEAIC PROBLE he fowad knematc elatons fo a heapod machne can be mathematcall fomulated n seveal was. Eve epesentaton of the poblem can have ts advantages and dsadvantages whch become emphased when a dffeent optmaton algothm s appled. A. he poston and oentaton of the moble platfom In ode to defne a fowad knematc poblem we have to epesent the actual heapod confguaton,.e. the actual poston and oentaton of the moble platfom. he most common appoach utles the thee postonal coodnates of the cente of the moble platfom and thee
angles that defne ts oentaton. he coodnates ae epesented b vecto t : t t t () t and the thee otatonal angles ae hee defned as ollptch-aw angles α, β and γ. he angle values epesent the consecutve otaton about the, and as, espectvel [8]. he heapod geomet s defned wth s vectos fo base and s vectos fo moble platfom, whch defne the s jont coodnates on each platfom: b p b b, p p,,..,. (3) he above vectos ae epesented n local coodnate sstems of the base and moble platfom and ae of constant value. he base and moble platfom ae pesumed to be plana, whch can be peceved fom the coodnate of the jont vectos. he stut vectos l can then be epessed as l b + t + R p,,..,, (4) whee R s the otatonal mat, calculated fom thee otatonal angles. If the poston and oentaton of the moble platfom s known, the length of each stut s q D b, t + R p,,..,, (5) ( ) whee D epesents the Eucldean dstance between the vecto pas. Fo an abta soluton to a fowad knematc poblem,.e. abta poston and oentaton of the moble, the eo can be epessed as the sum of squaes of dffeences between the calculated and actual length values. Havng stated the above elatons, we can defne the fst optmaton functon and the elated unknowns as F D( b t R p ) q, +. () X t t t α β γ [ ] B. he canoncal fomulaton of the fowad knematcs he dea behnd ths appoach [] s to use the elements of the otaton mat, athe than the angle values, to epesent oentaton: R [ n o a] n n n o o o a a a. (7) Wthout loss of genealt we can poston the ogns of the local coodnate sstems of the base and moble platfom at the stut jonts wth nde one, as shown n Fg., whch gves us the followng paamete values: b b p p b p. (8) Zb b b p Zp Yb p 5 p Fg. Postonng of coodnate sstems fo base and moble platfom Afte etensve smplfcatons, the fowad knematc can be epessed as a sstem of 9 equatons wth 9 unknowns. hee of those 9 equatons ae of lnea fom, whch can be used to educe the numbe of vaables wthout ntoducng addtonal complet n the sstem. hee of the s vaables t, n, o, t, n and o can be eplaced wth lnea combnatons of the othe thee, whch leaves us wth onl s unknowns. We used that appoach to defne anothe taget functon as F ( n + n + n ) + ( o + o + o ) ( t + t + t l ) + + + X 3 ( n o + n o + no ) ( n t + n t + n t A t A n A) b Yp Xb ot + o t + ot Bt B + B3o B4t B5n Bo [ n n o o o t ]. p 4 p b 5 n B p3 Xp b3 b (9) and the constants' values can be found n [] o []. In the scope of ths wok some othe poblem fomulatons have been used. hose fomulatons dd not, howeve, show an advantages ove the pevous defned two, so the ae omtted hee. oe nfomaton can be found n []. V. EXPERIENAL RESULS he fowad knematc poblem s pesented as fve (two of whch ae shown hee) optmaton functons fo whch the optmaton algothm has to fnd the mnmum, the value of the functons beng the eo of the estmated soluton. Seveal optmaton methods have been appled to each of the functons n ode to fnd an effectve combnaton whch would allow fo eal-tme applcaton. he algothms appled n ths wok ae Powell's method, Hooke-Jeeves', steepest descent seach, Newton-Raphson's (NR) method, NR method wth constant Jacoban and Fletche-Powell algothm. Solvng of fowad knematc was smulated n statc and dnamc condtons. he goal was to fnd the combnaton whch would eld the best esults consdeng the convegence, speed and accuac. he most pomsng combnatons wee tested n dnamc condtons, whee the
algothm had to tack a peset tajecto of the moble platfom wth as small eo and as lage samplng fequenc as possble. hose combnatons nclude Hooke- Jeeves' and Fletche-Powell algothm wth functon F, but the most successful optmaton method was Newton- Raphson's algothm appled to functon F. In dnamc smulaton, the statng heapod confguaton s known and seves as an ntal soluton. Dung the samplng peod the algothm has to fnd the new soluton, whch wll become the ntal soluton n the net ccle. Seveal heapod movements wee defned as tme dependant functons of the poston and oentaton of moble platfom. One of those tajectoes, heeafte denoted as A, s defned wth ( t) sn t, ( t). cos t, ( t) 7 +.5 sn( t α ( t) 5 sn(.8t () t β ( t) sn + 5 cos( 4t γ t 5 actan t 4, t 4 ( ) ( ). he esults of the dnamc smulaton ae pesented n the fom of a gaph whee eos n the thee otaton angles and thee poston coodnates of the moble ae pctued. he samplng peod was set to ms, whch equals to a H samplng fequenc. he eos shown epesent the absolute dffeence between the calculated and the actual heapod confguaton. Due to the lage numbe of ccles, the eo s defned as the bggest absolute eo value n the last ms, so the gaphs n each pont show the wost case n the last ms of smulaton. he eos ae pesented sepaatel fo angles, n degees, and poston coodnates. he eos fo movement A and Newton-Raphson algothm wth functon F ae shown n Fg. and Fg. 3. Absolute eo. 5. 3. 3 5. 4. 4 5. 5. 5 3 4 Ccle nde Fg. 7 Absolute angle eo (α, β, γ NR algothm wth F, movement A Absolute eo 5. 4. 4. 4 5. 5 3 4 Ccle nde Fg. 8 Absolute coodnate eo (,, NR algothm wth F, movement A he acheved level of accuac s ve hgh as the absolute eo does not eceed - both fo angles and coodnates. Anothe tajecto s deved fom the descbed one b enlagng some of the ampltudes n ( whch s denoted as movement B (the alteed values ae n boldface): ( t) sn t, ( t) ( t) 8 + 3 sn( t), α( t) 55 sn (.8t ) β t ( t) 3 sn + 5 cos( 4t. cos t,, ( t) 5 actan( t 4 t 4. () γ he movement B eos ae shown n Fg. 4. Whle stll low, the eo fo movement B has two dstnctve peaks at cetan ponts n smulated moton. What s the cause of those peaks? athematcal analss has shown ([9], [], []) that thee ma est up to 4 dstnctve solutons fo fowad knematc poblem fo Stewat platfom wth plana base and moble platfom fo the same set of stut lengths. Absolute eo. 9.... 3. 4 3 4 Ccle nde Fg. 9 Absolute angle eo (α, β, γ NR algothm wth F, movement B Let us suppose that n one heapod confguaton thee ests no othe fowad knematc soluton fo actual set of stut lengths, but that n some othe confguaton thee est seveal of them. If heapod n ts movement passes though those two confguatons, then at a cetan pont n between thee has to be a dvson pont whee the numbe of solutons nceases. In those dvson ponts the solvng algothm ma, unfotunatel, begn to follow an of the possble paths, because an of them epesents a vald fowad knematc soluton! hat s eactl the poblem that occus n movement B: the algothm ma o ma not follow the coect tajecto. If the latte s the case, than the absolute eo looks lke n Fg. 5. Absolute eo... 9. 3 4 Ccle nde Fg. Absolute angle eo (α, β, γ NR algothm wth F, movement B - dvson he algothm wll andoml follow ethe the coect tajecto o the equvalent one. It s mpotant to note that n both cases the optmaton functon emans ve low (app. -3 to - ) dung the whole pocess because both tajectoes depct a vald soluton to the fowad knematc poblem. he poblem s, onl one of them epesents the actual heapod confguaton n each pont of tme. Wthout an addtonal nfomaton about the heapod confguaton, such as ma be obtaned fom eta tanstonal dsplacement sensos, thee s unfotunatel no wa to detemne whch of the estent solutons to the fowad knematc poblem fo the same set of stut lengths descbes the actual heapod confguaton. Nevetheless, wth some assumptons we ma devse a stateg that
should keep the solvng method on the ght tack. If the change of the decton of movement s elatvel small dung a sngle peod, whch s n ths case onl ms, then we can t to pedct the poston of the moble platfom n the net ccle. We can use the solutons fom the past ccles to constuct a staght lne and estmate the ntal soluton n the net teaton. Let the soluton n the cuent teaton be P and the solutons fom the last two ccles P and P. hen we can calculate the new ntal soluton usng one of the followng methods: X P P, () X.5 P. 5 P, (3) ( ).5 P + P,.5 ( P + P. (4) X.5.5 he above methods wee tested n conjuncton wth NR algothm and functon F fo all the smulated tajectoes. he esults ae ve good: the solvng method was now able to tack the coect soluton dung the whole smulaton pocess fo all thee estmaton methods. he numbe of conducted epements was seveal hunded and eve tme the algothm's eo magn was below - both fo angles and coodnates. Howeve, the descbed algothm adaptaton wll onl be successful f the assumpton of a small decton change dung a few teatons s vald. o test the algothm's behavou, smulated movement B was acceleated b facto, 4 and 8, whle mantanng the same ccle duaton of ms. Onl b eachng the 8-fold acceleaton, when the total movement tme equals a ve unealstc half a second, dd the algothm poduce sgnfcant eos, whle stll holdng to the coect soluton. VI. CONCLUSION Poposed knematc evaluaton method s pue computatonal and heav tme consumng. odel must be fst defned and than evaluated. Howeve, n egad to most othe methods ths method gves mo e ealstc knematc paamete values, because t use not just one end effecto oentaton but most oentatons that can be asked fo n manufactung. ethod can be easl combned wth othes heapod evaluatons such as wokng aea o/and eo analss gvng moe poweful heapod desgn tool. Combnng seveal epesentatons of the fowad knematc poblem wth optmaton technques, an effcent method fo solvng the fowad knematc was found. he solvng method was able to detemne the eact poston and oentaton of the moble platfom wthn nsgnfcant eo magns (less than to the powe of of the mnmum heapod dmenson) and wth H samplng fequenc. he poblem of equvalent tajectoes was noted: because of the estence of multple solutons to fowad knematcs, thee ma est moe than one path that moble platfom can follow whle havng eactl the same stut lengths n eve pont of the wa. It has to be sad that eve such path epesents an equal coect soluton of the fowad knematcs, but onl one of them epesents the tue moble platfom tajecto. An empcal algothm was devsed whch can ncease the pobablt of fndng the ght soluton, and t poved tself successful n eve test case. Unfotunatel, t cannot be poven that t wll do so n eve magnable movement of the moble platfom. Howeve, the solvng method wll alwas fnd the ght soluton f the change n the poston o movng decton of the moble platfom s e latvel small dung a few samplng peods. VII. ACKNOWLEDGEN hs wok was caed out wthn the eseach poject "Poblem-Solvng Envonments n Engneeng", suppoted b the nst of Scence and echnolog of the Republc of Coata. VIII. REFERENCES [] R.S. Stoughton,. Aa, "A odfed Stewat Platfom anpulato wth Impoved Detet" IEEE ans. on Robotcs and Automaton, vol. 9, no., pp. -73, 993. []. Uchama, et al., "Pefomance Evaluaton of manpulatos usng the Jacoban and ts applcatons to tajecto plannng", n Poc. nd Int. Smp. Robotcs Res., Koto, Japan, 984. [3] K.H. Pttens, R.P. Podhoodesk, "A Faml of Stewat platfoms wth Optmal Detet", J. of Robotc Sstems, (4):43-479, 993. [4]. Huang, D.J. Whtehouse, J. Wang, "he Local Detet, Optmal Achtectue and Desgn Ctea of Paallel achne ools " Annals of the CIRP, vol. 47/, pp.347-35 998. [5] L. Jelenkovc, he Evaluaton And Analss Of Stewat Paallel echansms, Sc thess (n Coatan http://www.ems.fe.h/~leonj/,. [] B. Dasgupta,.S. uthunjaa, "A Canoncal Fomulaton of the Dect Poston Knematc Poblem fo a Geneal - Stewat Platfom", ech. ach. heo, Vol. 9, No., pp. 89-87, 994. [7] J. - P. elet, "Dect Knematc of Paallel anpulatos", IEEE ansactons on Robotcs and Automaton, Vol. 9, No., pp. 84-845, 993. [8] R. P. Paul, Robot anpulatos, he I Pess, Cambdge, 98. [9]. Raghavan, "he Stewat Platfom of Geneal Geomet has 4 Confguatons", Jounal of echancal Desgn, Vol. 5, pp. 77-8, June 993. [] Hust, "An Algothm fo Solvng the Dect Knematc Of Stewat -Gough-pe Platfoms", ftp://ftp.mccm.mcgll.edu/pub/techep/994/ci- 94-.pdf, 994. [] F. Wen, C. Lang, "Dsplacement Analss of the - Stewat Platfom echansms", echansm and achne heo, Vol. 9, No. 4, pp. 547-557, 994. [] D. Jakobovc, he Evaluaton of Poblem Solvng ethods fo Stewat Paallel echansm Knematcs, Sc thess (n Coatan http://www.ems.fe.h/~et/magstej/sc_thess.pdf,.