Forward Kinematics of a Stewart Platform Mechanism

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1 wd emcs f Sew Pfm Mechsm mgj Jkvć cu f ecc geeg d mug Usk 0000 Zge dmgjjkvc@feh sc Pe kemc mus sses sme hee dvges ve he se cues such s gd ecs d hghe wkd weve he mveme c s esced vese kemcs ecuse f he cme f he fwd kemcs es The m f hs wk s cme dffee mhemc eeses f he fwd kemcs em wh vus m ghms d fd sue cm h m e ued e-me evme dd we e he esece f equve jeces f he me fm d sugges d he svg mehd h hvg ssfed ce ssums s e successfu sve he fwd kemcs em e-me cds wh ve hgh ecs I INTOUTION The Sew fm mechsm m efeed s hed s e kemc sucue h c e used s ss f ced m wh degees f feedm (df) such s mufcug cesses d ecse muve sks The mechsm sef csss f s fm (se fm he se) d me fm h e ceced v s sus mued uves js The sus hve -u mechsm h ws chgg he egh f ech dvdu su The desed s d e f he me fm s cheved cmg he eghs f he s sus sfmg he s s df hee s d hee e es The eghs f he sus c f cuse e chged deede u such fsh h he hed csuc ws Pe mus hve eceved cesg e he s w decdes due he hee dvges ve he cve seque cues These cude mg hes me us mechc sucue d hghe se fequeces The segh mss s s fvu f e mus whch ws f eve ge wkds Whe he sg e se mechsms s ccumued hugh sees f ks hs s he cse wh e mus whch e cseque ce f efmg ve ecse sks O he he hd hese mechsms hve eve sm wksce med wh mmum su eghs d ge vues he js s we s he dmess weve he m dffcu wh e mus s he cme f cg he mveme The em f vese kemcs f hed e mus c e defed s fdg he su eghs eeded s he me fm ce s wh desed e The su hs em s deed cme d c e cmued ve sh me uheme he cmu f egh f ech su c e ced w deede e whch c dd seed u he cess Ths Le ud cu f ecc geeg d mug Usk 0000 Zge eud@feh cedue s used gude he me fm cg s mveme The fwd kemcs f e mu s fdg he s d e f he me fm whe he su eghs e kw Ths em hs kw csed fm su f he ms gee - fm f hed mu (wh s js he se d s he me fm) Ths cedue wud e vue cg he fce-feedck f he mu I wud s vde ew c sses f hed mechsm such s fceque ses s-e ses ec I hs wk seve mhemc eeses f he fwd kemcs em he fm f m fucs e cmed wh vus m ghms d d mehds de fd effce cedue h wud w f ecse fwd kemcs svg e-me cds I he e sec we defe he ms msg eeses f he em Om ghms d sme ds egdg he em hd e desced sec III I sec IV he eeme esus e eseed d csg emks e sed sec V II T OW INMTIS POLM The fwd kemcs es f hed mche c e mhemc fmued seve ws ve eese f he em c hve s dvges d dsdvges whch ecme emhsed whe dffee m ghm s ed I hs sec he fve mhemc eeses used s m ge fucs e desced The s d e f he me fm I de defe fwd kemcs em we hve eese he cu hed cfgu e he cu s d e f he me fm The ms cmm ch ues he hee s cdes f he cee f he me fm d hee ges h defe s e The cdes e eeseed vec : () d he hee ges e defed s -chw ges α β d γ The ge vues eese he csecuve u he d s esecve The hed geme s defed wh s vecs f se d s vecs f me fm whch defe he s j cdes ech fm:

2 0 0 () The ve vecs e eeseed c cde ssems f he se d me fm d e f cs vue The se d me fm e esumed e whch c e eceved fm he cde f he j vecs The su vecs c he e eessed s () whee s he m ccued fm hee ges If he s d e f he me fm s kw he egh f ech su s () whee eeses he ucde dsce ewee he vec s su fwd kemcs em e s d e f he me he e c e eessed s he sum f sques f dffeeces ewee he ccued d cu egh vues: f () vg sed he ve es we c defe he fs m fuc d he eed ukws s X γ β α () Ths fuc s hgh -e due he gmec fucs cuded he ccu f he m O he he hd s deve d s he ms cmm used e f he eese f he fwd kemcs em The cc fmu f he fwd kemcs The de ehd hs ch [] s use he eemes f he m he h he ge vues eese e: [ ] (7) The eemes f he m e edeede he fwg fsh: (8) (9) 0 (0) () Whu ss f gee we c s he gs f he c cde ssems f he se d me fm he su js wh de e s shw g whch gves us he fwg mee vues: 0 () Z X Y X Y Z g Psg f cde ssems f se d me fm fe eesve smfcs he fwd kemcs c e eessed s ssem f 9 equs wh 9 ukws: () () () 0 () (7) (8) (9) (0) () whee he css e () ()

3 () ) ( () The m fuc d he eed ves vec X e he defed s X () educed cc fmu The equs (9)-() e f e fm whch c e used educe he ume f ves whu ducg dd cme he ssem Thee f he s ves d c e eced wh e cms f he he hee whch eves us wh s ukws sce f we chse eme he fwg ves (7) we c defe he ge fuc s X (8) I s sse fuhe educe he ume f ukws s w s hee u wh eve cese equ cme [] Th w he fuh m fuc c e defed s [ ] ] X (9) whee he cs eemes e (0) Vec defed The e c e desced wh sge u uves vec [] If he vec whse egh equs e s deed wh k d he ge wh θ he we c use he cmes f he duc k θ () eese e su he fwd kemcs em []: X () The cves fm -ch-w ges vec d vce ves c e fud [] The m fuc s defed he sme w s he fuc he eemes f he m hve e ccued fm vec fuc evu III T OPTIMIZTION LGOITMS The fwd kemcs em s eseed s fve m fucs f whch he m ghm hs fd he mmum he vue f he fucs eg he e f he esmed su Seve m mehds hve ee ed ech f he fucs de fd effecve cm whch wud w f e-me c The ghms ed hs wk e Pwe's mehd ke-jeeves' seees desce sech New-hs's (N) mehd N mehd wh cs Jc d eche-pwe ghm ch f he ghms s fud e me sue f sme es f fucs h f hes whch c e eceved fm eeme esus e hs e smg he su ve em The ms educed cc fmu (fuc ) f fwd kemcs em cs hee deede ves The chce f hse ves c e eed he educ cess s s sse hve

4 dffee ve es hee-dmes educed fm weve sge su hee-dmes sce eeses e u whe fm f hed cfgus ecuse sge cfgu s fu desced s-dmes sce The fm f he se f hed cfgus cuded sge su s deemed he ves ese he emg hee eme f he educed fm su s eeseed wh ves d he e f he ves whch defe he me s e ves d e ese he su ece eve hed cfgu wh me fm e mchg he hee vues f d u wh s w e eeseed wh he sme su hee-dmes sce The ec s vues e esed hugh css' vues ce he sfm s-dmes sce s udeke Ths e s used effecve esme he fwd kemcs su esumg e f he me fm fm he evus s kw s IV XPIMNTL SULTS Svg f fwd kemcs ws smued sc d dmc cds The sc eemes cuded he cms f fve m fucs d s m ghms ed 0 es cfgus The g ws fd he cm whch wud ed he es esus csdeg he cvegece seed d ccuc The ms msg cms wee he esed dmc cds whee he ghm hd ck ese jec f he me fm wh s sm e d s ge smg fequec s sse Thse cms cude ke-jeeves' ghm wh fuc d eche-pwe mehd wh fucs d u he ms successfu m mehd ws New-hs's ghm ed fuc Th cm hs cheved he es esus dmc esg s we I dmc smu he sg hed cfgu s kw d seves s su ug he smg ed T he ghm hs fd he ew su whch w ecme he su he e cce Seve hed mvemes wee defed s me deed fucs f he s d e f me fm Oe f hse jeces heefe deed s s defed wh π () s π () cs () 7 s () α s 8 β γ () () 0 s cs () c( ) 0 The esus f he dmc smu e eseed he fm f gh whee es he hee ges d hee s cdes f he me e cued The smg ed T ws se ms whch equs 00 smg fequec The es shw eese he sue dffeece ewee he ccued d he cu hed cfgu ue he ge ume f cces he e s defed s he gges sue e vue he s 00 ms s he ghs ech shw he ws cse he s 00 ms f smu The es e eseed see f ges degees d s cdes The es f mveme d New-hs ghm wh fuc e shw g d g sue e ce de g sue ge e (α β γ ) N ghm wh mveme sue e ce de g sue cde e ( ) N ghm wh mveme The cheved eve f ccuc s ve hgh s he sue e s hghe h 0 - h f ges d cdes he jec s deved fm he desced e egg sme f he mudes () whch s deed s mveme (he eed vues e dfce): π () s π () cs () 8 s () α s 8 β γ () () 0 s cs () c( ) 0 The mveme es e shw g d g Whe s w he e f mveme hs w dscve eks ce s smued m Wh s he cuse f hse eks? Mhemc ss hs shw ([] [] []) h hee m es u 0 dscve sus f fwd kemcs em f Sew fm wh se d me fm Thse 0 sus e dvded w gus f 0 ewee whch he dffeece s he sg f he cde I he wds e hf f hse mhemc sus hve hsc meg e hse wh sve s vue u he esece f mue sus f he sme se f su eghs m ve s em f he svg mehd

5 sue e ce de g sue ge e (α β γ ) N ghm wh mveme sue e ce de g sue cde e ( ) N ghm wh mveme Le us suse h e hed cfgu hee ess he fwd kemcs su f cu se f su eghs u h sme he cfgu hee es seve f hem If hed s mveme sses hugh hse w cfgus he ce ewee hee hs e dvs whee he ume f sus ceses I hse dvs s he svg ghm m ufue eg fw f he sse hs ecuse f hem eeses vd fwd kemcs su! Th s ec he em h ccus mveme : he ghm m m fw he cec jec If he e s he cse h he sue e ks ke g d g 7 sue e ce de g sue ge e (α β γ ) N ghm wh mveme - dvs sue e ce de g 7 sue cde e ( ) N ghm wh mveme - dvs The ghm w dm fw ehe he cec jec he equve e I s m e h h cses he m fuc ems ve w ( ) dug he whe cess ecuse h jeces dec vd su he fwd kemcs em The em s e f hem eeses he cu hed cfgu ech f me The e ewee juc s g d g 7 cu shws he dsce ewee he w equve jeces The d mehd Whu dd fm u he hed cfgu such s m e ed fm e s dsceme sess hee s ufue w deeme whch f he ese sus he fwd kemcs em f he sme se f su eghs desces he cu hed cfgu Neveheess wh sme ssums we m devse seg h shud kee he svg mehd he gh ck If he chge f he dec f mveme s eve sm dug sge ed whch s hs cse ms he we c edc he s f he me fm he e cce We c use he sus fm he s cces csuc sgh e d esme he su he e e Le he su he cue e e P 0 d he sus fm he s w cces P d P The we c ccue he ew su usg e f he fwg mehds: X 0 P0 P () X 0 P0 0 P () T 0 P0 P T 0 ( P P ) (7) X 0 T T The ve mehds wee esed cjuc wh N ghm d fuc f he smued jeces The esus e ve gd: he svg mehd ws w e ck he cec su dug he whe smu cess f hee esm mehds The ume f cduced eemes ws seve huded mes d eve me he ghm's e mg ws ew 0 - h f ges d cdes weve he desced ghm d w e successfu f he ssum f sm dec chge dug few es s vd T es he ghms ehvu smued mveme ws cceeed fc d 8 whe mg he sme cce du f ms O echg he 8-fd ccee whe he mveme me equs ve uesc hf secd dd he ghm duce sgfc es whe s hdg he cec su I hs cse sme dffeeces c e eceved ewee he hee sed esm mehds If we se e mg f 0 - f ges ( degees!) d 0 - f cdes he he ecege f ekg hse mgs f he hee d mehds w e me equ hse shw Te I TL I T PNTG O ING O MGINS d mehd s () s () s (7) echg % % % ecege Thee s w f eg whu eesve mhemc ss f eve ed me fm mveme whehe hed w ss dvs s hes Th s wh he smg ed mus e

6 ke s w s sse ssue eve sm dec chges dug few cces V ONLUSION mg seve eeses f he fwd kemcs em wh m echques effce mehd f svg he em ws fud he uses f e-me smu seve me fm jeces wee mhemc edefed The sk f fwd kemcs svg mehd ws he ck he s d e f he me fm kwg he sg s The svg mehd ws e deeme he ec s d e f he me fm wh sgfc e mgs (ess h 0 he we f f he mmum hed dmes) d wh 00 smg fequec The em f equve jeces ws ed: ecuse f he esece f mue sus fwd kemcs hee m es me h e h h me fm c fw whe hvg ec he sme su eghs eve f he w The svg ghm m ude sme ccumsces sk equve jec ce dvs s I hs e sd h eve such h eeses equ cec su f he fwd kemcs u e f hem eeses he ue me fm jec emc ghm ws devsed whch c cese he f fdg he gh su d ved sef successfu eve es cse Ufue c e ve h w d s eve mge mveme f he me fm The svg mehd w ws fd he gh su f he chge he s mvg dec f he me fm s eve sm dug few smg eds If h cd s fufed he desced mehd c e used e-me fwd kemcs svg VI NOWLGMNT Ths wk ws ced u wh he esech jec "Pem-Svg vmes geeg" sued he Ms f Scece d Techg f he euc f VII NS [] sgu TS Muhuj " c mu f he ec Ps emcs Pem f Gee - Sew Pfm" Mech Mch The V 9 N [] J -P Mee "ec emcs f Pe Mus" I Tscs cs d um V 9 N [] P Pu Mus The MIT Pess mdge 98 [] M ghv "The Sew Pfm f Gee Geme hs 0 fgus" Ju f Mechc esg V 77-8 Jue 99 [] M us " ghm f Svg he ec emc Of Sew-Gugh-Te Pfms" f://fmccmmcgedu/u/eche/99/im- 9-0df 99 [] We Lg "sceme ss f he - Sew Pfm Mechsms" Mechsm d Mche The V 9 N

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