A Generalized Recursive Coordinate Reduction Method for Multibody System Dynamics

Transcription

1 eaoal Joual fo Mulscale Compuaoal Egeeg, 1(2& (23 A Geealzed Recusve Coodae Reduco Mehod fo Mulbody Sysem Dyamcs J. H. Cchley & K. S. Adeso Depame of Mechacal, Aeoaucal, ad Nuclea Egeeg, Resselae Polyechc sue, oy, New Yo, USA ABSRAC he mehod of ecusve coodae educo (RCR offes soluos o he fowad poblem of mulbody dyamcs a a cos whch he umbe of opeaos s lea boh he umbe of geealzed coodaes,, ad he umbe of depede algebac cosas, m (e.g., O( + m. Howeve, he RCR s pesely esced applcably (albe boad ad suscepble o fomulao sgulaes. hs acle develops wo mehods fo avodg fomulao sgulaes as well as a ecusve geeal coupled loop soluo ha exeds he RCR o he complee se of mulbody sysems. Applcao of hese echques ae fuhe llusaed wh a specal fve-ba lage. he exsg RCR coupled wh hese developmes cosue a geealzed ecusve coodae educo mehod ha should be used place of he adoal O( cosa echque (uly O( + m² + m³ fo supeo O( + m compuaoal pefomace. Addess all coespodece o J. H. Cchley; ames@mulbodydyamcs.com. Docume D# JMC ( /3/$5. 23 by Begell House, c.

2 182 CRCHLEY & ANDERSON F A F A F A A[] b ch[] D dep Des[] F F F M F F f G H H H H ; ; M NOMENCLAURE he spaal acceleao veco of body as measued fame F he ecusve spaal veco of acceleao compoes assocaed wh body measued fame F whch ae explc he sysem geealzed speed devaves ( u s. he spaal veco of acceleao compoes assocaed wh body measued fame F whch ae o explc he sysem geealzed speeds ( u s. he acesal body se of a body dex coespodg o he bachg body a coupled loop sysem he se of chld bodes of a body A abay depede coodae o local loop he se of depede coodaes o local loop he desceda body se of a body he spaal foce veco of body he ecusve spaal foce veco of body A colum max of he ecusve spaal foce vecos F wh depedes f hough l he spaal foce veco assocaed wh depede geealzed speed u he coodae educed equao of moo o local loop A spaal foce veco emedae quay assocaed wh body A coex depede fs coodae assocaed wh a local loop A ow max of spaal esos epeseg he depedece of spaal velocy o ouboad depede geealzed speeds A ow max of spaal esos epeseg he depedece o ouboad depede geealzed speeds spaal velocy s expesso ems of p he spaal eso eleme of H assocaed wh depede geealzed speed A ow max of spaal esos epeseg he depedece o ouboad depede geealzed speeds he h coodae educed equao of moo (ouboad of p A ow max of spaal eso emedae quaes A abay depede coodae o local loop he spaal ea eso of body he ecusve spaal ea eso of body A ecusve spaal ea eso assocaed wh depede geealzed speed u he coodae educed equao of moo o local loop A ecusve spaal ea eso assocaed wh depede geealzed speed u he coodae educed equao of moo o local loop A ecusve spaal ea eso assocaed wh depede geealzed speed u he coodae educed equao of moo exsg oly o commo bodes of wo coupled loops ad A colum max of he ecusve spaal ea esos ; wh depedes f hough l p M A colum max of he ecusve spaal ea esos ; wh depedes f hough l A spaal ea eso assocaed wh depede geealzed speed u he coodae educed equao of moo o local loop A spaal ea eso assocaed wh depede geealzed speed u he coodae educed equao of moo o local loop A spaal ea eso emedae quay assocaed wh depede geealzed speed u eaoal Joual fo Mulscale Compuaoal Egeeg

3 A GENERALZED RECURSVE COORDNAE REDUCON MEHOD 183 ; B BB / A spaal ea eso emedae quay assocaed wh depede geealzed speed u he ceal ea dyadc of body B ch[] Des[] d K l N M M M m m m B P ~ P p p[] Q R S S U U u u. u dex coespodg o a abay emac loop he body se of depede chlde of body he body se of depede descedas of body he se of depede coodaes o local loop dex coespodg o a abay body, coodae, o emac loop dex coespodg o a abay body o coodae umbe o local loop A dagoal max of he spaal paal velocy vecos assocaed wh depede geealzed speeds f hough l dex coespodg o a abay body o coodae umbe dex coespodg o a abay body o coodae umbe o local loop he las depede coodae assocaed wh a local loop he Newoa (o eal efeece fame he umbe of geealzed coodaes a mulbody sysem he umbe of depede coodaes ouboad of he vual emal body p o local loop he umbe of geealzed coodaes a local emac loop he ecusve agulazed mass max dagoal assocaed wh geealzed speed A squae max of he coupled bloc agulazed mass max ems whch ae M wh depedes f hough l he umbe of depede algebac cosa elaos a mulbody sysem he umbe of depede algebac cosa elaos he local loop he mass of a body B he spaal paal velocy veco epeseg he paal devave of N V wh espec o u he cosaed spaal paal velocy explcly accoug fo he cosa hough he dec embeddg of he cosas he equaos o moo dex coespodg o he vual emal body of he local loop he pae of a body A scala emedae quay assocaed wh he local emacs of loop ad body A spaal eso emedae quay assocaed wh he local emacs of loop ad body dex coespodg o a abay body o coodae umbe he spaal shf eso whch epeses he poso veco coss poduc couplg of lea ad agula elaoshps assocaed wh body ad s pae body he spaal shf eso whch epeses he poso veco coss poduc couplg of lea ad agula elaoshps bewee body ad body he spaal agulazao eso assocaed wh geealzed speed he spaal uy eso A uy dyadc he geealzed speed assocaed wh he moo of body elave o s pae body A max of he depede geealzed speeds f hough l he geealzed acceleao assocaed wh he moo of body elave o s pae body Volume 1, Numbe 2&3, 23

4 184 CRCHLEY & ANDERSON F V F V F V F B v he spaal velocy veco of body as measued fame F he ecusve spaal veco of velocy compoes assocaed wh body measued fame F whch ae explc he sysem geealzed speeds (u s he spaal veco of velocy compoes assocaed wh body measued fame F whch ae o explc he sysem geealzed speeds (u s he velocy of a body B s cee of mass as measued fame F W X Y A spaal eso emedae quay ecusvely elag he emacs of body, p[ ], ad hough he dec embeddg of cosas o loop A spaal eso emedae quay ecusvely elag he emacs of body, p[ ], ad hough he dec embeddg of cosas o loop A spaal eso elag he emacs of body, p, ad hough dec embeddg of cosas o loop Z A spaal eso elag he emacs of body, p, ad hough dec embeddg of cosas o loop he base body of loop whch baches o fom he loop F B ω he agula velocy of body b as measued fame f 1. NRODUCON he complexy of a mulbody sysem model s ofe smplfed o oba soluos a desed (o ealsc me scale fo applcaos such as ealme opeao o hadwae--he-loop smulao, obocs ad cool smulao, model-based pedcve cool, vual pooypg, desg opmzao, ad he le. he fdely of such applcaos s mos geeally compomsed by boh he model educo ad smplfcao equed o ealze smulao/aalyss me cosas. hs espec, apd compuao of he fowad dyamcs of mulbody sysems has a mpoa ole may aeas. hs has pomped eseaches whose eess le a wde vaey of felds o vesgae mehods fo ceasg he pefomace of mulbody fowad dyamcs mehods, whch had adoally compued soluos fo he depede coodae acceleaos ode ³ (O(³ opeaos oveall. 1983, Feahesoe [1] peseed a effce fomulao of exsg O( mehods (hose of Veesheg [2] ad Amsog [3], called he aculaed body algohm (ABA. Based o Wale ad O s effce O(³ fowad soluos [4], whch wee u deved fom ecusve O( soluos of he vese poblem [5,6], he ABA acheved lea cos vesus complexy (O( fo he fowad poblem hough he exploao of addoal ecusve elaoshps. Sce he ABA s oduco, myad O( mehods ha exed he applcably ad flexbly of he ecusve fomulao have followed. Mos oably ae he exesos volvg vaous classes of flexble bodes [7-9], egao wh opmal fleg ad cool [1,11], desg sesvy [12,13], ad paallel compug [14 17]. Suffce o say ha mpovemes o he udelyg ecusve O( fomulao beef all of hese aeas. he fs ecusve O( algohms could be appled oly o ee-ype mulbody sysems bu wee qucly exeded by Feahesoe [18] ad Bae ad Haug [19] o clude hose wh emac (closed loops. Howeve, hese ad mos ohe subseque closed loop soluos eque he fomao ad veso of a m m cosa max, volvg O(m² ad O(m³ opeaos, especvely (whee m s he umbe of depede algebac cosas. Because of he cubc aue of he cosa soluo, hese ode algohms ca be oupefomed by effce adoal O(³ mehods fo modes o heavly cosaed sysems. As a aleave o exsg O( closed-loop mehods, Valáše [2] ouled wo mehods fo he eame of mulbody loop sysems, whch he effecs of depede loop coodaes could be obaed ems of he depede loop coodaes a umbe of compuaos lea he umbe of depede coodaes. a depede developme, Adeso eaoal Joual fo Mulscale Compuaoal Egeeg

5 A GENERALZED RECURSVE COORDNAE REDUCON MEHOD 185 [21] fomulaed smla, hough moe geeal, equaos based o hs O( algohm, whch mplemes Kae s mehod [22]. Adeso espoused he poeal beefs of such a mehod, whch, apa fom offeg he mpoved compuaoal complexy ced by Valáše, also cluded supeo cosa sably elave o he adoal O( cosa fomulao. Moe ecely, he auhos [23] have exeded he fomulao, ow efeed o as ecusve coodae educo (RCR, o clude a boad aay of heavly coupled mulloop sysems. hs wo decly demosaes he pedced esuls of Adeso [21] ad deals he lmaos of he fomulao. hs acle peses modfcaos o he RCR fo he lmg cases of Adeso ad Cchley [23], whch pea o abay seleco of depede coodaes ad coupled loops wh dffee loop base bodes. Supassg hese lmaos esuls a geealzed ecusve coodae educo mehod capable of accommodag abay mulbody sysems. deed, hs appoach s ow beg appled by he auhos o model sysems wh such dffee applcaos (as well as spaal ad empoal scales as auomove dveles ad suspesos, mcoelecomechacal sysems (MEMS ad aspecs of he maufacug/assembly pocesses, ad dealed modelg of he dyamc behavo ad maeal chaacescs of aosucues ad molecula sysems. 2. PRELMNARES hs seco, specalzed mulbody oaos ae oduced ad used o llusae he ecusve elaoshps ad pocedues fudameal o he O( algohms, whch wll ad he subseque developmes Spaal eso Noao A spaal eso oao has bee adoped o gealy smplfy he fom of he equaos used o descbe popees of mulbody sysems. Spaal esos ae lea algebacally augmeed hee-dmesoal esos (vecos ad dyadcs used o combe aslaoal ad oaoal fomao o a commo uppecase scp symbol. Fo example, a spaal velocy eso V ad a spaal ea eso may be gve as N B ω N B V = N B v B B B B / * / * B B U B = = m B U m B U whee he lef supescp N s he fame of efeece, B s he body ame wh B* deog he cee of mass, B BB / s he ceal ea dyadc, m B s he body s mass, ad U s a uy dyadc. hus, a spaal momeum could be defed as he do poduc of spaal ea wh spaal velocy BB / B N B ω B N B V = N B mb v o o be cofused wh he eso mulplcao BN V B, whch would sead poduce wo hd-ode esos. Noe ha he aspose of a spaal quay mples a echage of bass vecos (a echage of dces o dyadc quaes Body Ses A geeal mulbody ee sysem, such as ha show Fıgue 1, allows a umbe of useful body ses o be defed. We oe ha each body he sysem: has a sgle pae body p[], excep he base body, whch has he eal efeece N as s pae has a se of chld bodes ch[], whch may be empy s sad o be a emal body f ch[] s empy has a acesal body se A[], whch coas he body s pae p[], ha body s pae p[p[]], ad subseque pae bodes bac o ad cludg he base body Volume 1, Numbe 2&3, 23

6 186 CRCHLEY & ANDERSON speeds u ad he devaves. u, especvely. he F V ad F A have he followg ecusve defos V = ( S V + P F F p[ ] u (3 A = ( S A + P F F p[ ] u (4 FGURE 1. Sysem schemac of a abay ee sysem. has a desceda body se deoed Des[], whch coas he body s chlde ch[], he chlde s chlde ch[ch[]], ad subseque chld bodes ou o ad cludg emal bodes 2.3. Recusve Kemacs May ecusve defos he emacs of mulbody sysems ae val, occug aually hough he use of elave coodaes. Howeve, befoe hese elaoshps ae expessed he spaal oao of Seco 1, he veloces ad acceleaos ae decomposed o poos, whch ae explc he coodae veloces, he devaves, ad all emag ems. Fuhemoe, obsevg lea asfomaos fom coodae veloces q. ad acceleaos q.. o geealzed speeds u ad he devaves u,. (as s commoly doe wh Kae s Mehod [24] allows ceased effcecy [25] whou loss of geealy. he velocy ad acceleao decomposos ae gve by F F F V = V + V (1 whee u s he coodae geealzed speed, P s he spaal paal velocy veco, ad S s he spaal shf dyadc, whch epeses he poso veco coss-poduc couplg of lea ad agula elaoshps. A abay spaal paal velocy P ca be cosuced as: P whee N S P V A = ( [ ] ( = u else A[] else ch[ ] p[ p [ p[ p[ ]] ]] p[ ] Ad U s he spaal uy U U = U (5 (6 We oe ha he quaes N A, N A, P, ad S fo all bodes ca be fomed O( opeaos fo a gve sysem sae. A = A + A, (2 F F F whee F V ad F A ae he poos of he velocy ad acceleao ha ae o explc he geealzed 2.4. Recusve Dyamcs he spaal fom of he equaos of moo fo a mulbody ee sysem coag oly sgle degee of feedom os s gve by Kae s Mehod [24] as eaoal Joual fo Mulscale Compuaoal Egeeg

7 A GENERALZED RECURSVE COORDNAE REDUCON MEHOD 187 ( P N ( A F = = 1 (7 whee F coas all foces ad momes o body appled abou he cee of mass as well as he eal foce cobuos ow ems of he sysem sae. he equao of moo (7 eadly educes o (8 by obsevg he paal velocy decomposo gve (5. N ( P S ( A F = (8 Des[ ] Oe may ow mae he assumpo ha he fom of he h equao of moo ca be expessed as N ( P ( A F = (9 hus, val bouday daa s exaced fom he equaos of moo assocaed wh emal bodes F = ad = F (1 feedom, ad subsug he pevous soluo (12 follows as p[ ] p[ ] N ( ˆ p[ ] [ ( ] Des[ p[ ]] p[ ] (14 p[ ] p[ ] N p[ ] p[ ] = ( P [( A p[ ] p[ ] + S ( Des[ p[ ]] p[ ] p[ ] N p[ ] [ ] = ( P [( A p p[ ] N A ] (15 N + S ( A F ] (16 ch[ p[ ]] p[ ] p[ ] N p[ ] [ ] = ( P [( A p p[ ] + S ( N p[ ] {( S A + P u } ] ch[ p[ ]] (17 p[ ] p[ ] N p[ ] [ ] = ( P [( A p p[ ] + S ( N p[ ] {( S A P ( M 1 ( P ch[ p[ ]] N p[ ] ( ( S A } ] (18 Subsug he ecusve fom fo N A of (4 o (1 gves N p[ ] ( P [ (( S A + P u F ] = (11 p[ ] p[ ] N p[ ] p[ ] = ( P ( A (19 p[ ] ad he ecusve emedae quaes ae obaed hough goupg of ems as wh Ad solvg fo. u yelds N p u = 1 [ ] M P S A ( ( ( ( (12 p[ ] p[ ] = + ( S (2 ch[ p[ ]] p[ ] F p[ ] = F + F (21 ch[ p[ ]] M P = ( P (13 = S U P M 1 ( ( P (22 Now pefomg smla opeaos o he equao of moo assocaed wh he pae body s degee of he O( soluo he volves calculag he ecusve emedae quaes (2 ad (21 wad fom he emal bodes, effecvely lowe agulazg Volume 1, Numbe 2&3, 23

8 188 CRCHLEY & ANDERSON he sysem mass max. he fowad subsug fo he soluo usg equaos (12 ad (4 og ha N A N =. 3. RECURSVE COORDNAE REDUCON he adoal O( cosa echque (whch acually pefoms as O( + m² + m³ whee m s he umbe of depede algebac cosa elaos [23] cus os o poduce ee opologes fom geeal mulbody cofguaos coag emac loops. By coas, he mehod of ecusve coodae educo geeaes a assocaed ee opology hough he oduco of phaom bodes, whch ae massless copes of he loop base body (he fs body ouboad of he eal efeece ha baches o fom he loop. hs cosuco s demosaed Fıgue 2 as well as a local loop umbeg scheme Kemacs Ope loop sysems esulg fom he addo of a phaom body o a pevously closed loop ae subec o he velocy ad acceleao cosas A = (24 he velocy cosa may be decomposed exacly as wh he ucosaed sysem, esulg V V V (25 = = + 1 = ( S V + P u + V (26 u s always a vald choce fo a depede coodae, ad heefoe ca be solved fo as u wh 1 = + Q ( P [( S V V ] (27 Q = [( P P ] 1 (28 Equao (27 ca be subsued bac o (26 o oba he 1s fom of he velocy cosa V = ( = R [( S V + V ] (29 P -1 P 3 Loop P +1 Depede Coodaes 2 N -2 N -1 N 1 Loop Base Body Phaom copy of FGURE 2. Opeg a closed loop usg a phaom body eaoal Joual fo Mulscale Compuaoal Egeeg

9 A GENERALZED RECURSVE COORDNAE REDUCON MEHOD 189 whee R = U P Q ( P (3 Mag a appopae seleco fo he ecusve fom of he h equao ad solvg follows as + 1 = R [( S V + V ] ( = R [( S P u + ( S Fo he depede bodes, we oe ha he usual V decomposo of (37 s vald o subsue (33, esulg (38 wh V = ( S V + P (37 1 u = W V + X V 1 (38 V 1 + V ] ( W = ( S + X ( S (39 Ad heefoe he geeal soluo X = P Q ( P S R + 1 (4 u + 1 = Q ( P S R [( S V + V ] ( he elaoshp (38 ca he be used o we addoal ecusve elaos expessg he depede veloces ems of he las depede coodae he loop p s obaed wh Q = [( P S R ( S P ] (34 V Y V Z V p = + (41 (ude he assumpo ha he las m cosecuve coodaes cosue a vald seleco of depede coodaes e.g., he fs p coodaes fom a vald se of depede coodaes. Sasfyg he 1s equao Y = W Y 1 Z = X + W Z 1 wh he val bouday daa (42 ( = R [( S V + V ] (35 eveals he ecusve elaoshp + 1 R = R [ U ( S + 1 PQ( P S R ] (36 p p Y = U ad Z = (44 hs same pocedue ca be followed wh he depede acceleaos esulg A Y A Z A p = + (45 Volume 1, Numbe 2&3, 23

10 19 CRCHLEY & ANDERSON 3.2. Dyamcs he cosaed fom of he equao of moo s gve by = ( P ( N A (46 = 1 whee ~ P ae he cosaed o-holoomc spaal paal veloces (whch explcly accou fo he cosa hough dec embeddg of cosas o he equaos of moo. Hee we oe he addoal emac defos A = ( S A + A N N (47 Subsuo of (5 o (49 esuls p 1 N = ( P ( ( S A + A + ( P = p N ( Y ( ( S A + A = p (51 Ad A fo he depede bodes s gve ems of p A by equao (45, whch allows us o ewe (51 as p ; N = ( P ( A + A wh = (52 V = ( S V + V N N (48 Subsug he acceleao elaoshp (47 o (46 ad escg ou aeo o he local loop we oba = ( P ( ( S = (49 N A + A he cosaed paal veloces ae he paal devaves of he veloces we ems of he depede coodaes as (41. hese paal devaves ae P P d ad d N V p = ( = Y P dep ad d u dep (5 whee d ad dep ae he ses of depede ad depede coodaees (especvely o local loop. ; F = ( S ( Y ( S = p p = p p = ( Y Y p = p = F = ( [ ] (53 (54 Y F Z A p = p = p (55 he coodae educed equao of moo (52 may he be solved ecusvely exacly he same fasho as he ucosaed equaos of Seco 4. ha s, subsuo of (5 o he h equao of moo (52 esuls p P ( S ; ( N A = = ( + A (56 eaoal Joual fo Mulscale Compuaoal Egeeg

11 A GENERALZED RECURSVE COORDNAE REDUCON MEHOD 191 Ad aga, he ecusve assumpo s ha he h equao s of he fom ; N = ( P ( A + A (57 ad val bouday daa s obaed fo he las depede coodae he loop p ; ; = = F = F p p p p p p (58 he ecusve fom of A fom (4 s he subsued o (57, ad he uow u. s solved fo ems of he dyamcs of he pae body. N u 1 ; = ( M ( P ( A + ( S p[ ] p A F (59 he equao of moo assocaed wh he pae of he h body (e.g., [56] wh eplaced by 1, he fs em ca be exaced fom he sum ad he emag sum eplaced hough appopae subsuo of he ecusve quaes (57. Fuhe subsug he soluo of. u ad goupg ems eveals he soluo fo he ecusve elaoshps ; = + 1 ; 1 ; 1 1 (6 = + ( S (61 F 1 1 = F + F (62 = S [ U P ( M 1 ( P ] (63 M P = ( P (64 A he loop base body we oe ha A =, heeby doppg he addoal em ad followg he defos fo ucosaed ee sysems (2 22. A addoal ecusve agulazao of equaos (o show s also equed ad vally obaed whe obsevg ha he paal devaves of (5 chage fom o he loop base body. he mos cosag lmao of hs, he RCR fomulao, s he assumpo ha he las m coodaes of a opeed loop cosue a vald seleco of depede coodaes. Fo hs easo, soluos fo may looped mulbody sysems cao be modeled by he peseed fom of he RCR, ad may moe have he possbly of eeg sgula cofguaos ad/o aeas of umecal ll codog dug smulao. wo mehods fo avodg sgulaes by aleg he depede coodae seleco wh he famewo of he RCR wll be descbed ad evaluaed he followg secos. 4. ARBRARY SELECON OF DEPENDEN COORDNAES he fs mehod of avodg sgulaes assocaed wh he RCR volves he specfcao of abay depede coodaes decly he fomulao of he RCR algohm. ohe wods, depede geealzed coodaes wll be cluded bewee he depede coodaes. We also oe ha he las coodae he loop ca ad wll always be chose as depede. hs s because o dog so would eque he phaom body o have mass popees, whch, alhough val, adds eedless addoal complexy o he fomulao (e.g., some seps ca be spped by og ha he spaal ea assocaed wh body s always zeo. Fıgue 3 llusaes a possble seleco of depede coodaes as well as addoal umbeg fomao ha wll be equed by he ew fomulao. he emac soluo (36, whch solves fo he behavo of a coodae a loop, volves he veso of a depede cosa elaoshp. Because hee ae exacly m depede cosa equaos assocaed wh loop, hs veso s o admssble fo he depede coodaes f hough l (whee f ad l ae he o coodaes assocaed wh bodes F ad L of Fıg. 3. sead, cobuos fom hese coodaes appea decly he emacs fomulao as Volume 1, Numbe 2&3, 23

12 192 CRCHLEY & ANDERSON P +1 Fs depede Ouboad of P P P -1 F Depede Coodaes Loop 3 2 N 1 FGURE 3. Body umbeg he seleco of abay depede coodaes. Las depede Ouboad of P L N -1 Phaom copy of Loop Base Body + 1 = R [( S V + V + ( S P u ] (65 Des[ ] whee Des s he se of depede descedas o he local loop (e.g., Des[ ] = Des[ ] d ad ( (68 D D D 1 D V = S V + PD u D D = W V D 1 D + X V + X Des[ D ] ( S P u (69 D R U ( S P Q ( P + 1 R = + 1 dep S R (66 + R 1 d whee W D ad X D ae gve exacly as (39 4. Ad he depede coodaes ca be cas a smla fom, V = ( S V + P (7 1 u Hece, he soluo fo he depede geealzed acceleaos ca be obaed by he acceleao fom of (65 ad ae gve by = W V 1 + X V + P u (71 u D D D D+ 1 D = Q ( P D S R [( S A + A + D 1 D ( u ] Des[ D ] S P (67 wh W = (S ad X =. Combg hese wo elaoshps fo a geeal body o he loop ad ouboad of p (e.g., Des[p ] esuls We ow oe ha he veloces of he bodes assocaed wh he depede coodaes D ae gve by 1 V = W V + X V + G K u (72 eaoal Joual fo Mulscale Compuaoal Egeeg

13 A GENERALZED RECURSVE COORDNAE REDUCON MEHOD 193 whee G U d 1 ( spaal = X ( ch [ ] l S ( S dep 1 ( spaal (73 f l K = dag ( Pf...( P l ( spaal (74 u = u f u l (75 1( scala he cosaed equaos of moo assocaed wh he las depede coodaes f hough l of he local loop ae gve by o = ( P ( A (82 = p + 1 N = ( P ( S = p + 1 ( N A + A (83 W X ( S d = 1 S + X S dep ( ( d = + 1 PQ ( P S R dep (76 (77 wh ch[ ] = ch[ ] d ad equal o he umbe of depede coodaes ouboad of p (e.g., he umbe of depedes f hough l. Oce aga he veloces may be we ems of p ad he ow poo of, bu wh a addoal cobuo fom all of he depede vaables lyg bewee hem (u. should be clea ha P = H P (84 whee H s he eleme of H coespodg o he cobuo of he h geealzed speed. Usg hs elaoshp, Eq. (83 becomes = ( P ( H ( ( S = p + 1 N A p + Y A + Z A + H Ku (85 = + + u (78 p V Y V Z V H K whee Y = W Y 1 (79 ad goupg ems esuls ; N = ( P ( H ( A = p + 1 p + + A H Ku F (86 Z = X + W Z 1 (8 wh H = G + W H 1 (81 ; = ( S (87 Volume 1, Numbe 2&3, 23

14 194 CRCHLEY & ANDERSON = Y (88 M P H = ( K (97 F = F Z A (89 whch may be we max fom as H = H (9 M K A A F M N p p u = ( + M M M (98 whch ca also be we as ; = ( P ( A + A N p + H Ku (91 p whee he F M, M,ad M ae appopae colum maces wh elemes ;,, ad F, especvely, ad M M s he squae max cosuced fom he M fo fom f o l. he soluo whee fo depede coodaes f hough l ; ; = ( H (92 = p + 1 u = ( M 1 K ( A + A p M M M N p M (99 ( (93 = p + 1 H = F = ( H F (94 = p + 1 s he obaed fo he ouemos depede geealzed acceleaos. u. Fo he emag depede coodaes o he loop ( p, he assocaed quaes ad equaos of moo ae almos hose of Seco 3, dffeg oly he defos of he ems assocaed wh he vual emal body p, whch ae sead H = ( H H (95 = p + 1 p ; p ; = ( Y H = p + 1 K M 1 ( K M M (1 Ad he soluos fo he depede geealzed acceleaos mus sasfy he se of lea scala equaos p p = Y H ( = p + 1 K M K 1 M p ( M (11 wh M ; P N u A = ( ( + A p (96 F p p = F ( Y H = p + 1 K M 1 ( K F (12 M Ad he ecuso poceeds exacly as befoe. M eaoal Joual fo Mulscale Compuaoal Egeeg

15 A GENERALZED RECURSVE COORDNAE REDUCON MEHOD 195 F G E D Reduda Jos H A C B E E Loop F G D Loop New Jo A H C A B FGURE 4. Abay sgula loop. FGURE 5. Nosgula epeseao of he loop of Fıg GENERAL COUPLED LOOPS A secod mehod of avodg sgulaes he RCR fomulao volves aleg he opology of he mulbody sysem so ha vald depede degees of feedom ae adace ad a he ed of all ope loops. he opology adusme volves he addo of fcous (alhough compleely vald os o he sysem ad eles o he ably o hadle sysems of coupled loops ha do o shae a commo base body. A abay closed loop wh a vald se of emal os s llusaed Fıgue 4. A val soluo o hs poblem, whch may be vald may cases, s o smply wo he loop he oppose deco. hus he pesece of sgula coodae seleco a boh eds, adused loops such as ha show Fıgue 5 ca be used o avod he sgulay. he compuao of wo coupled loops ad wh dffee base bodes ad eques wo dffee epeseaos of he spaal acceleao elave o eal fame fo bodes alog he eseco of boh loops. A = ( S A + A N N (13 Nog he elaoshps A = ( S A + A (15 N N A = ( S A + A (16 subsuo ad smplfcao of ehe esuls A = ( S A + ( S A + A N N (17 Usg elaoshp (17 o hadle coupled loops wh he RCR s smply a mae of booeepg. he local emacs ema uchaged, ad he dyamcs of he bachg body b fom whch he wo ope chas exed ae gve by ( b ( N = Des[ b] b P A F (18 N N A = ( S A + A (14 f he bachg body s assocaed geealzed coodae s a depede coodae (whch s o coupled loops esulg fom opology adusmes fo Volume 1, Numbe 2&3, 23

16 196 CRCHLEY & ANDERSON he puposes of avodg a sgula coodae seleco, he equao of moo s Pluggg he usual ecusve elaoshps (4 fo f A ad f A as well as he fom of he soluos fo u. f ad u. f gve (59, esuls a equao explc oly cobuos fom quaes assocaed wh bodes,, ad b. Howeve he fou spaal ea coeffces of N A, N A, A b, ad A b cao be used decly o fom ecusve spaal eas. sead he coeffces ae combed ad dsbued as dcaed by equaos (15 ad (16 o poduce hee coeffces of he spaal acceleaos N A, A, ad A b. Ad Usg (17 o decompose N A b oe may we b b N b b ( P ( A b N + ( P ( A F = (19 Des[ b] b ; ; b N A + b A b = ( P b b b b + A (111 Subsug fom he ecusve equao of moo (57 assocaed wh each of he fs bodes (f ad f he wo bachg opeed loops, ad aga assumg ha boh bodes have assocaed depede coodaes (whch oe s o he case of he opology adusme, oe obas b b N = ( P ( A b b F + ( P S b b f b ; ( f N A f + A f f F + ( P S b f b he hee-em epeseao of (111 oly apples o he bodes whch ae commo o boh loops. As o he base body of he e loop ( hs case A = A =, ad he educo bac o wo ems s auomac. he case of depede coodaes beg pese a he eseco of he wo loops, elaoshp (45 allows he hee ems o be cosuced a he fs depede coodae boad of b. Ad soluos volvg depede f ae smlaly solved. f N f f f ( ; A + A (11 6. DSCUSSON AND RESULS Oe of he smples mulbody cofguaos fo whch he RCR fomulao of Seco 3 wll bea dow s he plaa fve-ba lage epeseed Fıgue 6. hs sysem, he aco of coodaes q₁, q₂, ad q₃ ae ha of evolue os, ad q₄ ad q₅ epese he moo of elescopg psmac os. should be clea ha whe esced o a fowad sweep of he fve-ba, he RCR wll bea dow as a esul of he vald seleco of depede coodaes (3 5. FGURE 6. Plaa fıve-ba mechasm. eaoal Joual fo Mulscale Compuaoal Egeeg

17 A GENERALZED RECURSVE COORDNAE REDUCON MEHOD AUOLEV q1 (adas AUOLEV q4 (mees ACS q1 ad q4 GCL q1 ad q4 1.8 value me (secods FGURE 7. Fıve-ba smulao esuls. he abay coodae seleco (ACS mehod of Seco 4 does, howeve, exhb he expeced soluo (as obaed wh AUOLEV as demosaed by he depede coodae aecoes show Fıgue 7. hs example, coodaes q₂, q₃, ad q₅ ae used as he depede coodaes, all bas ae h ods of u legh (mees ad mass (logams, ad he lage s eleased fom es he vecal plae wh FGURE 8. Avalable fıve-ba ee opologes. Volume 1, Numbe 2&3, 23

18 198 CRCHLEY & ANDERSON he al codos q 1 = π 4, q 2 = π 2, q 3 = π 2, q 4 =, q 5 =. he geeal coupled loop (GCL soluo of Seco 5 fo he same fve-ba mechasm s also show Fıgue 7 (he moos ae decal. he case of hs smple mechasm, he wo mulbody ee opologes of Fıgue 8 ae possble o avod he sgulay. should be oed ha he sysem of Fıgue 8b s acually a exsg soluo of a ype demosaed Adeso ad Cchley [23] (e.g., coupled loops wh he same base body, he fom of whch s oly avalable fo plaa ad some specal spaal loops. Fo hs easo oly he soluo dcaed by Fıgue 8a coespodg o he geeal coupled loop pocedue s gve Fıgue 7. Alhough each of he wo mehods dscussed Secos 4 ad 5 oba he same esul, he compuaoal ode of he algohms s vey dffee. he complexy of he abay coodae seleco mehod s a cubc fuco of he umbe of depede coodaes locaed bewee he depede oes (p + 1 ad, owg o he veso of M M. hs espec, s bouded oly by he oal umbe of depede coodaes pacpag a gve loop ( m 1 ad should heefoe be eaed as a loop local O( ³ opeao. hs wos case behavo ca bes be obseved he case of spaal loops wh lage plaa poos adace o he loop base body. he addoal complexy of he coupled loop soluo, f appled ellgely, eve exceeds O(m. he low ode esulg fom he addo of mulple loops ha may be equed o avod seveal sgulaes s obaed because he ewly fomed loops eed eve be coupled o each ohe. Hece, he wos case behavo s a m coodae gowh he descpo of he loop, whch s solved lea me O( + 2m o smply O( + m. he coupled loop soluo also epeses he las pece of a geealzed ecusve coodae educo (GRCR mehod whch yelds O( + m pefomace fo abay mulbody sysem cofguaos. As such, he GRCR should be used o eplace he adoal ode cosa fomulao ay applcao save hose whee he cosa foces ae boh explcly equed ad few umbe. ha s, he cosa foces ca be exaced fom he GRCR he usual way, whch may be less effce oly he case of vey lghly cosaed sysems. should be oed ha he pesece of heavy local loop couplg, such as oe would fd a mulbody lace (o mesh, he local couplg wll poduce b spaal ea ems o each body a local loop. he compuaoal complexy of such mehods geeal s uclea, because opmal ee epeseaos of such sysems ae equed. some cases he esuls ca sll be obaed O( + m, ohes O( + b m s moe appopae, whle O(log b may be a appopae uppe boud. he applcao of RCR mehods o such sysems s a opc of cue ad fuue eseach. he GRCR peseed hs pape ca also be exploed o oba a paallel compuao algohm wh logahmc me complexy (O(log. hs fohcomg wo pomses a heoecal mmum ode of compuaos (O( o he heoecal mmum ode of pocessos (O( as well as ceased flexbly ad accuacy as a esul of he fom of he cosa eame. 7. CONCLUSONS he ecusve coodae educo mehod whch has pevously bee show o offe ue O( + m pefomace fo a lage class of mulbody sysems has bee expaded o accommodae he complee se of mulbody sysems. wo mehods have bee deved o cope wh sgulaes he ogal fomulao esulg fom he fxed seleco of depede coodaes. he abay coodae seleco mehod equg he vese of a max s locally cubc, O( ³, ad s supeo o he adoal O( closed-loop soluo f he sysem loops ae small o he coodaes eed oly be aleed slghly. he ohe soluo volvg he ceao of a coupled loop epeseao of he sgula loop offes O( + m complexy. he fully ecusve geeal coupled loop soluo peseed also complees a geealzed ecusve coodae educo mehod ha ca be appled o abay mulbody sysems esulg O( + m pefomace fo all bu he mos heavly coupled mulbody laces. As such, hs mehod should be used place of he adoal O( cosa mehod fo supeo compuaoal pefomace. ACKNOWLEDGMEN Suppo fo hs wo eceved ude Naoal Scece Foudao hough awad No s gaefully acowledged. eaoal Joual fo Mulscale Compuaoal Egeeg

19 A GENERALZED RECURSVE COORDNAE REDUCON MEHOD 199 REFERENCES 1. R. Feahesoe. he calculao of oboc dyamcs usg aculaed body eas. eaoal Joual of Robocs Reseach, 2(1:13 3, A. F. Veeshchag. Gauss pcple of leas cosa fo modelg he dyamcs of auomac mapulaos usg a dgal compue. Sove Physcs [Dolady], 2(1:33 34, W. W. Amsog. Recusve soluo o he equaos of moo of a -l mapulao. : Poceedgs of he Ffh Wold Cogess o he heoy of Maches ad Mechasms, volume 2, , J. S. Y. Luh, M. W. Wale, R. P. C. Paul. O-le compuaoal scheme fo mechacal mapulaos. Joual of Dyamc Sysems, Measuemes, ad Cools, 12:69 76, M. W. Wale, D. E. O. Effce dyamc compue smulao of oboc mechasms. Joual of Dyamc Sysems, Measuemes, ad Cools, 14: , J. M. Hollebach. A ecusve Lagaga fomulao of mapulao dyamcs ad a compaave sudy of dyamcs fomulao complexy. EEE asacos o Sysems, Ma, ad Cybeecs, SMC- 1(11:73 736, A. K. Baeee. Bloc-dagoal equaos fo mulbody elasodyamcs wh geomec sffess ad cosas. Joual of Gudace, Cool, ad Dyamcs, 16(6:192 11, A. Ja, G. Rodguez. Recusve flexble mul body sysem dyamcs usg spaal opeaos. Joual of Gudace, Cool, ad Dyamcs, 15(6: , K. S. Adeso. A effce fomulao fo he modelg of geeal mul-flexble-body cosaed sysems. eaoal Joual of Solds Sucues, 3(7: , G. Rodguez. Kalma fleg, smoohg, ad ecusve obo am fowad ad vese dyamcs. EEE Joual of Robocs ad Auomaao, RA-3(6: , K. Keuz-Delgado, A. Ja, G. Rodguez. Recusve fomulao of opeaoal space cool. eaoal Joual of Robocs Reseach, 11(4:32 328, K. S. Adeso, Y. H. Hsu. Aalyc full-ecusve sesvy aalyss fo mulbody dyamc cha sysems. Mulbody Sysems Dyamcs, 8(1:1 27, Y. H. Hsu, K. S. Adeso. Recusve sesvy aalyss fo cosaed mul-gd-body dyamc sysems desg opmzao. Sucual ad Muldscplay Opmzao, 24(4: , D. Bae, J. G. Kuhl, E. J. Haug. A ecusve fomao fo cosaed mechacal sysem dyamcs: Pa, Paallel pocessg mplemeao. Mechasms, Sucues, ad Maches, 16: , K. S. Adeso. A effce modelg of cosaed mulbody sysems fo applcao wh paallel compug. Zeschf Agewade Mahema ud Mecha, 73(6: , K. S. Adeso, S. Dua. A hybd paallelzable low ode algohm fo dyamcs of mul-gd-body sysems: Pa, Cha sysems. Mahemacal ad Compue Modelg, 3: , R. Feahesoe. A dvde-ad-coque aculaed body algohm fo paallel calculao of gd body dyamcs. Pa 1: Basc algohm. eaoal Joual of Robocs Reseach, 18(9: , R. Feahesoe. Robo Dyamcs Algohms. Kluwe, New Yo, D. S. Bae, E. J. Haug. A ecusve fomao fo cosaed mechacal sysem dyamcs: Pa, Closed loop sysems. Mechasms, Sucues, ad Maches, 15(4:481 56, V. Sesal, M. Valáše. Kemacs ad Dyamcs of Machey. Macel Dee, New Yo, K. S. Adeso. mpoved ode- pefomace algohm fo he smulao of cosaed mul-gdbody sysems. : Poceedgs of he hd Symposum o Mulbody Dyamcs ad Vbaos, ASME Desg Egeeg echcal Cofeece 21, (DEC1, DEC21/VB-21334, Psbugh, PA, K. S. Adeso. A ode- fomulao fo moo smulao of geeal cosaed mul-gd-body sysems. Compues ad Sucues, 43(3: , K. S. Adeso, J. H. Cchley. mpoved ode- N pefomace algohm fo he smulao of cosaed mul-gd-body sysems. Mulbody Sysems Dyamcs, 9(2: , R. Kae, D. A. Levso. Dyamcs: heoy ad Applcao. McGaw-Hll, New Yo, P. C. Mguy,. R. Kae. Moo vaables leadg o effce equaos of moo. eaoal Joual of Robocs Reseach, 15(5: , Volume 1, Numbe 2&3, 23