Symbolic Equation of Motion and Linear Algebra Models for High- Speed Ground Vehicle Simulations.

Transcription

1 Symbolc Equaon of Moon and Lnear Algebra Models for Hgh- Speed Ground Vehcle Smulaons. y: James. D. Turner, Ph.D., ADS and Smulaon Cener, 2401 Oakdale lvd., Iowa Cy, Iowa, Absrac. Synhec envronmen modelng for human-cenered research requres a hghlevel of model fdely o suppor he lluson of a real-world experence for he smulaor operaor. Ths requremen s me by developng mahemacal models ha capure he crcal dynamcal behavors for he modeled ground vehcles. Ths paper presens a physcs-based modelng approach ha makes exensve use of compuer-aded symbol manpulaon capables for generang hgh performance smulaon sofware. A general-purpose Lagrangan mehod s presened, ha explos nnovave sparse paral dervave algorhms for dramacally accelerang he generaon of he equaons of moon. A major benef of he Lagrangan mehod s ha all opology-based consrans are elmnaed from he problem formulaon. Mahemacal models are provded for analyzng user-specfed me-varyng and closed-loop opologcal consrans. Symbolc mehods and daa srucures are presened for he equaons of moon, lnear algebra algorhms, and Forran subroune generaon. The objec-orened symbolc envronmen consss of: (1) A general-purpose model buldng ool (poson, velocy, orenaon, and angular velocy models), (2) sparse paral dervave algorhms, (3) generalzed force algorhms, (4) knec energy paral dervave algorhms, and (5) sparse applcaon-specfc LDL decomposon symbolc lnear algebra algorhms. A scrp-based ool s presened ha bulds on he synax of he objec-orened ls-based compuer-aded mahemacs Macsyma program. Macsyma provdes he core capables for symbol manpulaon, dfferenaon, vecor algebra, and hghly opmzed Forran generaon. Example applcaons wll be presened for 10-body HMMWV model. ITRODUCTIO Synhec envronmen modelng for human-cenered research requres hghfdely physcs-based models for supporng he lluson of a real-world experence for he smulaor operaor. Expandng needs for hgh-fdely demand ha vehcle modelers nclude compuaonally nense models for vehcle subsysems and complex physcsbased re-sol neracons. These applcaons, and ohers, place a premum on mnmzng he compuaonal effor devoed o handlng he vehcle dynamcs par of he synhec envronmen compuaonal load.

2 Saemen Of The Problem Ths paper s concerned wh addressng he vehcle dynamcs par of he synhec envronmen-modelng problem. Tradonal mulbody modelng approaches have been developed wh he dea of supporng general-purpose smulaon capables. Ths approach has produced successful commercal mulbody producs such as DADS, DISCOS, ADAMS, and many oher powerful sofware producs. Orgnally hese ools were developed o avod he me-consumng error-prone process of developng an applcaon-specfc modelng ool ha mus be exhausvely valdaed and verfed. Ths goal guded he advanced mulbody developmen communy durng he 1980 s and 1990 s. Of course, hese developers were aware of symbolc ools, bu he general consensus among sofware developers a he me was ha supercompuers were requred for handlng all bu academc-scale problems. In he pas few years, however, he performance of PC s has ncreased o he level ha applcaon-specfc symbolc soluon mehods are now praccal for analyss wh access o PC and WorkSaon developmen sysems. Symbolc soluons combne he benefs of valdaed and verfed smulaon ools wh he added advanage of hgh-performance ha can be acheved hrough he developmen of an applcaon-specfc model. Oher symbolc mulbody ools exs; however, hey do no ake advanage of he marx srucure and sparse paral dervave algorhms avalable for Lagrange s mehod. Compuaonal Issues Wh Topology-ased Consrans Compuaonally, as more bodes are modeled n a mulbody sysem, he number of opology-based consrans ncreases very rapdly, leadng o expensve soluon algorhms. For example, assumng ha one s modelng a smplfed ground vehcle

3 conssng of: a base body wh sx degrees-of-freedom (DOF) and no consrans, four wheel assembles wh egh DOF and 20 consrans, and a seerng mechansm wh one DOF and 5 consrans. Ths noonal sysem has a oal of 15 DOF and 45 consrans, requrng a soluon for 15 acceleraon DOF and 45 Lagrange mulplers. For hs smple applcaon he number of opology-based consrans s 3x larger han he number of DOF. Alernavely, he proposed symbolc mulbody algorhm compleely elmnaes he 45 consrans from ever enerng he mahemacal model. In a convenonal mulbody algorhm, he requremen for explcly handlng he 45 Lagrange mulplers has a dramac mpac on he compuaonal effcency of commercal codes such as DADS, DISCOS, and ADAMS. Recursve problem formulaons parally address hs problem, however, by reformulang he soluon for he Lagrange Mulplers o sequenally solve a seres of low-order marx sysems. everheless, opology-based Lagrange mulplers reman an essenal par of he compuaonal burden. The proposed symbolc ool acceleraes he smulaon of mechancal sysems by compleely elmnang he opology-based consrans from he problem model. A Lagrange mulpler capably, however, s provded for compung userspecfed me-varyng consrans for evaluang consran loads, for answerng engneerng desgn quesons. Capables are also provded for handlng closed-loop opologcal srucures. As shown n wha follows, he full-scale mplemenaon of he classcal Lagrange dynamcs mehod has a sgnfcan mpac on he sze and srucure of he resulng equaons, and he models compuaonal effcency for smulang he response of lnked mechancal sysems.

4 Paper Conrbuon To hs end, hs paper presens an objec-orened symbolc mul-body modelng envronmen ha uses a scrp-based user-defned npu for buldng: (1) model daa, (2) processng he model geomery and knemacs, (3) he equaons of moon usng Lagrange s mehod, (4) he generalzed forces, (5) user-specfed me-varyng consran condons, (6) mass marx, (7) a sparse symbolc LDL lnear equaon for he sysem acceleraons, and (8) ransformng all mah models o opmzed Forran sofware for generang he sysem dynamc response. The goal of he symbolc modelng capably s o creae applcaon-specfc models ha elmnaes all: (1) opology-based Lagrange mulplers, (2) mahemacal operaons leadng o zero resuls, (3) neffcen algorhm srucures, (4) logc blocks, (5) do-loops, and (6) all oher compuer language facles ha ac o slow down he smulaon performance. An objec-orened ls-based compuer language s used o develop he sofware. The paper s presened n seven secons. The mahemacal model s presened n he frs secon. Ths secon presens he quas-coordnae ransformaons requred for buldng he equaons of moon usng Lagrange s equaons. The second secon presens he general form for he consran equaons and he ransformaons requred o map general hnge consran condons o generalzed coordnae form. The sysemlevel equaons of moon are presened n secon hree. The paral dervave models for evaluang he Lagrangan are presened n he forh secon. The generalzed force s presened n he ffh secon. Resuls of a 10-body HMMWV are presened n he sxh secon. Conclusons are presened n secon seven.

5 QUASI-COORDIATE FORMULATIO FOR EQUATIOS OF MOTIO The mahemacal modelng echnque used n hs paper s based on he work of Joseph-Lous Lagrange ( ), who publshed hs analycal dynamcs mehod n 1788 n Mécanque Analyque (1). In hs he lays down he law of vrual work, and from ha one fundamenal prncple, by he ad of he calculus of varaons, deduces he whole of mechancs, boh of solds and fluds. Hs mehod remans aracve oday for wo reasons. Frs, gven he sysem knec energy, he process of generang he equaons of moon (EOM) s reduced o performng mechancal dfferenaon of a scalar funcon. Second, hs mehod provdes an auomaed way o elmnae opologybased consran forces and orques. Unlke he earler force-based mehods of ewon and Euler, Lagrange s mehod focuses aenon on he physcal dsplacemens of he bodes. The recpe for consrucng he EOM by Lagrange s mehod consss of four seps. For all bodes n he model one needs o buld: he poson, velocy, orenaon, and angular velocy vecors; he sysem knec energy; he EOM by dfferenang he knec energy; and he generalzed forces. The only weakness of he mehod s ha hand calculaons become cumbersome, edous, and error-prone when many bodes and DOF are nvolved. Ths paper presens a symbolc envronmen for handlng he ransformaonal deals requred generang he EOM and buldng hgh-performance Forran smulaon producs. Lagrange s Mehod Lagrange s equaon s obaned by applyng he calculus of varaons for Hamlon s prncple (2,3):

6 ( T V ) d ( ) ( ) 2 δ δ δ 1 = 0; = 0; = where q denoes he vecor of generalzed coordnaes, he me nerval end pons are assumed o be fxed, and δ denoes he usual varaonal symbol. Applcaon of Hamlon s prncple leads o he lagrangan equaons: d L L = Q ; = 1,, n d q q (1) where L = T V denoes he scalar Lagrangan funcon, T = T(q,q) denoes he knec energy, V=V(q) denoes he poenal energy, and from he vrual work = δ he generalzed force s defned by = 1 δw Q q Q. Equaon (1) s vald for ndependen generalzed coordnaes. Many problems, however, are more naurally analyzed by nroducng dependen ses of generalzed coordnaes. The dependen coordnaes are referred as quas-coordnaes (2,3,4,5,6,7). In deed, for ground-vehcle smulaons, s very convenen o model he chasss wh body-fxed quas-coordnaes. Quas-Coordnae Transformaon Equaons A quas-coordnae verson of Eq. (1), for he base body roaon and ranslaon, s obaned by nroducng a change of varables of he form: where ψ rs ξs = ψ1sq 1 + ψ 2sq ψ nsq n; s = 1, 2,, n (2) s a known funcon of he ndependen generalzed coordnaes, q. The frs sep consss of generang he followng vecor versons of Eq. (1) for he base body roaon and ranslaon: d L L = Q (3) d α

7 and d L L Q = (4) r d v r where denoes an array of Euler angles, α denoes an array of Euler angle raes, denoes he neral poson vecor, and v denoes he neral velocy vecor. r Equaons (3) and (4) are evaluaed by nroducng a body knec energy of he form: T = 1 2 Γ 0 α J S Γ 0 0 C v S M 0 C α v (5) where J denoes he body nera ensor, S denoes cener of mass vecor expressed he form of a skew symmerc marx, M denoes he body mass, Γ( ) denoes a nonorhogonal ransformaon marx ha relaes Euler angle raes and body componens of angular velocy (.e., ω = Γα ), and C ( ) 1 denoes he drecon cosne marx ha maps neral vecor componens no body vecor componens (.e., v = Cv ). Inroducng a quas-coordnae change of varables n Eq. (5) leads o: T ω J S ω = 1 2 v S M where only he dynamc reference frame vecor componens appear. Roaonal Equaon Transformaon Equaon (3) s ransformed by observng ha Eq. (4) s a funcon of Euler angles and Euler angle raes (, ) v. Afer he ransformaon, he new varables are Euler angles and body componens of angular velocy (, ω ). Snce Γ = Γ( ) and C = C ( ) n

8 Eq. (5), follows ha he ransformaon mus consder boh he orenaon and velocy varables. For smplcy, we assume ha any poenal erms have been have been accouned for n he generalzed forces, so ha only knec energy parals mus be consdered. Equaons are requred for T, T, and d d T. The chan rule of calculus s used o complee he ransformaon. To hs end, he paral dervave for T follows as: ω ( v ) = + ω v T T T T C $ % ( v ) C$ % Γ T = ω Γ + ω v "! # "! # (6) Smlarly, he paral dervave for T s gven by: T ω T T = = Γ j ω ω j (7) where j = 1, 2, 3 denoes an mpled summaon. The me dervave of Eq. (7) s d T T d T = Γ + Γ d ω ω j j j d j (8) where he me dervave of -j h elemen of follows as: Γ j Γ j 1 Γ j = = Γ ω (1 x 3) (3 x 1) (9) 1 C denoes he drecon cosne marx defned by C [From][To], where s he neral frame and denoes he body frame and C = (C ) T.

9 ! where he Euler angle raes have been replaced wh body angular veloces. Inroducng Eqs. (6), (8), and (9) no Eq.(3), and mulplyng hrough by -, leads o " #! " # C Γ ω ( v ) C Γ $ % 1 1 C ω ( v ) C 2 $ % 2! Γ ω Γ " C # ( v ) C $ % 3 d T Γ T T + Γ Γ Γ Γ = Γ d ω ω v (3 x 3) (3 x 3) 3 Q (10) Roaonal Quas-Coordnae Equaon Of Moon Afer processng Eq. (10) symbolcally for all welve Euler angle sequences, one obans he ransformed roaonal EOM: &( ') d T T T + [ ω] + v = Γ Q (11) d ω ω v where he new marces have he followng form: [ ] *, + * + 0 ω ω, 0 v v, - * +, -, -. /, -. / -. / ; v v3 0 v1 2 ω 1 0 v2 v1 0 ω = ω ω = ω Many auhors (4,5,6,7) have denfed he frs marx of Eq. (11). Typcally he second erm has been assumed. The necessary condon presened n Eq. (10) for beleved o be an orgnal conrbuon. Translaonal Equaon Transformaon Equaon (4) s ransformed by observng ha Eq. (5) s only a funcon of he [v ] s ranslaonal velocy. Equaons are requred for T r, T v, and d d T v. The chan

10 rule of calculus s used o complee he ransformaon. To hs end, he ransformaon for T r follows as: T ( r ) T ( Cr ) T T = = = ( C ) (12) r r r r r r Smlarly, he paral dervave of he neral velocy vecor s gven by: from whch follows ha: ( v ) ( Cv ) T T T T = = = ( C ) (13) v v v v v v d T T d T = ( C ) + ( C ) (14) d v v d v Inroducng Eqs. (12) and (14) no Eq. (4), and mulplyng he equaon ) hrough by ( C, leads o he ransformed equaon d T T T + ( C ) ( C ) ( C ) ( C ) ( C ) Q = (15) r d v v r Ths equaon s furher smplfed by usng he knemac deny for he me dervave of he drecon cosne marx (2,3,5) gven by where [ ω ] s defned followng Eq. (11). [ ω] C = C (16) Translaonal Quas-Coordnae Equaon of Moon Inroducng Eq. (16) no Eq. (15), one obans he ransformed ranslaonal EOM: d T T T + [ ω] ( C ) Q = (17) r d v v r

11 Equaons (11) and (17) represen he desred quas-coordnae forms for Lagrange s equaon, for he base body roaon and ranslaon. Generalzed Coordnaes Afer he quas-coordnae ransformaon has been compleed, he me dervave of he generalzed coordnae vecor s COSTRAIT EQUATIOS ( ω 1, ω 2, ω 3, v 1, v 2, v 3, 7,, n ) q = q q (18) Capables are provded for supporng user-specfed fxed and me-varyng consrans. A sysem-level velocy consran s defned by: V = b (19) where denoes he global consran marx, V denoes he sysem velocy vecor, and b denoes he vecor of user-specfed consran raes. Ths equaon s vald for all of he veloces n he sysem model. The sysem-level velocy vecor, for an n-body sysem, s defned by V = [ ω, v, ω, v,..., ω, v ] (20) n n where he angular velocy and ranslaonal veloces are provded for each body. The srucure of s presened n he hnge knemac secon. To be useful for he quas-coordnae varables of Eq. (18), one needs a velocy ransformaon ha relaes Eqs. (18) and (20). The ransformaon equaon s defned by Gven Eqs. (18) and (20) where ω = ω (, ) and v v ( q, q) V j j q q = T q (21) j =, he ransformaon marx, T, s obaned usng sandard symbolc ules, so ha no specal programmng s requred. Inroducng Eq. (21) no Eq. (19), leads o he generalzed coordnae consran marx j

12 where Tq = q = b (22) = T. Generally and T have a very complcaed srucure, however, he produc T s very compac and smple. Hnge Knemacs The general velocy consran marx of Eq. (19) s obaned by consderng he consraned DOF a each hnge. A 6x1 knemac equaon for he me dervave of he hnge raes follows as v h = V + V q n p m (23) where denoes he Euler angle hnge raes, v denoes he hnge ranslaonal veloces, V = [ ω, v ] denoes he ouboard body velocy sae, V = [ ω, v ] denoes he nboard n n n body velocy sae, m m m p denoes he knemac ransformaon marx ha maps he ouboard body veloces o he hnge frame, and q denoes he knemac ransformaon marx ha maps he nboard body veloces o he hnge frame. Equaon (23) provdes dfferenal equaons for he free hnge varables and consran equaons for Eq. (22). Consraned Hnge Raes Assumng ha he consraned DOT of Eq. (23) have been denfed, one obans a noonal sysem-level consran equaon of he form: p, h 1 q, h p, h 2 q, h = : ; p, h m q, hm (24) where each row has only wo marx npus, he h k subscrp denoes he marx block conssng of he rows correspondng o he consraned hnge DOF, and m denoes he

13 number of hnges (m n-1 because of he possbly of closed-loops exsng n he model). The, T, and marces are symbolcally generaed durng he pre-processng sep of he model-buldng algorhm. The me dervave of Eq. (24) s no formulaed symbolcally, because he produc of marces defned by Eq. (22) represens a relavely small n c x n q marx ha s easly dfferenaed symbolcally, n c denoes he number of user-defned consrans (ypcally small,) and n q denoes he number of generalzed coordnaes n Eq. (18). Free Hnge Raes Assumng ha he free DOF of Eq. (23) have been denfed, knemac dfferenal equaons of he form ( ),, = V + Vm (25) h, f q f n p f are defned, where (*) denoes an array of free generalzed coordnae raes, h, f denoes he free hnge DOF, f denoes he marx block conssng of he rows correspondng o he free hnge DOF. Durng symbolc processng, Eq. (25) s saved as a srng varable(s) for wrng he Forran subroune for he sysem knemacs equaons. Closed-Loop Hnge Knemacs Closed loops are deeced n he npu opology daa by defnng an nboard body ls as b b b b n In:= [ 0, j, k,..., p ] (26) where b r denoes he r h body, he negers denoe he nboard bodes, and 0 denoes he neral frame for he base body (b 1 ). For example, usng Eq. (26), he nboard body for b 3 s In( b 3 ) = k. Ths array allows he enre sysem opology o be examned.

14 > Assumng ha hnge σ has been denfed as a cu-jon hnge, where he bodes lnkng he hnge are he ouboard body α and nboard body δ; one obans he cu-loop verson of Eq. (23) gven by 0<1 2 4<5 3 v σ = V + V q( α ) α p( δ ) δ (27) Equaon (26) s used o deec a shared body along he nboard pah of bodes for Eq. (27). Assumng ha he common body has been denfed, say π, hen Eq. (23) s used o defne a generc body velocy ransformaon of he form V = V v 1 n q( n) p( In(n) ) In(n) h (28) where In(n) s defned by Eq. (26). y repeaedly applyng Eq. (28) for Eq. (27), Eq. (27) s ransformed so ha V V and V V = =. Collecng he nermedae resuls α π δ π durng he ransformaon process for Eq. (27) leads o > = ( ) + V v σ π (29) where he subscrp Σ denoes a collecon of all he erms obaned n he ransformaon. The srucure of Eq. (29) dffers from Eq. (23) because only one body velocy appears, namely V π. Generalzed Closed-Loop Velocy Consran Equaon The user-specfed me-varyng consran of Eq. (19) s generalzed o accoun for closed-loop consrans of he form of Eq. (29), by denfyng he consraned DOF n Eq. (29) and developng he modfed sysem-level consran marx

15 H L F H C D I I E G A cl V = b cl where b = and bcl? =? cl C D E C D E H I J K (( ) ( ) σ,, ) c c c (30) and he consran block marx s gven by C b 1 b π b n-1 b C n H I H I c = 0 0 0, c (31) where he b j denoe he body numbers and only one block s none-zero, and M,c denoes he marx block marx of Eq. (29) ha corresponds o he consraned DOF for hnge σ. Inroducng he velocy ransformaon marx, T, of Eq. (21) he generalzed coordnae verson of Eq. (30) follows as q = b where = T (32) cl cl cl Knemac Dfferenal Equaon for he Cu-Loop Hnge Free DOF The knemac dfferenal equaon for he free cu-loop DOF of hnge σ n Eq. (29), are gven A J K v σ, f ( ) ( Vπ ) = +, f, f (33) where he subscrp f denoes he free DOF. Equaon (33) s saved as a srng varable(s) for he symbolc generaon of he knemac dfferenal equaons. SYSTEM-LEVEL EQUATIOS OF MOTIO The quas-coordnae form for he EOM are assembled as follows:

16 S U W T X D A E OQ PR d T T T [ ω] d ω ω v + + v = Γ Q d T T T + = d v v r [ ω] ( C ) Q r d T T = Q ; = 7,, n d q q (34) The symbolc processor evaluaes and ransforms hese equaons o smpler forms. The poenal energy s assumed o be zero. All loads are assumed o be provded by userspecfed force and orque models. Marx Form Of The Equaons Of Moon Afer processng Eq. (34), one has an EOM and consran equaons of he form: M q = F λ and q = b (35) q where λ denoes he consran Lagrange mulpler for boh me varyng and/or closedloop consran calculaons. The soluon for he consraned generalzed coordnaes s obaned by dfferenang he consran marx as q + q = b (36) and combnng Eqs. (35) and (36) n he form H I JYK J K F GD E D F q E M q = 0 λ b q (37) Ths equaon represens he classc descrpor form for he equaons of moon. Snce he number of me-varyng/closed-loop consrans s ypcally small, hs equaon s easly nvered symbolcally usng a sparse LDL (8) algorhm, whch rgorously elmnaes all zero operaons hrough ou he lnear algebra reducon process.

17 LAGRAGIA SPARSE PARTIAL DERIVATIVE ALGORITHM Compung paral dervaves of he sysem knec energy generaes he equaons of moon for Eq. (34). Two seps are requred. Frs, he knec energy parals mus be defned. Second, sparse algorhms are nroduced for accelerang he generaon of Eq. (34). Sysem Knec Energy For a collecon of lnked rgd bodes, he sysem knec energy s defned by: 1 T = ( M v v + ω J ω ) 2 = 1 (38) where ( ) denoes he vecor do produc, M denoes he mass of he h body, J denoes he nera ensor for he h body wh componens (referenced o he cener of mass), and v and ω denoe he h body ranslaonal and roaonal veloces. Paral dervaves of Eq. (38) are requred for generang he EOM. For engneerng-scale applcaons, buldng he knec energy and compung every paral dervave s very neffcen, because many of he paral dervaves vansh. Grea compuaonal effcency s realzed by denfyng he funconal dependences for he ranslaonal and roaonal veloces as a pre-processng sep. Ths allows a sparse paral dervave algorhm o be developed for handlng hese calculaons. Knec Energy for a Sngle ody For ndvdual bodes he knec energy paral dervave calculaons are smplfed by defnng he knec energy n he momenum form: T = h + v p ( ω ) / 2 (39)

18 where he body angular momenum s defned by h = J ω and he body lnear momenum s defned by p = M v. Frs-Order Sngle ody Knec Energy Parals The paral dervaves of he knec T ξ energy are defned as, q follows as T =. From Eq. (39), he frs-order paral dervave w.r.. ξ T, = ω, h + v, p q q q (40) Smlarly, he paral dervave of Eq. (39) wh respec o q s gven by T, = ω, h + v, p q q q (41) Tme Dervave of T, q Takng he me dervave of Eq. (41) leads o d ( T, ) T,, q T q,, q = q q j j + q q j j (42) d where he frs erm of Eq. (42) s gven by,,.,,. T q q v,,. = + q p + ( ω ) h ( ), q ( h,. ), (,. ) q q j j + v q p q q j j j j j q q j q q j q q j ω (43) and he second erm n Eq. (42) s gven by T,,. q = ω, h,. q + v, p,. q (44) ( ) ( ) q q j j q q j j q q j j where he knemac paral dervave deny ω,, = V,, = 0 has been used, and momenum parals are defned by, ξ, ξ, ξ, ξ q q q q j j h = J. ω and p = M v for ξ = q or q (45) The brackeed erms n Eqs. (40) hrough (44) ndcae ha a vecor-nner produc s evaluaed for he j h varables.

19 Avodng An Exponenal Exploson n he Sze of he Symbolc Equaons A key pon o observe n Eqs. (40) hrough (44) s ha he lnear and angular momenum appear (hgh-lghed n red). Symbolcally, hs creaes an opporuny o defne body level equaons. The ssue s ha h and p can be defned n wo ways durng a symbolc compuaon. Frs, he symbolcally generaed expressons for ω and v are used, where ω = f ( q, q ) and v = g ( q, q). The problem wh hs approach s ha each vecor can conss of ens o hundreds of symbolc erms n real-world applcaons. The problem becomes nracable when many producs of symbolc expressons are mulpled ogeher. Second, h and p can be reaed as numbers, when he parals of Eqs. (40) hrough (44) are symbolcally evaluaed and saved for generang opmzed Forran models. The second opon s he bes choce. The second opon acs o reduce an exponenal exploson n he number o erms appearng n he symbolcally generaed equaons of moon. Durng a mulbody smulaon, ha uses Eqs. (40) and (44), he lnear and angular momenum vecors are numbers no symbolc expressons. Ths observaon represens one of he key ssues ha separaes numercal algorhms from symbolc compuaonal sraeges; opporunes of hs ype mus be exploed a possble every sep n an algorhm. Mass Marx Calculaon From Eq. (42) he sysem mass marx s defned as n n M = h + V p ( ω,,,, j j ) j q q q q = 1 j= 1 (46)

20 where he momenum paral dervaves are defned by Eq. (45). The non-vanshng elemens of, j, and M j are saved as srng varables for symbolc processng. A Forran mass marx s wren ha only updaes he me-varyng erms all zero operaons are by passed. Sparse Paral Dervave Calculaons The knec energy paral dervaves requre calculaons of he general form: ψ, where ξ = q or q ξ These calculaons are smplfed by defnng he funcon relaonshps for ψ as a preprocessng sep. The pre-processng sep consss of scannng V n Eq. (20), and developng a ls-based daa srucure for sorng he sub-lss ha denfy he funconal dependences for each varable n V, leadng o b b b 1 2 Θ = [[ ϒ, ϒ ],[ ϒ, ϒ ],,[ ϒ, ϒ ]] ω ω ω 1 v1 2 v2 n vn n where b j denoes he j h body and ϒ denoes a ls conanng he generalzed coordnae raes ha appear n each varable. For example, assumng ha he roaonal and ranslaonal varables for he 3 nd body have he followng funconal dependences ω = ω ( ω,, ) and v = v ( ω, v,,, l ) hen he ls funcons for ϒ ω and ϒ 3 v are defned by 3 ϒ ω = [ ω,, ] and ϒ v = [ ω, v,,, l ] A pre-processng algorhm also deermnes he global sorage locaons for he daa conaned n Θ, defned as Ω. The srucure of Ω s dencal o Θ, excep ha he varables are replaced by neger locaons. Ths daa formaon s compued one me and used everywhere n he symbolc processng algorhm. The nformaon conaned n

21 S U W T X V Θ perms sparse paral dervaves o be compued. The nformaon conaned n Ω perms ndrec daa sorage algorhms o be developed. Ths daa perms a grea reducon n he number of paral dervaves ha need o be compued for evaluang Eq. (34). As an example, hese daa-srucures are used o reduce compuaons, consder he follow vecor do produc ha s summed over all bodes n n lengh ( Θ ) c = [ c, c,, c ] = a. b = a * b 1 2 n, q Θ ( r ) Θ ( r ), q r = 1 = 1 r ϒ whch replaces he n nner summaons wh lengh( Θ r ) summaons, whle compung he only non-vanshng paral dervaves. GEERALIZED FORCE The generalzed forces and orques are compued usng he followng equaon: Q v ω k k = Fk + Τk k = 1 q q where Q denoes he h generalzed force, F k denoes he k h body force acng a he (47) cener of mass, Τ k denoes he k h body orque acng a he cener of mass. The sparse paral dervaves used for compung he knec energy calculaons are re-used for he generalzed for calculaons. These equaons are saved as srng varables for generang Forran Sofware. 10-ody HMMWV MODEL AD SIMULATIO RESULTS TD, Daa defnon, Tmng Resuls, Tmng Comparsons for oher mulbody ools and lnear algebra compuaons, and smulaon resuls COCLUSIOS

22 TD REFERECES 1. Lagrange, J.L., Méchanque Analyque (Pars, 1788). Many laer Addons. 2. Whaker, E.T., Analycal Dynamcs of Parcles and Rgd odes, Dover Publcaons, ew York, 1944, pp Pars, L.A., A Trease on Analycal Dynamcs, Ox ow Press, Merovch, L., Mehods of Analycal Dynamcs, McGraw-Hll Publshng Company, ew York, Y (1970), pp Lkns, P.W., Analycal Dynamcs and onrgd Spacecraf Smulaon, Techncal Repor , July 15, 1974, Je Propulson Lab, Pasadena, CA. 6. Merovch, L. and Qunn, R.D., Equaons of Moon for Maneuverng Flexble Spacecraf, Journal of Gudance and Conrol, Vol. 10, o. 5, Sep.-Oc. 1987, pp Qunn, R. D., Equaons of Moon for Srucures n Terms of Quas-Coordnaes, J. Of Appled Mechancs, Vol. 57, o. 3, pp , Sep Engeln-Müllges, G, and Uhlg, F., umercal Algorhms wh Forran, Sprnger Verlag, 1996.