Modeling and computing constrained

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1 F EAURE A RICLE HE COMPUAION OF CONSRAINED DYNAMICAL SYSEMS: MACHING PHYSICAL MODELING WIH NUMERICAL MEHODS Reseaches have nvestgated modelng and computaton of constaned dynamcal systems, but scentsts and engnees sometmes ovelook consstent matchng of mathematcal modelng and numecal methods. he authos use multbody system softwae to study offhghway vehcles, automoble and andod cash smulaton, and planetay ephemedes. Modelng and computng constaned dynamcal systems a tme-vayng system consstng of bodes connected by jonts that can be epesented by constant equatons nvolves modelng eal-wold poblems and constuctng dffeental algebac equatons and numecal ntegaton. he pupose s to obtan genealzed coodnates, the tme devatves, and body foces fo dynamc analyss and smulaton. Reseaches have developed geneal appoaches and softwae packages fo multbody knematc and dynamc analyss, 1 4 yet specfc examnaton of systems fom fst pncples offe dffeng pespectves of old and new poblems and eveal featues that mght have ntally passed unnotced. Fo example, an ncompatblty between physcal modelng and numecal methods mght lead to naccuate conclusons about system desgn. Authos such as Edwad Haug, 5 Pavz Nkavesh, 2 and Ahmed Shabana 3,4 have substantally pusued plana and spatal multbody modelng and constaned vaatonal dynamcs. he multbody system equatons that esult fom the modelng ae n fact dffeental-algebac equatons (DAEs), because the system equatons ae augmented wth algebac constants that defne the connectvty of the system s components. (Fo nfomaton about addtonal DAE eseach, see the sdeba.) In ths atcle, we use ou own plana and spatal constaned vaatonal system softwae, Multbody System, 6 8 to compute offhghway vehcle dynamcs, cash smulatons, and planetay ephemedes and to ecod the ntegaton method s accuacy. Studyng foce modelng n multbody systems by nvestgatng pogessvely smoothe body nteacton foces should encouage scentsts and engnees to develop new methods to detect numecal poblems n system computaton. hese methods mght nvolve eplacng the foces, n pat wth the equvalent constant equatons, o usng stffness detecton technques to employ futhe physcal modelng nea the tme nteval concened /1/$1. 21 IEEE BUD FOX, LESLIE JENNINGS, AND ALBER Y. ZOMAYA Unvesty of Westen Austala Multbody system equatons he followng devaton follows the wok of Shabana and shows how we can fomulate the geneal equatons of dynamc equlbum fo 28 COMPUING IN SCIENCE & ENGINEERING

2 Related wok Multbody system equatons ae n fact dffeental-algebac equatons (DAEs), because the system equatons ae augmented wth algebac constants defnng the connectvty of the system s components. Bend Smeon and hs colleagues 1 povde a summay of the theoy, computaton, and applcatons of DAEs. Kathyn Benan and he colleagues ntoduce the basc types of DAEs, constaned vaatonal poblems, the theoy, solvablty and ndex concept, lnea and nonlnea systems, numecal methods, algothms, applcatons and DAE examples. 2 U Asche and Lnda Petzold have caefully oganzed the study of ODEs and DAEs and pesent methods fo the computaton. 3 Ahmed Shabana has studed tack vehcle dynamcs, 4 7 hs wok pusung a dffeent method of dynamc equatons computaton. he nvestgaton of constant complance eveals the dffculty encounteed n numecal computaton of the tack vehcle equatons. Contact foce modelng athe than the dffeentaton of the constant equatons appeas to be the cause of computatonal dffculty, and new methods and algothms mght be equed to detect numecal poblems n advance, to facltate ntegaton. (Fo futhe detals of tack vehcle dynamcs see the atcle by Bud Fox and hs colleagues. 8 ) We have also nvestgated automoble cash smulaton nvolvng two plana andod multbody system models. he fst s a smplfed andod system, and the second s a 46-body-pat andod muscle-skeletal and vetebal system. he soft andod tssue contacts esult n smoothe contact foce models and, hence, ae ease to compute. Fo futhe detals of automoble and andod modelng and computaton, see the atcle by Fox and hs colleagues. 9 Planetay ephemedes computed by JPL ae avalable n the Astonomcal Almanac. 1 We nvestgated the numecal accuacy of ntegatng smooth astophyscal systems by computng the moton of an atfcal satellte and ephemedes of the planets of the sola system. Fo futhe detals of astophyscal systems see the atcle by Fox and hs colleagues. 11 Refeences 1. B. Smeon, C. Fuhe, and R. Rentop, Dffeental-Algebac Equatons n Vehcle System Dynamcs, Suveys on Mathematcs fo Industy, 1991, pp K.E. Benan, S.L. Campbell, and L.R. Petzold, Numecal Soluton of Intal-Value Poblems n Dffeental-Algebac Equatons, Elseve Scence, New Yok, U.M. Asche and L.R. Petzold, Compute Methods fo Odnay Dffeental Equatons and Dffeental-Algebac Equatons, SIAM, Phladelpha, Penn., Nakansh and A.A. Shabana, Contact Foces n the Nonlnea Dynamc Analyss of acked Vehcles, Int l J. Numecal Computng Methods n Eng., Vol. 37, 1994, pp Nakansh and A.A. Shabana, On the Numecal Soluton of acked Vehcle Dynamc Equatons, Nonlnea Dynamcs, Vol. 6, No. 6, Dec. 1994, pp M.K. Sawa,. Nakansh, and A.A. Shabana, Chan Lnk Defomaton n the Nonlnea Dynamcs of acked Vehcles, J. Vbaton and Contol, Vol. 1, No. 2, May 1995, pp A.A. Shabana, Computatonal Dynamcs, John Wley & Sons, New Yok, B. Fox, L.S. Jennngs, and A.Y. Zomaya, Numecal Computaton of Dffeental Algebac Equatons fo Non-lnea Dynamcs of Multbody Systems nvolvng Contact Foces, to be publshed n ASME J. Mechancal Desgn, B. Fox, L.S. Jennngs, and A.Y. Zomaya, Numecal Computaton of Dffeental-Algebac Equatons fo Non-lnea Dynamcs of Multbody Andod Systems n Automoble Cash Smulaton, IEEE ans. Bomedcal Engneeng, Vol. 46, No. 1, 1999, pp he 1998 Astonomcal Almanac, US Govenment Pntng Offce, Washngton, DC, B. Fox, L.S. Jennngs, and A.Y. Zomaya, Numecal Computaton of Dffeental Algebac Equatons fo the Appoxmaton of Atfcal Satellte ajectoes and Planetay Ephemedes, ASME J. Appled Mechancs, Vol. 67, No. 3, Sept. 2, pp multbody systems usng genealzed Catesan coodnates and the vtual wok pncple. 4 hs pncple states that the vtual wok due to the neta foces s equal to the sum of the vtual wok due to the extenally appled foces and constant foces. hat s, δw I, = δw e, + δw c,. (1) he vtual wok of the extenally appled foces actng on the gd body s δ W e, e, = Q δq (2) whee Q e, s the vecto of genealzed extenally appled foces coespondng to the vecto of genealzed plana coodnates, JANUARY/FEBRUARY 21 29

3 q = [R x,, R y,, θ ]. (3) he vtual wok of the jont constant foces actng on the gd body s C(q, t) = (12) and dffeentatng twce wth espect to tme gves δ W c, c, = Q δq (4) Cq = Q = ( Cq ) q C q C. q d q q 2 qt tt (13) whee Q c, s the vecto of genealzed constant foces coespondng to the genealzed coodnates q. Fnally, the vtual wok due to the neta foces s gven as δwi, = ρ δdv V (5) Augmentng Equaton 11 wth Equaton 13, we obtan the complete system of 3NB + NC equatons n 3NB + NC unknowns, M Cq q Qe Cq = λ Qd (14) whee ρ and V ae the densty and volume of body, espectvely. he poston of an abtay pont on the body wth espect to the global coodnate system s Fo spatal systems, Eule paametes Θ = ( θ, θ, θ, θ ) (15) = R + Au = da R + u θ dθ (6) (7) athe than Eule angles ae used, esultng n the genealzed coodnates q R x R y R z θ θ1θ2 θ3 = [ ]. (16) and the otaton matx of the body wth angle of otaton θ s cosθ snθ A =. (8) snθ cosθ We can expess the vtual wok of the neta foces of body as δ W I, v, = [ Mq Q ] δq (9) whee Q v, s the vecto of centfugal neta foces. By substtutng Equatons 2, 4, and 9 nto Equaton 1, we obtan v, e, c, [ Mq Q Q Q ] δq =. We can show that the system equatons ae Mq + C λ = Q q e (1) (11) whch epesents a system of 3NB equatons n 3NB + NC unknowns. NB s the total numbe of bodes n the plana system, and NC s the total numbe of ndependent constant equatons. Because thee ae moe unknowns than equatons, the constant equatons ae adjoned to the system equatons. We can wte the constant equatons denotng body connectvty as We can show that the otaton matx s A = I+ sn V + ( V 2 2 θ θ 2 ) sn 2 (17) and the theoetcal constant equaton fo body s ΘΘ 1= he knetc enegy of body s 1 = dv 2 ρ V and we can show that 1 1 = Rm + m RR, R Θ ΘΘ, Θ. 2 2 (18) (19) (2) By substtutng ths expesson nto the Lagange equatons of moton d dt = Q q q (21) we ave at the equatons of moton fo body, whch ae 3 COMPUING IN SCIENCE & ENGINEERING

4 Body Y X θ Y j u p u pj p j X j pj R j Y R pj p p Suface body j Global coodnate system X Fgue 1. Body suface contact geomety ndcatng vecto nfomaton equed to complete the body s spng-dampe contact foce model. m RR, m R Q Θ = Q ω. (22) Augmentng these equatons wth the constant equatons yelds the spatal augmented system equatons fo body mrr, m ΘΘ, 2Θ [ 2Θ] Q er, = Q e, Θ Qv, + 2ΘΘ (23) fom whch we can constuct the complete system of equatons, epeated hee as M Cq q Qe Cq = Qd. λ 2G I + R ΘΘ, Θ ΘΘ, R Θ λ (24) Dffeental algebac equatons DAEs ae dffeental equatons augmented wth algebac constants. Lnda Petzold and he colleagues have publshed mateal on the solvablty and computaton of DAEs; the book gves an ntoducton to DAEs and efeences theen concenng the eale wok on the theoy and computaton. 9 U Asche and Petzold 5 pesent many mpotant numecal methods fo the computaton of odnay dffeental equatons (ODEs) ncludng ntal-value poblems and bounday-value poblems, and DAEs. A DAE mght be of the fom Fx,x (,) t = (25) he DAE of the multbody system consdeed hee s of the fom I M C q q v f q, t µ v Q e Cq,t = ( )= ( ) (26) whee µ = λ and v = q. Howeve, the matx on the left-hand sde s sngula. Dffeentatng the thd equaton of the matx vecto system gven by Equaton 26, wth espect to tme twce, JANUARY/FEBRUARY 21 31

5 (a) (b) Fgue 2. (a) Smple andod model whose genealzed body pats epesent clustes of ogans and s used n automoble accdent smulaton. (b) Vetebal andod model that shows all vetebae n the spnal column and s used to model andod dynamcs n both fowad and ea automoble accdent smulaton. yelds I M C q C q whch has a nonsngula leadng matx povdng C/ q s of full ank fo all tme. he followng defnton classfes a DAE wth espect to dffeentaton of the system equatons gven by Equaton 25. he mnmum numbe of tmes that all o pat of the DAE Fx,x,t ( )= must be dffeentated wth espect to t to detemne ẋ as a contnuous functon of x and t, fo t n some nteval, s the ndex of the DAE. 9 he ognal system has been dffeentated twce, and the substtuton of µ = λ can be consdeed as an addtonal dffeentaton. hs esults n the ODE n Equaton 27 and hence the ognal system Equaton 26 s a DAE of ndex thee. Note that hee t µ ()= t λ( τ) dτ q v fq,q q = (, t )= f (, v, t ), µ (27) Fgue 3. ack vehcle model showng the spocket, dle, olle, and tack lnks used fo offhghway vehcle modelng. s computed by the ODE softwae. o fnd λ, µ(t) must be dffeentated ths s, an unstable pocess. Asche and Petzold 1 and Kathyn Benan and he colleagues 9 dscuss the computatonal and numecal dffcultes that can ase as a esult of dffeentatng the constant equatons. he constant equatons mght not be satsfed as the ntegaton pogesses, and they do not ecommend excessve dffeentaton of the constants. Multbody models and contact foces Investgatng smple models eveals dynamcal nfomaton useful fo modelng lage systems. Fgue 1 epesents a body-suface contact model. We can obseve oscllatoy moton though ths fom of contact and change the spng and dampng coeffcents to smulate dffeng mateal compounds. he model employs a spng-dampe element between ponts p and p j, the fome esdng on the most penetated pat of the body and the latte on the suface. he magntude of ths foce s F c = kδ j j + c j δ (28) whee k and c ae the spng and dampng coeffcents, espectvely, and δ j = R y, R y,j. (29) Shabana 4 states that the vtual wok due to a foce wth magntude s p p p j j j δq δw = F c j δ p j j j q q q = Q δq + Q δq j j F cj (3) 32 COMPUING IN SCIENCE & ENGINEERING

6 (a) (b) (c) (d) (e) (f) (g) (h) Fgue 4. A font mpact of 15 N s appled to a vetebal model of an automoble occupant estaned as shown. Images ae captued at.25-second ntevals. whee Q and Q (31a) (31b) he poston vecto wtten n the global coodnate system s whee and j = = R + Au R A u pj p pj p j j pj u p QR = Q c, = F θ QRj = Q c, j = F θ 3π cos θ = 2 3π sn θ 2 R R x, jx, u p = j I up A j, θ, u I A p j, j θ, j (32) (33a). (33b) We can use othe contact models such as the pogessve pseudo-dampng contact foce 7 pj pj pj pj F = kδ + cδ δ cj j j j (34) to model othe physcal systems such as those n Fgue 2. ack vehcle modelng (see Fgue 3) uses the contact foce models we showed eale, and changng the spng and dampng coeffcents to smulate soft body contacts can appoxmate the behavo of hade bodes. Employng specalzed constants n the contact egon can allevate numecal computaton dealng wth stffe systems. Results Off-hghway vehcle dynamcs, cash smulaton, and astophyscal modelng all nvolve spase systems of equatons and eque Gaussan elmnaton to be pefomed at each tme step n the computaton and ntegaton. Spatal systems may eque Eule paametes as the oentaton vaables athe than Eule angles, because accodng to Shabana, 3 the body-otaton matx mght not be well defned fo cetan body oentatons. Also, fo cetan values of tme and angula velocty ω, the submatx m θθ, may be sngula and hence the mass matx M alone may be sngula, although the leadng coeffcent matx s stll nonsngula. he followng matx n- JANUARY/FEBRUARY 21 33

7 veson technque fo the computaton of the system equatons should not be used, because t eques that the uppe left block A be nvetble (fo all tme steps). AB C D (35) whee = D CA 1 B, E = A 1 B, and F = CA 1. We should also be awae that thee ae fou Eule paametes that s, fou angula genealzed coodnates pe body. hee should be no moe angula constant equatons pe body than thee We used the vtual ae angula coodnates pe body to avod a dmensonally wok pncple n nconsstent system of equatons. conjuncton wth the Fgue 4 shows a typcal smulaton slde-set of a augmentaton multbody system. We can vsbly detect the fludty of method to deve a set moton and pefom the followng constant complance of plana DAEs. test to detemne the accuacy of the esults; h s the tmestep sze used n the ntegaton scheme, and C(q,t) ae the constant equatons: ( ) ( ) < Cq, t Cq, A + E F E = 1 1 F h. (36) he tack vehcle dynamcs study nvolvng contact foces exploes ntegaton of the system equatons usng dffeent ntegaton schemes. Studyng constant complance eveals that nonlnea, nondffeentable contact foces of hgh magntude (asng fom the body-nteacton modelng method) pesent numecal dffcultes fo softwae that expects almost analytcally smooth data. he twce-dffeentated constant equatons mght poduce some numecal naccuacy, but we egad the nondffeentable contact foce model as the numecal poblem s eal cause. he automoble cash smulaton nvolves numecally dffcult computaton due to the contact and constant foces. he contact foce modelng uses a pogessve dampng foce, whch s a functon of both the body penetaton dstance and ts tme devatve (see Equaton 34), and s not dffeentable eveywhee n tme. he smplfed model lagely pedcts the body dynamcs behavo; howeve, a moe nvolved vetebal model eveals geate consttuent body-pat detal, ncludng the whplash effect on the cevcal vetebae. 7 he thd system we nvestgated nvolves the spatal modelng of a planetay system usng Eule paametes. hs system has data that s analytcally smooth, because thee s no contact between any of the bodes. System equaton ntegaton s commenced usng avalable ntal helocentc ectangula equatoal genealzed coodnates and the tme devatves and contnues ove a tme nteval to geneate the planetay ephemedes and atfcal satellte tajectoes. We nvestgated constant complance to establsh the computaton s accuacy wthn a ange of thee to fve sgnfcant fgues of the data povded by the Astonomcal Almanac afte smulaton tmes of 2 and 36 days. he constant equatons ae the theoetcal constants ntoduced by the Eule paamete method we descbed eale. Lessons leaned In multbody modelng nvolvng wtng code fo numecal computaton of equatons, eseaches should pay caeful attenton to mathematcal modelng, numecal methods, and softwae. We found that f the mathematcal modelng and numecal methods ae matched n a way that cates to the dynamcs of the poblem beng studed, the esultng softwae development wll povde a useful eusable tool fo poblem nvestgaton n the futue. Mathematcal modelng Coodnate selecton must be caefully made, usually between genealzed Catesan coodnates and elatve coodnates. Inevtably, the spasty of the system equatons obtaned usng the genealzed coodnates s lost f we decde on the elatve descpton; howeve, ths leads to a smalle set of equatons we must solve. A decson between Eule angles and Eule paametes may be equed; nvetblty of system matces should be nvestgated by those wokng on the poblem to pevent the nadvetent computaton of a sngula system. A geneal multbody and computatonal dynamcs method mght allow ease genealzaton of a system than a moe specfc o patcula appoach, such as the Denavt Hatenbeg method commonly found n obotcs wok. Reseaches should caefully pusue foce modelng wth an emphass on contnuty and dffeentablty of the appopate functons so that the assocated numecal methods of the co- 34 COMPUING IN SCIENCE & ENGINEERING

8 ect ode ae selected. Hghe-ode schemes can make use of the avalable hghe-ode devatves and lead to moe accuate computaton. Numecal methods Reseaches need to consde the stablty of the poblem beng computed and the stablty of the numecal methods employed to compute the system of equatons. he eseache should nvestgate the natue of the egenvalues of the leadng system matx and defne a stablty constant to ad the dscusson of the esults. (See the Asche and Petzold book fo moe detals. 1 ) he method s effcency mght be at the pogammes dsceton. If they use ndustal softwae (fo example, the Lvemoe Solve of Odnay Dffeental Equatons [LSODAR] lne of ntegatos o the dffeental algebac softwae Dffeental Algebac Equatons System Solve [DASSL]), then automatc method swtchng wll match the appopate method to the natue of the poblem. he constant complance eo n system computaton should be quoted. Fo example, n the cuent wok, the constant complance gven by Equaton 36 s made fo each tme step, and the computaton s temnated should ths dffeence be too lage. In cetan cases, eseaches should nvestgate the local and global eos and note theoem 3.1 n the Asche and Petzold book, 1 whch elates consstency, stablty, and convegence. Requng too lttle accuacy of LSODAR lets the softwae skp ove the egons whee the data mght not be able to be ntegated accuately. he esultant tajectoes ae then naccuate, but the use does not know ths. Askng fo hgh accuacy means that LSODAR knows when t cannot ntegate ove a egon, so t stops wth an ndcated eo condton. he use then does not have tajectoes fo the full tme nteval but knows that what has been computed s accuate. Softwae Softwae tools such as Maple, Mathematca, and Matlab ae common packages that we can use alone o, n the case of Matlab, wth the C and Fotan pogammng languages. hese tools mght povde nsght nto the poblem unde nvestgaton. Poject selecton and evoluton often centes aound the avalablty of qualty and nfomatve softwae that s sutable fo lage, ongong pojects. Hee, we conveted the DAE gven by Equaton 26 to the undelyng ODE, Equaton 27. We used LSODAR to numecally ntegate the system. Howeve, t mght be moe sutable to dectly use DASSL and, n so dong, avod the tme consumng analytc dffeentaton of the constant equatons. Reseaches should also caefully consde selectng the appopate softwae fo the exstng hadwae wth a vew to potng the esultng poject code to othe platfoms n advance. Matchng physcal popetes wth numecal methods when modelng constaned dynamcal systems lets scentsts and engnees pefom accuate analyses of system behavo. We used the vtual wok pncple n conjuncton wth the augmentaton method to deve a set of plana DAEs. We deved the spatal system of DAEs by substtutng an expesson fo body knetc enegy nto the Lagange equatons of moton. he DAEs wee cast as ODEs though dffeentaton of the constant equatons. Contact foce modelng and ntegaton schemes must be caefully chosen and constant complance should be nvestgated to ecod the computatonal accuacy. he dffculty of ntegatng ove foces such as these n the ght-hand sde of the ODE can only be esolved f we employ new contact models o nondffeentablty detecton methods. We used the numecal ntegato LSODAR to ntegate ODEs, wheeas DASSL dectly computes DAEs. In ths wok, we dffeentated the constant equatons to cast the DAE as the undelyng ODE. Benan and he colleagues do not ecommend ths pocedue. 9 It mght esult n educed accuacy, but we have shown that the natue of the ght-hand sde, moe so than the dffeentated constant equatons, s the cause of numecal poblems, especally n systems nvolvng mpacts. Refeences 1. R.R. Ryan, Adams Multbody System Analyss Softwae, Multbody Systems Handbook, Spnge Velag, Beln, P.E. Nkavesh, Compute Aded Analyss of Mechancal Systems, Pentce-Hall, Uppe Saddle Rve, N.J., A.A. Shabana, Dynamcs of Multbody Systems, John Wley & Sons, New Yok, A.A. Shabana, Computatonal Dynamcs, John Wley & Sons, New Yok, E.J. Haug, Compute Aded Analyss and Optmzaton of Mechancal System Dynamcs, NAO ASI Sees, Vol. F9, Spnge-Velag, Beln, JANUARY/FEBRUARY 21 35

9 Scentsts get the stuff they need I need an actuato! (1) Go to Physcs oday s onlne Buyes Gude, the Yellow Pages fo the physcal scences B. Fox, L.S. Jennngs, and A.Y. Zomaya, Numecal Computaton of Dffeental Algebac Equatons fo Non-lnea Dynamcs of Multbody Systems nvolvng Contact Foces, to be publshed n ASME J. Mechancal Desgn, B. Fox, L.S. Jennngs, and A.Y. Zomaya, Numecal Computaton of Dffeental-Algebac Equatons fo Non-lnea Dynamcs of Multbody Andod Systems n Automoble Cash Smulaton, IEEE ans. Bomedcal Engneeng, Vol. 46, No. 1, 1999, pp B. Fox, L.S. Jennngs, and A.Y. Zomaya, Numecal Computaton of Dffeental Algebac Equatons fo the Appoxmaton of Atfcal Satellte ajectoes and Planetay Ephemedes, ASME J. Appled Mechancs, Vol. 67, No. 3, Sept. 2, pp K.E. Benan, S.L. Campbell, and L.R. Petzold, Numecal Soluton of Intal-Value Poblems n Dffeental-Algebac Equatons, Elseve Scence, New Yok, U.M. Asche and L.R. Petzold, Compute Methods fo Odnay Dffeental Equatons and Dffeental-Algebac Equatons, SIAM, Phladelpha, Penn., (2) Seach the poduct ndex fo actuato (3) Vew moe than 2, potental supples actuatos Amecan Pezo Ceamcs, Inc. Enegen Inc. EREMA Poducts, Inc. Industal Devces Cop. Kek Moton Poducts Inc. Newpot Cop. OptoSgma Cop. (4) Choose a company Newpot Cop. P.O. Box Deee Ave, Ivne CA 9266 el: (949) oll fee el: (8) Fax: (949) Poduct Categoes acoustc nose baes actuatos autocollmatos ball sldes... Aveage s 13 supples fo each of 2,2 poduct categoes! (5) Contact the company by phone, fax, o e-mal... and get you stuff! Bud Fox s a eseach assocate n the Paallel Computng Reseach Laboatoy n the Depatment of Electcal and Electonc Engneeng at the Unvesty of Westen Austala. He eceved hs BSc and PhD fom the Unvesty of Westen Austala. Hs eseach nteests ae n multbody dynamcs, computatonal scence, and mathematcal modelng. Contact hm at the Paallel Computng Reseach Laboatoy, Dept. of Electcal & Electonc Eng., Unv. of Westen Austala, Nedlands, Peth Westen Austala 697; budfox@ee.uwa. edu.au. Lesle Jennngs s an assocate pofesso n the Depatment of Mathematcs at the Unvesty of Westen Austala. Cuently, he s wokng on the nteface of optmal contol, numecal analyss, and softwae engneeng. He eceved a BSc fom the Unvesty of Adelade, Adelade, South Austala and a PhD n numecal analyss fom the Austalan Natonal Unvesty, Canbea, Austala. Hs eseach nteests le n numecal analyss and n the applcaton of optmal contol to human movement modelng, multbody systems, chemcal engneeng, and flte desgn. Contact hm at the Dept. of Mathematcs, Unv. of Westen Austala, WA, 697, Austala; les@maths.uwa.edu.au. Albet Y. Zomaya s a pofesso and Deputy-Head n the Depatment of Electcal and Electonc Engneeng at the Unvesty of Westen Austala, whee he also leads the Paallel Computng Reseach Laboatoy. He s an assocate edto fo the IEEE ansactons on Paallel and Dstbuted Systems and IEEE ansactons on Systems, Man, and Cybenetcs. He s the foundng edto-n-chef of the Wley book sees on Paallel and Dstbuted Computng. In Septembe 2, he was awaded the IEEE Compute Socety s Metoous Sevce Awad. Hs eseach nteests ae n hgh-pefomance computng, paallel and dstbuted algothms, computatonal machne leanng, scentfc computng, adaptve computng systems, moble computng, data mnng, cluste computng, megacomputng, and weless netwoks. He eceved hs PhD fom the Depatment of Automatc Contol and Systems Engneeng, Sheffeld Unvesty, Unted Kngdom. Contact hm at the Paallel Computng Reseach Laboatoy, Dept. of Electcal and Electonc Eng., Unv. of Westen Austala, Nedlands, Peth, Westen Austala 697; zomaya@ee. uwa.edu.au; COMPUING IN SCIENCE & ENGINEERING