Abstract: 1. Introduction: components such as damping from the water viscosity and different masses for large manipulated objects.

Transcription

1 Dynamc Smulaon for ero-gravy Acves Norman I. Badler, Dmrs N. Meaxas, Gang Huang, Ambarsh Goswam, Sueung Huh Unversy of Pennsylvana, Phladelpha, USA Absrac: Worng and ranng for space acves s dffcul n erresral envronmens. We approach hs crucal aspec of space human facors hrough D compuer graphcs dynamcs smulaon of crewmembers, her ass, and physcs-based movemen modelng. Such vrual crewmembers may be used o desgn ass and analyze her physcal worload o mze success and safey whou expensve physcal mocups or parally realsc neural-buoyancy ans. Among he sofware ools we have developed are mehods for fully arculaed D human models and dynamc smulaon. We are developng a fas recursve dynamcs algorhm for dynamcally smulang arculaed D human models, whch comprses nemac chans -- seral, closed-loop, and ree-srucure -- as well as he neral properes of he segmens. Moon plannng s done by frs solvng he nverse nemac problem o generae possble raecores, and hen by solvng he resulng nonlnear opmal conrol problem. For example, he mnmzaon of he orques durng a smulaon under ceran consrans s usually appled and has s orgn n he bomechancs leraure. Examples of space acves shown are zero-gravy self orenaon and ladder raversal. Energy expendure s compued for he raversal as. Keywords: dynamc smulaon, D compuer graphcs smulaon, arculaed body, human model, opmal conrol, moon plannng.. Inroducon: Worng and ranng for space acves s dffcul n erresral envronmens. There s a grea demand for human facors regardng he dynamc ably of he human body n space. In compuer graphcs (CG and vrual realy (VR such human facors would be helpful o creae realsc human anmaons, and desgn ass o mze success and safey whou always havng o resor o expensve physcal mocups or parally realsc neural-buoyancy ans. The laer accommodaes some of he -gravy experence bu adds unrealsc dynamc componens such as dampng from he waer vscosy and dfferen masses for large manpulaed obecs. There are many forward and nverse dynamcs smulaon echnques for smulang human behavor ha have been mplemened by CG researchers [4][9][], bu hey are no compuaonally effcen and smulaons canno be done n real me. Some conrol echnques have also been used, such as feedbac conrol o follow he desred raecores generaed by nverse nemacs [4] and he spaceme consran mehod based on opmzaon heory []. However, hey do no address explcly eher he orque opmzaon problem or dynamcs ha deal wh he varous opologcal confguraons of he arculaed body. Ths paper frs presens an effcen way o conver he geomery of an arculaed body o a dynamc ree srucured model, whch s suable for dynamc smulaon and moon plannng. In he dynamc ree, each on has only one degree of freedom, whch gves only one paren ln and one chld ln. Each ln has s own roo on, and up o hree branch ons. The ln can be a dummy ln, whch has no mass, lengh and nera. The roo of he dynamc ree usually s a base on fxed o he reference frame. Each leaf of he dynamc ree mus be ln, hough he ln can be a dummy ln. Jon Paren Ln Jon Chld Ln Ln Roo Jon Jon Ln Branch Jon Ln Ln Branch Jon Ln Branch Jon Fgure. The Relaonshp beween Jons and Lns Secon of he paper shows a smple ype of funcon ha we used for smulang he moons, and s properes for nemac and dynamc smulaon. All he dynamc smulaons n hs paper use hs funcon, whch we call nemac moon funcon, o specfy he moon of each on or he body.

2 In Secon an effcen recursve dynamcs mehod for he dynamc ree based on Feahersones mehod [] s suded and used o dynamcally smulae asronau self-orenaon (Secon 4 and ladder raversal n -gravy (Secon 5. Furhermore, n Secon we presen an opmal conrol algorhm based on hs effcen recursve dynamcs mehod and graden mehod o fnd he local mnmum on-orque moon under specfed consrans []. From he bomechancs pon of vew, he mnmum onorque moon s he opmal moon under ceran condons for human body, for example he moon o do wegh lfng n seconds. Fnally, n Secon 7 we oulne some energy expendure models ha can be evaluaed based on he compued on orques.. Dynamc Model for an Arculaed Body: Physcal properes: he number of degrees freedom of he on, he on ype, (such as -on, -on, and -on, ec. he on lms, he on orque lms, he dampng coeffcens for he on. Noe: each on has s own on coordnae frame called on prvae coordnae frame, whch s dfferen from he on local coordnae frame defned n he ree srucure, n whch he on roaon axs s always he z-axs. For example: n he on prvae coordnae frame, he z-axs and he y-axs are on roaon axes, and he x- axs s defned n he rgh-hand sense. In he on local coordnae frame boh roaon axes are defned as he z-axs. By roang he frs z-axs 9 degrees concdes wh he second z-axs. (See Fg.. Jon The physcally based smulaon of human moon s obaned from nemac and dynamc calculaons. The effcency of he compuaon depends on he algorhms used for hese calculaons. O Jon Type: -Jon An arculaed body can be represened opologcally as a ree whose lns represen s maor pars. Bascally, hs means here are no nemac loops and ha no par of he arculaed body s enrely dsconneced from he res. An mmoble ln (or an appropraely chosen ln f none of he lns s mmoble s consdered as he roo of he dynamc ree, and he ouermos lns are s leaves. Two lns conneced by a on are he paren and chld lns of he on, dependng on wheher he ln s more proxmal or dsal o he roo. The on s called he roo on of s chld ln. The ons wh he same paren ln are called he branch ons of he ln. The base ln s he ln connecng he fxed pon o he roo on of he dynamc ree. Every ln excep he base ln has exacly one roo on, bu may have up o hree branch ons f modelng humans. We represen he arculaed body as a ree srucure. Each on has he followng properes: Tree srucure properes: he ndex for he on, he ndex for s paren and chld lns. Noe he base ln s defned as he null ln, and he on conneced o he null ln s on. (See Fg.. O 9 Fgure. Jon Prvae Coordnae frame O for -on, Jon Local Coordnae frame O for o- z he roaon axs, and O for o-y he roaon axs Each ln has he followng geomerc and dynamc properes: Tree srucure properes: he ndex for he ln, he ndex for s roo on, he number of he branch ons, he ndex for each branch on. Noe a leaf ln has no branch on, we defne as he null on. Dynamc properes: he mass, he cener of mass n he lns roo on coordnae frame, he prncpal neral marx, and he roaon marx from he lns roo on coordnae frame o s prncpal coordnae frame. Denav-Harenberg noaon [] parameers for each par of he lns roo We consder only mulple-degree-of-freedom on wh orhogonal roaon axs.

3 and branch on (See Fg. : he lengh of he common normal a ; he dsance beween he orgn O and he pon H ; he angle α beween he lns roo on axs and branch on axs n he rgh-hand sense; he angle θ beween he lns roo on coordnae frame x -axs and he common normal, whch s he x-axs of he lns branch on coordnae frame, measured abou s roo on z-axs n he rgh-hand sense. If a lns roo on s a mulple-degree-offreedom on, he lns roo on z-axs s he las roaon axs n he on prvae coordnae frame. For example f he on ype s -on, he y-axs n he on prvae coordnae frame s he lns roo on z-axs we are usng. In he same manner, f a lns branch on s a mulple-degree-of-freedom on, he lns branch on axs s he frs roaon axs n he on prvae coordnae frame. Jon - Ln - H z Jon O y a x θ Ln Fgure. Denav-Harenberg Noaon y O α z Jon x Jon The numberng scheme we use for he arculaed body s ha each ln has he same ndex number as he lns roo on. The ndex of a lns roo on s less han he ndex of he lns lefmos branch on. The ndex of a lns lef branch on s less han he ndex of he lns rgh branch on. (The numberng on Fg. 4 follows hs scheme. Thus, we defne he arculaed body as a lef-rgh ree-le srucure. Base Null Ln J J: Jon L: Ln : D on : D on : D on L J Fgure 4. Numberng Scheme for Arculaed Tree L We now presen a mehod o effcenly represen an arculaed ree usng a dynamc ree represenaon. Mulple-degree-of-freedom ons can be synheszed from he approprae number of sngle-degree-of-freedom ons. A movng arculaed body could be reaed as a dynamc ree srucure by nroducng a fcous, unpowered on beween he roo and some fxed pon. If he roo of he dynamc ree has complee moon freedom hen he on has sx degrees of freedom. The arculaed body defned as an arculaed ree srucure s convered o a dynamc ree by represenng every mulple-degree-of-freedom on wh he approprae number of sngledegree-of-freedom ons. Based on our mehod, we add dummy lns no a mulple-degree-offreedom on, such ha every on n he dynamc ree s a sngle-degree-of-freedom on. The dummy ln s defned as he ln wh no lengh, no mass, and no nera. Beween he wo consecuve generaed sngle-degree-of-freedom ons here s a roaon marx, whch defnes he ransformaon marx beween he wo new on local coordnaes. Therefore a wo-degreeof-freedom on wll generae wo sngle-degreeof-freedom ons, such ha he z-axes n boh of he on local coordnaes are parallel o he orgnal on roaon axes. The numberng scheme for he dynamc ree s he same for he arculaed ree. (See Fg. 5. The on coordnae frame s aached o ln. J L 4 L 7 J 4 L J J 7 J 5 L L 5 J L Srcly speang, he par s he sngle-degree-of-freedom roo on and he sngle-degree-of-freedom branch on.

4 (snce Base Null Ln J J: Jon L: Ln L J J L L J J 5 J 4 L 4 L J 7 L 7 J L L5 J 8 L 8 J 9 L 9 J L mos human moon sars wh zero velocy and zero acceleraon, and ends wh zero velocy and zero acceleraon as well. To sasfy hese boundary condons, we choose he smples ype of funcon. Is dervave (namely, velocy funcon s n he form of: x( C ( ( e C s a consan ha can be calculaed from he nal and endng posons, s he sarng me, and e s he endng me. J Fgure 5. Numberng Scheme for Dynamc Tree, whch s generaed from Arculaed ree n Fg. 4. Jon O Jon Type: -Jon Jon + O + Dummy Ln O L T + Fgure. Two-degree-of-freedom -on convered o wo sngle-degree-of-freedom ons and one dummy ln. (Compare wh Fg.. If on n arculaed ree can generae on, dummy ln, and on + n dynamc ree, T s he ransformaon marx beween he on + local coordnae frame and he on + local coordnae frame.. Knemac Moon Funcon: In our smulaon we would le o assess he feasbly of connuous moon. Therefore, we frs need o generae hese moons and hen use nverse dynamcs o compue he on orques. The human moon smulaon s carred ou based on nemac calculaons, and he on orques are obaned by nverse dynamcs. The ssue s wha nd of funcon we should use o specfy he moon of ons or bodes. There are an nfne number of funcons we could choose for he moon smulaon, for example lnear funcons and quadrac funcons. However, For smplcy, we se, e T, and x (, x ( T. The nemac moon funcon s x( C ( T Ths funcon obvously sasfes he velocy and acceleraon consrans. Namely, x(, x( T, x(, x( T. We negrae he velocy funcon o oban he moon funcon: x( C T T + 5 x T, we ge From ( C T 5 So he funcon for he moon smulaon s x ( ( T 5T + ( 5 T The varable x ( can represen a poson vecor (he hree componens have dfferen parameers of he nemac moon funcon or on θ for a revolue on. varable (for example ( Now we sudy a more complcaed moon problem, n whch he moon funcon should sasfy he consran ha he mum speed durng he moon should have an upper bound V. Frs we solve he funcon for he moon smulaon as above, and hen chec f 5 V x( T he nemac moon 8T funcon reaches s mum a he mddle of he moon. If s sasfed, he funcon we calculaed s he fnal funcon for he moon smulaon. If s no sasfed, we wll change he moon o hree phases. The frs phase s he acceleraon phase, n whch he moon speeds up o V n seconds. The second phase s he consan speed phase, n whch he moon eeps he speed a V for seconds. The hrd phase

5 s he deceleraon phase, n whch he moon slows down o zero velocy n seconds. Thus, T, T V V x x V ( (, [, ( V (, [, + x + 4 ( ( T T ( + ( + ( T ( + ( + T + V ( [ T ],, Noe T Self Orenaon: People re-oren hemselves easly on earh snce gravy assures a force componen and herefore an expecaon of frcon wh he supporng surface. A space waler can move and mae urns f equpped wh SAFER (Smplfed Ad for Exravehcular acvy Rescue, whch produces sx degrees of freedom n movemen. Bu how can an asronau maes urns when floang nsde he vehcle wh no exernal force and orque acng on her body and no hand or foo resrans? In hs secon, we descrbe he dynamc heory -- based on he conservaon of momenum and Feahersones nverse recursve dynamcs -- whch allows he asronau o do a self-orenaon. Then we do he smulaon and calculae he on orques durng he smulaed moon. A -gravy self-orenaon moon can be compleed n hree seps: leg lfng, leg wsng, and leg closng. Fgure 7. Sandng Posure Fgure 8. Leg Lfng Fgure 9. Leg Twsng Fgure. Leg Closng For he purpose of hs smulaon, we use a smplfed arculaed body whch ncludes wo ons (he lef and rgh hp ons and hree lns (he orso ln, he lef and rgh leg lns, whch can be convered o a dynamc ree wh sx ons and seven lns. (See Fg.. The O frame s he reference frame. The orso ln coordnae frame O s aached o he cener of mass of he orso. The rgh hp on prvae coordnae frame O and he lef hp on prvae coordnae frame O are aached o he rgh and lef leg lns, respecvely. C and C are he cener of mass of he rgh and lef legs, respecvely. O Torso Ln O O O J4 J5 L L L L C C Jon has degree of freedoms, and s defned a he cener of he mass Rgh Leg Ln Lef Leg Ln of he orso ln J L5 L4 L J J J Fgure. Coordnae frames for Arculaed Body and s Dynamc Tree. Noe: L, L, L4, and L5 are dummy lns Snce he asronau s floang nsde he vehcle, here s no exernal force and orque exered on he asronau, and so spaal momenum (.e. lnear and angular momenum of he human body s conserved durng he moon. The dynamc noaons we use n hs paper are based on hose n Feahersones boo. The conservaon of momenum s expressed as: I ˆ vˆ + Iˆ vˆ + Iˆ vˆ cons ˆ Î ( s he spaal nera for each ln n he reference frame. vˆ ( s he spaal velocy for each ln n he reference frame. The spaal veloces ˆv and ˆv of he lef and rgh leg lns can be wren as:

6 J vˆ vˆ + Sˆq + Sˆ q + Sˆq vˆ vˆ + Sˆ q + Sˆ q + Sˆ q J5 ] fˆ Iˆ aˆ + vˆ ˆ Iˆ vˆ aˆ aˆ + Sˆq + Sˆ q + Sˆ q + vˆ ˆ Sˆq + Sˆ q + Sˆ q + Sˆ q ˆ Sˆ q + Sˆ q + Sˆ q ˆ Sˆ q ( ( The on orque abou each on axs s hus: Q SˆS fˆ, Q Sˆ S fˆ, L4 -Jon L J7 -Jon L5 J -Jon Fgure. Smplfed Arculaed Tree wh degrees of freedom (All he frame are on prvae coordnae frames. ( for he orso ln. d ˆ fˆ I vˆ Iˆ aˆ + vˆ ˆ Iˆ vˆ d In order o ge he lef-hp on orque, he spaal force fˆ s compued as -Jon L Le fˆ be he resulan spaal force exered on he orso ln hrough he rgh and lef hp ons. fˆ gves he overall rae change of momenum J J4 ( -Jon L can ge he poson and orenaon of he orso ln and he spaal acceleraon of as well. J Iˆ Sˆq + Sˆ q + Sˆ q + Iˆ Sˆ4 q4 + Sˆ5 q5 + Sˆ q By negrang v or ang s dervaves, we -Jon The spaal velocy v s obaned by solvng he above equaons: vˆ Iˆ + Iˆ + Iˆ [( L J q ( ( s he specfed funcon for moon smulaon durng each sep, compued from (. S ( s he spaal vecor for he axs of on. ( -Jon -Jon L A dynamc ree wh ons and lns can be generaed from he arculaed ree easly. We add a dummy ln o he lef wrs z-on f he rgh wrs y-on s he roo on for he dynamc ree. In order o smplfy he smulaon, we assume here are no dynamc closed loops durng he moon. Namely, f he rgh hand grabs he bar hen he lef hand mus be free, hs s smplfcaon s based on how asronaus raverse ladders. We can dvde he moon no wo nds of moon perods: Sarng/endng moon perod, Mddle moon perod. The sarng moon perod begns wh boh hands on he bar, bu only one grabbng (say rgh hand, and ends wh he free hand (lef hand on he fron bar. The endng moon perod s he reverse of he sarng moon perod. The mddle moon perod sars wh one hand grabbng bar (say rgh hand and he oher hand (lef hand on bar (-, and ends wh he oher hand (lef hand on bar (+. (See Fg.,4. Q SˆS fˆ. 5. Ladder Traversal: Snce an asronau floas n space (-gravy, s hard o eep balance among he lmbs when raversng a ladder. By leng he legs floa free and only usng he hands o grab he bars, he asronau can conrol her body moon. In hs Secon we do he dynamc smulaon and calculae he on orques, so we can compue he mum possble worload for an asronau raversng a ladder. In he fuure, we plan o use compuer vson echnques o capure an acual asronaus movemen for dealed comparson. The arculaed ree we used o smplfy he asronau body consss of egh ons and 7 lns. (See Fg.. Fgure. Endng Posure, whch s he same as sarng posure (a (b (c (d (e (f

7 (g Fgure 4. Anmaon Sequence of Ladder Traversal n one perod The nemac moon funcons are compued n wo levels. The global moon plannng level gves he smulaed moon for he orso ln by he mehod we descrbed n Secon. The perodc moon plannng level gves he smulaed moons for each on durng ha perod. These funcons are calculaed by gvng he nal and fnal posons for on-varables and he mum speed consrans. The nal and fnal on-varables are compued based on nverse nemacs, wh he poson and orenaon of he orso ln specfed, and he hands posons are nown a each perods nal and fnal sages. The velocy of he orso ln (obaned from he perodc moon plannng level n he drecon of moon should be conssen wh he one compued from he global moon plannng level. Once we have he funcons of he smulaed moon for each on, we calculae he on orques. The above compues nemacs. In order o compue he on orques we use an effcen calculaon scheme, whch s smlar o Feahersones and can be obaned by performng hese calculaons pernen o ln n on -coordnae frame. The equaons n on -coordnae frame, whch s aached o ln, s as follows: vˆ ˆ vˆ + Sˆ q, ( v ˆ. paren. paren. paren. paren fˆ ˆ a v ˆ I ˆ + ˆ ˆ Ivˆ, ˆ. paren aˆ ˆ aˆ + vˆ ˆ Sˆ q + Sˆ q, ( a ˆ fˆ ˆ ˆ fˆ, (f { chld} f + {. chld } S Sˆ fˆ Q.. Opmal Conrol: ˆ. paren., fˆ fˆ The above smulaons are done n real me. Bu we also neres n opmzed moon of he human moon under ceran specfed consrans, whch can no be done n real me. The opmal conrol problem s formulaed by mnmzng he on orques, snce s suable n space applcaons. The problem can be saed as []: Opmze: DOF f mn [ τ (, q ] d q Subec o all he gven consrans DOF s he oal degrees-of-freedom of τ, s he on orque a he arculaed body. ( q me when he funcons of he smulaed moon are expressed n B-Splne funcons [] wh he conrol pons q. The nonlnear programmng mehod requres he explc calculaon of he graden. We defne he opmzaon funcon based on a gven se of conrol pons q and he wegh coeffcens λ for all he consrans: F DOF f n ( q, λ [ τ (, q ] d + λc (, q n s he oal number of consrans. We opmze F ( λ mehod: q ( l ( l δq [ F( q, λ ] ( + ( + m n m n δλ ( m+ n DOF N, by usng he graden ( l ( l ( F( q, λ ( m+ n m, N s he number of B-splne conrol pons for each ons smulaed moon funcon n B-splne form. Lo and Meaxas [] gve he deals on how o compue he graden F, λ. By of he opmzaon funcon ( solvng he above lnear equaon sysem, we oban δ q and δλ. And we have he new se of he conrol pons and he wegh coeffcens: ( l ( l q + q + δq, ( l + ( l λ δλ. λ + If he eraon ndex l s larger han he specfed mum eraon mes, or δ q δ << and δλ <<, q δλ we choose ( l + q as he fnal conrol pons for each ons B-splne nemac moon funcon. 7. Energy Expendure: I s commonly suggesed ha slled human movemens opmze ceran crera, whch are relaed o he energy expendure []. Among he suggesed crera, we eleced o measure he vrual human's energy expendure based on he followng quanes: q

8 a he weghed sum of he absolue values of he on orque, b he change of he oal mechancal energy of he body, c he mechancal power generaed or ransferred n he ons, and d he rae of sudden change n movemen (er. The weghed sum of he absolue values of he on orque The me negral of he on orque has been suggesed as a useful measure of he energy expendure when s appropraely weghed for he negave and posve wor condons [,]. Among many suggesons, Wllams[4] and Perrynows e al. [5] have repored ha he negave wor s approxmaely hree mes more effcen han he posve wor. For a gven on, he wor s posve when he orque and he angular velocy have he same drecon, and negave oherwse. The posve on wor corresponds o he concenrc muscle acvy and he negave on wor corresponds o he eccenrc muscle acvy. So he sum of he weghed absolue on momens s used as our energy expendure measure he absolue on momen s defned as ρ τ f τ *ω o τ oherwse ω τ refers o he orque of he on and refers o he angular velocy of he on. Fg. 5 shows he weghed sum of he absolue on orque whou any shoulder consran he hgh pch denoes he me when he asronau ouches he ladder. Fgure 5. Weghed Sum of he absolue on orques The change of he oal mechancal energy of he body E TKE + RKE ( refers o he Translaonal Knec TKE Energy of he on and RKE he Roaonal Knec Energy of he on. The mechancal power generaed or absorbed a he ons P τ ( ω ω + τ refers o he orque of he on and refers o he angular velocy of he on. ω The mechancal power ransferred a he ons P ι ω ι p τ * f τ *ω τ τ * ω p oherwse o refers o he orque of he on and ω refers o he angular velocy of he on. Ths creron ncorporaes boh concenrc and eccenrc muscle acves. The rae of sudden change n movemen(jer A J J refers o he me dervave of acceleraon of he on (er. The er s shown o be mnmzed for a ceran class of acves [7]. 8. Concluson: Dynamcally correc human moon smulaon requres a proper dynamc ree model effcenly creaed from a smplfed arculaed ree. I also requres he generaon of an arculaed body movemen paern from a sarng posure o a fnal posure. The man focus of hs paper s o smulae self-orenaon and ladder raversal of an asronau n -gravy. These and oher dynamc smulaons may be used for boh vsualzaon and analyss of - or mcro-gray ass. We mplemened he mehods developed here o generae he dynamc ree, smulae he

9 desred moons, and compue he energy expendure funcons. These procedures wll be useful n he fuure analyss and safey evaluaon of novel space acves. Acnowledgmens: Ths research s parally suppored by NASA NRA NAG 5-99 and ONR IP N o he second auhor. Bblography: [] G. Engeln-Mullges, F. Uhlg. Numercal Algorhms wh C. Sprnger, 99 [] R. Feahersone. Robo Dynamcs Algorhm. Kluwer Academc Publshers, Boson, 987 [] R. Flecher. Praccal Mehods of Opmzaon. John Wley & Sons, Ld. 98 [4] J.K. Hodgns, W.L. Wooen, D.C. Brogan, and J.F. OBren. Anmaon of Human Ahlecs. In Proc. SIGGRAPH 95, pages 7-78, Augus 995 [5] K.W. Llly. Effcen Dynamc Smulaon of Roboc Mechansms. Kluwer Academc Publshers, Boson, 99 [] J. Lo. Ph.D. Thess: Recursve Dynamcs and Opmal Conrol Technques for Human Moon Plannng. Mechancal Engneerng and Appled Mechancs, Unversy of Pennsylvana, 998 [7] J. Lo and D. Meaxas. Effcen Human Moon Plannng Usng Recursve Dynamcs and Opmal Conrol Technques. Proc. Of he Second Symposum on Mulbody Dynamcs and Vbraon a he 7 h Bennal Conference on Mechancal Vbraon and Nose, Las Vegas, NV, Sep. -5, 999 [8] J. Lo and D. Meaxas. Recursve Dynamcs and Opmal Conrol Technques for Human Moon Plannng. Proc. Of Compuer Anmaon Conference, CA99. Geneva, Swzerland, May -9, 999 [9] A.J. Sewar and J.F. Cremer. Beyond eyframng: An algorhmc approach o anmaon. In Proc. Of Graphcs Inerface, pages 7-8, 99 [] H.W. Sone. Knemac Modelng, Idenfcaon, and Conrol of Roboc Manpulaors. Kluwer Academc Publshers, Boson, 987 [] J. Wlhelms and B. Barsy. Usng Dynamc Analyss o Anmae Arculaed Bodes such as Humans and Robos. In Graphcs Inerface, 985 [] A. Wn and M. Kass. Spaceme Consrans. ACM Compuer Graphcs, (4, 988 [] R.N. Marshall, G.A. Wood and L.S. Jennngs. Performance obecves n human movemen: A revew and applcaon o he sance phase of normal walng. Human Movemen Scence 8 ( [4] K.R. Wllams and P.R. Cavanagh. A model for he calculaon of mechancal power durng dsance runnng. Journal of Bomechancs, 5-8, 98 [5] M.R. Perrynows, D.A. Wner and R.W. Norman. Transfers of mechancal energy whn he oal body and mechancal durng readmll walng. Ergonomcs, 98, 47-5 [] J.G. Andrews. Bomechancal measures of muscular effor. Medcne and Scence n Spors and exercse 5, 99-7 [7] N. Hogan. An organzng prncple for a class of volunary movemens. Journal of Neuroscence 4,