Optimization of stiffness and damping properties of below-knee prosthesis

Transcription

1 Opmzaon of sffness and dampng properes of below-knee proshess GIL SERRANOLÍ MASFERRER Deparmen of Appled Mechancs Dvson of Dnamcs HALMERS UNIVERSIY OF EHNOLOGY Göeborg, Sweden Maser s hess :9

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3 MASER S HESIS :9 Opmzaon of sffness and dampng properes of below-knee proshess GIL SERRANOLÍ MASFERRER Deparmen of Appled Mechancs Dvson of Dnamcs HALMERS UNIVERSIY OF EHNOLOGY Göeborg, Sweden

4 Opmzaon of sffness and dampng properes of below knee proshess GIL SERRANOLÍ MASFERRER GIL SERRANOLÍ MASFERRER, Maser s hess :9 ISSN 6-87 Deparmen of Appled Mechancs Dvson of Dnamcs halmers Unvers of echnolog SE- 96 Göeborg Sweden elephone: 6 ()-77 halmers Reproservce Göeborg, Sweden

5 Opmzaon of sffness and dampng properes of below knee proshess GIL SERRANOLÍ MASFERRER Deparmen of Appled Mechancs Dvson of Dnamcs halmers Unvers of echnolog ABSRA Nowadas, proshec manufacurers have developed new desgns of below-knee proshess whch are anhropomorphc and safe. Bu more research should be done. hs projec res o fnd ou opmzed parameers of below-knee proshess o walk wh as low energ as possble. A bomechancal opmal problem s formulaed o fnd ou suable sffness and dampng properes of below-knee proshess. hese properes mean he relaon beween he orque on he ankle jon of he proshec leg wh ankle angle and s veloc. he creron used has been mnmzng energ of boh on he healh par of he bod as well as eernal energ wh whch he proshess baer s suppled. In order o sae he opmal problem, frsl an anhropomorphc human ga has been looked for b usng compuer smulaons. I has been chosen a mechancal human model ha conss of wo legs (wh hgh and shank), he runk and wh jons: a ankle, knee and hp. Foo has been modeled as fooprn. he ssem has 7 degrees of freedom. he moon has been parameerzed b usng polnomal and Fourer seres and her parameers have been chosen so ha he moon was anhropomorphc. onrol orques a jons and ground reacon forces have been deermned b usng Lagrange equaons. Afer ha, he opmzaon problem has been consdered usng energ of he healh par of he bod as cos funcon, whle energ consumed b he proshess has been calculaed n each varance. Algorhm of soluon of opmzaon problem has been mplemened n MALAB so ha opmzed knemac parameers of he ga as well as sffness and dampng parameers of he below-knee proshess are deermned auomacall. fmncon funcon s used o solve he opmal problem and ode funcon o solve he dfferenal equaon o fnd ou knemacs of he runk. he analses of he obaned opmzed values of sffness and dampng parameers of a below-knee proshess as well as knemacs of he opmzed ga are presened. Fuure work whn he projec opc s oulned. Ke words: below-knee proshess, human ga, Lagrange equaons, opmzaon, cos funcon, ankle, knee, hp. I

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7 onens ABSRA ONENS PREFAE NOAIONS I III V VI INRODUION MEHANIAL AND MAHEMAIAL MODELS. Revew of bomechancal cos funcons. os funcons seleced 7. Saemen of he problem 7 INIIAL MODEL. Leg knemacs.. Effecs on he varaon of he parameers.. Leg knemac plos 7. runk knemacs 9.. runk knemacs plos 9. Leg dnamcs.. Dnamc plos OPIMIZAION PROBLEM. Malab Program RESULS 8. Knemac resuls 9. Dnamc resuls 9. Oher resuls 6 ONLUSIONS 6. Fuure work 7 REFERENES 6 HALMERS, Appled Mechancs, Maser s hess :9 III

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9 Preface hs Maser s hess was carred ou a he Dvson of Dnamcs a he Deparmen of Appled Mechancs on halmers Unvers of echnolog durng he sprng of. I s m fnal projec (n Erasmus program) of m degree, Mechancal Engneerng, carred ou n Barcelona (aalona) n Unversa Polècnca de aaluna (ESEIB-UP). M supervsor and eamner was Professor Vkor Berbuk from he Dvson of Dnamcs. I would lke o hank hm for all gudance and suppor. Also, I am graeful o Håkan Johansson from he Dvson of Dnamcs who helped me wh MALAB quesons and he was ver paen n several momens n order o each me a lo of secres of hs program. Fnall, I also would lke o hank all m faml and frends for her suppor. Göeborg, June Gl Serrancolí Masferrer HALMERS, Appled Mechancs, Maser s hess :9 V

10 Noaons erms: α β ψ A a b p E h E p G J a J b K K p L M m a m b m f O p q r r a r b R R u X R Y R Z angle from vercal poson o hgh angle from vercal poson o shank angle from vercal poson o runk ankle jon lengh of he hgh lengh of he shank dampng of he below-knee proshess on he ankle jon Energ of he healh par of he bod Energ consumed b he below-knee proshess cenre of mass of he runk momen of nera of he hgh relave o he Z as a pon O momen of nera of he shank relave o he Z as a pon K knee jon sffness of he below-knee proshess on he ankle jon lengh of he sngle sep oal mass of he bod mass of he hgh mass of he shank mass of he foo hp jon orque on he ankle jon orque on he hp jon dsance from he suspenson pon O of he legs o he cenre of mass of he runk dsance from O o he cenre of mass of he hgh dsance from K o he cenre of mass of he shank horzonal componen of he ground reacon vercal componen of he ground reacon duraon of he sngle sep orque on he knee jon horzonal as horzonal poson of he hp horzonal poson of he proshec foo horzonal poson of he applcaon pon of he ground reacon force horzonal poson of he healh foo vercal as vercal poson of he hp vercal poson of he proshec foo vercal poson of he applcaon pon of he ground reacon force vercal poson of he healh foo Z as, perpendcular o he plane of moon VI HALMERS, Appled Mechancs, Maser s hess :9

11 Inroducon A below-knee proshess s a proshec leg wh shank, ankle jon and proshec foo. Knee jon s usuall healh,.e. knee belongs o paen. Somebod who has los hs shank b accden or b llness can wear below-knee proshess n order o be able o walk and o do hs lfe more comforable. here are some man requremens ha lower lmb proshess should fulfl. I should have a cheap prce, a long lfe, safe and naurall moons. hs hess focuses on more specfc and echncal backgrounds: energ opmzaon, such as energ consumpon of he healh par of he bod and eernal energ consumed b baer. I would be uncomforable o wear a proshess f he paen had o do more effor han normal. I would be non-vable f he paen had o walk n a nonanhropomorphc wa. I would be a useless proshess f sops runnng durng s ga,.e. he baer has o be able o keep enough energ n order o suppl he devce properl. o sum up, a proshess whch runs anhropomorphcall wh as lle energ as possble should be acheved. here are some models whch mnmze he energ used b a healh paen n a walkng dsance whn a gven me. Bu s no common o look for he mnmum energ consumpon of he proshess and he mnmum energ consumpon of he healh par of he paen bod a he same me. he man arge of he maser hess s o fnd ou sffness and dampng properes of a below-knee proshess, whch avod wasng energ: eernal power and healh power. hs means lookng for he suable relaon beween dnamcs (orque) and knemacs (angular poson and veloc) on he proshess ankle ha could be useful o desgn a new below-knee proshess. Keepng hs purpose n mnd, several pons need o be handled. Frs of all, a mahemacal model of human bod moon n a sance phase on a leg wh belowknee proshess has o be found. Anhropomorphc knemacs and dnamcs for nal parameers also need o be found. Ever resul has o be compared wh he leraure and o check f hese resuls are anhropomorphc or no. Once an anhropomorphc moon s obaned, energ consumpon of human bod n a sance phase has o be evaluaed. I could also be esed for dfferen sffness and dampng properes and dfferen leg knemacs. Fnall, o accomplsh our goal, our cos funcon wll be mnmzed, energ of he bod, wh suable resrcons and opmzed values of sffness and dampng properes wll be obaned. MALAB wll be used as sofware supporer. I wll help o solve comple relaons, dfferenal equaons and he opmal problem. I wll also help o see dfferen plos o undersand he moon. All hs work would be amed o desgn a new and more effcen pe of lower lmb proshess. A fuure work would be he mechancal desgn of he proshess wh sffness and dampng properes found n hs projec. HALMERS, Appled Mechancs, Maser s hess :9

12 hs repor consss of hree man pars: an eplanaon of he mechancal and mahemacal models, a descrpon of he nal model found and a developmen of he opmal soluon. I ends up dscussng he resuls obaned. HALMERS, Appled Mechancs, Maser s hess :9

13 Mechancal and mahemacal models A model as smple as possble bu ha represens he human ga as real and anhropomorphc as possble should be chosen. As human ga could be consdered perodc, he double sep could be suded, sance phase and swng phase of each leg. Bu snce he goal s o look for suable sffness and dampng properes of he ankle proshess, s enough o ake onl one sep, wh a sance phase on he leg wh below knee-proshess. In our model, s consdered ha durng swng phase, he ankle orque s null. I has been chosen a mechancal model wh a fooprn. he foo s represened as a pon, Fgure.. Y G Ψ O α K K A Z β A X Fgure. Mechancal model he ssem conans a runk (GO) and wo legs. Each leg consss of wo elemens (OK and K A ). Our model has several smlares wh one used b Berbuk () []. I shows enough pons o represen all he daa needed o compue n our equaons. Proshess leg wll be represened as leg whch s on sance phase. he ssem would have 7 degrees of freedom, bu ankle jon of he proshess leg s fed (A ), so he ssem has degrees of freedom. In addon o he weghs of he runk, hghs and shanks he ground reacon forces and he conrol momens a he jons of he legs ac n he ssem. he ssem moves hrough he X as over a horzonal surface (he X-Z plane). o descrbe he se of generalzed coordnaes, followng noaons wll be used:,, Ψ, α, β, (are represened n he Fgure.); m s he mass of he runk; r s he dsance from he suspenson pon O of he legs o he cenre of mass of he runk; J s he nera s momen of he runk relave o he Z as a pon O; m a, r a, a, J a are he mass, he dsance from O o he cenre of mass, he lengh and he momen of nera of he hgh relave o he Z as a pon O, respecvel; m b, r b, b, J b are he mass, he dsance from K o he cenre of mass, he lengh and he momen of nera of he shank relave o he Z as a pon K, respecvel; m f s he mass of he foo. HALMERS, Appled Mechancs, Maser s hess :9

14 Lagrange equaons of he second knd are used o fnd ou he equaons of moon of he ssem: d d q V q q Q q (.) Where q s a generalzed coordnae wh correspondng generalzed force Q q. s knec energ and V s poenal energ. he equaons of moon are wren as follows []: B as: ' ' ' ' ' ' [ K a ( α cos α ) K b ( β cos β ) ] K r ( ψ cos ) R R M ψ (.) B as: ' ' [ a sn b ] K r ( ψ sn ) R R ' ' ' ' ( g) K ( α α ) K ( β sn β ) ψ (.) M B Ψ angle: ( cosψ snψ ) g K snψ q q J ψ K (.) B α angle: J α r r b ( ) g Ka snα ( R cosα R snα ) ( cosα snα ) a K β cos( α β ) β sn( α β ) Ka B β angle: J c β K q u a (.) b ( cos β sn β ) a K α cos( α β ) α sn( α β ) b ( ) g K b sn β ( R cos β R sn β ) u p b (.6) q, u and p are he orques on hp, knee and ankle respecvel. Where: a b a b M m m m m m (.7) K r a m r (.8) a a ( m m ) K m r a (.9) K b b b b f f m r b m (.) a ( m m ) J J a b f (.) J c J b b m f (.) wh, HALMERS, Appled Mechancs, Maser s hess :9

15 . Revew of bomechancal cos funcons Some auhors have wren abou cos funcons used n bomechancal opmzaons. he am here s o fnd he bes suable funcon o acheve our objecve. o represen he suable ga paern, dfferen energ cos funcons have been suded. hen, some cos funcons o evaluae opmal energ are presened: Fague cos funcon. M. Ackermann, A. J. van den Boger, 9 [] proposed a faml of cos funcons ha represened dfferen ga paerns, conss of weghed muscle acvaons: J m ω ω a p ( ) d (.) where m s he number of muscle groups, a s he muscle acvaon, and p and ω are he eponen of a and weghng facors, respecvel. Dependng on he muscle acon, 8 dfferen cos funcons can be dfferenaed, snce could be p,, or, and here are wo ses of weghng facors ω. Meabolc energ cos funcon. Frank. Anderson and Marcus G. Pand [] hpoheszed ha he suable moor paern should be found mnmzng he meabolc energ ependure per un dsance moved. he cos funcon s as follow: E d X cm ( f. M f. oal E B m f ) X cm (). A X cm m ( f. M m ) X. S cm m (). W m d (.) where X cm () and X cm ( f ) denoe he poson of he cenre of mass of he. model a he begnnng and a he end of he smulaed ga ccle. B s he basal meabolc hea rae of he whole bod, Ȧ m, Ṁ m, Ṡ m, m are he acvaon, manenance, shorenng and mechancal hea raes of each muscle. Smlar o he prevous one, hpohess of H. Haze and J. D. Bus [] conss of mnmzng he meabolc energ, epressed as: W.. E g h s w r (.) where ġ s he acvaon hea rae, ḣ s he manenance hea rae, ṡ s he shorenng hea rae, ẇ s he work rae and ṙ s he rae of hea dsspaed n he parallel srucures. HALMERS, Appled Mechancs, Maser s hess :9

16 A. E. Mne and R. McN Aleander [] consdered ha he meabolc cos funcon of muscle acv (meabolc power) was as follow: P α Φ ωma ω (.6) ωma where α represens a fracon of he muscle s fbres acvaed, s he orque on he muscle jon, ω s he angular veloc and he defned a funcon Φ (ω/ω ma ) from epermenal resuls: Φ,,6 ω ω ω ma ma (.7) ω ma, ω ω ma Muscular force cos funcon,8 ω ω,6 ω ω ma,6 ω ω A. Pedo, V. V. Krshman and L. Sark [6] hpoheszed ha he had o mnmze he oal muscular force o fnd ou he suable human locomoon paern. her cos funcons were as follows: ma J J J J ma F F F F F F ma (.8) where J s a performance creron relaed o he nal force requred o produce he se of orques. J also mnmzes oal muscular force bu penalzes large ndvdual muscle force severel. J s smlar o J bu emplos he muscles more effcenl b demandng large force producon from he large muscles; ndeed, akes no accoun he nsananeous sae of each muscle, snce F ma depends upon he nsananeous lengh of muscle as well as s veloc. J s a performance creron whch uses muscles more effcenl whle keepng her level of acvaon as low as possble. Mechancal energ cos funcon. Vkor Berbuk, Anders Bosröm, Bogdan Lwun, and Bo Peerson [] hpoheszed ha he mechancal energ cos funcon depended on he orques of he arculaons jons: hp, knee, ankle and meaarsal jons and her angular veloces: 6 HALMERS, Appled Mechancs, Maser s hess :9

17 E L [ q ( α ( ) ψ ( ) ) u ( α ( ) β ( ) ) p ( β ( ) γ ( ) ) w ( γ ( ) ε ( ) )] d (.9) where q, u, p and w are he orques ha ac n he hp, he knee, he ankle and he meaarsal jons respecvel. ψ () s he angular veloc beween he runk and he hp, α () s he angular veloc beween hp jon and each hgh, β () s he angular veloc beween knee jon and each shank, γ () s he angular veloc beween ankle jon and each foo and ε () s he angular veloc beween meaarsal jon and each se of oes. L s he lengh of he sep. Vkor Berbuk, Bogdan Lwn and Mroslav Demduk [7] ook no accoun ver smlar mechancal energ cos funcon, bu he hpoheszed ha mechancal energ onl depends on orques on he hp and knee jons (snce he consdered he conrol npus were orques acuaors acng onl a hp and knee jons), on he veloces of hese jons and on he lengh of he sep. hs s as follow: E L [ q ( α ( ) ψ ( ) ) u ( α ( ) β ( ) )] d (.). os funcons seleced he focus wll be on he problem of denfng he ga paern b mnmzng a couple of cos funcons. hese funcons wll be he energ cos funcon of he healh par of he bod (E h ), cos funcon of he energ consumed b he proshess (E p ). hese funcons are defned as follows: E h L [ q ( α ( ) ψ ( ) ) u ( α ( ) β ( ) )] d ph ( β h ( ) ) { }d (.) L E p L { p p ( p ( ) )}d β (.). Saemen of he problem Our ssem could have wo phases on one leg: sance phase [, ), when proshess leg s on he floor and healh leg s no; and swng phase [, ), when he proshess leg has no conac wh he floor. Bu as menoned, onl he frs phase wll be represened. Snce human moon s perodc, followng boundar condons could be consdered: HALMERS, Appled Mechancs, Maser s hess :9 7

18 (,ψ ) f ( ) f ( ) f () f ( ) f (.) () ( ) L () ( ) () () ( ) L ( ) (.) (.) where s he duraon of a sngle sep. α and β are perodc and her perod s. As proshess leg s on he floor and healh leg s n swng phase, followng boundar condons can be defned: ) ( ) [, ) (.6) a( a a a ( ) [, ) (.7) a and a are he coordnaes of he pon A (Fgure.). Whou an resrcon of he general he followng addonal condons for and are gven b: L (.8) a a a ( ) L a ( ) (.9) (.) a a a ( ) L a ( ) (.) In order our model o be anhropomorphc some angular dsplacemens consrans could be used. hese wll be aken from epermenal resuls from leraure. θ ( ) µ Θ ( ) (.) k k k θ ( ) µ Θ ( ) (.) a a a θ ( ) µ Θ ( ) (.) Where: µ ( ) α ( ) ψ ( ) (.) k µ ( ) α ( ) β ( ) (.6) a π µ ( ) β ( ) (.7) 8 HALMERS, Appled Mechancs, Maser s hess :9

19 wh, In addon, here are wo logcal resrcons so ha he human ga s anhropomorphc: [, ],(,) α ( ) β ( ), (.8) [ ] ( ),, (.9) here are also dnamc resrcons, reacon force has o be alwas posve and he ankle proshess orque has o be defned as a funcon of β and β. [ ] R ( ),, (.) p π ) K p β( ) p β ( ) K p [, ] (.) ( Afer defnng our mechancal model, a cos funcon needs o be chosen. hen he opmzed moon wll be found. In he followng secon wll be dscussed whch s he bes cos funcon for us. Le Z ) {,,,,,,,, ψ, ψ, α, α, β, β,,} ( be a vecor of he phase sae and U ( ) { p, q, u, p, } be a vecor of he conrollng smul of he ssem. Once hese vecors are defned, he followng problem can be saed. Problem A. Assume ha he sep lengh L s gven, and he duraon of he sngle * * suppor phase. I s requred o deermne he conrol process [ Z ( ), U ( ) ], for [, ], whch mnmze he cos funcon seleced, E h and E p (.) (.), subjec o dfferenal consrans (.) (.6), boundar condons (.) (.), anhropomorphc consrans (.8) - (.9) and dnamc resrcons (.) (.). In he ne secon, frs some funcons wll be parameerzed and hen wll follow an aemp o fnd ou an nal moon whch s anhropomorphc. hs moon wll be used laer as an nal guess n he opmal problem. HALMERS, Appled Mechancs, Maser s hess :9 9

20 HALMERS, Appled Mechancs, Maser s hess :9 Inal Model Before opmzng he model, an nal model ha s anhropomorphc mus be found. Mahs and logc can be used o search for a suable nal model. Lookng for a suable leg knemacs and dnamcs ha looks anhropomorphc comes frs.. Leg knemacs Our mechancal model has o sasf our equaons and consrans. Bu he moon has o be also anhropomorphc. A shape for our man varables needs o be chosen and wll lead o he oher varables as a funcon of hem. Fourer seres wll be used o defne our man varables, hese wll be (), (), (), (). So, hese funcons wll be se as follows: ) ( N n n n n b n a * * sn cos π π (.) ) ( N n n n n b n a * * sn cos π π (.) ) ( * * sn cos N n n n n b n a π π (.) ) ( * * sn cos N n n n n b n a π π (.) here are some of hese parameers whch could be defned a pror because of he boundar condons. If aken no accoun ha our moon s perodc, () should sasf followng condons: ) ( () ) ( () L (.) herefore, and wll be defned as: ( ) L (.6) ( ) (.7)

21 HALMERS, Appled Mechancs, Maser s hess :9 Smlar boundar condons are defned for (): ) ( () ) ( () (.8) And we could ge and as: ( ) (.9) ( ) (.) Boundar condons for () are as follows: ) ( () ) ( () () L (.) So, and are defned as: N n a n (.) ( ) L (.) ( ) (.) here are smlar boundar condons for (): ) ( () ) ( () () (.) So, and are defned as: N n a n (.6) ( ) (.7) ( ) (.8)

22 In order o deal wh all oher parameers ha could change he followng vecor wll be defned: [ ] (.9) where: [,, a b ], n, n where n (...N ) (.) [,, a b ], n, n where n (...N ) (.) [,,, a b ], where n (...N ) (.), n n [,,, a b ],, where n (...N ) (.) n n Frs of all, all fed parameers ha ake par n he moon equaons (. o.) mus be defned. hese are aken from []. See able.. able. Fed daa.7 s r.9 m J 7.96 Nm L.76 m m 6.7 kg a.7 m r a.8 m J a.7 Nm a.7 m r a.8 m J a.7 Nm b. m r b. m J b.6 Nm b. m r b. m J b.6 Nm m a 8.9 kg m f. kg m a 8.9 kg m f. kg m b. kg m b. kg.. Effecs on he varaon of he parameers How each varable parameer affecs n coordnaes defned b polnomal parameers and Fourer Seres (,, and ) s suded. I s mporan knowng how he moon changes f Fourer parameers are modfed. ( ) n m ( ) n m / s (.) Evaluang above funcons, he concluson s he followng: - Inal veloc s defned b: L ( ) (.) HALMERS, Appled Mechancs, Maser s hess :9

23 - and he nal acceleraon b he epresson: ( ) (.6) So varng our parameers, dfferen funcon shapes are obaned: For eample, varng. See able.. able.. ( ), ẋ ( ) and ẋ ( ) rals for dfferen, and for. ( ) ẋ ( ) ẋ ( ) Zero acceleraon Noe ha onl polnomal parameers are vared, no snus and cosnes parameers. For, wll be enough usng polnomal parameers. As sgn of parameers are equal, all parameers wll affec n a smlar wa. I could be also noed ha f >, wll have a srong effec snce s eponen s hgher han ohers (n absolue value): on veloc and on acceleraon; HALMERS, Appled Mechancs, Maser s hess :9

24 HALMERS, Appled Mechancs, Maser s hess :9 whle f <, ma have a srong effec snce s eponen s lower han ohers (as much as s close o ): on veloc and on acceleraon epresson. I could be se ha ) ( ẋ has o be alwas posve: s m ) ( > and could be defned a mamum possble veloc s m ) ( <. See able.. able.. Epressons ha sasfes he resrcon ) ( > ẋ ( ) L > ) ( < ẋ ( ) L < For an nal opmzaon, and are chosen. s m n m n / ) ( ) ( (.7) hs coordnae has a specal consran, because has o be posve, wll be anhropomorphcall wrong f he foo would nroduce under he ground. As a frs appromaon, he funcon () has been se as a parabola. herefore mamum hegh s n he mddle of he sep. In addon, he nal veloc wh posve value and fnal veloc wh negave have also been se. A mamum hegh ( ) m. ) ( < and a mamum veloc ( ) s m / ) ( < can be defned. See able.. able.. Epressons ha sasfes he resrcons ) ( ẏ (sngular pon a he mddle) 6 ) ( < ẏ (sngular pon has o be a mamum) > () ẏ and ) ( ẏ m <. ) ( ( ) <, s m / < ( ) < For an nal opmzaon 9., 8 and s chosen.

25 ( ) n m ( ) n m / s (.9) Regardng (), s smlar han ( ), bu an nal value has o be chosen. hs nal value s gven b, represens he suaon of he hp a. So, m < <. 76 m (.) Arbrar bu logcall,. s chosen. Smlar resrcons lke ẋ ( ) are se now. See able.. able.. Epressons ha sasfes he resrcons L ẋ ( ) > ẋ < L > < ( ) For an nal opmzaon wll be chosen. So, veloc wll be consan L ( ) n m ( ) n m / s.m / s and acceleraon wll be zero. Hp should be beween a hegh of.7m < ( ) < m. So, (.).7m < < m (.).9 s chosen. Snus and cosnes parameers are used, nsead of polnomal parameers, because oherwse, he acceleraon of hp would no be anhropomorphc. o sar jus a couple of parameers are aken, so N. * wll be *, because s a parabola beween and. ẏ() should be slower han ẏ ( ), herefore s se ( ) <. m. See able.6. able.6. Epressons ha sasfes he resrcons π π.7m < ( ) < m.7m < a cos b sn < m π π π π ( ) <.m / s a sn b cos <.m / s For an nal opmzaon. 9, a, b. 8 wll be chosen. HALMERS, Appled Mechancs, Maser s hess :9

26 In order o defne angular moons of he lnks, our model needs o be aken no accoun (Fgure.) and followng feaures of he moon (see Fgure.). Onl one sep s suded. Leg s supposed o wear he below knee proshess. Foo moves one sep whle foo s suppored on he floor. Ne sep would be almos he same. Now, angles from healh leg would be he same lke proshess leg n he las sep and vce versa. O O A K K A L K K L A Fgure.. Defnon of he suded moon. So, o fnd ou angular moons, he followng ssem of equaons has o be solved: ( ) ( ) b sn β a sn α ( ) ( ) b cos β a cosα (.) Angles wll be as follows: α ( ) ( ) arcsn ( ) ( ) b sn β (.) a ( ) ( ) a b ) b β arcsn arcan (.) ( ) ( ) Veloc and acceleraon of hese angles wll be her me dervaon. As menoned on (.8) n order o be an anhropomorphc model, angle β < α. In secon.., all leg knemacs parameers are shown. See Fgure.. 6 HALMERS, Appled Mechancs, Maser s hess :9

27 .. Leg knemac plos Fgure.. Inal legs knemacal soluon, wh N N N. N (m).8 (m) (s) (s) v (m/s).6 v (m/s) (s) (s) (m) (m) (s) (s)..... v (m/s). v (m/s) (s) (s) HALMERS, Appled Mechancs, Maser s hess :9 7

28 alfa (rad).. alfa (rad) (s) (s) valfa (rad/s) valfa (rad/s) (s) (s) bea (rad) bea (rad) (s) (s) vbea (rad/s) vbea (rad/s) (s) (s) alfa-bea (rad).. alfa-bea (rad) (s) (s) 8 HALMERS, Appled Mechancs, Maser s hess :9

29 . runk knemacs Before lookng for runk moon defned b ψ angle, a p orque value mus be assgned, snce wll be needed o fnd ou ψ. p s he ankle jon orque of he proshess leg and s waned ha depends on β angle and s angular veloc, β, relaed wh sffness and dampng properes. So, followng assumpon s done: π p( ) K p β( ) p β( ) K p (.6) Once defned β ( ) and β ( ) suable values for K p and p have o be chosen. As n [], p reaches up o - M [Nm], and sars a p Nm, where M s he mass of he whole bod n kg. In our model, M 7,8 kg, so f chosen p 8, Nms/rad and K p 6, Nm/rad, a suable soluon smlar han he one found ou n [] wll be obaned. Our p sars a Nm and reaches - Nm, see Fgure.. Once defned p, ψ could be looked for. ombnng equaons. o. a dfferenal equaon whose unknowns are onl ψ, ψ and ψ could be acheved. In he followng secon he runk knemacs plos can be seen. See Fgure.. As nal guess ψ.rad and ψ.rad / s have been chosen... runk knemacs plos Fgure.. ψ, ψ and ψ plos. ps (rad) (s) vps (rad/s) (s) aps (rad/s ) (s) HALMERS, Appled Mechancs, Maser s hess :9 9

30 Our knemacs resuls can be summarzed n a plo where can be seen whole sngle sep. I seems anhropomorphc bu one also assumes ha s mprovable n order o be more naural, snce he runk moon s srange. I looks lke uncomforable and unsable. Mabe, hs problem wll be able o be fed when he opmal problem s solved..6.. Y (m) X (m) Fgure.. Draf of sngle sep of hs nal model.. Leg dnamcs In secon. has alread been seen how o fnd p. Now, all oher dnamcs varables (p, u, u, q, q, R, R, R and R )can be obaned. Frs of all, some of hese varables are beween and, snce one foo s n swng phase and does no ouch he floor. hese varables are p snce he ankle has been represened as a fooprn and does no ouch he floor, so s orque s null. For he same reason, reacon force on he floor of healh leg (leg ) s null oo, so R R. Reacon force has wo coordnaes: and, because does no work n a fed pon bu moves along he mplc foo. R and R are coordnaes ha defne he poson of he applcaon pon. I has been supposed R. See Fgure.. HALMERS, Appled Mechancs, Maser s hess :9

31 , p R R, R R Fgure.. Implc foo (n our model L, R ). Subsung known varables n equaon (.), R s obaned. And he same wh R n equaon (.). ombnng equaons (.) and (.6) hp orques can be found ou: q and q ; and knee orques u and u. In he followng secon dnamc plos can be seen. See Fgure.6. HALMERS, Appled Mechancs, Maser s hess :9

32 .. Dnamc plos Fgure.6. Dnamc plos R (N) - R (N) (s) (s) q (Nm) (s) q (Nm) (s) 6 8 u (Nm) 6 u (Nm) (s) (s) 6 - p (Nm) (s) HALMERS, Appled Mechancs, Maser s hess :9

33 Opmzaon problem In order o acheve our man goal, whch s fndng ou sffness and dampng properes of he below-knee proshess, our opmal problem has o be desgned accurael. As menoned n. and., wo cos funcons need o be mnmzed a he same me (Problem A). hs would lead o dscuss whch he bes soluon s. here are 8 or more varng parameers o opmze our cos funcons. or more of hese parameers belongs o leg knemac parameers, of hem belongs o runk knemac parameers and of hem o conrol parameers. hese are: - Vecor [ ], as has been defned n (.)-(.). hese are or more leg knemac parameers, depends on wheher snus and cosnes parameers are aken no accoun or no. - ψ and ψ, whch are nal condons of he dfferenal equaon o fnd ou ψ moon (runk knemac parameers). - K p and p, sffness and dampng properes of he below-knee proshess. Fndng hem ou s he man goal of our work. I s requred o proceed sep b sep. Frsl jus few parameers wll be vared opmzng jus one cos funcon: energ of he healh par of he bod. Once opmzed, hese wll be used as nal values. hen he cos funcon wll be agan opmzed wh hese values and few more parameers. All parameers need o be opmzed. Sep b sep and f s necessar new resrcons are nroduced. Ne dagram represens our followed seps. Mn Eh Varables {Kp, p} Resrcons: ψ <,rad Kp,p > ψ, ψ, Kp,p as nal parameers Mn Eh Varables {Kp, p,,, } Resrcons: ψ <, rad Kp,p > m < <,m ẏ ( / ) ψ, ψ, Kp,p,,, as nal parameers HALMERS, Appled Mechancs, Maser s hess :9

34 Mn Eh Varables {Kp, p,,,,, b } Resrcons: ψ <, rad Kp,p > m < <,m ẏ ( / ) ψ ( ) ψ ( ) <, rad,9 m Mn Eh Varables {Kp, p,,, } Resrcons: ψ <, rad Kp,p > m < <,m ẏ ( / ),9< ψ ( ) ψ ( ) <, rad ψ, ψ, Kp,p,,, as nal parameers < < m <,9 ψ, ψ, Kp,p,,,,, b as nal parameers Mn Eh Varables {Kp, p,,,,, b, ψ,ψ } Resrcons: ψ <, rad Kp,p > m < <,m ẏ ( / ) ψ ( ) ψ ( ) <, rad,9 m,9< < < m <,9 ψ, ψ, Kp,p,,,,, b, as nal parameers Mn Eh Varables {Kp, p,,,,, b, ψ,ψ,.. } Resrcons: ψ <, rad Kp,p > m < <,m ẏ ( / ) ψ ( ) ψ ( ) <, rad,9 m,9< < < m <,9 Fgure.. Organzaon char of our opmal problem. 6 HALMERS, Appled Mechancs, Maser s hess :9

35 Varances, and could var slghl f snus and cosnes parameers are used nsead of polnomal parameers. In each varance several aspecs need o be aken no accoun. Frs of all he soluons gven mus be anhropomorphc; f hese are no, s no worh connung wh ha. In each varance he energ consumed b he proshess needs o be checked. Boh cos funcons have o be mnmzed a he same me. he soluons should have low value of energ of he healh par of he bod and a he same me a low value of he energ consumed b he proshess. Also whch s our knemacs and dnamcs n each varance should be aken no consderaon compared wh prevous varances. Fnall, he fnal moons found, should be anhropomorphc. herefore, hese should be compared wh epermenal moons from leraure.. Malab Program Malab code has been wren o solve he opmal problem (Problem A) whch has been se n prevous secon. I had o be feasble bu also eas o use as well as manageable. Our Malab program can be spled n several pars. On he frs one, our fed (lengh of bod pars, me sep...) and varable parameers are se. Here changng he value of ever varable parameer s eas. On he ne one, nal leg and runk knemacs are found. hen, dnamcs of he healh par of he bod as well as orque on he ankle of he proshess leg are obaned. Ne par s he man secon; s where he opmal problem s gong o be solved. We call fmncon funcon and Malab should opmze our parameers o mnmze he cos funcon. So, our cos funcon as well as resrcons, consrans and boundar condons need o be nroduced. A sngle cos funcon (nsead of wo cos funcons Pareo fron) s opmzed n order o know how he value of he consumed energ changes whle some parameers are vared. Once opmzed parameers are obaned, hese resuls mus be evaluaed. New knemacs and dnamcs of he ssem wll be found. hen he plos ha are needed should be seen. In fgure. all pars of he program are presen. HALMERS, Appled Mechancs, Maser s hess :9

36 bsman.m SEING PARAMEERS fedparameers.m varableparameers.m bskn.m ode soluonps.m INIIAL KINEMAIS legskn.m ploskn.m legskn.m bkpdn.m plodn.m quadenergh.m quad energh.m OPIMIZAION fmncon fedparameers.m bskn.m fedparameers.m bskn.m confun.m bkpdn.m bkpdn.m legskn.m legskn.m quad fenerg.m ode soluonps.m fedparameers.m EVALUAION bkpdn.m legskn.m legskn.m plos ploskn.m plodn.m repmove.m Fgure.. Organzaon char of he Malab program. bkpdn.m: fnds ou all dnamcs varables. bskn.m: hs program fnds ou all knemacs varables. bsman.m: s he man program whch calls each subprogram. confun.m: here are resrcons, equales and nequales for parameers whch have o be opmzed. Wh quadenergh.m, boh are npu daa for fmncon funcon. 6 HALMERS, Appled Mechancs, Maser s hess :9

37 daa.m: ask f user wans o see plos of knemacs, dnamcs or ohers. energh.m: s he defnon of he power consumed b he healh par of he bod. epdaa.m: s where he epermenal daa has o be nroduced fenergp.m: s he power consumed b he below-knee proshess. If he negral s appled, he energ consumed b he proshess durng one sngle sep wll be obaned. fedparameers.m: all fed parameers are defned:, a, a, b, b, L, m, m a, m b, m f, m a, m b, m f, r, r a, r a, r b, r b, J a, J a, J b, J b and J. legskn.m: hs program fnds ou all leg knemacs, n bskn.m hs subprogram s used o oban hese varables as vecors, wh he same me seps lke he soluon of he dfferenal equaon. plodn.m: makes dnamcs-me plos. ploskn.m: makes leg knemacs-me plos. ploohers.m: makes dfferen knds of plos. K p versus p, N (number of varance) versus E p, ψ versus ψ, N (number of varance) versus E h. quadenergh.m: s he cos funcon of he healh par of he bod. I s an mplemenaon of he negral of he power consumed b he healh par of he bod durng he me of a sngle sep. repmove.m: makes a lle move wh he ga moon wh he ad of geframe command and hen, a move fle wh moveav command can be obaned. soluonps.m: hs program has ψ dfferenal equaon. Ever varable has o be defned as a scalar n order o solve he dfferenal equaon, so legskn.m fnds ou,,,, α, β and her veloces and acceleraons as scalars. varableparameers.m: parameers whch could be changed n opmzaon phase are defned n hs fle. hese are sffness and dampng values (K p and p ), vecor [ ] and nal values of ψ : ψ and ψ. varableparameers.m... varableparameers7.m: varable parameers ha have been opmzed durng opmal problem n each varance. HALMERS, Appled Mechancs, Maser s hess :9 7

38 Resuls Once one has worked wh MALAB, resuls can be looked for. he man goal can be forgoen: lookng for he opmal sffness and dampng properes of a below-knee proshess, wh whch paen use as less energ as possble, as well as he proshess use he mnmum energ as possble. However, our model has o be anhropomorphc and conssen. So, knemac and dnamc resuls of our model need o be compared wh leraure models. All eernal daa ha has been used n hs secon comes from [8] (Davd A. Wner, 99). hs epermenal daa was esed from healh people. I consss of rals carred ou n a laboraor. he dd rals walkng n hree dfferen veloces, beween 8 o seps/mn: naural walkers wh an average cadence of, seps/mn wh a srde lengh of, m; fas walkers wh an average of cadence of, seps/mn and a srde lengh of,6 m; slow walkers wh an average cadence of 86,8 seps/mn and a reduced srde lengh of,8 m. hese daa conss of ankle, knee and hp angles, so α, β and ψ angles from our model have been able o oban; ankle, knee and hp orques as well as ground reacon (horzonal and vercal). o solve he opmal problem varances menoned a Secon have been followed. Soluons alread had a low value of our cos funcon (energ of he healh par of he bod). In ne able, hese varances can be seen wh some of man resuls, opmzed values of sffness and dampng properes (K p and p ), mnmum values of he healh par of he bod (E h ) and energ consumed b he below-knee proshess wh ha confguraon. Varance Varance Varance Varance Varance Varance 6 K p (Nm/rad) 6,8 89,9 77,6, 6,6, p (Nms/rad),8 66, 6,8, 6,, E h (J/m) 66, 6,7 7,,,7,6 E p (J/m),96 6,6,,,9,6 able.. Opmal sffness and dampng values (K p and p ), and energ values (E h and E p ) for each opmal confguraon. hen, lookng no our resuls one can check f he are anhropomorphc enough. Knemacs resuls are shown n ne secon. 8 HALMERS, Appled Mechancs, Maser s hess :9

39 . Knemac resuls oordnaes and (moon of he hp) seems o be anhropomorphc enough, as one can see n Fgure (m) v (m/s)..6. (m). Varance o (s).99 Varances o.98 Varance Varance.97 Varance v (m/s) Varance o (s). Varance o. Varance Varance. Varance (s) Fgure.. Moon of he hp.,, and (s) Horzonal moon of he hp has a consan veloc, sars a, m and fnshes when has gone hrough a lengh equal o one sep (L). Vercal poson, coordnae, decreases along opmal seps. I could be guessed ha f pah s shorer, spends less global energ. Followng, plos of he healh ankle foo pah can be seen, durng s swng phase. See fgure. (a) and (b) (m).8.6 v (m/s)... Varances o Varance (s) Fgure..(a) Varance o Varance (s) HALMERS, Appled Mechancs, Maser s hess :9 9

40 ..8 Varance Varance Varance Varance Varance Varance Varance Varance Varance Varance Varance Varance 6 (m).6. v (m/s) (s) (s),, and. Fgure. (b). Moon of he healh ankle n s swng phase. Is horzonal coordnae haven been opmzed unl varance 6, where can be seen ha here s a change n s veloc. Is hegh ncrease durng opmal process, herefore vercal veloc ncrease oo. Epermenal daa for angles from leraure [8] s avalable, so hese wll be compared wh our resuls. Noe ha epermenal resuls ha are avalable belongs o double sep, bu our resuls are from onl one sep; when he proshess leg s on s sance phase and he healh leg s on s swng phase. Wh he ad of [], one can conclude ha our sngle sep ook place beween % and % of he double sep (suppor phase on boh heel and meaarsal jon) for proshess leg and from 6% o % (swng phase of he foo over he surface) for healh leg. Before proceedng o compare our resuls wh leraure resuls, one assumpon has o be eplaned whch s supposed n order o ge daa. In [8], he mechancal model used s lke n Fgure.. As can be seen, s dfferen han our model (Fgure.), snce angle of he foo θ f angle beween horzonal and a lne along he boom of he foo measured from he dsal end ( h meaarsal phalangeal jon) have no been consdered. In order o ge β angle, foo angle has o be appromae. Fgure.. Mechancal model of [8]. θ a s he ankle angle. HALMERS, Appled Mechancs, Maser s hess :9

41 o appromae hs angle, he me plo of he double sep (Fgure.) has been aken no accoun. Lookng he poson of he heel and meaarsal durng he sep (see Fgure.), some angles durng he double sep have been decded, see able.. hen, nerval mes beween hese values have been appromaed lneall. Fgure.. me plo of he double sep on one leg, Daa from [8]. Fgure.. me plos of dsplacemens of heel and meaarsal. me (%) Angle ( ) able.. Appromaed values of foo angle. Sandard Devaon of hese daa null has been assumed. HALMERS, Appled Mechancs, Maser s hess :9

42 Accordng Fgure., he followng relaons have been assumed: β θ f θ a 8 (.) α θ θ θ 8 (.) f a k ψ θ θ θ θ 8 (.) f a k h So, once α, β and ψ epermenal daa are found, one can compare our resuls wh hem. β angle on he leg whch has he proshess s compared, see Fgure bea (rad) -. Varance o Varance -. Varance Varance Bound Bound Ep. Mean (%) Fgure.. β angle on he proshec leg wh ppe of he epermenal resuls. In he plo, bound means he value of magnude calculaed plus or mnus one sandard devaon. Almos all of our resuls are nsde he ppe of he epermenal resuls. In Fgure.6, resuls of β can be seen. here are no specal commens n hs resuls, jus noe ha sep 6 changes a lle he shape of he plo; sars wh a lower value han he oher varances and fnshes wh hgher value. Epermenal resuls from hs daa are no avalable, so he can be compared... Varance o Varance Varance Varance 6 vbea (rad/s) (s) Fgure.6. Plo of β on he proshec leg. HALMERS, Appled Mechancs, Maser s hess :9

43 Now, α angle on he leg whch has he proshess s gong o be compared, see Fgure.7... alfa (rad) -. - Ep. Mean Bound Bound Varance o Varance Varance Varance (%) Fgure.7. α angle on he proshec leg wh ppe of he epermenal resuls. Our resuls are close o he ppe ha deermnes he mean of epermenal resuls plus and mnus one sandard devaon ( α ± s. dev ). One can conclude ha a he end of he sngle sep hgh of he proshec leg s no as fleed as n epermenal resuls. I can be also seen ha n frs varance he fleon of he hgh was hgher han laes varances. In Fgure.8. One can see he plo of veloc of alpha angle,α valfa (rad/s) Varance o Varance - Varance Varance (s) Fgure.8. Veloc of he α on he proshec leg. HALMERS, Appled Mechancs, Maser s hess :9

44 Lke n β (Fgure.6) on he same leg, here are no specal commens n hs resuls, jus noe ha sep 6 changes a lle he shape of he plo. Regardng swng phase on he healh leg, mos of our resuls are under he ppe of he epermenal resuls. See β angle on hs leg, Fgure.9. On varance 6, fleon of he healh shank s larger..6.. bea (rad) Ep. Mean Bound Bound Varance Varance Varance Varance Varance Varance (%) Fgure.9. β angle on he healh leg. Epermenal resuls from hs daa are no avalable, so our resuls can be compared wh hem. However one can see n order ha shank of he healh leg ges more fleon, s veloc s also hgher. See Fgure.. vbea (rad/s) Varance Varance Varance - Varance Varance Varance (s) Fgure.. Plo of β on he healh leg. HALMERS, Appled Mechancs, Maser s hess :9

45 If α angle resuls are compared wh leraure, one also can see ha a he begnnng of he swng phase sars a almos he same angle, bu hen, epermenal resuls ncrease more han our resuls. See Fgure.... alfa (rad) Ep. Mean Bound -. Bound Varance Varance - Varance Varance Varance Varance (%) Fgure.. Plo of α angle on he healh leg. Regardng α angular veloc on he healh leg, lke angular veloces on he proshec leg, no specal commens can be saed, because no epermenal daa s avalable as a reference. However one can see ha varance 6 dffers from oher resuls... Varance Varance Varance Varance Varance Varance 6 valfa (rad/s) (s) Fgure.. Plo of α on he healh leg. Las angle ha s lef o be analzed s ψ. In Fgure.., our resuls can be seen wh epermenal daa. One can sae ha our resuls a he begnnng of he sngle sep are close o be anhropomorphc. Epermenal resuls fle he runk more han ours. HALMERS, Appled Mechancs, Maser s hess :9

46 As s known, f he runk s n a vercal poson, he ga wll be more safe and sable. In [8], one also can learn ha n human ga as a frs appromaon he runk could be consdered almos vercal (acuall s based slghl forward of vercal). In our resuls, runk s close o be vercal, especall varances, and. Bu epermenal resuls show ha n anhropomorphc ga here s a movemen more forward and backward. I should be also aken no accoun ha foo angle has been appromaed; probabl real anhropomorphc ppe has a smooher shape ps (rad) daa daa daa Varance Varance Varance Varance Varance Varance (%) Fgure.. runk angle (ψ ) plo, wh ppe of he epermenal resuls. Regardng ψ, from he frs unl ffh varance, resuls sar nearl a he same veloc. Varance 6 sars wh a lower veloc (n absolue value), bu hen, durng he sep, ncreases s veloc more han oher varances. See fgure.... Varance Varance Varance Varance Varance Varance 6 vps (rad/s) (%) Fgure.. Veloc of he runk angle, ψ. 6 HALMERS, Appled Mechancs, Maser s hess :9

47 In fgure. an neresng plo can be seen, conss of he relaon beween ψ and ψ. From he rd varance a new resrcon s nroduced: ψ ( ) ψ ( ) <, rad so ha, runk angle a he begnnng and a he end of our sep were almos he same. One can bear ou hs fac n hs plo. Also, as menoned few lnes above, runk angle would be beer as much vercal s (or slghl forward of vercal). So, one can consder varances, or more anhropomorphc, n addon, he have almos he same ψ and ψ on he bounds.. vps(rad/s) Varance Varance Varance Varance Varance Varance ps(rad) Fgure.. Relaon ψ versus ψ. Knemacs resuls can be seen n one plo where he sngle sep s represened. Plos from varances, and are avalable, whch are he mos anhropomorphc. Fgure Y (m).8 Y (m) X (m) X (m) HALMERS, Appled Mechancs, Maser s hess :9 7

48 .6.. Y (m) X (m) Fgure.6. Varance (above lef), Varance (above rgh) and Varance (down a he mddle). One can see ha varances and are almos he same, lke has been seen n Fgure.. In varance he runk s slghl forward o vercal. As a frs appromaon seems more anhropomorphc and naural han varances and. o fnsh comparng and dscussng knemacs plos, R coordnae s suded, horzonal poson of he pon where ground reacon acs. In Fgure.7 s emporal progress can be seen. In Fgure. he mplc foo have been defned and how o calculae hs coordnae R s known, R m have also been defned, herefore s eas o fnd ou R. Lookng hs plo and Fgure. one can guess ha p, he orque of he proshess leg, wll be negave, snce ground reacon force s appled n fron of he ankle ( R >.76 m). hese resuls are anhropomorphc, as hs pon of applcaon would be on a real foo, snce he mamum value of R s n varance ( R.88) and would be a a reasonable dsance from ankle jon:,88-,76, m. Dsance beween ankle jon and oes s hgher han,m, so our resuls could be consdered anhropomorphc R (m) Varance Varance Varance.76 Varance Varance Varance (s) Fgure.7. Horzonal poson of he pon applcaon of he ground reacon. 8 HALMERS, Appled Mechancs, Maser s hess :9

49 . Dnamc resuls o sar wh dnamcs, resuls of he orque on he ankle of he proshec leg are compared wh he ankle orque of a healh bod from epermenal resuls. I s assumed ha won be he same, bu he should be smlar n order ha he paen can walk n an anhropomorphc wa. In Fgure.8 one can see ha he shape of p s compleel dfferen from epermenal resuls, however, values from he begnnng of he sep are nsde of he epermenal ppe. One also can sae ha dependng on varances, he shape of our p dffers a lle. - - p (Nm) Ep. mean Ep. up Ep. down Varance Varance Varance Varance Varance Varance (s) Fgure.8. orque on he ankle jon of he proshec leg, p. As foo has been consdered as a fooprn, orque ankle of he healh leg durng s swng phase wll be null. As regards hp orque on he proshec leg (q ), n Fgure.9 one can see ha he mos resuls are nsde or close o he ppe of he epermenal resuls. Varance 6 has a sgnfcan dfference, because a he end of he sep, s hp orque goes up fas. hs fac can be anhropomorphc. I s a resul of he opmal problem; mabe f new resrcons are added n hs varance, he problem wll be able o be solved. HALMERS, Appled Mechancs, Maser s hess :9 9

50 q (Nm) 8 6 Ep. Mean Bound Bound Varance Varance Varance Varance Varance Varance (%) Fgure.9. Hp orque on he proshec leg, q. Knee orque on he leg ha has he proshess (u ) has some pars ha can be consdered anhropomorphc, bu a he end of he sep, he ge ou of he anhropomorphc ppe. Lke n hp orque on he same leg, varance 6 goes up a he end of he sep unreasonabl. See Fgure.. u (Nm) (%) Fgure.. Knee orque on he proshec leg, u. Ep. Mean Bound Bound Varance Varance Varance Varance Varance Varance 6 Resuls abou hp orque on he healh leg (q ) don have srong commens o be saed. Jus lke q and u, varance 6 dffers from oher varances. See Fgure.. HALMERS, Appled Mechancs, Maser s hess :9

51 q (Nm) Ep. mean Bound Bound Varance Varance Varance Varance Varance Varance (%) Fgure.. Hp orque on he healh leg, q. he same for he knee orque on he healh leg. See Fgure.. u (Nm) 8 6 Bound Ep. Mean Bound Varance Varance Varance Varance Varance Varance (%) Fgure.. Knee orque on he healh leg, u. Ground reacon force s close o ppe of he epermenal resuls. As menoned, varance 6 s no anhropomorphc, more consrans should be pu. Neverheless, oher resuls seem o be anhropomorphc. If one look on horzonal componen (R ) on Fgure., one can see ha frsl decelerae he bpedal ssem (when R <) and hen accelerae (when R >). Regardng vercal componen (R ), a he begnnng and a he end of our suded perod has he hghes values. See Fgure.. HALMERS, Appled Mechancs, Maser s hess :9

52 - R (N) Ep. Mean Bound Bound Varance Varance Varance - Varance Varance Varance (%) Fgure.. Horzonal componen of he ground reacon force. R. R (N) 8 6 Ep. Mean Bound Bound Varance Varance Varance Varance Varance Varance (%) Fgure.. Vercal componen of he ground reacon force. R. Once knemacs and dnamcs of all varances have been dscussed, could be saed ha, akng no accoun all commens ha have been sad sad, as a frs appromaon all of hem are anhropomorphc enough ecep varance 6. HALMERS, Appled Mechancs, Maser s hess :9

53 . Oher resuls Now, focus on resuls ha wll help o r o solve Problem A. Hence, f one akes a look on Fgure., one can see ha energ of he healh par of he bod (E h ) decreases when more parameers o opmze are nroduced. From varances o no, because no parameers were nroduced, jus new resrcons. 7 6 Varance Varance Varance Varance Varance Varance 6 Eh 6 Number of varance Fgure.. Evoluon of energ of he healh par of he bod (E h ) durng varances n opmal problem. he evoluon of E h could have been guessed, bu one doesn know anhng abou he evoluon of E p, he energ consumed b he below-knee proshess. I could be supposed ha f one spends less healh energ, energ consumed b he proshess would be hgher. If Fgures. and.6 are compared, from o varance one could sae ha s rue, bu also depend on whch s he confguraon of all parameers o opmze, snce varances - and -6 our saemen s no rue. Ep Varance Varance Varance Varance Varance Varance 6 6 Number of "sep" Fgure.6. Evoluon of he energ consumed b he below-knee proshess (E p ) durng varances n opmal problem. HALMERS, Appled Mechancs, Maser s hess :9

54 In Fgure.7 he relaon E p versus E h can be seen and one can conclude ha range of E h s from,6 o 66, J/m and range of E p s from,6 o,96 J/m. Ep (J/m) Varance Varance Varance Varance Varance Varance Eh (J/m) Fgure.7. Relaon beween E h and E p. Fnall, he mos mporan relaon can be ploed, dampng ( p ) versus sffness (K p ) properes of resuls ha have been obaned. Our resuls shows ha for obaned opmzed knemacs of he moon sffness values are beween, Nm/rad and 89,9 Nm/rad and dampng resuls are beween, Nms/rad and,96 Nms/rad. See Fgure p (J/m) Varance Varance Varance Varance Varance Varance Kp (J/m) Fgure.8. Relaon beween dampng and sffness properes (p versus Kp). HALMERS, Appled Mechancs, Maser s hess :9

55 6 onclusons hs maser hess has objecves o conrbue n desgnng a new pe of acvel conrolled below-knee proshess whch spends as low energ as possble. he knowledge abou suable sffness and dampng properes of he below-knee proshess s needed. In he repor an opmzaon problem has been formulaed whch ncludes esmaon of opmzed sffness and dampng parameers of he below-knee proshess as well as deermnaon of knemacs of human bod moon mnmzng energ consumpon of he healh par of he bod (E h ). Numercal algorhm has been developed and MALAB (fmncon funcon) has been used o solve he comple opmzaon problem. Apar from one varance n our opmzaon problem, sasfacor resuls have been acheved, snce obaned resuls are almos anhropomorphc. A he same me low values of E h have been obaned, whle E p value has been aken no accoun n order ha doesn ncrease oo much. If one looks on able., one could sae ha a suable confguraon would be among varances o, because he have a low value of E h whle he keep a low value of E p ; especall and would be he moons more sable, wh a vercal runk poson. Fnall, has been managed o fnd ou a reasonable range of values for sffness (K p ) and dampng ( p ) properes. hs fac means ha possble relaons beween orque appled on he ankle jon of he below-knee proshess leg wh s angle and angular veloc have been obaned. However, our model dffer from epermenal daa [8] ha have been used, snce he angle of foo has been appromaed n order o ge epermenal daa ha was needed. So, comparsons ha has been done aren eremel accurae, because real moon of he angle foo s no avalable. hs s a pcal problem when a mechancal ssem s smulaed. 6. Fuure work Our model s wo-dmensonal, ha s he mechancal ssem moon s consdered onl n a sagal plane. A possble ne sep of he work would be smulaon of human moon and esmaon of sffness and dampng properes of he below-knee proshess whn he frame of D model. One also could spend more me refnng MALAB code, lookng for dfferen resrcons or changng nal condons n order o fnd new anhropomorphc moon whch consumes a lower energ value and hus fndng ou new combnaons of K p and p possble values. o end our work, has o be sad ha hs s a par of a conrbuon o desgn a new pe of proshess. Our research could be connued desgnng he proshess wh acvel conrolled ankle jon. o buld hs new pes of proshess a suable confguraon of mechancal elemens: sprng, damper, shaf, bearngs. should be desgned. HALMERS, Appled Mechancs, Maser s hess :9