Constant Power Model in Arm Rotation—A New Approach to Hill’s Equation

This is an elec onic ep in o he o iginal a icle. This ep in may di e om he o iginal in pagina ion and ypog aphic de ail. Au ho (s): Ti le: Yea : Ve sion: Please ci e he o iginal e sion: All ma e ial supplied ia JYX is p o ec ed by copy igh and o he in ellec ual p ope y igh s, and duplica ion o sale o all o pa o any o he eposi o y collec ions is no pe mi ed, excep ha ma e ial may be duplica ed by you o you esea ch use o educa ional pu poses in elec onic o p in o m. You mus ob ain pe mission o any o he use. Elec onic o p in copies may no be o e ed, whe he o sale o o he wise o anyone who is no an au ho ised use . Cons an Powe Model in A m Ro a ion—A New App oach o Hill’s Equa ion Rahikainen, Ah i; Vi ma i a, Mikko Rahikainen, A., & Vi ma i a, M. (2014). Cons an Powe Model in A m Ro a ion—A New App oach o Hill’s Equa ion. Wo ld Jou nal o Mechanics, 4(6), 157-169. h ps://doi.o g/10.4236/wjm.2014.46018 2014 Wo ld Jou nal o Mechanics, 2014, 4, 157-169 Published Online June 2014 in SciRes. h p://www.sci p.o g/jou nal/wjm h p://dx.doi.o g/10.4236/wjm.2014.46018 How o ci e his pape : Rahikainen, A. and Vi ma i a, M. (2014) Cons an Powe Model in A m Ro a ion—A New App oach o Hill’s Equa ion. Wo ld Jou nal o Mechanics, 4, 157-169. h p://dx.doi.o g/10.4236/wjm.2014.46018 Cons an Powe Model in A m Ro a ion—A New App oach o Hill’s Equa ion Ah i Rahikainen, Mikko Vi ma i a Neu omuscula Resea ch Cen e , Depa men o Biology o Physical Ac i i y, Uni e si y o Jy äskylä, Jy äskylä, Finland Email: [email p o ec ed] Recei ed 19 Ma ch 2014; e ised 18 Ap il 2014; accep ed 16 May 2014 Copy igh © 2014 by au ho s and Scien i ic Resea ch Publishing Inc. This wo k is licensed unde he C ea i e Commons A ibu ion In e na ional License (CC BY). h p://c ea i ecommons.o g/licenses/by/4.0/ Abs ac The pu pose o his s udy was o u he de elop he cons an powe model o a p e ious s udy and o p o ide he inal solu ion o Hill’s o ce- eloci y equa ion. Fo ea m and whole a m o a- ions o h ee di e en subjec s we e pe o med downwa ds (elbow and shoulde ex ension) and upwa ds (elbow and shoulde lexion) wi h maximum eloci y. These a m o a ions we e eco d- ed wi h a special came a sys em and he heo e ically de i ed model o cons an maximum powe was i ed o he expe imen ally measu ed da a. The momen o ine ia o he a m sec o s was calcula ed using imme sion echnique o de e mining accu a e alues o ic ion coe icien s o elbow and whole a m o a ions. The expe imen s o he p esen s udy e i ied he conclusions o a p e ious s udy in which heo e ically de i ed equa ion wi h cons an maximum powe was in ag eemen wi h expe imen ally measu ed esul s. The esul s o he p esen s udy we e compa ed wi h he mechanics o Hill’s model and a u he de elopmen o Hill’s o ce- eloci y ela ionship was de i ed: Hill’s model was ans o med in o a cons an maximum powe model consis ing o h ee di e en componen s o powe . I was concluded ha he e a e h ee di e en s a es o mo- ion: 1) he s a e o low speed, maximal accele a ion wi hou ex e nal load which applies o he hypo hesis o cons an momen ; 2) he s a e o high speed, maximal powe wi hou ex e nal load which applies o he hypo hesis o cons an powe and 3) he s a e o maximal powe wi h ex e nal load which applies o Hill’s equa ion. This is a new app oach o Hill’s equa ion. Keywo ds A m Mo emen , Fo ce-Veloci y Rela ionship, Muscle Powe , Hill’s Equa ion 1. In oduc ion Hill’s o ce- eloci y ela ionship o skele al muscle (Figu e 1) [1] [2] is one o he mos essen ial equa ions o A. Rahikainen, M. Vi ma i a 158 Figu e 1. Hill’s o ce- eloci y ela ionship wi h co esponding powe - eloci y cu e (dashed g ay cu e), whe e F0 is maximum isome ic o ce o o ce wi h ze o eloci y, 0 is he highes possible eloci y, a and b a e cons an o ce and cons an eloci y. Maximum powe P0 is ypically ound a abou 30 % o 0 [4]. In o a ional mo emen o - que M co esponds o o ce F and angula eloci y co esponds o eloci y in Hill’s equa ion. muscle mechanics and i has been an objec o biomechanical s udies o yea s (e.g., [3]-[6]). In muscle me- chanics, his ela ionship is o en p esen ed by Hill’s equa ion ( )( ) ( ) 0 F a b bF a+ += + , whe e F is cu en muscle o ce a cu en sho ening eloci y o con ac ion, a is cons an o ce and b is cons an eloci y, F0 is he maximum isome ic muscle o ce, i.e., he maximum o ce ha muscle can de elop a a gi en cons an leng h, and is eloci y [1] [2]. Fo compa ison be ween di e en muscles b is bes exp essed in e ms o b/l0, whe e l0 is he s anda d leng h o muscle. This equa ion was based on labo a o y measu emen s wi h a Le in-Wyman e - gome e in which he ac i a ed muscle was eleased a a sui able speed in an isola ed muscle condi ion. The ob- ained cons an eloci y was hen plo ed agains he obse ed ension. Fo ce measu ed om skele al muscle du ing maximum ension depends on se e al in e nal and ex e nal ac o s which ha e been ho oughly e iewed (e.g., [3] [7] [8]). Ma sumo o [9] men ioned ha because almos all he iso onic da a ha e been es ic ed o one muscle leng h l0, he maximum leng h wi h almos no es ing ension, and he eloci ies measu ed a e hose ini- ial alues when he load begins o mo e. Examining he leng h egion, l ≤ l0, o an iso onically con ac ing muscle, Ma sumo o [9] ound ha he cons an s a/F0 and b/l0 emained ixed h oughou he ange o leng hs o e which he sho ening akes place. In con as o Hill’s isola ed muscle p epa a ions, o ce (F) o he in- ol ed muscles in o a ional mo emen c ea es a momen (M = × F) abou he join . The leng h o he muscle’s momen a m ( ) changes as he o a ional mo emen p oceeds abou he join axis. This o a ion mo emen is he combined e ec o he o ces o se e al di e en muscles. Howe e , i is di icul o de e mine he con ibu ion o each muscle on o ce p oduc ion due o many di e en ac o s, and also o de e mine he o que abou he join . Se e al ex enso and lexo muscles we e used by Raiko a [10] in he model o he lexion-ex ension mo- ion in he elbow join . Fu he mo e, he o ce o a skele al muscle is an accumula ion o o ces gene a ed by ac- i e mo o uni s belonging o his muscle [11]. Raiko a e al. [12] men ioned ha access o each mo o uni (MU) is impossible, and he ec ui men and o ce de eloping p ope ies o all indi idual MUs canno be known. In his pape [12] he p ocess o lea ning as elbow lexion in he ho izon al plane was simula ed and he esul was compa ed wi h expe imen ally measu ed da a. P e ious expe imen s o Rahikainen e al. [13] we e based on he heo y o a m o a ions including ou phases: 1) s a o he mo ion; 2) phase o cons an maximum o a ional momen ; 3) phase o cons an maximum muscula powe ; 4) s opping o he mo ion. I was assumed ha he muscula sys em is able o ans e only a ce ain amoun o chemical ene gy du ing he ime o con ac ion and he e o e, a m o a ion mus ha e maxi- mum powe ha canno be exceeded. I was also assumed ha he maximum powe ac s wi hin a ce ain ange o eloci y which was conside ed as cons an maximum powe and his is possible only when he eloci y is F/P F 0 0 a b P 0 A. Rahikainen, M. Vi ma i a 159 high enough. The eloci y o he mo ion inc eases o he poin whe e he maximum powe occu s and ac ing o- a ional momen is less han he maximum momen . Consequen ly, powe emains cons an as he angula eloc- i y inc eases and he momen dec eases. The p esen s udy con inues he expe imen s o Rahikainen e al. [13] and u he de elops i s heo y o me- chanics. The p e ious s udy p esen ed a cons an powe model and i s alidi y and accu acy o esul s we e as- sessed. In he p esen s udy new a m o a ion expe imen s o h ee subjec s we e cap u ed wi h wo di e en e- co ding sys ems. Mo e accu a e calcula ions o momen s o ine ia and a new mo e e ec i e de e mina ion o he ma ched ange o measu ed and heo e ical angula eloci y cu es we e used. Also a new app oach o Hill’s equa ion was p esen ed. In he le side o Hill’s equa ion ( )( ) ( ) 0 F a b bF a+ += + he cons an a has he dimensions o o ce and b he dimensions o eloci y, o he wise addi ion is impossible. The e o e, (F + a) is o ce and ( + b) is eloci y, and o ce mul iplied by eloci y is powe as can be seen in Figu e 1. The e m (F + a) ( + b) is muscles’ o al powe including F , which is he powe o mo ing he ex e nal load. The igh side o he equa ion, b (F0 + a), includes only cons an s in his ega d he equa ion can be conside ed as a cons an powe model. Howe e , he cons an powe o Hill’s equa ion p esen ed in his pape is no he abo e men ioned powe o Hill’s o iginal cu e as i is usually conside ed in biomechanics, bu i is he sum o h ee di e en powe componen s (see Discussion and Conclusions). The cons an powe model o his s udy ac s du ing high speed mo emen s wi h no ex e nal load, whe e Hill’s equa ion does no seem o i he expe imen al poin s ([2] p. 32, Figu e 3) e y well. As an explana ion o his misma ch Hill men ioned ha “sha p ise a he end o he cu e in he egion o e y low ension was due o he p esence o a limi ed numbe o ibe s o high in insic speed and no such equa ion could i he obse ed poin s below P/P0 = 0.05”. The p esen model is based on he muscula sys em’s abili y o ans e chemical ene gy and, he e o e, i is no necessa y o know he con ibu ion o he indi idual muscles in ol ed. The pu pose o he p esen s udy was o examine how he heo e ical cons an powe model which was i s used in linea mo ion [14] and la e applied o o a ional mo ion [13] i s he measu ed angula eloci y cu es o a m o a ion expe imen s. The u he pu pose o he s udy was o de e mine how Hill’s equa ion unc ions as a cons an powe model and o compa e Hill’s equa ion wi h he model o he p esen s udy. 2. Ma e ials and Me hods In he p esen s udy, he measu emen s o a m o a ions (Subjec s S1 and S2) we e eco ded by a special came a sys em [15] [16]. Figu e 2 shows he p inciple o he sys em whe e angula eloci y was calcula ed wi h he Fo mula: SRT ϕ =∆∆  (1) whe e R is a m leng h, ΔS is he dis ance be ween wo successi e measu ing poin s and ∆T is ime inc emen . Addi ional elbow lexion and ex ensions we e pe o med o one Subjec S3 by using Vicon mo ion analysis sys em wi h 8 came as. This sys em made i possible o use highe ame a es (300 Hz). Subjec s we e no mal heal hy people ep esen ing di e en age and physical ac i i y backg ound (S1: 62 yea s, 1.80 m, 82 kg; S2: 35 yea s, 1.69 m, 73 kg, aikido aining, weigh li ing and S3: 25 yea s, 1.83 m, 70 kg, high jumping). The accu acy o he special came a sys em has been desc ibed in e e ences [13] [15]-[17]. The angula e- loci y- ime cu es we e d awn (wi h a line hickness o 0.5 mm) on he squa ed pape , whe e 1 mm co es- ponded o an angula eloci y o 0.1 ad/s and ime o 1 ms gi ing he accu acy o eloci y cu es. The heo e i- cal and measu ed angula eloci y cu es coincided wi hin he dis ances o 35 - 70 mm, which was enough o he e i ica ion o he cons an powe model in he a m o a ion expe imen s. Sligh oscilla ions a he beginning o he mo emen did no exis wi hin he cons an powe phase and, he e o e, he e i ica ion o he cons an powe model was possible in all expe imen s. The accu acy o he Vicon eco ding de ice, e i ied by calcula - ing he oo -mean-squa e e o o each came a, anged om 0.06 o 0.17 mm du ing calib a ion. 2.1. Model o A m Ro a ion The model used in he p esen s udy is cons uc ed acco ding o New on’s II law and i was i s used in linea mo ion [14] and hen applied o o a ional mo ion [13] by Rahikainen e al. The heo y o a m o a ion is as ol- lows: A he beginning o he mo emen , angula eloci y is na u ally ze o and i akes some ime o gene a e o ce. A his ea ly phase o mo ion, elas ic p ope ies o muscle- endon complex has in luence on he mo ion, bu a he s a e o ull ension, hese elas ic elemen s ha e no u he dynamic e ec . The ea e i can be A. Rahikainen, M. Vi ma i a 160 Figu e 2. Example o elbow lexion. Ro a ion an- gles be ween wo a m images (0 - 200 ms) a e gi en in 20 ms inc emen s acco ding o ligh ma ks. I is no ewo hy ha he momen a m o he muscles in ol ed in elbow lexion changes du ing he mo emen and, he e o e, he muscles’ con ibu ion o powe changes as well. assumed ha a maximum muscle o ce akes ac ion and wi hin o a ional mo ion maximum o a ional momen ac s as well. A his phase o mo ion he e is a cons an alue o glide ic ion ac ing. Because he muscle sys em is able o ans e only a ce ain quan i y o chemical ene gy du ing he ime o con ac ion, he e mus be a con- s an maximum powe , which he muscle is able o gene a e wi hin a ce ain ange o eloci y. As he eloci y inc eases he mo ion eaches he poin whe e he maximum powe akes ac ion and ac ing o a ional momen is less han he maximum momen . This way powe emains cons an as he angula eloci y inc eases and momen dec eases. Now, liquid ic ion, di ec ly p opo ional o eloci y, is ac ing. The cons an glide ic ion dec eases as o ces a he join dec ease and i becomes negligible. Du ing he cons an powe phase o he model o a ion- al momen is momen o ine ia mul iplied by angula accele a ion which equals he momen gene a ed by mus- cle o ce minus he momen gene a ed by inne ic ion o muscle. The e ec o g a i a ional o ce is added a - e wa ds (see Pa ag aph 2.3). The model o a m o a ion is he equa ion o mo ion: d d P IC T ϕϕ ϕ = −   (2) whe e Momen o ine ia in a m o a ion I Angula eloci y ϕ  Powe gene a ed by a m muscles P Time T Momen gene a ed by muscle o ce P ϕ  Momen gene a ed by inne ic ion o muscle C ϕ  Coe icien o ic ion C Inne ic ion o muscle is liquid ic ion inside muscle, which is di ec ly p opo ional o eloci y. The same liquid ic ion was also used in he s udy o Rahikainen e al. [17] which was ini ially adop ed om Alonso and Finn [18]. I was assumed ha , ini ially, mo emen p oceeds a a cons an maximum momen and hen momen gene a ed by he muscle o ce ( P ϕ  in Equa ion 2) is cons an . I was also assumed ha , as eloci y inc eases, 200 180 160 140 120 100 80 60 40 20 0 A. Rahikainen, M. Vi ma i a 161 mo emen p oceeds a cons an maximum powe a a ce ain ange o eloci y and hen he powe P in Equa ion 2 is cons an . In o de o de e mine he alidi y o his hypo hesis, Equa ion 2 was sol ed o angula eloci- y- ime unc ion and his equa ion was employed o alidi y de e mina ion. The solu ion o Equa ion 2 om he p e ious s udy [13]: 2 1e CT I P C ϕ −  = −     (3) 2.2. Calcula ion o Momen o Ine ia The mass dis ibu ion o he subjec ’s a m sec o s di e ed om he a e age alues o mass ables in he li e a u e. The e o e, he mass dis ibu ion o he a m sec o s we e de e mined by sinking he a m sec o s in o wa e . The masses o he a m sec o s we e calcula ed by mul iplying he o e lowed wa e olume wi h he co esponding a m sec o densi y. The leng h o he subjec ’s whole a m was measu ed wi h is clenched, and he leng hs o a m sec o s (hand, o ea m and uppe a m) we e measu ed. Acco ding o Win e [6] he a m sec o densi ies (kg/l) we e as ollows: hand 1.16, o ea m 1.13, uppe a m 1.07. De ini ion o momen o ine ia: 2 dI m=∫ (4) whe e dm is o a ing mass and is he dis ance o o a ing mass om he o a ional axis. De i a ion o he mo- men o ine ia abou he end o a m sec o assuming e en mass dis ibu ion: 2 22 2 1 00 0 d d dd d3 LL L m m mL I m L = = = = ∫∫ ∫ (5) whe e m is he mass o o a ing a m, and L is he leng h o o a ing a m. De i a ion o he momen o ine ia abou he cen e o he a m sec o assuming e en mass dis ibu ion: 233 2 2 233 2 d3 12 22 L L m L L mL I m L −  = = +=   ∫ (6) Because he dis ibu ion o mass in he a m sec o is no e en, he momen o ine ia o an addi ional mass was calcula ed wi h Fo mula: 2 3 I m = (7) whe e m is addi ional mass o he a m sec o and is he es ima ed dis ance o he cen e o g a i y o addi ional mass om he o a ion axis. Subjec s’ a m sec o dis ances, leng hs and masses a e p esen ed in Table 1. These alues a e hen subs i u ed in he abo e men ioned Equa ions 4-7 o calcula e he inal momen s o ine ia o o- ea m and whole a m o a ions abou he elbow and shoulde join (example o S1 summa ized in Table 2). Momen o ine ia o o ea m o a ion Fo ea m Momen o ine ia o majo o ea m mass abou he elbow join (m = 1.0 kg, L = 0.28 m): 22 2 11 1.0 0.28 0.0261kg m 33 mL I× = = = ⋅ Momen o ine ia o addi ional o ea m mass abou he elbow join (m = 0.24 kg, = 0.07 m): 22 2 12 0.24 0.07 0.0012 kg mI m ==×= ⋅ Hand Momen o ine ia o he hand sec o abou he cen e o mass (m = 0.58 kg, L = 0.11 m): 22 2 13 0.58 0.11 0.0006 kg m 12 12 mL I× = = = ⋅ Momen o ine ia o he hand sec o abou he elbow join (m = 0.58 kg, = 0.34 m): 22 2 14 0.58 0.34 0.0670 kg mI m ==×= ⋅ A. Rahikainen, M. Vi ma i a 162 Table 1. The es ima ed mass dis ibu ion o a m sec o s. Fo ea m’s addi ional mass is due o he muscle’s mass dis ibu ion o he o he end o a m sec o . Dis ance om elbow join (m) Dis ance om shoulde join (m) Leng h (m) Mass (kg) Subjec S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 Hand 0.11 0.11 0.12 0.58 0.52 0.58 cen e o mass 0.34 0.32 0.34 0.64 0.60 Fo ea m 0.28 0.26 0.27 1.00 0.90 1.00 cen e o mass 0.44 0.41 addi ional mass 0.07 0.06 0.07 0.37 0.34 0.24 0.22 0.24 Uppe a m 0.30 0.28 0.33 2.14 2.53 Ba e y 0.37 0.35 0.67 0.63 0.26 0.29 0.29 Table 2. Summa ized in o ma ion o calcula ion o momen o ine ia o o ea m o a ion (uppe pa ) and whole a m o a- ion (lowe pa ) o one subjec . Momen o ine ia is calcula ed abou he elbow join , cen e o mass (com) and shoulde join . Segmen m (kg) L (m) (m) Abou Momen o ine ia I (kg⋅m2) Fo ea m 1.00 0.28 elbow 2 11 3 mL I= 0.0261 Fo ea m (addi ional mass) 0.24 0.07 elbow 2 12 I m = 0.0012 Hand 0.58 0.11 com 2 13 12 mL I= 0.0006 Hand 0.58 0.34 elbow 2 14 I m = 0.0670 Ba e y 0.26 0.37 elbow 2 1b I m = 0.0356 To al 0.131 Segmen m (kg) L (m) (m) abou momen o ine ia I (kg⋅m2) Uppe a m 2.14 0.30 shoulde 2 21 3 mL I= 0.0642 Fo ea m 1.00 0.28 com 2 22 12 mL I= 0.0065 Fo ea m 1.00 0.44 shoulde 2 23 I m = 0.1936 Fo ea m (addi ional mass) 0.24 0.37 shoulde 2 24 I m = 0.0329 Hand 0.58 0.11 com 2 25 12 mL I= 0.0006 Hand 0.58 0.64 shoulde 2 26 I m = 0.2376 Ba e y 0.26 0.67 shoulde 2 2b I m = 0.1167 To al 0.652 Ligh ma ke ba e y Momen o ine ia o he ba e y abou he elbow join (m = 0.26 kg, = 0.37 m): 22 10.26 0.37 0.0356 kg m b I=×= ⋅ To al momen o ine ia o o ea m o a ion abou elbow join 2 0.0261 0.0012 0.0006 0.0670 0.0356 0.131kg m++++= ⋅ A. Rahikainen, M. Vi ma i a 163 Momen o ine ia o whole a m o a ion Uppe a m Momen o ine ia abou he shoulde join (m = 2.14 kg, L = 0.30 m): 22 2 21 2.14 0.30 0.0642 kg m 33 mL I× = = = ⋅ Fo ea m Momen o ine ia o he majo o ea m mass abou he cen e o mass (m = 1.0 kg, L = 0.28 m): 22 2 22 1.0 0.28 0.0065 kg m 12 12 mL I× = = = ⋅ Momen o ine ia abou he shoulde join (m = 1.0 kg, = 0.44 m): 22 2 23 1.0 0.44 0.1936 kg mI m ==×= ⋅ Momen o ine ia o addi ional mass abou he shoulde join (m = 0.24 kg, = 0.37 m): 22 2 24 0.24 0.37 0.0329 kg mI m ==⋅= ⋅ Hand Momen o ine ia o he hand sec o abou he cen e o mass (m = 0.58, L = 0.11 m): 22 2 25 0.58 0.11 0.0006 kg m 12 12 mL I× = = = ⋅ Momen o ine ia o he hand sec o abou he shoulde join (m = 0.58 kg, = 0.64 m): 22 2 26 0.58 0.64 0.2376 kg mI m ==×= ⋅ Ligh ma ke ba e y Momen o ine ia o he ba e y abou he shoulde join (m = 0.26 kg, = 0.67 m): 22 20.26 0.67 0.1167 kg m b I=×= ⋅ To al momen o ine ia a whole a m o a ion abou shoulde join 2 0.0642 0.0065 0.1936 0.0329 0.0006 0.2376 0.1167 0.652 kg m++++++= ⋅ Acco ding o he abo e men ioned calcula ions, he co esponding momen s o ine ia o o ea m o a ion and whole a m o a ion o S2 we e 0.110 and 0.551 kg⋅m2, espec i ely. As he o al leng h o o ea m and hand was he same o Subjec s S1 and S3, i was assumed ha he momen s o ine ia o hese segmen s we e also he same and, he e o e, he momen o ine ia o o ea m o a ion abou he elbow join o S3 was 0.131 kg⋅m2. The mea- su emen s wi h Vicon came a sys em o S3 we e done wi h e lec i e ma ke s and he co esponding momen o ine ia wi hou he ba e y was 0.095 kg⋅m2. 2.3. E ec o G a i a ional Fo ce on he Mo emen In o ea m o a ion, he e ec o g a i a ional o ce is mino compa ed wi h maximum muscle o ces and he momen induced by g a i y mgΣ× was omi ed om he mo ion model (Equa ion 2). In whole a m o a- ion, his momen is added o he mo ion mechanics in he ollowing manne : The powe gene a ed by his g a i a ional momen is ( ) mg ϕ Σ×  , whe e mg is g a i a ional o ce o a m segmen s, is dis ance o he cen e o g a i y o segmen s om he o a ion axis and ϕ  angula eloci y o a m o a ion. The heo e ical angula eloci y (Equa ion 3) and he measu ed angula eloci y ma ch wi hin a e y na ow eloci y ange and he powe induced by g a i y can be calcula ed as a cons an ac o . I is included in he powe P ac- co ding o he p e ious s udy [13]. 2.4. De e mining he Ma ched Range o Measu ed and Theo e ical Angula Veloci y Cu es Figu e 3 shows he echnique ha was used o de e mine he ma ched ange o measu ed and heo e ical angula A. Rahikainen, M. Vi ma i a 164 Figu e 3. Example o he echnique o ind he ma ched ange (A - B) o measu ed and heo e ical angula eloci y. The ze o poin o ime o he heo e ical angula eloci y is a he in e sec ion o he ime-axis and he b oken-line cu e (see ex o mo e in o ma ion). The heo e ical angula eloci y cu e (b oken line) coincides wi h he measu ed cu e (solid line) be ween A and B. eloci y cu es. F ic ion coe icien alues (C) and powe alues (P) we e ob ained by i ing he heo e ical an- gula eloci y cu es o he measu ed ones. These wo cu es coincide only i ce ain C and P alues a e used in he i ing p ocess. The measu ed angula eloci y alues a e shown as poin s on he eloci y cu e and he heo e ical angula eloci y (Equa ion 3) is shown as a b oken line. The ma ched ange was ound by using he ollowing i e a ion p ocedu e: based on p e ious expe imen s, he andomized ini ial alues (see Figu e 3) wi hin he ma ched ange we e selec ed o angula eloci y a poin 1 ( ) 16.8 ad s ϕ =  . A ze o poin on he ime axis o he heo e ical angula eloci y was selec ed 0.050 s be o e poin 1, which co esponds o 0.060 s on he ime axis o he igu e. The ea e , he di ec ion o i e a ion p ocess was obse ed and a e some i e a ion, he inal heo e ical angula eloci y was d awn acco ding o Equa ion 3 o ma ch he measu ed eloci y cu e. The co ec heo e ical angula eloci y was ob ained wi h he ze o poin a 0.073 s on he ime axis. The a io o powe and ic ion coe icien in Figu e 3 is 2 2 1e CT I P C ϕ − = −  (8) The inal heo e ical angula eloci y was d awn wi h momen o ine ia I = 0.135 kg⋅m2, ic ion coe icien alue C = 2.92 kg⋅m2/s and a io o powe and ic ion coe icien P/C = 354 1/s2. 3. Resul s A m o a ion expe imen s eco ded by he came a sys em o Rahikainen [16] a e p esen ed in Figu es 4-7 and expe imen s wi h Vicon mo ion analysis sys em in Figu e 8. The heo e ical angula eloci y cu es in Figu es 4-8 a e ma ked wi h b oken lines and hey coincide wi h he measu ed angula eloci y cu es (solid lines) be- ween he poin s A - B, whe e mo emen p oceeds a cons an powe . Ini ially, mo emen p oceeds a cons an accele a ion, hen liquid ic ion becomes in luen ial and accele a ion dec eases jus be o e he sec ion A - B (cons an powe ), which is inally ollowed by s opping o he mo emen . In gene al, he sec ions A - B a e long enough o e i y he exis ence o he cons an powe model. The measu emen s in Figu e 8 wi h Subjec S3, made by he Vicon mo ion analysis sys em, did no ha e a clea sec ion o cons an accele a ion a he beginning o he mo emen . High oscilla ion in ha sec ion u ned i inde ini e. In he elbow ex ension, he oscilla ion is weak and he usual cons an accele a ion sec ion can be dis inguished a he mo emen ini ia ion. I also seems ha he cons an accele a ion sec ion in Figu e 6 has a simila oscilla ion as in Figu e 8. The measu ed da a in Figu es 4-8 ha e been smoo hed by he 6 h o de polynomial cu e i ing. The used ic ion coe icien alues a ied be ween 2.8 - 3.1 kg⋅m2/s in o ea m o a ions and be ween 3.6 - 3.8 kg⋅m2/s in 50 200150100 T(ms) ( ad/s) I0.135 kgm2 C2.84 kgm2/s P/C 321 1/s2 I0.135 kgm2 C2.92 kgm2/s P/C 354 1/s2 16.8 ad/s 1 A B