T and V be the total kinetic energy and potential energy stored in the dynamic system. The Lagrangian L, can be defined by

Transcription

1 From MEC '05 Itrgratg Prosthtcs ad Mdc, Procdgs of th 005 MyoElctrc Cotrols/Powrd Prosthtcs Symposum, hld Frdrcto, Nw Bruswc, Caada, ugust 7-9, 005. EECROMECHNIC NYSIS OF COMPEE RM PROSHESIS (EMS) Prmary uthor: Stwart Scotlad Hll Othr uthors:. D B, D.. Gow* Napr Uvrsty, Edburgh, 0 Colto Road, Edburgh, Scotlad *Rhabltato Egrg Srvcs, otha Prmary Car rust bstract: charactrsato of a complt arm prosthss s cssary to dvlop ffctv cotrol. hs s a dscrpto of th us of agrag mthodology to dscrb th systm ad to optms for moto cotrol. h agrag quatos of moto ar drvd from th Nwtoa quatos of moto. agrag aalyss dscrbs th systm trms of Ktc () ad Pottal rgs (V). h Ktc rgy () s foud through a gralsd co-ordat systm, whr s a fucto of th co-ordats ad tm drvats. I th ocosrvatv prosthtc arm, pottal rgy (V) s foud from th gralsd forcs. hs dscrptos compass both lctrcal ad mchacal rgs, whch ar th usd to provd th optmum cotrol sttgs. hs aalyss mthod allows multpl trmal aalyss pots to b combd, allowg a lctrcal twor wth losss, ad a mchacal twor wth losss, combd by a couplg twor. hus th aalyss allows for mchacal ad lctrcal trmals th twor. hs twor approach lds tslf to a complt prosthtc arms systm, whr trmals th twor ca rag from dvdual fgrs to shouldr ots. Itroducto: altratv to th Nwto-Eulr formulato s th agraga formulato. Whras th Nwto-Eulr formulato mght b sad to b a forc balac approach to dyamcs, th agraga formulato s a rgy-basd approach to dyamcs []. Gralsd Co-ordats: h gralsd co-ordats q...q, compltly locats th dyamc systm. t ad V b th total tc rgy ad pottal rgy stord th dyamc systm. h agraga, ca b dfd by ( q, ) V () Sc th Pottal ad Ktc rgs ar fuctos of q ad,(,..., ), so thrfor s th agraga. Mchacal: h agraga quatos of moto of dyamc systms ar gv by d dt Q,..., () Dstrbutd udr a Cratv Commos ttrbuto-nocommrcal-no Drvatv Wors 3.0 Utd Stats cs by UNB ad th Isttut of Bomdcal Egrg, through a partrshp wth Du Uvrsty ad th Op Prosthtcs Proct.

2 From MEC '05 Itrgratg Prosthtcs ad Mdc, Procdgs of th 005 MyoElctrc Cotrols/Powrd Prosthtcs Symposum, hld Frdrcto, Nw Bruswc, Caada, ugust 7-9, 005. Whr Q s th gralzd forc corrspodg to th gralsd co-ordat q. Cosdrg th vrtual wor do by o-cosrvatv forcs actg o th systm ca dtfy th gralsd forc. h tc rgy of a l s gv by r r r r mvc vc + ω I ω (3) Whr v s th vlocty vctor of th ctr of th lag ad ω s th agular vlocty vctor wth rfrc to th bas co-ordat fram. m s th mass of th l ad s th rta tsor at th ctr xprssd th bas co-ordats. I h total tc rgy stord th whol arm lag s th gv by (4) h xprsso for th tc rgy s wrtt trms of th vlocty ad agular vlocty of ach l mmbr, whch ar ot dpdt varabls. h abov quato ca b r-wrtt trms of a dpdt ad complt st of gralsd co-ordats, spcfcally ot dsplacmts q q, q,..., ]. [ h acoba matrx of DOF sral mapulator s gv by q (5) ad ar colum vctors of th acoba matrx rspctvly assocatd, th lar vlocty of th d- wth th lar ad agular vlocts. Usg vctor ffctor ca b wrtt as: v &... + & q + q (6) Smlarly, th agular vlocty of th d-ffctor was xprssd as a lar combato of th colum vctors ω (7) ad ω by rgardg th l as a d- h sam mthod ca b appld to ffctor. vc ( ) ( ) ( ) vc q & (8) ( ) ( ) ( ) ω ( ) (9) Dstrbutd udr a Cratv Commos ttrbuto-nocommrcal-no Drvatv Wors 3.0 Utd Stats cs by UNB ad th Isttut of Bomdcal Egrg, through a partrshp wth Du Uvrsty ad th Op Prosthtcs Proct.

3 From MEC '05 Itrgratg Prosthtcs ad Mdc, Procdgs of th 005 MyoElctrc Cotrols/Powrd Prosthtcs Symposum, hld Frdrcto, Nw Bruswc, Caada, ugust 7-9, 005. h moto of l dpds o oly ots through. hrfor, ( ) ( ) ( ) [ ] (0) ( ) ( ) ( ) [ ] () Substtutg xprssos for vc ad ω to th quato for tc rgy gvs, ( m ( ) ( ) + ( ) I ( ) ) H () H s gv as a X matrx. ( ) ( ) ( ) ( ) H ( m + I ) l (3) h matrx H corporats all th mass proprts of th whol arm lag, as rflctd to th ot axs, ad s rfrrd to as th mapulator rta tsor. h mapulator rta tsor has proprts smlar to thos of dvdual rta tsors. h quadratc form assocatd wth th mapulator rta tsor rprsts tc rgy, ad th mapulator rta tsor s postv dft sc tc rgy s always strctly postv ulss th systm s at rst. Howvr, th mapulator rta tsor volvs acoba matrcs that vary wth arm cofgurato. Hc th mapulator rta tsor s cofgurato-dpdt. t H b th [, ] compots of th mapulator rta tsor H, th th total tc rgy ca b rwrtt a scalar form so that H (4) Whr H s a fucto of q,...q. I addto to th computato of th tc rgy, th pottal rgy V ad gralzd forcs ar rqurd to b foud ordr to drv agrag s quato of moto. t g b th vctor rprstg th acclrato of gravty wth rfrc to th bas co-ordat fram. h th pottal stord th whol arm lag s gv by V r m g c r 0 (5) Whr 0 r s th posto vctor of th ctr of th l, whch s dpdt o th arm cofgurato. hus th pottal fucto s a fucto of q,..., q []. c r Dstrbutd udr a Cratv Commos ttrbuto-nocommrcal-no Drvatv Wors 3.0 Utd Stats cs by UNB ad th Isttut of Bomdcal Egrg, through a partrshp wth Du Uvrsty ad th Op Prosthtcs Proct.

4 From MEC '05 Itrgratg Prosthtcs ad Mdc, Procdgs of th 005 MyoElctrc Cotrols/Powrd Prosthtcs Symposum, hld Frdrcto, Nw Bruswc, Caada, ugust 7-9, 005. By substtutg gravty torqu, rta torqus, ad corols ad ctrfug ffcts to th orgal agrag quatos gvs th mchacal dfto for a complt arm systm. H + h qq & & + G q,,..., (6) Whr: G m g ( ) (7) ad h H H (8) Elctrcal: h frst stp aalysg a complcatd lctromchacal systm by a cosrvato of rgy approach s to rduc th systm cotag lctromchacal couplg trms to a mmum. o do ths, sparat out all purly lctrcal parts ad all purly mchacal parts of th systm cludg losss. hs sparato procdur s carrd out to th xtt that ach lctrcal trmal par s coupld to o rgy stor, thr magtc or lctrcal. y tral trcoctos btw crcuts that ar coupld to dffrt rgy storags ar cludd th xtral lctrcal twor. h mchacal varabls rprstd by th mchacal trmal pars ar thos, whch affct rgy storag th lctrc ad magtc flds. h sparato procdur rsults th gral cosrvatv lctromchacal couplg twor Fgur whch thr ar lctrcal trmals ad m mchacal trmals pars. Each lctrcal trmal par wll b coupld to thr magtc fld rgy storag or lctrc rgy fld storag. h total stord rgy W th couplg twor s Elctrcal Ergy Elctrcal Ntwor v Q Couplg Ntwor Mchacal Ergy Mchacal Ntwor Fgur. Elctromchacal Systm wth Couplg Ntwor Rprstd by W l q ψ W + W (9) q Ψ Whr W s th rgy stord lctrc flds ad W s rgy stord magtc flds. It s assumd that W s a stataous cofgurato of th systm. Cosdr a lctrcal trmal par coupld to th lctrcal fld storag. Wh th q ad q ar spcfd dpdtly, th currt th th trmal s dq dt ad th voltag v at th th trmal s gv by th tral costrats. Nxt, cosdr a Dstrbutd udr a Cratv Commos ttrbuto-nocommrcal-no Drvatv Wors 3.0 Utd Stats cs by UNB ad th Isttut of Bomdcal Egrg, through a partrshp wth Du Uvrsty ad th Op Prosthtcs Proct.

5 From MEC '05 Itrgratg Prosthtcs ad Mdc, Procdgs of th 005 MyoElctrc Cotrols/Powrd Prosthtcs Symposum, hld Frdrcto, Nw Bruswc, Caada, ugust 7-9, 005. lctrcal trmal par that s coupld to magtc fld storag. Wh th ψ ad ar spcfd dpdtly, th voltag th th trmal s v dψ dt ad th currt at th th trmal s gv by th tral costrats. It should b mtod that stad of spcfyg th q ad ψ th voltags v ad th currts could hav b cosdrd as dpdt. h xt problm s to fd th gralzd forc du to th lctromchacal couplg. Sc th m mchacal trmal pars ar charactrsd by dpdt varabls, t s possbl to cosdr ach mchacal trmal par dvdually to fd th forc. t us df th gralzd forc Q as th forc appld to th th mchacal co-ordat by th couplg twor. ca b foud by cosdrg that a arbtrary placmt dq of th th mchacal co-ordat durg th tm dt tas plac. ll othr mchacal co-ordats ar fxd ad th lctrcal varabls may chag accordac to th tral costrats du to th lctrcal twor. hs mas that oly o lctrcal varabl at ach lctrcal trmal ca b chagd arbtrarly. ll losss ar sad to b thr part of th mchacal twor or th lctrcal twor. Stp by Stp mplmtato ad Coclusos: Mchacal Ntwor: Slct a sutabl co-ordat systm to rprst th mchacal cofgurato f th systm. Obta th Ktc rgy () Obta th pottal rgy (V), as a fucto of th co-ordats, f a cosrvatv systm. If th systm s ocosrvatv, fd th gralsd forcs th form of Q. Elctrcal Ntwor: Us th currts to form th gralsd co-ordats. Obta th total lctrc rgy co-ffct, as a fucto of th mchacal ad lctrcal co-ordats. Calculat th powr quatts. Df th xtdd agraga. Q x l l + W V (0) h agraga mthod of modllg allows for a complt aalyss mthod trms of rgy. Udrstadg th rgs ad losss of a systm allows for a mor complt mplmtato for cotrol systm dsg. agrag offrs o mthod for modllg a systm such as a prosthtc arm. By combg th lctrcal ad mchacal sgmts for aalyss, allows for a mor -dpth udrstadg of a prosthtc arm, tha has prvously b th cas. Rfrcs:. Scavcco,, Sclao, B, Modllg ad Cotrol of Robot Mapulators, Sprgr, ISBN ta, K,R,E, Dyamc alyss, Dsg ad Cotrol of a Idustral Paralll Robot Uvrsty of Glasgow, hss 000. Dstrbutd udr a Cratv Commos ttrbuto-nocommrcal-no Drvatv Wors 3.0 Utd Stats cs by UNB ad th Isttut of Bomdcal Egrg, through a partrshp wth Du Uvrsty ad th Op Prosthtcs Proct.