Statics. Consider the free body diagram of link i, which is connected to link i-1 and link i+1 by joint i and joint i-1, respectively. = r r r.

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1 Statcs Th cotact btw a mapulato ad ts vomt sults tactv ocs ad momts at th mapulato/vomt tac. Statcs ams at aalyzg th latoshp btw th actuato dv tous ad th sultat oc ad momt appld at th mapulato dpot wh th mapulato s at st. Cosd th body dagam o lk, whch s coctd to lk - ad lk by jot ad jot -, spctvly. m g N N a oc actg at O -, appld to lk by lk - a oc actg at O, appld to lk by lk Gavty oc actg at th ctod C o lk Momt appld to lk by lk - Momt appld to lk by lk N N

2 Balac o la ocs g m,,, 0, Balac o momts wth spct to th ctod C wh ( ) ( ) ( ) N N C C,,, 0, 3x posto vcto om O - to O w..t th bas coodat am C 3x posto vcto om O to th ctod C w..t th bas coodat am

3 0 0 ad N pst th couplg oc ad momt, spctvly, appld to lk by th bas. ad N pst th couplg oc ad momt, spctvly, appld to th vomt by th d-cto (lk ). Fo covc, th vomt s gadd as a addtoal lk, umbd. Two uatos, la ocs ad momts, o all th lk mmbs. Th total umb o vcto uatos s th, whas th umb o couplg ocs ad momts volvd s (). Thus () >. Two mo o th couplg ocs ad momts d to b spcd; othws th uatos a ot solvabl. Th al couplg oc ad momt a th oc ad momt that th mapulato am appls to th vomt. Ths balac th acto oc ad momt xtd to th last lk by th vomt as - ad - N

4 Euvalt jot tous Each jot s dv by a dvdual actuato that xts a dv tou o oc btw th adjact lks. t τ b th dv tou o oc xtd by th -th actuato dvg jot. Fo a psmatc jot, th dvg oc τ s a la oc xtd alog th jot axs -. Assumg that th jot mchasm s ctolss, th jot oc τ ad th la couplg oc btw lks - ad ca b latd by T τ b ( psts a poduct.) wh b s a ut vcto potg th dcto o th jot axs. Ths mpls that th actuato b must ba oly th compot o whch s th dcto o th jot axs, ad that th compots all th oth dctos a suppotd by th jot stuctu. Ths compots o th couplg oc a tal ocs, whch do ot poduc wok.

5 Fo a volut jot, τ psts a dv tou ath tha a la oc. Th dv tou s balacd wth th couplg tou compot o N whch s th dcto o ts jot axs: τ b T N Oth compots o th couplg tou N a suppotd by th jot stuctu, whch a woklss costat momts. All th jot ocs ad tous a combd to d th -dmsoal jot tou ad oc vcto τ. τ τ τ M τ

6 Thom: Assum that th jot mchasms a ctolss, th th jot tous τ that a ud to ba a abtay dpot oc F a gv by τ T F Wh s th 6x mapulato acoba latg tsmal jot dsplacmt d to tsmal d-cto dsplacmt dp: dp d Th jot tou vcto τ psts th t tous that balac th dpot oc ad momt. τ s calld th uvalt jot tous cospodg to th dpot oc F.

7 Poo o τ T F basd o th pcpl o vtual wok Dto o vtual dsplacmt : abtay dsplacmt o a mchacal systm that coom to th gomtc costats o th systm δ δ x, τ : vtual dsplacmts at dvdual jots,,, δφ : postoal ad otatoal dsplacmts, spctvly, at th d-cto : jot tous at dvdual jots,,,, N : dpot oc ad momt actd o th d-cto by th vomt Th, th vtual wok do by th ocs ad momts s gv by o T T δwok τ δ τ δ τ δ δx N δwok T τ δ F T δp δφ Th am s ulbum ad oly th vtual wok o abtay dsplacmts. δwok vashs

8 acoba vstd 6 R d dx dp φ [ ] A A A T R R 6 v p dt dp ω dx dφ 3 R : tsmal d-cto taslato ( ) 3 R : tsmal d-cto otato ( ) Cosdg th tsmal tm cmt dt, Wh Tho, : la vlocty o th d-cto : agula vlocty o th d-cto A A A v ω

9 I jot s psmatc, t poducs a la vlocty at th d-cto th sam dcto as th jot axs but dos ot gat a agula vlocty. b d A 0 I jot s volut, t otats th compost o dstal lks om lks to at th agula vlocty ω ι gv by ω b θ Ths agula vlocty poducs a la vlocty at th d-cto as wll as a agula vlocty. ω ( b ) θ ω b θ A To summaz: o a psmatc jot b A 0 o a volut jot b A b