Lecture 9-3/8/10-14 Spatial Description and Transformation

Transcription

1 Letue 9-8- tl Deton nd nfomton Homewo No. Due 9. Fme ngement onl. Do not lulte Otonl et edt hot oof tht = - Homewo No. egned due 9 tud eton.-.. olve oblem: ee lde 6 7. e Mtlb on. f oble. On. otton mt lo le to velot veto. OG Ẑ Ẑ OG : otton & nlton.. Eq..8 Deton of the oton nd oentton of veto oton veto ˆ nt veto - Oentton n nl e [ ˆ ] = ˆ ˆj ˆ j ˆ whee [ ] the detonl one of on X Y nd e. [ ˆ ] = the detonl unt veto of. ˆ ˆ

2 Oentton of the unt veto n fme {} = [ ˆ ] otton mt of {} w..t. {} the ojeton of eh of {} onto eh of {}. dot odut of two unt veto ml the deton one of the ngle between the veto. = [ Ẑ ]= ˆ ˆ ˆ ˆ ˆ ˆ. he ow e the otted unt veto n {} eeed eltve to the efeene fme {}. he olumn e the unt veto of the efeene fme {} eeed eltve to the otted {}. unt veto the mgntude of eh ow nd olumn = nd theefoe - = = I.6 = - =.7 Fme Deton nd Mng {} = { OG }.8 Fme {} m be ontuted b hftng t ogn fom tht ondng wth efeene fme {} b veto OG nd b ottng t e b. Fme hft onl: Veto n {} eeed wth eet to {} though veto ddton: = + OG.9 he otton mt ont of thee unt olumn veto o thee unt ow veto: = [ Ẑ ] = [ Ẑ ]. Fme otton onl: If OG = tht the ogn of fme {} nd {} onde then =. Wth both fme hft nd otton veto n {} gven b Homogenou nfom Mt = + OG.7 omote tnfomton mt eeentng both fme hft nd otton: =

3 OG.9 oton nlton If veto fom the ogn of {} tnlted b veto Q lo fom the ogn defne = + Q. = D Q q.5 he tnlton mt oeto D Q n be defned D Q q =[ q q q.6 wo te nfomton If fme {} defned eltve to {} nd {} defned eltve to {} then tnfomton mt n be deved fom Eq =. OG OG. ` he nvee of n.9: OG.6 nd eonng fo the oton veto = OG OG OG Deton = - otton =

4 Denvt-Htenbeg mete fom hte α - - d θ α - =Ln twt o ±9º -=Ln Length d =X offet θ = otton bout X Ln nfomton lng the Denvt-Htenbeg mete n equentl tnfomton otte then tnlten{} { -} ottethen tnlten d D D = d d d.6 nfom Equton ueve tnfomton of oton veto m be efomed n fowd nd evee deton. o fnd : ng Q Q Q {} {} {} {}

5 Othe W of otton Deton Due to the oete of Othonoml otton e ew mmet Mt ottonl mte e oe othonoml mte the detemnnt e. uh ew mt n whh + = et fo eh ottonl mt. = I - I +.56 h thee mete. hu m be eeented b =[ ]. Fo lton n jont ngul velot nl t n be hown tht fo gven odut of nd t devtve fom ew mmet mt 5.7 nd h the fom: =.57 Gven n oton veto h followng oet: 5.7 whee veto o odut nd eeent the ngul velot of ottng. he lton n be etended to the e of otton bout genel K ˆ. X-Y- Fed ngle Fme {} nd {} e ondent. otte {} bout b γ the bout Y b β nd bout Ẑ b α. X Y =.6 =.6 = he oluton n be found ung tgonomet dentte um of que of ne nd one ngle nd the tngent ngle on the tem n.6:

6 tn tn tn.66 -Y-X Eule ngle Fme {} nd {} e ondent. otte {} bout Ẑ b α then bout b β nd then b γ. X Y =.7 =.7 he eult the me.6. he eult fom otton bout thee fed e the me the eult fom otton bout the movng fme ten n evee ode. -Y- Eule ngle lble when thee e nteet n the e of the wt jont whee w th nd oll eome togethe Jont nd 5 e othogonl nd Jont 5 nd 6 e ol. Fme {} nd {} e ondent. otte {} bout Ẑ b α then bout b β nd then b Ẑ b γ. = 7 tn tn tn.7 Othe ngle-et onventon totl oble. Found n end D otton bout Equvlent ngle- Eule ngle Fme {} nd {} e ondent. otte {} bout veto Kˆ b θ followng the ght hnd ule.

7 X θ = Y θ = θ = K θ =.8 K =.8 = o n ˆ K.8 ` Eq..8. olve: ˆ ot = whee = Y whee [K K K ] olumn veto eeentng the ojet of K onto the nl e of {}. Ẑ Ẑ Equvlent ngle otton Eq..8 K X

8 te Me fme {} to be otted onde wth fme {}. lt {} w fom {} o tht wll onde wth K the equvlent. otte {} bout b θ. Me the otton e tht t w done wth efeene to fed of {} ot ˆ ee oblem.9 fo mlt mt. 5 Me ubttuton ung b b b b b b E Eule mete fo Equvlent ˆ nd θ. oted wth equvlent nd ngle of otton K Eule mete nt Qutenon = [ ] whee n n n o.89 hen.9.9 oblem: Vef tht.9 nd.8 eld the me otton. oluton: ele the Eule mete n.9 wth the tem n.89 to ve t.8. e the elton o o n Fo gven otton mt the deton veto Kˆ nd θ e found b ettng the Eule mete:.9