OTATION IN 3D WOLD IGID BODY MOTION
igid Bod Motion Simultion igid bod motion Eqution of motion ff mmvv NN ddiiωω/dddd Angulr velocit Integrtion of rottion nd it s eression is necessr.
Simultion nd Eression of ottion igid bod movements cn be considered s combintion of trnsltion nd rottion. Ho to reresent the orienttion/rottion of rigid bod?
Toolog of rottion D osition One oint on line D rottion One oint on circle 3D osition One oint on 3D sce 3D rottion????
Eression of ottion ottion mtri Euler ngles Ais nd Angle Quternion There re mn eressions, becuse ech eression hs dvntges nd disdvntges. Let s see the rottion, differentil nd integrl.
Euler Angles For emle, rotte coordinte is in order of is 3 An orienttion cn be reresented b 3 ngles Order of rottions cn be vried. roll itch is one oulr order. Euler ngle cn be understood s joint ngles of mniultor nd orienttion of the end effector.
Euler Angles nd ottion Mtri The folloing three bsic rottion mtrices rotte vectors round the,, or is -- Euler ngles -- Euler ngles,,,, 3 3,, 3 3
ottion order is imortnt in 3D E.g., rotte Euler ngles ith the sme,, -- Euler ngles rottion: -- Euler ngles rottion: Comrison If the order is different, the result is different.
Chrcteristics of Mtri Multiliction ABC ABC in generl, it is correct, but AB BA is incorrect in most cses. The results deend on the rottion orders X is 3 Y is 3 Z is 3 Y is 3 X is 3 Z is 3 esult different orienttions
Problems on Mtri/Euler ngles bsed Eression Euler ngles hve some gulrities., 8, is sme orienttion for n b,, b is sme orienttion for n b. Comrison of orienttions re not es. E.g., 8, 3, 8, 3 6, 8, 6 33 mtri hs 9 elements nd inconvenient ottion is eressed not b ll 33 mtrices, but b 33 mtrices hose bsis re orthogonl ech other, sies re nd determinnt is. Comuttionl error m cuse violtion of these constrints. Differentil oertion is strnge form.
Is integrtion ossible? Integrl of elements is meningless, becuse ddition is not lloed. t t t t t t dt dt dt dt dt dt t ω ω ω ω ω ω, t if ω integrl t t t : t comosite:
Comrison of Position nd ottion osition: velocit: v direction: 3 d dt 3D Vector SO3 ngulr velocit : ω d dt ω ω ω ω ω ω 3 3 3 rottion 3 ω mtri Secil orthogonl orthonormlied mtri ith determinnt of t t ω t t d t t t t dt
Ais nd Angle bsed Eression As n intuitive rottion eression; otte rd round this is Cn eressed b 3D vector 3 : rottion ngle : direction of rottion is
Ais nd Angle bsed Eression The rnge of is vector is ~π π π - Inside of bll ith rdius π Normlition of vector if eceeds π ' ', ' hen ' ', ' hen ', hich stisfis ' Find l l l l l l Z n n l l < < π π π π π
Ais nd Angle bsed Eression Angulr velocit is differentil of the is nd the ngle. d ω dt The integrl of ngulr velocit is meningless. ngulr velocit : ω t ngulr velocit integrl: I ω if ω t ω t <, t < t ω t dt do not eress orienttion of t ωdt Addition hs no mening!, fter rottion of ω t
Ais nd Angle bsed Eression Addition of is nd ngle is imossible drn for 9 nd 9
Ais nd Angle bsed Eression Norml differentil is ngulr velocit. Addition hs no menings, comosition is troublesome For comosition, Find corresonding rottion mtri nd multil them. Integrl hs no mening
Quternion Quternion Euler rmeter Euler ngles Eress ngle ith four rel numbers Quternion for rottion round is u u ngle is q q,, u. Also cn be ritten in q v u, v
Quternion s oertion Tke ijk s n ension of imginr unit i Multiliction ijk k j i k j i q kk kj ki k jk jj ji j ik ij ii i k j i k j i k j i q q This is the rel form of quternion
Quternion s oertion Multiliction k j i kk kj ki k jk jj ji j ik ij ii i k j i q q - -k k Multiliction of quternion results in comosition of rottion q q q
Quternion s oertion ottion of osition vector k j i k j i ˆ The quternion hich corresonds * ˆ ' ˆ q q q k j i q k j i q * q q quternion the conjugtion of is * ' is
Quternion nd ottion Ais For quternion,,, v u Comre to Ais ngle reresenttion,, ' ' ' u Their forms re similr. But in quternion; Comosition of to rottions rottion of oint ottion is unit vector re done b the four rules of rithmetic ithout clculting trigonometric functions.
Quternion nd ottion For q v Quternion ith rottion round is, v u * q qq v u The rottion quternion sie is. u u is Shericl shell ith rdius of in 4,,, dimensionl sce
Quternion nd ottion q v,, v u u u nd < π cn eress n rottion. <, < cn eress n rottion. v q nd q eress the sme rottion. ight hemishere cn eress n rottion Comosition of quternion m result in left hemishere. Such quternions hve different quternions in left hemishere. Sme rottion eressed b different quternions cn led bugs. -q q
Summr of ottion Eression Ais nd Angle Differentil is ngulr velocit vector. Quternion Comuttion is fst onl bsic rithmetic oertion is needed for comosition nd rottion of oints. The rottion is is embedded. Euler ngles Es to understnd. Singulrit: one orienttion is reresented b infinite Euler ngles. ottion Mtri Clcultion is es. Corresond to bsis vectors.
Trnsformtion ithout She Chnge ottion ottion mtri orthonormlied mtri ith determinnt of three bsis vectors re orthogonl ech other nd sie is The determinnt of orthonormlied mtri is or - Set of 3D rottion mtri is nmed secil orthogonl grou SO3 SO is set of rottion mtri in D Confirmtion rottion mtri rottion round n is Confirm. n rottion round n is cn be eressed b rottion mtri.. An rottion eressed b rottion mtri cn be reresented b rottion round n is. ottion round n is ottion mtri A mtri hich eresses rottion round is ngle Proof is in the net ge You cn confirm tht this mtri is orthonormlied nd determinnt
Trnsformtion ithout She Chnge ottion Point is the result of rotting, in the ngle, round n is. O v Define v to stisf,v nd re orthogonl ech other. ' v v v Thus, e cn mke 33 rottion mtri b rotting three bsis vectors.
Trnsformtion ithout She Chnge ottion ottion mtri ottion round n is A quternion is ner to rottion mtri thn is nd ngle. So, e strt from quternion.
Quternion Mtri ottion mtri for rottion,in ngle, round is Quternion for rottion,in ngle, round is E:, element of the mtri α α α α α α The rottion mtri cn be eressed s は
Mtri Quternion m m m 3 m m m 3 m m m 3 3 33 m m m m m 33 3 4 m 33 3 4 m3 m3 m 3 m3 4, 4, cn be found in the sme 4 There should be ±. Hoever, in quternion, -q eress the sme rottion to q. So e choose onl. To quternions re corresond to one rottion mtri m m 3 m m 3 m Finll, checking m3 m3 m3 m3 m m m m m33 4 m m m33 4 comes to hen the mtri belongs SO3 33
Mtri Quternion With constrints of rottion mtri orthonormlied nd determinnt, e cn sho the sie of the quternion is. Quternion hve is nd ngle. So, trnsformtion of n rottion mtri cn be eressed b rottion round n is.