Chapter 2: Rigid Body Motions and Homogeneous Transforms
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1 Chater : igi Bo Motion an Homogeneou Tranform (original lie b Stee from Harar) ereenting oition Definition: oorinate frame Aetn n of orthonormal bai etor anning n For eamle When rereenting a oint we nee to eif a oorinate frame With reet to o : With reet to o : i j 5 6 an are inariant geometri entitie But the rereentation i eenant uon hoie of oorinate frame
2 D rotation ereenting one oorinate frame in term of another otation ereenting one oorinate frame in term of another Where the unit etor are efine a: [ ] o in in o o in Thi i a rotation matri in o otation matrie a rojetion Projeting the ae of from o onto the ae of frame o Alternate aroah Projeting the ae of from o onto the ae of frame o π π o o o o
3 3 Proertie of rotation matrie Inere rotation: Or another interretation ue o/een roertie: Proertie of rotation matrie Inere of a rotation matri: ( ) ( ) T in o in o o o o o π π The eterminant of a rotation matri i alwa ± if we onl ue right-hane onention ( ) ( ) o in et o in
4 4 Proertie of rotation matrie Summar: Column (row) of are mutuall orthogonal Column (row) of are mutuall orthogonal Eah olumn (row) of i a unit etor The et of all n n matrie that hae thee roertie are alle the Seial Orthogonal grou of orer n et( ) T ( ) SO n ( ) SO n 3D rotation General 3D rotation: ( ) 3 SO Seial ae Bai rotation matrie o in o in in o o in in o o in in o
5 Proertie of rotation matrie (ont ) SO(3) i a grou uner multiliation. Cloure: if SO ( 3 ) SO ( 3 ). Ientit: I SO( 3) 3. Inere: T 4. Aoiatiit: ( ) 3 ( 3 ) Allow u to ombine rotation: a abb In general member of SO(3) o not ommute otational tranformation Now aume i a fie oint on the rigi objet with fie oorinate frame o The oint an be rereente in the frame o ( ) again b the rojetion onto the bae frame u w ( u w ) ( u w ) ( ) u w u w u w u w 5
6 otating an Objet otating a etor Another interretation of a rotation matri: otating a etor about an ai in a fie frame E: rotate about b π/ 6
7 otation matri ummar Three interretation for the role of rotation matri: ereenting the oorinate of a oint in two ifferent frame. ereenting the oorinate of a oint in two ifferent frame. Orientation of a tranforme oorinate frame with reet to a fie frame 3. otating etor in the ame oorinate frame 7
8 Similarit tranform All oorinate frame are efine b a et of bai etor Thee an n E: the unit etor i j In linear algebra a n n matri A i a maing from n to n A where i the image of uner the tranformation A Thin of a a linear ombination of unit etor (bai etor) for eamle the unit etor: e [... T ]... e [... ] T n If we want to rereent etor with reet to a ifferent bai e.g.: f f n the tranformation A an be rereente b: A T AT Where the olumn of T are the etor f f n A i alle the imilarit tranformation. Similarit tranform otation matrie are alo a hange of bai If A i a linear tranformation in o an B i a linear tranformation in o then the are relate a follow: E: the frame o an o are relate a follow: B ( ) A 8
9 w/ reet to the urrent frame E: three frame o o o Comoition of rotation Thi efine the omoition law for ueie rotation about the urrent referene frame: ot-multiliation Comoition of rotation E: rereent rotation about the urrent -ai b followe b about the urrent -ai o in in o in o in o o o in o in o in o in in in o What about the reere orer? 9
10 Comoition of rotation w/ reet to a fie referene frame (o ) Let the rotation between two frame o an o be efine b Let the rotation between two frame o an o be efine b Let be a eire rotation w/ reet to the fie frame o Uing the efinition of a imilarit tranform we hae: [( ) ] Thi efine the omoition law for ueie rotation about a fie referene frame: re-multiliation Comoition of rotation E: we want a rotation matri that i a omoition of about ( ) an then about ( ) the eon rotation nee to be rojete ba to the initial fie frame ( ) Now the ombination of the two rotation i: [ ]
11 Comoition of rotation Summar: Coneutie rotation w/ reet to the urrent referene frame: Pot-multiling b ueie rotation matrie w/ reet to a fie referene frame (o ) Pre-multiling b ueie rotation matrie We an alo hae hbri omoition of rotation with reet to the urrent an a fie frame uing thee ame rule Parameteriing rotation There are three arameter that nee to be eifie to reate arbitrar rigi bo rotation We will eribe three uh arameteriation:. Euler angle. oll Pith Yaw angle 3. Ai/Angle
12 Parameteriing rotation Euler angle otation b about the -ai followe b about the urrent -ai then b t th t i about the urrent -ai ZYZ Parameteriing rotation oll Pith Yaw angle Three oneutie rotation about the fie rinial ae: Three oneutie rotation about the fie rinial ae: Yaw ( ) ith ( ) roll ( ) XYZ
13 3 Parameteriing rotation Ai/Angle rereentation An rotation matri in SO(3) an be rereente a a ingle rotation about a An rotation matri in SO(3) an be rereente a a ingle rotation about a uitable ai through a et angle For eamle aume that we hae a unit etor: Gien we want to erie : Intermeiate te: rojet the -ai onto : Where the rotation i gien b: Parameteriing rotation Ai/Angle rereentation Thi i gien b: Thi i gien b: Inere roblem: Gien arbitrar fin an
14 4 igi motion igi motion i a ombination of rotation an tranlation g Define b a rotation matri () an a ilaement etor () the grou of all rigi motion () i nown a the Seial Euliean grou SE(3) Conier three frame o o an o an orreoning rotation matrie an ( ) 3 3 SO ( ) ( ) 3 SO 3 SE n Let be the etor from the origin o to o from o to o For a oint attahe to o we an rereent thi etor in frame o an o : ( ) Homogeneou tranform We an rereent rigi motion (rotation an tranlation) a matri g ( ) multiliation Define: Now the oint an be rereente in frame o : Where the P an P are: H H P H P H Where the P an P are: P P
15 Eamle Fin the homogeneou tranformation matri (T) for the following oeration: otation about OX ai Tranlation of a along OX ai Tranlation of along OZ ai otation of about OZ ai Anwer : T T T T T I a 4 4 Homogeneou Tranformation Comoite Homogeneou Tranformation Matri A A i A i? Tranformation matri for ajaent oorinate frame A A A Chain rout of ueie oorinate tranformation matrie 5
16 b Eamle 8 For the figure hown below fin the 44 homogeneou tranformation i matrie A i an A i for i n a n a e F A 3 3 n a a e a 4 4 Can ou fin the anwer b oberation bae on the geometri interretation of homogeneou tranformation matri? Homogeneou tranform The matri multiliation H i nown a a homogeneou tranform an we note that H SE( 3) Inere tranform: T T H 6
17 7 Homogeneou tranform Bai tranform: Three ure tranlation three ure rotation b a b a Tran Tran ot ot Tran γ γ γ γ γ ot
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