T f. Geometry. R f. R i. Homogeneous transformation. y x. P f. f 000. Homogeneous transformation matrix. R (A): Orientation P : Position

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1 Homogeneous transformaton Geometr T f R f R T f Homogeneous transformaton matr Unverst of Genova T f Phlppe Martnet = R f 000 P f 1 R (A): Orentaton P : Poston 123 Modelng and Control of Manpulator robots

2 124 Modelng and Control of Manpulator robots Phlppe Martnet Unverst of Genova Homogeneous transformaton Geometr Consder a 3D pont n space R f R 1 f T 1 = R f 1 Then f P f R R f R + =

3 Homogeneous transformaton Geometr T f Homogeneous transformaton matr Dfferent representatons R : Orentaton s, n, a Cosnus drectors RPY angles (Roll (), Ptch(), Yaw()) Brant angles (,,) Euler angles (,,) u.θ, u.sn(θ), u.sn(θ/2 θ/2), P : Poston Cartesan coordnates Clndrcal coordnates Sphercal coordnates Quaternon λ 1, λ 2, λ 3, λ 4 Unverst of Genova Phlppe Martnet 125 Modelng and Control of Manpulator robots

4 Homogeneous transformaton Geometr T f Homogeneous transformaton matr Dfferent representatons (.e) R : Orentaton s, n, a Drector Cosnus P : Poston Cartesan coordnates R s = s s n n n a a a P = P P P No rotaton No translaton P=(0,0,0) T Unverst of Genova Phlppe Martnet 126 Modelng and Control of Manpulator robots

5 Homogeneous transformaton Geometr T f Homogeneous transformaton matr Man propertes of the rotaton matr R : Orentaton s, n, a Drector Cosnus s = n = a = s n = 0 s 1 n = a s R= s s n n n a a a s a a n 1 T R = R = = 0 0 n a a s = = s n Unverst of Genova Phlppe Martnet 127 Modelng and Control of Manpulator robots

6 Geometr Homogeneous transformaton: Rotaton matr R f Rotaton matr Rot(,θ ) Matr to change the frame for one vector Unverst of Genova Phlppe Martnet 128 Modelng and Control of Manpulator robots

7 Geometr Homogeneous transformaton: Rotaton matr R f Rotaton matr Rot(,θ ) Unverst of Genova Phlppe Martnet 129 Modelng and Control of Manpulator robots

8 Geometr Homogeneous transformaton: Rotaton matr R f Rotaton matr Rot(,θ ) Unverst of Genova Phlppe Martnet 130 Modelng and Control of Manpulator robots

9 Homogeneous transformaton propertes Geometr T f Homogeneous transformaton matr Prop. 1) Prop. 2) Unverst of Genova Phlppe Martnet 131 Modelng and Control of Manpulator robots

10 Homogeneous transformaton propertes Geometr T f Homogeneous transformaton matr Prop. 3) Prop. 4) Unverst of Genova Phlppe Martnet 132 Modelng and Control of Manpulator robots

11 Homogeneous transformaton propertes Geometr T f Homogeneous transformaton matr Prop. 5) Prop. 6) Unverst of Genova Phlppe Martnet 133 Modelng and Control of Manpulator robots

12 Homogeneous transformaton propertes Geometr T f Homogeneous transformaton matr T s defned n R Prop. 7) T s defned n R j Unverst of Genova Phlppe Martnet 134 Modelng and Control of Manpulator robots

13 Homogeneous transformaton propertes Geometr T f Homogeneous transformaton matr Unverst of Genova Phlppe Martnet 135 Modelng and Control of Manpulator robots

14 Homogeneous transformaton propertes Geometr T f Homogeneous transformaton matr Prop. 8) Unverst of Genova Phlppe Martnet 136 Modelng and Control of Manpulator robots

15 Rgd bod pose parameteraton: poston Cartesan coordnates Unverst of Genova Phlppe Martnet 137 Modelng and Control of Manpulator robots

16 Rgd bod pose parameteraton: poston Clndrcal coordnates Sngulart : P=P=0 Unverst of Genova Phlppe Martnet 138 Modelng and Control of Manpulator robots

17 Rgd bod pose parameteraton: poston Sphercal coordnates f or f f or f Unverst of Genova Phlppe Martnet 139 Modelng and Control of Manpulator robots

18 Rgd bod pose parameteraton: orentaton Euler angles or Remarks: Euler angles can be defned also b (,,) Unverst of Genova Phlppe Martnet 140 Modelng and Control of Manpulator robots

19 Rgd bod pose parameteraton: orentaton RPY angles (,,) A RPY or Unverst of Genova Phlppe Martnet 141 Modelng and Control of Manpulator robots

20 Rgd bod pose parameteraton: orentaton Brant angles (,,) or Unverst of Genova Phlppe Martnet 142 Modelng and Control of Manpulator robots

21 Rgd bod pose parameteraton: orentaton Orentaton (u,θ) Rodrgues formula Unverst of Genova Phlppe Martnet 143 Modelng and Control of Manpulator robots

22 Rgd bod pose parameteraton: orentaton Orentaton (u.θ) Rodrgues formula, Unverst of Genova Phlppe Martnet 144 Modelng and Control of Manpulator robots

23 Orentaton (u.θ) T 2 (, θ ) = I u sθ + uˆ 3 ˆ..( 1 cθ ) T (, θ ) A( u, θ ) = 2.ˆ. u sθ A u A u 2 ( A( u, θ )) = Trace I + u sθ + uˆ 3 ˆ..( 1 cθ ) 2 ( A( u, θ )) = 3+ ( 1 cθ ). Trace( uˆ ) ( A( u, θ )) = 3 + ( 1 cθ )(. 2) ( A( u, θ )) = 1+ 2 cθ Trace Trace Trace Trace. Rgd bod pose parameteraton: orentaton ( ) [ u. sθ ] = [ u. sθ ] = (, θ ) A( u, θ ) A u 2 ( A( u, θ )) = 1+ 2 cθ Tr( A( u, θ )) 1 Tr. cosθ = 2 T s s s n n n a a a s n a n a Unverst of Genova s s n = s a s snθ = ± n a n n s 0 a Phlppe Martnet a a s n 0 s - n a - s n - a ( s n ) 2 + ( a s ) 2 + ( n a ) 2 = 2 u sθ = 2 u sθ = 2 u sθ n 1 a 2.snθ s a s n 145 Modelng and Control of Manpulator robots u =

24 Rgd bod pose parameteraton: orentaton Quaternon (λ 1,λ 2,λ 3,λ 4 ) Unverst of Genova Phlppe Martnet 146 Modelng and Control of Manpulator robots

25 Case of seral manpulator robot R 0 R 1 C 0 C 1 C 2 Mult-Rgd bodes R k R k+1 C k C n Consder a robot wth n+1 rgd bodes C k We assocate n+1 frames C 0 s the base of the robot (fed) R e Unverst of Genova Phlppe Martnet 147 Modelng and Control of Manpulator robots

26 Case of seral manpulator robot Mult-Rgd bodes The problem to solve s to obtan the poston and orentaton of the end effector frame R e n the fed frame R 0 k k k Rk + 1 Pk + 1 Tk + 1 = Elementar frame transform 0 T e = 0 T 1 1 T 2 2 T 3 L n 1 T n n T e Unverst of Genova Phlppe Martnet 148 Modelng and Control of Manpulator robots

27 Rgd bod knematcs Crcular moton v: tangental velct ω: angular veloct Unverst of Genova Phlppe Martnet 149 Modelng and Control of Manpulator robots

28 Rgd bod knematcs Rotatng frame R f : fed frame (orgn O fed) R m : moble frame just n rotaton w.r.t R f ω: angular veloct Unverst of Genova Phlppe Martnet 150 Modelng and Control of Manpulator robots

29 R f D Rgd bod knematcs R m d O m P O f R f : fed frame R m : Moble frame P : one pont n R m V ( P) d dt R f = V ( O ) + V ( P) + ω m d ( d) = D& + ( d) + ω dt R R f R m f R m d O m P ω : angular veloct of R m relatve to R f Unverst of Genova Phlppe Martnet 151 Modelng and Control of Manpulator robots

30 Knematc Rgd bod knematcs Usng ths relaton we can establshed The knematc evoluton of a mult-rgdbod robot See net slde Remarks : V ( P) R f = V ( O ) + V ( P) + ω m R f R m O m P If D=0 then V ( P) R f = V ( P) + ω R m O m P If D=0 and V ( P ) 0 then Unverst of Genova R = m = Om P ω 0 ω = ~ ~ ω ω d ω = ω 0 ω = [ ω] = [ ω] = [ ˆ ω AS ] ω ω 0 Phlppe Martnet V ( P) R f = ω O m 152 Modelng and Control of Manpulator robots P

31 Mult-Rgd bodes knematcs Angular veloct of R w.r.t R 0 epressed n R Veloct of O +1 w.r.t R epressed n R Veloct of O w.r.t R 0 epressed n R Veloct of O +1 w.r.t R 0 epressed n R Unverst of Genova Phlppe Martnet 153 Modelng and Control of Manpulator robots

32 Mult-Rgd bodes knematcs C -1 C C +1 = Angular veloct of R w.r.t R -1 epressed n R Angular veloct of R w.r.t R 0 epressed n R Angular veloct of R -1 w.r.t R 0 epressed n R -1 Unverst of Genova Phlppe Martnet 154 Modelng and Control of Manpulator robots

33 Mult-Rgd bodes knematcs C -1 C C +1 = Angular veloct of R w.r.t R -1 epressed n R Angular veloct of R w.r.t R 0 epressed n R Angular veloct of R -1 w.r.t R 0 epressed n R -1 Unverst of Genova Phlppe Martnet 155 Modelng and Control of Manpulator robots

34 Mult-Rgd bodes knematcs Consderng two frames R a and R b rgdl lnked (case for R n and R E ) twst Unverst of Genova Phlppe Martnet 156 Modelng and Control of Manpulator robots

35 Mult-Rgd bodes knematcs Consderng two frames R and R j and a twst V =(v,ω ) T epressed n O We wsh to compute the correspondng twst V j =(v j,ω j ) T epressed n O j Projecton wth Unverst of Genova Phlppe Martnet 157 Modelng and Control of Manpulator robots

36 Dfferental translaton and rotaton of frames Consder a dfferental translaton vector dp epressng the translaton of the orgn of frame R, and of a dfferental rotaton vector δ, equal to u.dθ, representng the rotaton of an angle dθ about an as, wth unt vector u, passng through the orgn O. The dfferental transformaton matr s defned as [Paul 81] Unverst of Genova Phlppe Martnet 158 Modelng and Control of Manpulator robots

37 Ra a T b = a R b Unverst of Genova a t b R b Phlppe Martnet =a Ṙ b. a R T b R= a R b R.R T =I3 Ω b/a a T Ṙ.R T +R.Ṙ T =0 Ṙ.R T = R.ṘT = Ṙ.R T Ṙ.R T =S(t) Ṙ=S(t).R p a (t)=r(t).p b (t) ṗ a (t)=ṙ(t).p b (t) ṗ a (t)=s(t).r.p b (t) ṗ a (t)=ω p a (t) ṗ a (t)=[ω] ṗ a (t)=ω (R.p b (t)).r.p b (t) S(t)=[Ω] Ωb/a a Consderp b (t)constant Angular veloct propertes 159 Modelng and Control of Manpulator robots

38 b T a = b R a Angular veloct propertes b t a b R a representstheorentaton b t a representstheposton Ra R b Ω b/a a a t b a T b = a R b Ω a/b b Ω b/a a Ω b/a a = Ω a/b a Ω b/a a = a R b.ω b/a b Ωb/a a Ω a/b b Ω a/b b = Ω b/a b Ω a/b b = b R a.ω a/b a Ωa/b b =a Ṙ b. a Rb T =b Ṙ a. b Ra T Unverst of Genova Phlppe Martnet 160 Modelng and Control of Manpulator robots

39 Representaton of forces (wrench) A collecton of forces and moments actng on a bod can be reduced to a wrench F at pont O, whch s composed of a force f at O and a moment m about O: Note that the vector feld of the moments consttutes a screw where the vector of the screw s f. Thus, the wrench forms a screw. Consder a gven wrench F, epressed n frame R. For calculatng the equvalent wrench j F j, we use the transformaton matr between screws such that: Unverst of Genova Phlppe Martnet 161 Modelng and Control of Manpulator robots