handout4_Jacobian - Movie Segment Locomotion Gates with...

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1 Movie Segment Locomotion Gates with Polypod, Mark Yim, Stanford University, ICRA 1994 video proceedings Instantaneous Kinematics • Linear & Angular Motion • Velocity Propagation J a c o b i a n •D ifferent ia l ±Mot ion •Exp l ic it±Form • Static Forces Differential Motion {0} {n} {n} Forward Kinematics Instantaneous Kinematics x x δ θδ θ ++ x Relationship: δθ x ± ± x Linear Velocity Angular Velocity ii i qd ε θε =+ Joint Coordinates i i revolute : d prismatic coordinate i Joint coordinate-i: 0 revolute 1 prismatic i = with 1 = and Joint Coordinate Vector: 12 ( .... ) T n qq =
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2 Jacobians: Direct Differentiation () ; x fq = x x x f q fq mm 1 2 1 2 ## F H G G G G I K J J J J = F H G G G G I K J J J J δ x f q q f q q x f q q f q q n n m n n 1 1 1 1 1 1 1 =+ + + " # " x f q f q f q f q q n n = L N M M M M M O Q P P P P P 1 1 1 1 " ### " . (1 ) ( ) ) n n x Jq q ×× × = Jacobian ) ) nn x q × = ±± where ij i j f q q = ) ) n n x q × = xJ θδθ = Example l 1 l 2 (x,y) x lc yl s l s 11 21 2 2 δθ δδ θ x l s l s l s cl c l c =− + + 2 1 2 2 2 1 2 2 X x y s xl c = L N M O Q P = −− L N M O Q P F H G I K J 2 2 1 2 J xx yy F H G G G I K J J J ∂θ 12 θθ = ± ± 2 2 y ls x ⎡⎤ = ⎢⎥ ⎣⎦ Stanford Scheinman Arm d 2 d 3 x 0 z 0 y 0 θ 1 {0} θ 2 x 6 y 6 z 6 θ 4 θ 5 θ 6 i α i-1 a i-1 d i θ i 1 000 θ 1 2 -90 0 d 2 θ 2 3 90 0 d 3 0 4 00 0 θ 4 5 -90 0 0 θ 5 6 90 0 0 θ 6 d 2 d 3 x 0 z 0 y 0 θ 1 {0} θ 2 x 6 y 6 z 6 θ 4 θ 5 θ 6 C C CCS SC SSS S SCS CC SCC C S S C S S S C 1 2 4 56 4 6 256 1 4 5 64 6 2 5 61 [( ) ]( ) ) ] −+ + + + + ++ SCCS SC CSS 4 5 6 24 5 6 4 6 2 5 6 CC CCC SS SSC SSCC CS SC CCC SS CSCC CS SCCC SS CSC 1 2 456 46 1 456 12 4 5 5 4 5 6 5 6 4 6 2 5 6 ) ] ( ) ) ] ( ) + + + +− CCCS SC 4 5 2 5 1 4 5 4 5 2 5 1 4 5 245 25 + CSd Sd SSd Cd 12 3 23 1 2 3 P x r xr r ⎛⎞ ⎜⎟ = = ⎝⎠
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3 Stanford Scheinman Arm x csd sd ssd cd p = + L N M M M O Q P P P 12 3 1 2 23 ± ± ± ± ± ± q q q q q q Q L N M M M M M M M O Q P P P P P P P 1 2 3 4 5 6 (3 1) (3 6) (6 1) () p p xx x Jq q ×× = ±± ± ± ± x y z F H G G I K J J = L N M M M 2 Linear Velocity V ± x p = Position x rq R = L N M M M O Q P P P 1 2 3 ± ± ± ± ± ± ± ( ( x r r r r q r q r q r q r q r q q q q R x x x = F H G G I K J J = F H G G G G G G I K J J J J J J F H G G G G I K J J J J 1 2 3 91 ) 1 1 1 6 2 1 2 6 3 1 3 6 96 1 2 6 61 ) " " " # ± ± x J qq RX R = Orientation: Direction Cosines C C CCS SC SSS S SCS CC SCC C S S C S S S C 1 2 4 56 4 6 256 1 12 4 5 64 6 2 5 [( ) ]( ) ) ] −+ + + + + ++ SCCS SC CSS 4 5 6 24 5 6 4 6 2 5 6 CC CCC SS SSC SSCC CS SC CCC SS CSCC CS SCCC SS CSC 456 46 1 456 12 4 5 5 4 5 6 2 4 5 6 25 6 ) ] ( ) ) ] ( ) −− + + + +− CCCS SC 4 5 2 5 1 4 5 4 5 2 5 1 4 5 245 25 R x = ± ± ± ± ± ± ± ( ( x r r r r q r q r q r q r q r q q q q R x x x = F H G G I K J J = F H G G G G G G I K J J J J J J F H G G G G I K J J J J 1 2 3 ) 1 1 1 6 2 1 2 6 3 1 3 6 1 2 6 ) " " " # Representations X x x P R = L N M O Q P •Cartesian •Spherical •Cylindrical •….
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handout4_Jacobian - Movie Segment Locomotion Gates with...

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