Kinematics-1 - Movie Segment Passive Walking, Tad McGeer,...

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1 Movie Segment Passive Walking, Tad McGeer, Fraser Univ, British Columbia, ICRA 1991 video proceedings Kinematics • Transformations • Representations Spatial Descriptions •Task ±Descr ipt ion Manipulator Prismatic Joint L in k i Revolute Joint Base End-Effector n moving link 1 fixed link Links: Joints: Revolute (1 DOF) Prismatic (1 DOF) Configuration Parameters A set of position parameters that describes the full configuration of the system. 9 parameters/link Generalized coordinates A set of independent configuration parameters Degrees of Freedom Number of generalized coordinates
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2 Generalized Coordinates n moving links: 6n parameters Generalized Coordinates 6 parameters { 3 positions 3 orientations 6 parameters { 3 positions 3 orientations Generalized Coordinates n 1 d.o.f. joints: 5n constraints 5 constraints n moving links: 6n parameters d.o.f. (system): 6n - 5n = n 123 (, , , , ) m xxx x A set of m parameters: that completely specifies the end-effector position and orientation with respect to {0} End-Efector Configuration Parameters 1 0 n + Operational Coordinates A set of independent configuration parameters 0 m 0 12 ,,, m x xx number of degrees of freedom of the end-effector. 0 : m Operational point 1 : n O + 1 0 n + Joint Coordinates θ 2 θ 3 θ 1 θ 1 θ 2 θ 3 θ Joint Space α () x y α x y Operational Coordinates Operational Space
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3 0 nm > Redundancy Degrees of redundancy: 0 A robot is said to be redundant if Position of a Point O P With respect to a fixed origin O, the position of a point P is described by the vector OP or simply by p. p Rigid Body Configuration E p q Q p OP = JJJG qO Q = O Euclidian Space Cartesian Frame y 0 P z p P Coordinate Frames Rigid Body Configuration A P Y A {A} ^ Z A ^ X A ^ Z B ^ {B} Y B ^ X B ^ Orientation: { A X B , A Y B , A Z B } ^ ^ ^ Position: A P Describes rotations of {B} with respect to {A} Rotation Matrix Rotation Matrix ˆ ˆ AB B B A B R X X = A P Y A {A} ^ Z A ^ X A ^ Z B ^ {B} Y B ^ X B ^ Y ^ Z B ^ {B} Y B ^ X B ^ 1 0 0 ˆ A B A B R X ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0 1 0 ˆ A B A B R Y ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0 0 1 ˆ A B A B R Z ⎡⎤ ⎢⎥ = ⎣⎦ ˆˆˆ AAAA B BB B R XYZ = 11 12 13 21 22 23 31 32 33 A B rrr Rrrr =
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4 Rotation Matrix Dot Product ˆˆ . . . ˆ B A B BA B A A X Y X X X X Z ⎡⎤ ⎢⎥ = ⎣⎦ ˆˆ ˆˆ ˆˆ ... BABA BA A BB A B A B A BA BA XX YX ZX R XY YY ZY XZ YZ ZZ = B A T X ˆˆˆ AAA A B BB B R XYZ = Y A {A} ^ Z A ^ X A ^ Z B ^ {B} Y B ^ X B ^ Inverse of Rotation Matrices Orthonormal Matrix AAAA B R = ˆ ˆ ˆ ˆ BT A T B TB B B B T AA A A A A X YX Y ZR Z == = 1 A T B RR R 1 A AT R R = Rotation Matrix A R R = Example Y A ^ Z A ^ X A ^ {A} O {B} Y B ^ Z B ^ X B A R =− F H G G I K J J 10 0 00 1 01 0 B A T B A T B A T X Y Z
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This note was uploaded on 05/12/2011 for the course ME 427 taught by Professor Vedattemiz during the Spring '11 term at Işık Üniversitesi.

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Kinematics-1 - Movie Segment Passive Walking, Tad McGeer,...

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