hw3 - HW 3 Twists and Screws Due 1 Homogeneous matrices and...

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HW 3: Twists and Screws Due: 10/02/12 1 Homogeneous matrices and screws Consider the rigid body transformation shown in the figure below. The rotation is π / 2 around the y -axis. x a y a z a x b y b z b 2 4 2 Figure 1: Change of coordinates and screws. (a) Compute the homogeneous 4 × 4 matrix g ab that describes this transformation. (b) As any rigid body transformation, also this transformation can be written as the exponent of a twist, i.e. as g ab = e ˆ ξθ = e ˆ ωθ I e ˆ ωθ ω × v + ωω T v θ 0 1 (1) where e ˆ ωθ SO (3) can be computed using Rodrigues’ formula, and ˆ ξ is the 4 × 4 matrix equivalent to the vector ξ R 6 (a unit twist) and defined as ˆ ξ := v ω = ˆ ω v 0 0 R 4 × 4 (2) For the homogeneous matrix g ab computed in part (a), compute the equivalent unit twist ξ and scalar θ R , that is, the ξ and θ such that g ab = e ˆ ξθ . (c) For the resulting twist ξθ , compute its screw parameters in frame Ψ a , that is, the line parameters ( ω , r ), magnitude M , and pitch h such that ξθ = M r × ω + h ω ω (3) Give a geometric interpretation/description of this screw and the screw motion that takes Ψ a to Ψ b . 1
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a b Ψ 0 x y z 0 . 2 rad/s (a) Rotation around the el- bow. a b Ψ 0 x y z 180 mph 170 mph (b) Nascar racing.
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