9 parameter i s revolute joint joints variable i di

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Unformatted text preview: arameter i s Revolute Joint Joints variable αi di ai Constant Constant or zero Constant Prismatic Joints Constant Joint variable Constant or zero Constant or zero Consecutive axes intersect ai = 0 Consecutive joint axes parallel di or di+1 can be set to zero For prismatic joint axis i, Position is arbitrary while direction fixed Can set ai-1 or a1 = 0 and di-1 or di+1= 0 Homogeneous Continued…. V V XY XY ⎡V N ⎤ ⎢ O⎥ V = H⎢ A ⎥ ⎢V ⎥ ⎢⎥ ⎢1⎥ ⎣⎦ ⎡n x ⎢n =⎢ y ⎢n z ⎢ ⎣0 The (n,o,a) position of a point relative to the current coordinate frame you are in. ox ax oy ay oz az 0 0 Px ⎤ ⎡V N ⎤ ⎥⎢ O ⎥ Py ⎥ ⎢V ⎥ Pz ⎥ ⎢V A ⎥ ⎥⎢ ⎥ 1 ⎦⎢ 1 ⎥ ⎣⎦ V X = n x V N + o x V O + a x V A + Px The rotation and translation part can be combined into a single homogeneous matrix IF and ONLY IF both are relative to the same homogeneous matrix IF and ONLY IF both are relative to the same coordinate frame. Homogeneous Matrices in 3D H is a 4x4 matrix that can describe a translation, rotation, or both in one O matrix Y N P X A Translation without rotation Z Y O N X Z A Rotation without translation ⎡1 ⎢0 H=⎢ ⎢0 ⎢ ⎣0 0 0 Px ⎤ 1 0 Py ⎥ ⎥ 0 1 Pz ⎥ ⎥ 0 0 1⎦ ⎡n x ⎢n H=⎢ y ⎢n z ⎢ ⎣0 ox ax oy oz ay az 0 0 0⎤ 0⎥ ⎥ 0⎥ ⎥ 1⎦ Rotation part: part: Could be rotation around zaxis, x-axis, y-axis or a combination of the three. Perspective transformation transformation Position Vector ROTATION ROTATION MATRIX SCALING ⎡* * ⎢* R 3*3 6 ⎢ T0 = ⎢* * ⎢ ⎣* F1*3 * *⎤ ⎥ * P*1 ⎥ 3 * *⎥ ⎥ * 1*1⎦ Rotation ⎡ ⎢ matrix =⎢ ⎢ ⎢ . . ⎣ perspect− transform Position ⎤ ⎥ vector ⎥ ⎥ ⎥ scaling⎦ Perspective transformation is useful in computer vision and and the calibration of camera ⎡ nx ⎢n 6 ⎢y T0 = ⎢ nz ⎢ ⎣0 sx ax sy ay sz az 0 0 px ⎤ ⎥ nsa py ⎥ ⎡ =⎢ p z ⎥ ⎣0 0 0 ⎥ 1⎦ p⎤ ⎥ 1⎦ Inverse of a homogeneous matrix ⎡ nx ⎢ sx −1 T =⎢ ⎢a x ⎢ ⎢ ⎣0 ⎡ ⎢ T R3*3 =⎢ ⎢ ⎢ ⎢0 0 ⎣ ny sy ay 0 − n p⎤ T⎥ − s p⎥ T⎥ −a p ⎥ 1⎥ ⎦ T nz sz az 0 − n p⎤ T⎥ − s p⎥ T⎥ −a p ⎥ 0 1⎥ ⎦ T USE...
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This document was uploaded on 03/12/2014 for the course MECHANICAL 214 at University of Manchester.

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