HW1_sol - Texas A&M University Department of Mechanical...

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Texas A M University Department of Mechanical Engg. MEEN-408 Intro. to Robotics 1. Solution to Problem 1 We have to show the following R x ( θ ) = 1 0 0 0 cosθ - sinθ 0 sinθ cosθ R y ( φ ) = cosφ 0 sinφ 0 1 0 - sinφ 0 cosφ (a) Since we are rotating about X-axis, the X coordinates doesn’t change. Let the length of the projection of point in Y-Z plane be l . Therefore the coordinates of point P in Y-Z plane are y = lcosα (1) z = lsinα (2) Rotating the point P about X-axis by angle θ ,we have coorinates of new point P’ as y = lcos ( α + θ ) (3) z = lsin ( α + θ ) (4) Simplifying equations above, we get y = l ( cosαcosθ - sinαsinθ ) = ycosθ - zsinθ (5) z = l ( sinαcosθ + cosαsinθ ) = zcosθ + ysinθ (6) Therefore we have x y z = 1 0 0 0 cosθ - sinθ 0 sinθ cosθ x y z (7) (b) Similarly when we rotate about Y-axis, the Y coordinate of point P does not change. x = l 0 cosβ (8) z = l 0 sinβ (9) We rotate the projection of point P in X-Z plane about Y axis by angle φ x = l 0 sin ( β + φ ) (10) z = l 0 cos ( β + φ ) (11) 1
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Simplifying equations above, we get x = l 0 ( sinβcosφ + cosβsinφ ) = xcosφ + zsinφ (12) z = l 0 ( cosβcosφ - sinβsinφ ) = zcosφ - xsinφ (13) Therefore we have x y z = cosφ 0 sinφ 0 1 0 - sinφ 0 cosφ x y z (14) 2. Solution to Problem 2 The sequence of rotation is: Rotate about X-axis by angle π/ 4 Rotate the new coordinates about Y-axis by angle π/ 3 (a) The relation b/w particle coordinates when rotated about X are x y z = 1 0 0 0 cos ( π/ 4) - sin ( π/ 4) 0 sin ( π/ 4) cos ( π/ 4) x 1 y 1 z 1 (15) Now rotate P X 1 Y 1 Z 1 about Y axis x 1 y 1 z 1 = cos ( π/ 3) 0 sin ( π/ 3) 0 1 0 - sin ( π/ 3) 0 cos ( π/ 3) x 2 y 2 z 2 (16) Therefore, the rotation matrix is as follows x y z = cos ( π/ 3) 0 sin ( π/ 3) 0 1 0 - sin ( π/ 3) 0 cos ( π/ 3) 1 0 0 0 cos ( π/ 4) - sin ( π/ 4) 0 sin ( π/ 4) cos ( π/ 4) 1 0 0 (17) x y z = 0 . 5 0 - 0 . 8660 (b) For the reversal of the sequence, we have the coordinates as x
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HW1_sol - Texas A&M University Department of Mechanical...

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