homework1.pdf - ECE 470: I R H 1 S S B O 10, 2009 Solution...

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Unformatted text preview: ECE 470: I R H 1 S S B O 10, 2009 Solution (Problem 2.1) Not depending on choice of frames implies that for any rotation, the dot product between two vectors will remain constant. Thus, we need to show for any R that is a rotation matrix that v T 1 v 2 = ( Rv 1 ) T ( Rv 2 ) = v T 1 R T Rv 2 = v T 1 R − 1 Rv 2 = v T 1 v 2 This implies that the projection of one vector onto another depends only on the relative orientation of the vectors. Solution (Problem 2.2, 2.3) Notice that bardbl v bardbl 2 = v T v ⇒ bardbl v bardbl = + √ v T v . Therefore, bardbl Rv bardbl = + radicalbig ( Rv ) T Rv = √ v T R T Rv = √ v T v = bardbl v bardbl Problem 2.3 follows from Problem 2.2 with v = p 1 − p 2 . Solution (Problem 2.10, 2.11, 2.12, 2.13) Postmultiply to rotate around the current frame. Premultiply to rotate around the world frame. 2.10: R = R y ,ψ R x ,φ R z ,θ 2.11: R = R z ,θ R x ,φ R x ,ψ 2.12: R = R z ,α R x ,φ R z ,θ R x ,ψ 2.13: R = R z ,α R z ,θ R x ,φ R x ,ψ Solution (Problem 2.15) 1 R 2 3 = R 2 1 R 1 3 where R 2 1 = ( R 1 2 ) T = 1 1 / 2 √ 3 / 2 − √ 3 / 2 1 / 2 . Therefore, R 2 3 = 1 1 / 2 √ 3 / 2 − √ 3 / 2 1 / 2 0 0 − 1 0 1 1 0 = − 1 √ 3 / 2 1 / 2 1 / 2 − √ 3 / 2 Solution (Problem 2.16) The roll, pitch, yaw rotation matrix is...
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homework1.pdf - ECE 470: I R H 1 S S B O 10, 2009 Solution...

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