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If I want to rotate a vector $V$ from coordinate system $A$ to $B$, I could use the rotation matrix by $V_B=R\cdot V_A$, where $R$ is the rotation matrix. There are many rotation sequences for $R$, such as $xyz$ and $yzy$. But after constructing those matrices, I found that different rotation sequence are not the same, e.q. $R_{xyz}V_A$ does not equal $R_{yay}V_B$, why is that? I thought if the Eulerian angles are the same, so the rotation matrix, no matter what rotation sequence it takes, should be the same. I know there is something wrong, could you please tell me where? Thanks!

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3 Answers 3

You can rotate $(1,0,0)$ to $(0,1,0)$ by rotating a quarter-turn counterclockwise around the $z$-axis, or by rotating $(1,0,0)$ around the $y$-axis up to $(0,0,1)$ and then down around the $x$-axis to $(0,1,0)$. The first way, $(0,1,0)$ winds up at $(-1,0,0)$; the second way, it winds up at $(0,0,-1)$. So just knowing where one vector goes doesn't tell you where other vectors go.

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Neat and leading 1. +1 –  B. S. Feb 27 '13 at 16:28

Without seeing what the problem is, no, I can't tell you what it is. It is true that the rotation matrix from one coordinate system to another should be independent of how you describe the rotations. If you give the specifics of the rotation sequences we might be able to help.

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Thanks! Guys. Let me make an example. –  Ray Feb 27 '13 at 14:45
    
The Euler angles between coordinate system A and B is angle = 0.8000 1.2000 0.2000 (rad) If we make the rotation matrix from xyz sequence, we have rotm1 = 0.3551 0.7937 -0.4939 -0.0720 0.5500 0.8321 0.9320 -0.2599 0.2525 and we also can have another matrix from yzy –  Ray Feb 27 '13 at 14:49
    
sequence, which is rotm2 = 0.3551 0.7937 -0.4939 -0.0720 0.5500 0.8321 0.9320 -0.2599 0.2525 If we have one vector defined in coordinate system A, V_A= 0.9000 2.3000 -7.0000 Then V_A*rotm1=[-6.3702, 3.7989, -0.2980], but V_A*rotm2=[-6.6716, 0.3594, -3.2343], In the same coordinate system B, should they be the same? I want to have the same vector after rotation from different rotation matrix. Thanks! –  Ray Feb 27 '13 at 14:55

$3$D rotations do not commute in general. Look up phenomena like Gimbal lock to see what can happen with Euler angles.

http://en.wikipedia.org/wiki/Gimbal_lock#Gimbal_lock_in_applied_mathematics

Also consider switching to representing your rotations by unit quaternions, you remove all the problems you get using Euler angles and they are very tidy.

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