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Homework Help: Linear transformation: Rotations in R3

  1. Mar 9, 2010 #1
    1. The problem statement, all variables and given/known data
    A robot arm in a xyz coordinate system is doing three consecutive rotations, which are as follows:
    1) Rotates (Pi/4) rad around the z axis
    2) Rotates (Pi/3) rad around the y axis
    3) Rotations -(Pi/6) rad around the x axis

    Find the standard matrix for the (combined) transformation T.

    2. Relevant equations

    3. The attempt at a solution
    I (think) I have gotten as far as finish part 1. By projecting the robot arm down to the xy plane, and by applying trigonometry, I find that the standard matrix for the rotation in 1) to be as follows (it's a 3x3 matrix, I don't know how to format properly):
    [cos (Pi/4) | -sin(Pi/4) | 0]
    [sin (Pi/4) | cos(Pi/4) | 0]
    [0 | 0 | 1] [||1]

    which is:
    [1/2[tex]\sqrt{}2[/tex] | -1/2[tex]\sqrt{}2[/tex] | 0]
    [1/2[tex]\sqrt{}2[/tex] | 1/2[tex]\sqrt{}2[/tex] | 0]
    [0 | 0 | 1]

    Hopefully that is the right answer to question 1), but my answer really is how do I go from here? I've found the first standard matrix, but how do I go forth in trying to find the standard matrix for the entire set of 3 rotations?
     
  2. jcsd
  3. Mar 9, 2010 #2
    You need to find the two other rotation matrices using the exact same method.

    The standard matrix for T is found by simply multiplying the three matrices together.
     
  4. Mar 9, 2010 #3
    I am partially with you. But do I rotate the already rotated arm (ie, finding standard matrix for the rotated arm), or do I find the two other standard matrices from rotating based on the e1, e2, e3 vectors?
     
  5. Mar 9, 2010 #4

    cronxeh

    User Avatar
    Gold Member

    You have 3 matrices:


    Rotation in z = [cos(pi/4) -sin(pi/4) 0; sin(pi/4) cos(pi/4) 0; 0 0 1]
    Rotation in y = [cos(pi/3) 0 sin(pi/3); 0 1 0; -sin(pi/3) 0 cos(pi/3)]
    Rotation in x = [1 0 0; 0 cos(-pi/6) -sin(-pi/6); 0 sin(-pi/6) cos(-pi/6)]


    [ 0.3536 -0.9186 0.1768 ]
    [ 0.3536 0.3062 0.8839 ]
    [-0.8660 -0.2500 0.4330 ]
     
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