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Materials : Rotation Matrix transformation using misorientation

  1. Jan 20, 2005 #1
    I have this materials mathematical problem(its not homework) and I figured people here are good at rotation/quaternions so you may be able to help. What Im trying to do:

    You are given an orientation(in RotMatrix form, or Euler angles)
    You must rotate this orientation to a final rotation, also in matrix form or Euler angles.
    Imagine you have N-1 steps to do this in. (in the end there will be N orientations, so N-1 steps)
    The change in orientation must be a linear change of the MISorientation between initial and final. This is the tricky part. Its not a change in the rotation about the axis, but depends on the MISorientation between the matrices. If you dont know what a misorientation angle is I have it below.
    So if N(the number of final orientations) is say 10, then theres going to be 10 orientations from initial to final that rotate in such a way that their misorientation is linear in incrementation.

    Here are the equations I have right now that I'm working with.
    Ill start with Gi, end with Gf, the initial and final orientation matrices.

    DG = Gf*Gi^(-1)
    DG is the required rotation matrix to move Gi to Gf.

    The misorientation angle is defined as:
    Cos(misori) == Trace[DG] - 1 == (DG11+DG22+DG33-1)

    So the change in misorientation angle between the steps is misori/(N-1) because there are 10, so there will be 9 rotations to make the initial one the final. Each of these rotations must have the misorientation angle of misori/9.

    What I need to do is get Mathematica up and running (the licence is in use right now) and try to get it to solve this for me.

    So far I don't think it is easily possible. I can make the declaration that only 2 Euler angles can vary. We can keep one constant. This should help some.
    But I think theres a lack of information in trying to determine a Rotation Matrix from a misorientation angle, because there is no unique solution. What sort of constraints can I put on it?

    Basically I have to solve the equation :
    Cos(misori/(N-1)) == Trace[DG] - 1
    for DG.
    Its really quite tough, and I'll be working on it for a few more hours. I just want to get outside opinions on it.

    I can do this in quaternions too, but the method for getting misorientation in quaternion is, though less computative computer-wise, more work by hand.

    I appreaciate any help at all. Thanks a lot. And If you know latex and feel like rewriting my equations in LaTex please do so. Thanks again!

    If this is not the place for this please let me know.
     
  2. jcsd
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