# Materials : Rotation Matrix transformation using misorientation

1. Jan 20, 2005

### K.J.Healey

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.