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What is the derivative of a skew symmetric matrix?

  1. Sep 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Need to prove that the derivative of a rotation matrix is a skew symmetric matrix muktiplied by that rotation matrix. Specifically applying it on the Rodrigues’ formula.


    2. Relevant equations


    3. The attempt at a solution
    I have shown that the cubed of the skew symmetric matrix is equal to the opposite of the skew symmetric matrix. I;m supposed to use this to finish the proof of the above problem. But I get to a point where I have something like the skew matrix multiplied by sin, so I'm thinking of just doing the product rule here but i'm unsure what the derivative of the skew matrix is. I went with "derivative of skew symmetric matrix is a skew symmetric matrix" but that didn't work out
     
  2. jcsd
  3. Sep 14, 2017 #2

    Orodruin

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    Derivative with respect to what? In order to have a derivative at all there must be one or more parameters that the object depends on.
     
  4. Sep 14, 2017 #3
    So the rotation matrix is R, rotated about an arbritrary axis k, and the rotation angle is theta. The derivative is with respect to theta. Here let me write the Rodrigues formula the best I can on my phone:

    R(k, theta) = I + {u}*sin(theta) + {u}^2 * (1-cos(theta))

    Where I is the identity matrix and {u} is a skew symmetric matrix, which would have the form of something like this:

    {u} = [0 -z y; z 0 -x; -y x 0]

    So I need to take the derivative of the rotation matrix R and show that it equals {u}*R
     
    Last edited by a moderator: Sep 14, 2017
  5. Sep 14, 2017 #4
    Weird, apparently (open bracket) u (close bracket) isn't showing up in my post, so I switched it to {u}
     
  6. Sep 14, 2017 #5

    Mark44

    Staff: Mentor

    This is because [u] is interpreted by a browser as the starting underscore tag -- the browser consumes this special character, which is why it seemed to disappear, and also why so much of your post was underscored.

    I fixed your previous post, but you should take a look at it again to see if it says what you meant.
     
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