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Linearization of product of rotation matrices

  1. May 8, 2012 #1

    I have the following function:

    [itex]\textbf{h}[/itex]([itex]\textbf{x}[/itex]) = [ [itex]\textbf{C}[/itex][itex]^{n}_{b2}[/itex] ][itex]^{T}[/itex] [itex]\textbf{C}[/itex][itex]^{n}_{b1}[/itex] [itex]\textbf{C}[/itex][itex]^{b1}_{b2}[/itex]


    [itex]\textbf{x}[/itex] = [ [itex]\textbf{ε}[/itex][itex]_{1}[/itex][itex]\;[/itex] [itex]\textbf{ε}[/itex][itex]_{2}[/itex] ][itex]^{T}[/itex] containing Euler angles [itex]\mathbf{ε}[/itex] such that direction cosine matrix [itex]\textbf{C}[/itex][itex]^{n}_{b1}[/itex] is a function of [itex]\textbf{ε}[/itex][itex]_{1}[/itex] and [itex]\textbf{C}[/itex][itex]^{n}_{b2}[/itex] is a function of [itex]\textbf{ε}[/itex][itex]_{2}[/itex] (through the relationship linking Euler angles and their corresponding cosine matrix), and [itex] b1, b2, n [/itex] are different reference frames.

    I want to linearize [itex]\textbf{h}[/itex]([itex]\textbf{x}[/itex]) with respect to [itex]\textbf{ε}[/itex][itex]_{1}[/itex] and [itex]\textbf{ε}[/itex][itex]_{2}[/itex], which should give me the following (3 x 6) matrix:

    [itex] \mathbf{H} = \left[ \frac{\partial \textbf{h}(\textbf{x})}{\partial \textbf{ε}_{1}} \frac{\partial \textbf{h}(\textbf{x})}{\partial \textbf{ε}_{2}} \right][/itex]

    Could anyone give me hints on how I could solve this problem, i.e. compute the elements of [itex]\mathbf{H}[/itex]? Thank you in advance for your suggestions :smile:
    Last edited: May 8, 2012
  2. jcsd
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