# Linearization of product of rotation matrices

1. May 8, 2012

### nicki84

Hello,

I have the following function:

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

with

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

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

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

Could anyone give me hints on how I could solve this problem, i.e. compute the elements of $\mathbf{H}$? Thank you in advance for your suggestions

Last edited: May 8, 2012