# About linearisation of equations of motion rigid body

I am modeling a system of solid bodies. Consider $\theta \approx 0$ and $\chi \approx 0$. At a certain moment I get the following formula for the angular velocity: $$\omega = \begin{bmatrix} 0 &-\sin(\theta) &\cos(\chi) \cos(\theta)\\ 0 &\cos(\theta) &\cos(\chi) \sin(\theta)\\ 1 &0 &\sin(\chi)\end{bmatrix} \cdot \begin{bmatrix} \dot{\theta}\\ \dot{\chi} \\ \dot{\phi} \end{bmatrix}$$

and I am interested in a linearisation of the equations. Then I write $$\omega \approx \begin{bmatrix} 0 &-\theta &1 \\ 0 &1 & \theta\\ 1 &0 & \chi\end{bmatrix} \cdot \begin{bmatrix} \dot{\theta}\\ \dot{\chi} \\ \dot{\phi} \end{bmatrix}$$ but the equation is still nonlinear. Also consider the inertia matrix in body coordinate $I_B$ and in world coordinate $I_C = R^T I_B R$. The equations of motion are non-linear especially when writing the angular momentum $L = I_C \cdot \omega$ and it's derivative $\dot{L} = \omega \times (I_C \omega) + I_C\ddot{\omega}$. How can I obtain the linerized equations of motion? Is it ok if I proceed as follows: $$\omega \approx \begin{bmatrix} \dot{\theta}\\ \dot{\chi} \\ \dot{\phi} \end{bmatrix} \hspace{1cm} \dot{L} \approx I_B\cdot \begin{bmatrix} \ddot{\theta}\\ \ddot{\chi} \\ \ddot{\phi} \end{bmatrix}$$ That is pretty much ignoring everything ... and saying that $\omega$ and $\dot{L}$ have those coordinates in world frame?

• How about transforming your co-ordinate basis and diagonalizing the matrix? Then, after solving the problem in the principal axis system, you may (or may not) want to transform back to the older co-ordinates? – 299792458 Mar 11 '17 at 8:21
• I have actually a system of 4 rigid bodies each with it's own $\omega$. They all have different coordinate systems, and I want to obtain a, sort of, global equation of motion, in the parameters like $\chi, ...$ – C Marius Mar 11 '17 at 8:25