# Jacobian of Euler's rotation equations

I propagate the attitude of a satellite in a free-floating, torque free environment from $$t_k$$ to $$t_{k+1}$$ by integrating Euler's equation of motion:

$$\dot{\omega} = [J]^{-1}([\omega]_{\times}[J]\:\omega)$$

with $$[J]\in\mathbb{R}^{3\times3}$$ being the (full) inertia matrix and $$\omega \in \mathbb{R}^3$$ the angular velocity vector, and $$[\cdot]_{\times}$$ being the skew-symmetric operator.

If we consider an Euler integration step of $$dt$$, we can therefore write:

$$\omega_{k+1} = \dot{\omega}_k\cdot dt = [J]^{-1}([\omega_k]_{\times}[J]\:\omega_k)\cdot dt$$

I would like to compute the analytical Jacobian $$\textbf{F}_\omega = \frac{\partial\omega_{k+1}}{\partial \omega_k}$$, or an approximation of it, in order to propagate the covariance from $$t_k$$ to $$t_{k+1}$$. Is there any known closed-form solution? With the cross-product, I really don't know how to tackle the problem.

Any help is welcome. Thank you.

• Based on the comment you make near the end of your question: does it help that the derivative of (a cross b) is a’ cross b + a cross b’ ? May 25, 2021 at 17:21
• It does, thank you very much! May 26, 2021 at 8:39

$$\frac{\partial\dot\omega}{\partial\omega} = [J]^{-1}\frac{\partial([\omega]_{\times}[J]\omega)}{\partial\omega}$$

$$= [J]^{-1} \frac{\partial(\omega\times[J]\omega)}{\partial\omega}$$

$$= [J]^{-1} \left (\frac{\partial\omega}{\partial\omega}\times[J]\omega + \omega \times [J]\frac{\partial\omega}{\partial \omega}\right)$$

$$= [J]^{-1} \left (1\times[J]\omega + [\omega]_\times[J] \right)$$

Using the two definitions:

1. $$a \times b = [a]_{\times}b = [b]^{\intercal}_{\times}a$$
2. $$[x]^{\intercal}_{\times} = -[x]_{\times}$$

on the term $$1\times[J]\omega$$, we find the correct answer: $$\frac{\partial\dot\omega}{\partial\omega} = [J]^{-1} \left ( [\omega]_{\times}[J] - \left [[J]\omega \right]_{\times} \right )$$