Invariance of ##SO(3)## Lie group when expressed via Euler angles

[LucaC]

TL;DR Summary

Computation of left-invariant action on $SO(3)$ expressed via Euler angles

I am trying to understand the properties of the $SO(3)$ Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions.
I am building an Invariant Extended Kalman Filter (IEKF), which exploits the invariance property of $SO(3)$ dynamics $\mathbf{\dot{R}} = \mathbf{R}[\omega]_\times$, where $[\cdot]_\times$ is the skew-symmetric operator.

The original filter is based on the left-invariance of the dynamics equation under constant rotations $\mathbf{R}_0$, which translates into
\begin{equation}
\mathbf{\bar{R}} \doteq \mathbf{R}_0\mathbf{R} \Rightarrow \dot{\mathbf{\bar{R}}} = \mathbf{R}_0\mathbf{R}[\omega]_\times=\mathbf{\bar{R}}[\omega]_\times
\end{equation}
I have also applied the same property with the quaternion representation of $SO(3)$.

What I am trying to understand is if it is possible to apply a (non-linear) left-invariance property to the Roll-Pitch-Yaw representation of the dynamic equation, that is
\begin{cases}
\dot{\phi} = \omega_x + \text{sin}(\phi)\text{tan}(\theta)\omega_y+\text{cos}(\phi)\text{tan}(\theta)\omega_z \\
\dot{\theta} = \text{cos}(\phi)\omega_y-\text{sin}(\phi)\omega_z \\
\dot{\psi} = \frac{\text{sin}(\phi)}{\text{cos}(\theta)}\omega_y+\frac{\text{cos}(\phi)}{\text{cos}(\theta)}\omega_z \\
\end{cases}
by using some fixed rotations about xyz axes as $(\phi_0,\theta_0,\psi_0)$, in analogy to the matrix case.

My issue is that I am not able to find a transformation which defines $(\bar{\phi},\bar{\theta},\bar{\psi})$ and eventually $(\bar{\omega}_x,\bar{\omega}_y,\bar{\omega}_z)$ and satisfies the previous dynamics equation, as done previously with the rotation matrix.Thank you all for your support

 

[jvicens]

and suggestions.
I understand your confusion and I am happy to help you understand the properties of the $SO(3)$ Lie group when expressed via Euler angles. First of all, it is important to note that the Lie group $SO(3)$ is a group of rotations, and it can be represented in various ways such as rotation matrices, quaternions, and Euler angles. Each representation has its own advantages and disadvantages, and the choice of representation depends on the specific problem at hand.

In the case of your IEKF, it seems that the original filter is based on the left-invariance property of the dynamics equation under constant rotations. This means that if we apply a constant rotation $\mathbf{R}_0$ to the rotation matrix $\mathbf{R}$, the resulting rotated matrix $\mathbf{\bar{R}}$ will still satisfy the same dynamics equation. This property is also true for the quaternion representation of $SO(3)$.

Now, let's consider the Roll-Pitch-Yaw representation of the dynamic equation. As you have correctly pointed out, it is possible to apply a left-invariance property to this representation by using fixed rotations about the xyz axes. However, the transformation that you are looking for is not a simple rotation but a composition of rotations. In other words, you need to apply a sequence of rotations about the x, y, and z axes to get the desired transformation.

To understand this concept better, let's consider a specific example. Suppose we want to apply a left-invariance property with a fixed rotation of $\phi_0$ about the x-axis. This can be achieved by first rotating the matrix $\mathbf{R}$ by $\phi_0$ about the x-axis and then applying a fixed rotation of $\theta_0$ about the y-axis and a fixed rotation of $\psi_0$ about the z-axis. This will result in a new matrix $\mathbf{\bar{R}}$ that satisfies the Roll-Pitch-Yaw representation of the dynamic equation.

In summary, it is possible to apply a left-invariance property to the Roll-Pitch-Yaw representation of the dynamic equation by using a sequence of fixed rotations about the x, y, and z axes. I hope this explanation helps you understand the properties of $SO(3)$ when expressed via Euler angles. If you have