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- 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

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