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

In summary, the invariance of the SO(3) Lie group expressed via Euler angles is significant because it allows for a clear understanding of the symmetry and rotations in three-dimensional space. This concept is essential in various fields, including physics, mathematics, and engineering. The SO(3) Lie group can be expressed using Euler angles by representing rotations as a sequence of rotations around three different axes. Its invariance can be mathematically demonstrated using the Lie algebra of the group. This invariance is closely related to the concept of symmetry and has many practical applications in fields such as computer graphics, robotics, and aircraft navigation. It is also crucial in understanding physical systems and phenomena, making it essential in scientific research and development.
  • #1
LucaC
1
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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
 
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  • #2
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
 

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

1. What is the significance of the Invariance of SO(3) Lie group when expressed via Euler angles?

The Invariance of SO(3) Lie group when expressed via Euler angles refers to the fact that the rotational transformations of a rigid body can be represented by a set of three angles, known as Euler angles, without changing the physical properties of the body. This means that the orientation of the body remains the same regardless of which set of Euler angles is used to represent it.

2. How does the Invariance of SO(3) Lie group affect the calculation of rotations using Euler angles?

The Invariance of SO(3) Lie group simplifies the calculation of rotations using Euler angles, as it allows for a consistent representation of the rotational transformations without any loss of information. This means that the same rotation can be described using different sets of Euler angles, making it easier to perform calculations and compare different rotations.

3. Can the Invariance of SO(3) Lie group be extended to other Lie groups?

Yes, the concept of invariance can be extended to other Lie groups, such as the special unitary group SU(2). In general, the invariance of a group refers to the property that the group elements remain unchanged under certain transformations, and this can be applied to various mathematical structures, including Lie groups.

4. What are the limitations of using Euler angles to represent SO(3) rotations?

While Euler angles are a convenient way to represent rotations in three-dimensional space, they do have some limitations. One limitation is that there are certain orientations that cannot be represented using Euler angles, known as gimbal lock. Additionally, the use of Euler angles can lead to numerical instability and difficulties in performing calculations involving multiple rotations.

5. How does the Invariance of SO(3) Lie group relate to the concept of symmetry in physics?

The Invariance of SO(3) Lie group is closely related to the concept of symmetry in physics. Symmetry refers to the property that a system remains unchanged under certain transformations, and the Invariance of SO(3) Lie group implies that the physical properties of a rigid body remain unchanged under rotations described by Euler angles. This concept of symmetry is fundamental in many areas of physics, including classical mechanics, quantum mechanics, and electromagnetism.

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