The Euclidean norm ||r|| of a rotation

Please let me know what is the Euclidean norm of a rotation vector; and the difference from Euler angles.

I met these terms in the following text ; although your answers do not need to respect the writer's intention at all , because you do not know whether the writer uses the terms correctly or not.

Thank you very much.

Rotation Vector The first three 16bit data values are a directed rotation vector presentation of the rotation velocity and are NOT three Euler angles as might be assumed from the previous section “How Data should be used”. The Euclidean norm ||r|| of the rotation vector r = (data[0], data[1], data[2]) is the rotation velocity about the vector r.

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The question is about attitude parameterisation - a rotation vector is one method of describing an attitude; Euler angles are another.

The ultimate description of a rotation is generally considered to be a 3x3 rotation matrix (in 3-space). However, since there are only 3 degrees of freedom in a rotation matrix, then there is considerable redundancy in its representation. Additionally, a rotation matrix does not really provide an intuitive description of what the rotation looks like.

So, what are some ways we can represent the rotation matrix? We can:

• Perform 3 successive rotations, each around a defined axis (Euler Angles); or
• Perform a single rotation about a fixed axis (Rotation vector)

Hence for Euler angles, we can create our rotation matrix as: $\mathbf{R} = \mathbf{R}_x(\phi) \mathbf{R}_y(\theta) \mathbf{R}_z(\psi)$, which consists three successive rotations; a rotation about Z by angle $\psi$, followed by a rotation about the (new) y-axis by $\theta$ followed by rotation about the new x-axis by $\phi$. Classically, these are roll ($\phi$), pitch ($\theta$) and yaw ($\psi$) in the aerospace world.

A rotation vector $\mathbf{\rho}$ consists of a rotation about axis $\frac{\mathbf{\rho}}{\parallel\mathbf{\rho}\parallel}$ by angle $\parallel\mathbf{\rho}\parallel$, except where $\parallel\mathbf{\rho}\parallel = 0$, in which the rotation matrix is simply the identiy matrix. To recover the rotation matrix, the matrix exponential is used: $\mathbf{R} = \operatorname{exp}(\left[\mathbf{\rho}\right]_\times)$ where $\left[\mathbf{\rho}\right]_\times$ is a skew symmetric matrix constructed as $\left[\mathbf{\rho}\right]_\times = \begin{bmatrix} 0 & -\rho_{z} & \rho_{y} \\ \rho_{z} & 0 & -\rho_{x} \\ -\rho_{y} & \rho_{x} & 0\end{bmatrix}$.

Fortunately, there is a shortcut to calculating the matrix exponential called the Rodrigues Rotation Formula.

In short: Euler angles are not the same as a rotation vector (except for very small angles, but that's another story). They cannot be used interchangeably.

There are many many more representations of attitude. Unit Quaternions are common. If you need a cure for insomnia, take a read of this survey paper on attitude representations.

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Thank you very much for the answer. – seven_swodniw Aug 25 '11 at 8:23

The euclidean norm for a 3-dimensional vector is defined as:

$\|\mathbf{r}\|=\sqrt{r_x^2+r_y^2+r_z^2}$

No matter what the interpretation of this vector is. In this case it seems the author uses a compact version of the axis-angle representation, where the vector $\mathbf{r}$ is the axis of rotation and the angle (or in this case angular velocity) is decoded in the norm of the vector (as for the rotation axis a direction/unit vector suffices).

So the rotation is a rotation about the axis $\mathbf{u}=\frac{\mathbf{r}}{\|\mathbf{r}\|}$ with the angle/velocity $\theta=\|\mathbf{r}\|$.

Euler angles on the other hand are three consecutive angles representing three consecutive rotations about coordinate axes.

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Thank you very much for the answer. – seven_swodniw Aug 25 '11 at 8:23

My interpretation would be that something is spinning about the central axis defined by $\langle \textrm{data}[0],\textrm{data}[1],\textrm{data}[2]\rangle$, and that it is rotating at an angular velocity of $v$ radians per unit of time (in the counterclockwise direction), where $$v=\|r\|=\sqrt{\textrm{data}[0]^2+\textrm{data}[1]^2+\textrm{data}[2]^2}.$$

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Thank you very much for the answer. – seven_swodniw Aug 25 '11 at 8:23