quat2rod

Convert quaternion to Euler-Rodrigues vector

Syntax

rod=quat2rod(quat)

Description

rod=quat2rod(quat) function calculates the Euler-Rodrigues vector, rod, for a given quaternion quat. The quaternion input and resulting Euler-Rodrigues vector represent a right-hand passive transformation from frame A to frame B.

Aerospace Toolbox uses quaternions that are defined using the scalar-first convention. This function normalizes all quaternion inputs.

Examples

Determine Euler-Rodrigues Vector from Quaternion

This example shows how to determine the Euler-Rodrigues vector from the quaternion.

q = [-0.7071 0 0.7071 0]

q = 1×4

   -0.7071         0    0.7071         0

r = quat2rod( q )

r = 1×3

         0   -1.0000         0

Input Arguments

quat — Quaternion

M-by-4 array

M-by-4 array of quaternions. quat has its scalar number as the first column.

Data Types: double

Output Arguments

rod — Euler-Rodrigues vector

M-by-3 matrix

M-by-3 matrix containing M Euler-Rodrigues vectors.

Algorithms

An Euler-Rodrigues vector $$\stackrel{\rightharpoonup }{b}$$ represents a rotation by integrating a direction cosine of a rotation axis with the tangent of half the rotation angle as follows:

$$\overrightarrow{b}=\left[\begin{array}{ccc}{b}_x& b_y& b_z\end{array}\right]$$

where:

$$\begin{array}{l}{b}_x=\tan \left(\frac{1}{2}\theta \right)s_x,\\ b_y=\tan \left(\frac{1}{2}\theta \right)s_y,\\ b_z=\tan \left(\frac{1}{2}\theta \right)s_z\end{array}$$

are the Rodrigues parameters. Vector $$\stackrel{\rightharpoonup }{s}$$ represents a unit vector around which the rotation is performed. Due to the tangent, the rotation vector is indeterminate when the rotation angle equals ±pi radians or ±180 deg. Values can be negative or positive.

References

[1] Dai, J.S. "Euler-Rodrigues formula variations, quaternion conjugation and intrinsic connections." Mechanism and Machine Theory, 92, 144-152. Elsevier, 2015.

Version History

Introduced in R2017a