rod2angle

Convert Euler-Rodrigues vector to rotation angles

Syntax

[rotationAng1,rotationAng2,rotationAng3] = rod2angle(rod)

[rotationAng1,rotationAng2,rotationAng3] = rod2angle(rod,rotationSequence)

Description

[rotationAng1,rotationAng2,rotationAng3] = rod2angle(rod) function calculates the set of rotation angles, rotationAng1, rotationAng2, and rotationAng3, for a given Euler-Rodrigues (also known as Rodrigues) vector, rod, using the default rotation sequence of 'ZYX' (yaw, pitch, roll).. The Rodrigues angles represent a passive transformation from frame A to frame B. The resulting rotation angles represent a series of right-hand intrinsic passive rotations from frame A to frame B.

[rotationAng1,rotationAng2,rotationAng3] = rod2angle(rod,rotationSequence) function calculates the set of rotation angles for a given Rodrigues vector and a specified rotation sequence, rotationSequence. The rotationSequence parameter also specifies the order of the three rotation angles.

Examples

Determine Rotation Angles from One Vector

This example shows how to determine rotation angles from vector, [.1 .2 -.1]

r = [.1 .2 -.1];
[yaw, pitch, roll] = rod2angle(r)

yaw = 
-0.1651

pitch = 
0.4074

roll = 
0.1651

Input Arguments

rod — Rodrigues vectors

m-by-3 matrix

m-by-3 matrix containing m Rodrigues vector.

Data Types: double

rotationSequence — Rotation sequence

'ZYX' (default) | 'ZYZ' | 'ZXY' | 'ZXZ' | 'YXZ' | 'YXY' | 'YZX' | 'YZY' | 'XYZ' | 'XYX' | 'XZY' | 'XZX'

Rotation sequence, specified as a scalar. For sequences like 'ZXZ', there are two rotations about the same axis, such as two different yaw angles in 'ZXZ'.

Data Types: char | string

Output Arguments

rotationAng1 — First rotation angles

m-by-1 array

First rotation angles, specified as an m-by-1 array, in radians. The rotationAng1 argument represents the first rotation in rotationSequence.

Data Types: double | single

rotationAng2 — Second rotation angles

m-by-1 array

Second rotation angles, specified as an m-by-1 array, in radians. The rotationAng2 argument represents the second rotation in rotationSequence.

Data Types: double | single

rotationAng3 — Third rotation angles

m-by-1 array

Third rotation angles, specified as an m-by-1 array, in radians. The rotationAng3 argument represents the third rotation in rotationSequence.

Data Types: double | single

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