angle2rod

Convert rotation angles to Euler-Rodrigues vector

Syntax

rod = angle2rod(rotationAng1,rotationAng2,rotationAng3)

rod = angle2rod(rotationAng1,rotationAng2, rotationAng3,rotationSequence)

Description

rod = angle2rod(rotationAng1,rotationAng2,rotationAng3) function converts the rotation described by the three rotation angles, rotationAng1, rotationAng2, and rotationAng3, into an M-by-3 Euler-Rodrigues (Rodrigues) matrix, rod, using the default rotation sequence of 'ZYX' (yaw, pitch, roll). The rotation angles represent a series of right-hand intrinsic passive rotations from frame A to frame B. The resulting Euler-Rodrigues vector represents a right-hand passive rotation from frame A to frame B.

rod = angle2rod(rotationAng1,rotationAng2, rotationAng3,rotationSequence) function converts the rotation described by the three rotation angles and a rotation sequence, rotationSequence, into an M-by-3 Euler-Rodrigues array, rod, that contains the M Rodrigues vector. The rotation sequence parameter also specifies the order of the three rotation angles.

Examples

Determine the Rodrigues Vector from One Rotation Angle

Determine the Rodrigues vector from rotation angles.

yaw = 0.7854;
pitch = 0.1;
roll = 0;
r = angle2rod(yaw,pitch,roll)

r = 1×3

   -0.0207    0.0500    0.4142

Determine Rodrigues Vectors from Multiple Rotation Angles

Determine the Rodrigues vectors from multiple rotation angles.

yaw = [0.7854 0.5];
pitch = [0.1 0.3];
roll = [0 0.1];
r = angle2rod(pitch,roll,yaw,'YXZ')

r = 2×3

    0.0207    0.0500    0.4142
    0.0885    0.1381    0.2473

Input 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

rotationSequence — Rotation sequence

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

rod — Euler-Rodrigues vector

3-element vector

Euler-Rodrigues vector determined from rotation angles.

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