Rodrigues to Rotation Angles

Convert Euler-Rodrigues vector to rotation angles

• Library:

• Aerospace Blockset / Utilities / Axes Transformations

• 

Description

The Rodrigues to Rotation Angles block converts the three-element Euler-Rodrigues vector into rotation angles. For more information on Euler-Rodrigues vectors, see Algorithms.

Ports

Input

rod — Euler-Rodrigues vector

three-element vector

Euler-Rodrigues vector determined from rotation angles.

Data Types: double

Output

R1,R2,R3 — Rotation angles

three-element vector

Rotation angles, in radians, from which to determine the Euler-Rodrigues vector. Quaternion scalar is the first element.

Data Types: double

Parameters

Rotation order — Rotation order

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

Rotation order for three wind rotation angles.

For the 'ZYX', 'ZXY', 'YXZ', 'YZX', 'XYZ', and 'XZY' rotations, the block generates an R2 angle that lies between ±pi/2 radians (±90 degrees), and R1 and R3 angles that lie between ±pi radians (±180 degrees).

For the 'ZYZ', 'ZXZ', 'YXY', 'YZY', 'XYX', and 'XZX' rotations, the block generates an R2 angle that lies between 0 and pi radians (180 degrees), and R1 and R3 angles that lie between ±pi (±180 degrees). However, in the latter case, when R2 is 0, R3 is set to 0 radians.

Programmatic Use

Block Parameter: rotationOrder

Type: character vector

Values: 'ZYX' | 'ZYZ' |'ZXY' | 'ZXZ' | 'YXZ' | 'YXY' | 'YZX' | 'YZY' | 'XYZ' | 'XYX' | 'XZY' | 'XZX'

Default: 'ZYX'

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.

Extended Capabilities

C/C++ Code Generation
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