Rotation Angles to Quaternions

Calculate quaternion from rotation angles

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Libraries:
Aerospace Blockset / Utilities / Axes Transformations

Description

The Rotation Angles to Quaternions block converts the rotation described by the three rotation angles (R1, R2, R3) into the four-element quaternion vector (q₀, q₁, q₂, q₃), where quaternion is defined using the scalar-first convention. Aerospace Blockset™ uses quaternions that are defined using the scalar-first convention. The block Rotation Order parameter specifies both the rotation sequence as well as the order of the rotation angles in the input. For example, if Rotation Order has a value of ZYX, the block inputs are in the rotation order z-y-x (yaw, pitch, and roll). The rotation angles represent a series of right-hand intrinsic passive transformation from frame A to frame B. The resulting quaternion represents a right-hand passive rotation from frame A to frame B. For more information on quaternions, see Algorithms.

Limitations

• The limitations for the ZYX, ZXY, YXZ, YZX, XYZ, and XZY implementations generate an R2 angle that is between ±90 degrees, and R1 and R3 angles that are between ±180 degrees.

• The limitations for the ZYZ, ZXZ, YXY, YZY, XYX, and XZX implementations generate an R2 angle that is between 0 and 180 degrees, and R1 and R3 angles that are between ±180 degrees.

Ports

Input

[R₁,R₂,R₃] — Rotation angles

3-by-1 vector

Rotation angles, specified as a 3-by-1 vector, in radians.

Data Types: double

Output

q — Quaternion

4-by-1 vector

Quaternion, returned as a 4-by-1 vector.

Data Types: double

Parameters

Rotation order — Rotation order

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

Specifies the rotation order for the three input rotation angles.

Programmatic Use

Block Parameter: rotationOrder

Type: character vector

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

Default: 'ZYX'

Algorithms

A quaternion vector represents a rotation about a unit vector $$\left({\mu }_x,{\mu }_y,{\mu }_z\right)$$ through the angle θ. A unit quaternion itself has unit magnitude, and can be written in the following vector format:

$$q=\left[\begin{array}{l}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]=\left[\begin{array}{c}\cos (\theta /2)\\ \sin (\theta /2){\mu }_x\\ \sin (\theta /2){\mu }_y\\ \sin (\theta /2){\mu }_z\end{array}\right]$$

An alternative representation of a quaternion is as a complex number,

$$q=q_0+i{q}_1+j{q}_2+k{q}_3$$

where, for the purposes of multiplication:

$$\begin{array}{l}{i}^2=j^2=k^2=-1\\ ij=-ji=k\\ jk=-kj=i\\ ki=-ik=j\end{array}$$

The benefit of representing the quaternion in this way is the ease with which the quaternion product can represent the resulting transformation after two or more rotations.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2007b