Rotation Angles to Quaternions

Calculate quaternion from rotation angles

Library:
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 rotation used in this block is a passive transformation between two coordinate systems. 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

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`[R₁,R₂,R₃]` — Rotation angles
3-by-1 vector

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

Data Types: double

Output

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`q` — Quaternion
4-by-1 vector

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

Data Types: double

Parameters

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

Specifies the output rotation order for three wind 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 $(μ_{x}, μ_{y}, μ_{z})$ through the angle θ. A unit quaternion itself has unit magnitude, and can be written in the following vector format:

$q = [\begin{array}{l} q_{0} \\ q_{1} \\ q_{2} \\ q_{3} \end{array}] = [\begin{matrix} \cos (θ / 2) \\ \sin (θ / 2) μ_{x} \\ \sin (θ / 2) μ_{y} \\ \sin (θ / 2) μ_{z} \end{matrix}]$

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:

$\begin{array}{l} i^{2} = j^{2} = k^{2} = - 1 \\ i j = - j i = k \\ j k = - k j = i \\ k i = - i k = j \end{array}$

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.