Quaternion Rotation

Rotate vector by quaternion

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Libraries:
Aerospace Blockset / Utilities / Math Operations

Description

The Quaternion Rotation block calculates the resulting vector following the passive rotation of initial vector vec by quaternion q and returns a final vector, the rotated vector or vector of rotated vectors. Aerospace Blockset™ uses quaternions that are defined using the scalar-first convention. This block normalizes all quaternion inputs. For the equations used for the quaternion, initial vector, and final vector, see Algorithms.

Examples

Developing the Apollo Lunar Module Digital Autopilot

Develop Apollo Lunar Module digital autopilot using Simulink^® and Aerospace Blockset.

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Ports

Input

q — Quaternion

quaternion | vector

Quaternions in the form of [q₀, r₀, ..., q₁, r₁, ... , q₂, r₂, ... , q₃, r₃, ...], specified as a quaternion or vector of quaternions.

Data Types: double

vec — Initial vector

vector | vector of vectors

Initial vector or vector of vectors in the form of [v₁, u₁, ... , v₂, u₂, ... , v₃, u₃, ...].

Data Types: double

Output

vec_rot — Final quaternion

rotated quaternion | vector of rotated quaternions

Final vector or vector of rotated vectors.

Data Types: double

Algorithms

The normalized quaternion has the form of

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

The vector has the form of

$$v=i{v}_1+j{v}_2+k{v}_3.$$

The Aerospace Blockset defines a passive quaternion rotation of the form:

$$v^{\prime }=q^{-1}\otimes \left[\frac{0}{v}\right]\otimes q\text{,}$$

where Ⓧ is the operator of a quaternion multiplication.

The final vector has the form of

$$v^{\prime }=\left[\begin{array}{c}{v}_1{}^{\prime }\\ v_2{}^{\prime }\\ v_3{}^{\prime }\end{array}\right]=\left[\begin{array}{ccc}(1-2q_2^2-2q_3^2)& 2(q_1{q}_2+q_0{q}_3)& 2(q_1{q}_3-q_0{q}_2)\\ 2(q_1{q}_2-q_0{q}_3)& (1-2q_1^2-2q_3^2)& 2(q_2{q}_3+q_0{q}_1)\\ 2(q_1{q}_3+q_0{q}_2)& 2(q_2{q}_3-q_0{q}_1)& (1-2q_1^2-2q_2^2)\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right]$$

References

[1] Stevens, Brian L., Frank L. Lewis. Aircraft Control and Simulation, Second Edition. Hoboken, NJ: Wiley–Interscience.

[2] Diebel, James. "Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors." Stanford University, Stanford, California, 2006.

Extended Capabilities

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

Version History

Introduced before R2006a