Quaternion Rotation

Rotate vector by quaternion

• Library:

• 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.

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

Model Examples

Developing the Apollo Lunar Module Digital Autopilot

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

Open Live Script

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