Quaternions to Direction Cosine Matrix

Convert quaternion vector to direction cosine matrix

Libraries:
Aerospace Blockset / Utilities / Axes Transformations

Description

The Quaternions to Direction Cosine Matrix block transforms a four-element unit quaternion vector (q₀, q₁, q₂, q₃) into a 3-by-3 direction cosine matrix (DCM). The outputted DCM performs the coordinate transformation of a vector in inertial axes to a vector in body axes. Aerospace Blockset™ uses quaternions that are defined using the scalar-first convention. This block normalizes all quaternion inputs. The quaternion input and the resulting direction cosine matrix represent a right-hand passive transformation from frame A to frame B. For more information, see Algorithms.

Ports

Input

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q — Quaternion
4-by-1 vector

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

Data Types: double

Output

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DCM_be — Direction cosine matrix
3-by-3 matrix

Direction cosine matrix, returned as a 3-by-3 matrix.

Data Types: double

Algorithms

Using quaternion algebra, if a point P is subject to the rotation described by a quaternion q, it changes to P′ given by the following relationship:

$\begin{array}{l} P^{'} = q P q^{c} \\ q = q_{0} + i q_{1} + j q_{2} + k q_{3} \\ q^{c} = q_{0} - i q_{1} - j q_{2} - k q_{3} \\ P = 0 + i x + j y + k z \end{array}$

Expanding P′ and collecting terms in x, y, and z gives the following for P′ in terms of P in the vector quaternion format:

$P^{'} = [\begin{matrix} 0 \\ x^{'} \\ y^{'} \\ z^{'} \end{matrix}] = [\begin{matrix} 0 \\ (q_{0}^{2} + q_{1}^{2} - q_{2}^{2} - q_{3}^{2}) x + 2 (q_{1} q_{2} - q_{0} q_{3}) y + 2 (q_{1} q_{3} + q_{0} q_{2}) z \\ 2 (q_{0} q_{3} + q_{1} q_{2}) x + (q_{0}^{2} - q_{1}^{2} + q_{2}^{2} - q_{3}^{2}) y + 2 (q_{2} q_{3} - q_{0} q_{1}) z \\ 2 (q_{1} q_{3} - q_{0} q_{2}) x + 2 (q_{0} q_{1} + q_{2} q_{3}) y + (q_{0}^{2} - q_{1}^{2} - q_{2}^{2} + q_{3}^{2}) z \end{matrix}]$

Since individual terms in P′ are linear combinations of terms in x, y, and z, a matrix relationship to rotate the vector (x, y, z) to (x′, y′, z′) can be extracted from the preceding. This matrix rotates a vector in inertial axes, and hence is transposed to generate the DCM that performs the coordinate transformation of a vector in inertial axes into body axes.

$D C M = [\begin{array}{l} (q_{0}^{2} + q_{1}^{2} - q_{2}^{2} - q_{3}^{2}) & 2 (q_{1} q_{2} + q_{0} q_{3}) & 2 (q_{1} q_{3} - q_{0} q_{2}) \\ 2 (q_{1} q_{2} - q_{0} q_{3}) & (q_{0}^{2} - q_{1}^{2} + q_{2}^{2} - q_{3}^{2}) & 2 (q_{2} q_{3} + q_{0} q_{1}) \\ 2 (q_{1} q_{3} + q_{0} q_{2}) & 2 (q_{2} q_{3} - q_{0} q_{1}) & (q_{0}^{2} - q_{1}^{2} - q_{2}^{2} + q_{3}^{2}) \end{array}]$

Extended Capabilities

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

Version History

Introduced before R2006a