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Quaternions to Direction Cosine Matrix - Convert quaternion vector to direction cosine matrix

Library

Utilities/Axes Transformations

Description

The Quaternions to Direction Cosine Matrix block transforms the 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.

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:

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

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.

Dialog Box

Inputs and Outputs

Input	Dimension Type	Description
First	4-by-1 quaternion vector	Contains the quaternion vector.

Output	Dimension Type	Description
First	3-by-3 direction cosine matrix.	Contains the direction cosine matrix.

See Also

Direction Cosine Matrix to Rotation Angles

Direction Cosine Matrix to Quaternions

Rotation Angles to Direction Cosine Matrix

Rotation Angles to Quaternions

Quaternion Rotation

Quaternions to Rotation Angles

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