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Quaternions to Direction Cosine Matrix

Convert quaternion vector to direction cosine matrix


Utilities/Axes Transformations


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.


Inputs and Outputs

InputDimension TypeDescription


4-by-1 quaternion vectorContains the quaternion vector.
OutputDimension TypeDescription


3-by-3 direction cosine matrix.Contains the direction cosine matrix.

Introduced before R2006a