Noel Hughes
Aerospace Engineer
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Quaternion to DCM

Quaternion to Direction Cosine Matrix Conversion

Noel Hughes

9/27/2009

 

 

 

 

            The columns of a direction cosine matrix are the components of unit vectors along the axes of an othogonal coordinate that has been rotated by the rotation which it describes.  Therefore, constructing the direction cosine matrix corresponding to a quaternion is accomplished by rotating each of the orthogonal unit vectors by the quaternion and placing them in the appropriate columns of the matrix.

 

The rotation operation is:

 

        
  
         eqn 15

 

 

(All references are to my paper, Quaternion to Euler Angle Conversion for Arbitrary Rotation Sequence Using Geometric Methods)

 

Example:

c2 is a coordinate frame rotated by the quaternion

 

 = 0.360423  0.439679  0.391904  0.723317                    eqn 17

 

The three vectors to be rotated are:

v1  = 1  0  0

v2  = 0  1  0

v3  = 0  0  1

 

After rotation, the three vectors are:

 

v1R  = 0.306185853     0.8838825       -0.35355216

v2R  = -0.250000803   0.433011621   0.866024084

v3R  = 0.918557021     -0.176776249  0.353553866

 

Each of these vectors becomes a column of the direction cosine matrix:

 

 

 

0.306185853   -0.250000803  0.918557021

0.8838825       0.433011621   -0.176776249

-0.35355216    0.866024084   0.353553866

 
 


DCM =          

 

 

 

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