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direction cosine matrix to axis angle of rotation (Definition)

The angle of rotation can be found from the trace of the direction cosine matrix to axis angle of rotation matrix

$\displaystyle A_{11} + A_{22} + A_{33} = 3cos(\alpha) + (1 - cos(\alpha))(e_1^2 + e_2^2 + e_3^2) $

Noting that the axis of rotation is a unit vector and has a length of 1 means

$\displaystyle e_1^2 + e_2^2 + e_3^2 = 1 $

therefore

$\displaystyle A_{11} + A_{22} + A_{33} = 1 + 2cos(\alpha) $

rearranging gives

$\displaystyle \alpha = cos^{-1}( \dfrac{1}{2} (A_{11} + A_{22} + A_{33} - 1))$ (1)

Inverse cosine is a multivalued function and there are 2 possible solutions for $\alpha$. Normally, the convention is to choose the principle value such that $0 < \alpha < \pi $

As long as $\alpha$ is not zero, the unit vector is given by

$\displaystyle \left[ \begin{array}{c} e_1 \ e_2 \ e_3 \end{array} \right] =... ...2 sin(\alpha)} \ \dfrac{(A_{12} - A_{21})}{2 sin(\alpha)} \end{array} \right]$ (2)

Above equation should be proved at some time...



"direction cosine matrix to axis angle of rotation" is owned by bloftin.
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Cross-references: function, unit vector, matrix, trace

This is version 2 of direction cosine matrix to axis angle of rotation, born on 2005-08-28, modified 2005-08-29.
Object id is 89, canonical name is DirectionCosineMatrixToAxisAngleOfRotation.
Accessed 2137 times total.

Classification:
Physics Classification: 45.40.-f (Dynamics and kinematics of rigid bodies)
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