Recall from Section 3.2.3 that given a rotation matrix
(3.43), the yaw, pitch, and roll parameters could be directly
determined using the function. It turns out that the
quaternion representation can also be determined directly from the
matrix. This is the inverse of the function in
(4.20).^{4.9}

For a given rotation matrix (3.43), the quaternion parameters can be computed as follows [210]. The first component is

and if , then

and

(4.27) |

If , then the previously mentioned equator problem occurs. In this case,

(4.28) |

(4.29) |

and

(4.30) |

This method fails if or or . These correspond precisely to the cases in which the rotation matrix is a yaw, (3.39), pitch, (3.40), or roll, (3.41), which can be detected in advance.

Steven M LaValle 2012-04-20