Any three-dimensional rotation can
be described as a sequence of yaw, pitch, and roll rotations.
One of the simplest ways to parameterize 3D rotations is to construct them from ``2D-like'' transformations, as shown in Figure 3.7. First consider a rotation about the -axis. Let roll be a counterclockwise rotation of about the -axis. The rotation matrix is given by
The upper left of the matrix looks exactly like the 2D rotation matrix (3.13), except that is replaced by . This causes yaw to behave exactly like 2D rotation in the plane. The remainder of
looks like the identity matrix, which causes to remain unchanged after a roll.
Similarly, let pitch be a counterclockwise rotation of about the -axis:
In this case, points are rotated with respect to and while the coordinate is left unchanged.
Finally, let yaw be a counterclockwise rotation
of about the -axis:
In this case, rotation occurs with respect to and while leaving unchanged.
Steven M LaValle