Now consider the rotation of a 3D rigid body. Recall from Section 3.3 that Euler's rotation theorem implies that every 3D rotation can be described as a rotation about an axis
though the origin. As the orientation of the body changes over a short period of time , imagine the axis that corresponds to the change in rotation. In the case of the merry-go-round, the axis would be
. More generally, could be any unit vector.
The 3D angular velocity is therefore expressed as a 3D vector:
which can be imagined as taking the original from the 2D case and multiplying it by the vector . This weights the components according to the coordinate axes. Thus, the components could be considered as
. The , , and components also correspond to the rotation rate in terms of pitch, roll, and yaw, respectively. We avoided these representations in Section 3.3 due to noncommutativity and kinematic singularities; however, it turns out that for velocities these problems do not exist . Thus, we can avoid quaternions at this stage.
Steven M LaValle