Figure 3.7:
Any three-dimensional rotation can
be described as a sequence of yaw, pitch, and roll rotations.
![\begin{figure}\centerline{\psfig{file=figs/yawpitchroll.eps,width=1.7in}}\end{figure}](img219.gif) |
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
![$\displaystyle R_z(\gamma) = \begin{bmatrix}\cos\gamma & -\sin\gamma & 0 \sin\gamma & \cos\gamma & 0 0 & 0 & 1 \end{bmatrix} .$](img220.gif) |
(3.16) |
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:
![$\displaystyle R_x(\beta) = \begin{bmatrix}1 & 0 & 0 0 & \cos\beta & -\sin\beta 0 & \sin\beta & \cos\beta \end{bmatrix} .$](img223.gif) |
(3.17) |
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:
![$\displaystyle R_y(\alpha) = \begin{bmatrix}\cos\alpha & 0 & \sin\alpha 0 & 1 & 0 -\sin\alpha & 0 & \cos\alpha \end{bmatrix} .$](img225.gif) |
(3.18) |
In this case, rotation occurs with respect to
and
while leaving
unchanged.
Steven M LaValle
2020-11-11