Yaw, pitch, and roll

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}

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 $ z$-axis. Let roll be a counterclockwise rotation of $ \gamma $ about the $ z$-axis. The rotation matrix is given by

$\displaystyle R_z(\gamma) = \begin{bmatrix}\cos\gamma & -\sin\gamma & 0  \sin\gamma & \cos\gamma & 0  0 & 0 & 1  \end{bmatrix} .$ (3.16)

The upper left of the matrix looks exactly like the 2D rotation matrix (3.13), except that $ \theta $ is replaced by $ \gamma $. This causes yaw to behave exactly like 2D rotation in the $ xy$ plane. The remainder of $ R_z(\gamma)$ looks like the identity matrix, which causes $ z$ to remain unchanged after a roll.

Similarly, let pitch be a counterclockwise rotation of $ \beta$ about the $ x$-axis:

$\displaystyle R_x(\beta) = \begin{bmatrix}1 & 0 & 0  0 & \cos\beta & -\sin\beta  0 & \sin\beta & \cos\beta  \end{bmatrix} .$ (3.17)

In this case, points are rotated with respect to $ y$ and $ z$ while the $ x$ coordinate is left unchanged.

Finally, let yaw be a counterclockwise rotation of $ \alpha$ about the $ y$-axis:

$\displaystyle R_y(\alpha) = \begin{bmatrix}\cos\alpha & 0 & \sin\alpha  0 & 1 & 0  -\sin\alpha & 0 & \cos\alpha  \end{bmatrix} .$ (3.18)

In this case, rotation occurs with respect to $ x$ and $ z$ while leaving $ y$ unchanged.

Steven M LaValle 2016-12-31