Yaw, pitch, and roll rotations

A 3D body can be rotated about three orthogonal axes, as shown in Figure 3.8. Borrowing aviation terminology, these rotations will be referred to as yaw, pitch, and roll:

1. A yaw is a counterclockwise rotation of $\alpha$ about the $z$-axis. The rotation matrix is given by

   $$R_z(\alpha) = \begin{pmatrix}\cos\alpha & -\sin\alpha & 0 \sin\alpha & \cos\alpha & 0 0 & 0 & 1 \end{pmatrix} .$$ (3.39)

   
   
   Note that the upper left entries of $R_z(\alpha)$ form a 2D rotation applied to the $x$ and $y$ coordinates, whereas the $z$ coordinate remains constant.

2. A pitch is a counterclockwise rotation of $\beta$ about the $y$-axis. The rotation matrix is given by

   $$R_y(\beta) = \begin{pmatrix}\cos\beta & 0 & \sin\beta 0 & 1 & 0 -\sin\beta & 0 & \cos\beta \end{pmatrix} .$$ (3.40)

   
   

3. A roll is a counterclockwise rotation of $\gamma$ about the $x$-axis. The rotation matrix is given by

   $$R_x(\gamma) = \begin{pmatrix}1 & 0 & 0 0 & \cos\gamma & -\sin\gamma 0 & \sin\gamma & \cos\gamma \end{pmatrix} .$$ (3.41)

   
   

Each rotation matrix is a simple extension of the 2D rotation matrix, (3.31). For example, the yaw matrix, $R_z(\alpha)$, essentially performs a 2D rotation with respect to the $x$ and $y$ coordinates while leaving the $z$ coordinate unchanged. Thus, the third row and third column of $R_z(\alpha)$ look like part of the identity matrix, while the upper right portion of $R_z(\alpha)$ looks like the 2D rotation matrix.

The yaw, pitch, and roll rotations can be used to place a 3D body in any orientation. A single rotation matrix can be formed by multiplying the yaw, pitch, and roll rotation matrices to obtain

$$\begin{split}R(\alpha,& \beta,\gamma) = R_z(\alpha) R_y(\b... ...\sin\gamma & \cos\beta \cos\gamma \end{pmatrix}. \end{split}$$ (3.42)

It is important to note that $R(\alpha,\beta,\gamma)$ performs the roll first, then the pitch, and finally the yaw. If the order of these operations is changed, a different rotation matrix would result. Be careful when interpreting the rotations. Consider the final rotation, a yaw by $\alpha$. Imagine sitting inside of a robot ${\cal A}$ that looks like an aircraft. If $\beta = \gamma = 0$, then the yaw turns the plane in a way that feels like turning a car to the left. However, for arbitrary values of $\beta$ and $\gamma$, the final rotation axis will not be vertically aligned with the aircraft because the aircraft is left in an unusual orientation before $\alpha$ is applied. The yaw rotation occurs about the $z$-axis of the world frame, not the body frame of ${\cal A}$. Each time a new rotation matrix is introduced from the left, it has no concern for original body frame of ${\cal A}$. It simply rotates every point in ${\mathbb{R}}^3$ in terms of the world frame. Note that 3D rotations depend on three parameters, $\alpha$, $\beta$, and $\gamma$, whereas 2D rotations depend only on a single parameter, $\theta$. The primitives of the model can be transformed using $R(\alpha,\beta,\gamma)$, resulting in ${\cal A}(\alpha,\beta,\gamma)$.