Roll-Pitch-Yaw

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The roll, pitch and yaw angles are three angles defined in regard of absolute transformation to describe the orientation of an object, generally vehicles, in three-dimensional space. In the following the common convention will be used, so the three angles can be described as follows (in the order they are applied):

• Yaw: Rotation around the vertical axis of the object or vehicle, respectively
• Pitch: Rotation around the lateral axis
• Roll: Rotation around the longitudinal axis (what is generally the movement axis of a vehicle)

There are different notations to define the axes of an object. Usually and in recent publications the vertical axis is the z-axis, the longitudal axis is x and then the lateral axis is the y-axis and directed to the left.

So the roll-pitch-yaw transformation matrix of the orientation is defined as follows:

$\begin{align} RPY(\phi,\theta,\psi)&=Rot(x,\phi)Rot(y,\theta)Rot(z,\psi) \\ &= \left[\begin{array}{cccc} \cos{\psi}\cos{\theta} & -\cos{\theta}\sin{\phi} & \sin{\theta} & 0\\ \cos{\psi}\sin{\phi}+\cos{\phi}\sin{\psi}\sin{\theta} & \cos{\phi}\cos{\psi}-\sin{\phi}\sin{\psi}\sin{\theta} & -\cos{\theta}\sin{\psi} & 0\\ \sin{\phi}\sin{\psi}-\cos{\phi}\cos{\psi}\sin{\theta} & \cos{\phi}\sin{\psi}+\cos{\psi}\sin{\phi}\sin{\theta} & \cos{\psi}\cos{\theta} & 0\\ 0 & 0 & 0 & 1 \end{array}\right] \end{align}$

Example: Orientation of a car Let the roll, pitch and yaw angles for a car be defined as follows: $\begin{align} &yaw: &\psi&=-&30^\circ \\ &pitch: &\theta&=&20^\circ \\ &roll: &\phi&=&35^\circ \end{align}$ Then the rotation matrix then results as: $\begin{align} RPY(35^\circ,20^\circ,-30^\circ)&=Rot(x,35^\circ)Rot(y,20^\circ)Rot(z,-30^\circ) \\ &= \left[\begin{array}{cccc} 0.8138 & 0.4698 & 0.3420 & 0\\ -0.2397 & 0.8075 & -0.5390 & 0\\ -0.5294 & 0.3566 & 0.7698 & 0\\ 0 & 0 & 0 & 1 \end{array}\right] \\ &= \left[\begin{array}{cccc} \vec{\mathbf{x}} & \vec{\mathbf{y}} & \vec{\mathbf{z}} & \vec{\mathbf{p}}\\ 0 & 0 & 0 & 1 \\ \end{array}\right] \end{align}$ The figure below shows how the three transformations are applied step by step. First the car is rotated around the z-axis about the yaw angle. Then the rotation around the y-axis follows and finally the car is rotated around the x-axis. The final orientation can be seen in the last view. If the matrix and the final orientation of the car are compared, it can be seen, that the vectors $\vec{\mathbf{x}}, \vec{\mathbf{y}}$ and $\vec{\mathbf{z}}$ correspond to the three local coordinate axes of the car. Position vector $\vec{\mathbf{p}}$ is a zero vector in this case.