Pose formats

XYZABC format

The XYZABC format is used to express a pose by 6 values. $XYZ$ is the position in millimeters. $ABC$ are Euler angles in degrees. The convention used for Euler angles is ZYX, i.e., $A$ rotates around the $Z$ axis, $B$ rotates around the $Y$ axis, and $C$ rotates around the $X$ axis. The elements of the rotation matrix can be computed by using

$$\begin{split}r_{11} & = \cos{B}\cos{A}, \\ r_{12} & = \sin{C}\sin{B}\cos{A}-\cos{C}\sin{A}, \\ r_{13} & = \cos{C}\sin{B}\cos{A}+\sin{C}\sin{A}, \\ r_{21} & = \cos{B}\sin{A}, \\ r_{22} & = \sin{C}\sin{B}\sin{A}+\cos{C}\cos{A}, \\ r_{23} & = \cos{C}\sin{B}\sin{A}-\sin{C}\cos{A}, \\ r_{31} & = -\sin{B}, \\ r_{32} & = \sin{C}\cos{B}, \text{and} \\ r_{33} & = \cos{C}\cos{B}. \\\end{split}$$

Note

The trigonometric functions $\sin$ and $\cos$ are assumed to accept values in degrees. The argument needs to be multiplied by the factor $\frac{\pi}{180}$ if they expect their values in radians.

Using these values, the rotation matrix $R$ and translation vector $T$ are defined as

$$\begin{split}R = \left(\begin{array}{ccc} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33} \end{array}\right), \qquad T = \left(\begin{array}{c} X \\ Y \\ Z \end{array}\right).\end{split}$$

The transformation can be applied to a point $P$ by

$$P' = R P + T.$$

XYZ+quaternion format

The XYZ+quaternion format is used to express a pose by a position and a unit quaternion. $XYZ$ is the position in meters. The quaternion is a vector of length 1 that defines a rotation by four values, i.e., $q=(\begin{array}{cccc}a & b & c & w\end{array})^T$ with $||q||=1$. The corresponding rotation matrix and translation vector are defined by

$$\begin{split}R = 2 \left(\begin{array}{ccc} \frac{1}{2} - b^2 - c^2 & a b - c w & a c + b w \\ a b + c w & \frac{1}{2} - a^2 - c^2 & b c - a w \\ a c - b w & b c + a w & \frac{1}{2} - a^2 - b^2 \end{array}\right), \qquad T = \left(\begin{array}{c} X \\ Y \\ Z \end{array}\right).\end{split}$$

The transformation can be applied to a point $P$ by

$$P' = R P + T.$$

Note

In XYZ+quaternion format, the pose is defined in meters, whereas in the XYZABC format, the pose is defined in millimeters.