KUKA XYZ-ABC format

KUKA robots use the so called XYZ-ABC format. $XYZ$ is the position in millimeters. $ABC$ are angles in degrees, with $A$ rotating around $z$ axis, $B$ rotating around $y$ axis and $C$ rotating around $x$ axis. The rotation convention is $z$-$y'$-$x''$ (i.e. $x$-$y$-$z$) and computed by $r_z(A) r_y(B) r_x(C)$.

Conversion from KUKA-ABC to quaternion

The conversion from the $ABC$ angles in degrees to a quaternion $q=(\begin{array}{cccc}x & y & z & w\end{array})$ can be done by first converting all angles to radians

$$\begin{split}A_r = A \frac{\pi}{180} \text{,} \\ B_r = B \frac{\pi}{180} \text{,} \\ C_r = C \frac{\pi}{180} \text{,} \\\end{split}$$

and then calculating the quaternion with

$$\begin{split}x = \cos{(A_r/2)}\cos{(B_r/2)}\sin{(C_r/2)} - \sin{(A_r/2)}\sin{(B_r/2)}\cos{(C_r/2)} \text{,} \\ y = \cos{(A_r/2)}\sin{(B_r/2)}\cos{(C_r/2)} + \sin{(A_r/2)}\cos{(B_r/2)}\sin{(C_r/2)} \text{,} \\ z = \sin{(A_r/2)}\cos{(B_r/2)}\cos{(C_r/2)} - \cos{(A_r/2)}\sin{(B_r/2)}\sin{(C_r/2)} \text{,} \\ w = \cos{(A_r/2)}\cos{(B_r/2)}\cos{(C_r/2)} + \sin{(A_r/2)}\sin{(B_r/2)}\sin{(C_r/2)} \text{.}\end{split}$$

Conversion from quaternion to KUKA-ABC

The conversion from a quaternion $q=(\begin{array}{cccc}x & y & z & w\end{array})$ with $||q||=1$ to the $ABC$ angles in degrees can be done as follows.

$$\begin{split}A &= \text{atan}_2{(2(wz + xy), 1 - 2(y^2 + z^2))} \frac{180}{\pi} \\ B &= \text{asin}{(2(wy - zx))} \frac{180}{\pi} \\ C &= \text{atan}_2{(2(wx + yz), 1 - 2(x^2 + y^2))} \frac{180}{\pi}\end{split}$$