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

A_r = A \frac{\pi}{180} \text{,} \\ B_r = B \frac{\pi}{180} \text{,} \\ C_r = C \frac{\pi}{180} \text{,} \\

and then calculating the quaternion with

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{.}

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.

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}