Kawasaki XYZ-OAT format

The pose format that is used by Kawasaki robots consists of a position $XYZ$ in millimeters and an orientation $OAT$ that is given by three angles in degrees, with $O$ rotating around $z$ axis, $A$ rotating around the rotated $y$ axis and $T$ rotating around the rotated $z$ axis. The rotation convention is $z$-$y'$-$z''$ (i.e. $z$-$y$-$z$) and computed by $r_z(O) r_y(A) r_z(T)$.

Conversion from Kawasaki-OAT to quaternion

The conversion from the $OAT$ 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

$$\begin{split}O_r = O \frac{\pi}{180} \text{,} \\ A_r = A \frac{\pi}{180} \text{,} \\ T_r = T \frac{\pi}{180} \text{,} \\\end{split}$$

and then calculating the quaternion with

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

Conversion from quaternion to Kawasaki-OAT

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

If $x = 0$ and $y = 0$ the conversion is

$$\begin{split}O &= \text{atan}_2{(2(z - w), 2(z + w))} \frac{180}{\pi} \\ A &= \text{acos}{(w^2 + z^2)} \frac{180}{\pi} \\ T &= \text{atan}_2{(2(z + w), 2(w - z))} \frac{180}{\pi}\end{split}$$

If $z = 0$ and $w = 0$ the conversion is

$$\begin{split}O &= \text{atan}_2{(2(y - x), 2(x + y))} \frac{180}{\pi} \\ A &= \text{acos}{(-1.0)} \frac{180}{\pi} \\ T &= \text{atan}_2{(2(y + x), 2(y - x))} \frac{180}{\pi}\end{split}$$

In all other cases the conversion is

$$\begin{split}O &= \text{atan}_2{(2(yz - wx), 2(xz + wy))} \frac{180}{\pi} \\ A &= \text{acos}{(w^2 - x^2 - y^2 + z^2)} \frac{180}{\pi} \\ T &= \text{atan}_2{(2(yz + wx), 2(wy - xz))} \frac{180}{\pi}\end{split}$$