# 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

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