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

O_r = O \frac{\pi}{180} \text{,} \\ A_r = A \frac{\pi}{180} \text{,} \\ T_r = T \frac{\pi}{180} \text{,} \\

and then calculating the quaternion with

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{.}

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

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}

If z = 0 and w = 0 the conversion is

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}

In all other cases the conversion is

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}