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

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