The pose format that is used by FANUC robots consists of a position \(XYZ\) in millimeters and an orientation \(WPR\) that is given by three angles in degrees, with \(W\) rotating around \(x\)-axis, \(P\) rotating around \(y\)-axis and \(R\) rotating around \(z\)-axis. The rotation order is \(x\)-\(y\)-\(z\) and computed by \(r_z(R) r_y(P) r_x(W)\).

Conversion from FANUC-WPR to quaternion

The conversion from the \(WPR\) 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

\[\begin{split}W_r = W \frac{\pi}{180} \text{,} \\ P_r = P \frac{\pi}{180} \text{,} \\ R_r = R \frac{\pi}{180} \text{,} \\\end{split}\]

and then calculating the quaternion with

\[\begin{split}x = \cos{(R_r/2)}\cos{(P_r/2)}\sin{(W_r/2)} - \sin{(R_r/2)}\sin{(P_r/2)}\cos{(W_r/2)} \text{,} \\ y = \cos{(R_r/2)}\sin{(P_r/2)}\cos{(W_r/2)} + \sin{(R_r/2)}\cos{(P_r/2)}\sin{(W_r/2)} \text{,} \\ z = \sin{(R_r/2)}\cos{(P_r/2)}\cos{(W_r/2)} - \cos{(R_r/2)}\sin{(P_r/2)}\sin{(W_r/2)} \text{,} \\ w = \cos{(R_r/2)}\cos{(P_r/2)}\cos{(W_r/2)} + \sin{(R_r/2)}\sin{(P_r/2)}\sin{(W_r/2)} \text{.}\end{split}\]

Conversion from quaternion to FANUC-WPR

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

\[\begin{split}R &= \text{atan}_2{(2(wz + xy), 1 - 2(y^2 + z^2))} \frac{180}{\pi} \\ P &= \text{asin}{(2(wy - zx))} \frac{180}{\pi} \\ W &= \text{atan}_2{(2(wx + yz), 1 - 2(x^2 + y^2))} \frac{180}{\pi}\end{split}\]