The incorrect way to convert eye velocity to position

A good guess would to triplicate the previous circuit, i.e. have one integrator for each dimension.

If velocity were translation velocity this would work.

But this is angular velocity, and this will not work.

To see that this does not work, recall what happened in the case where the head was pitched down while the eye was looking to the right , center or down.

When looking to the right or left, beside rotating the eye up, the eye was also turned torsionally.

Here in this circuit. torsional eye position is never effected by pitching the head down. The only thing pitching the head down affects is the vertical eye position.