The mechanism consists of 2 movable links and 3 screw joints.
In 1 rev of the blue crank:
The nut’s displacement s = h3.(h1-h2)/(h3-h2)
The nut’s rotation φ = (h1-h2)/(h3-h2) revs.
h1: pitch of the screw joint of the blue screw and the base.
h2: pitch of the screw joint of the blue screw and the green nut.
h3: pitch of the screw joint of the green nut and the base.
h1, h2, h3 carry negative sign in case of left-handed thread and vice-versa.
For this case, h1 = 2, h2 = 3 and h3 = 4 so s = -4 φ = -1
Wanted s and φ values can be obtained by combination of appropriate h1, h2, h3 and the thread direction.