The ring motion up and down was not very smooth, again due to friction on the centering pins.
So, after centering the rings using the pins and securing the rings to the translation stage, I removed all pins.
Now the motion up and down is very smooth.
I still have to fine tune the amount of steps that are needed to go up and down.
However, initial tests don't show a good repeatability of the positioning. My main suspect is that the vibration caused by the picomotor cause the disks to slip on the silicon lens. Indeed, when the disks are sitting on the rings, one can clearly hear them "rattle".
This afternoon I installed the picomotor and the translation stage that will be used to move the retaining rings up and down. No partciular problem: I only had to add some small aluminum foil shims between the ear of some rings and the square plate, to make the rings as horizontal as possible.
I tested the motion: with 300000 steps it's possible to move the rings all the way from the parked (down) position, to the up position. I also checked that when the rings are up, I can place four substarates and they fall properly into the alignment groove. Since the maximum speed of the picomotor is 2000 steps/s, it takes 150 seconds to move up and down the ring.
Finally, positive steps means that the rings are moving up, negative that they're moving down.
I raeligned the optical levers to the position I obtained by centering the samples with the rings. I haven't tested the repeatability yet.