Since my experiment with coil and magnets didn't work out very well, here's a new concept for the motion of the four retaining rings (all together) using a translation stage and a picomotor. This follows the same idea put forward by Steve Penn. The translation stage is a Newport 9066-COM-V and the picomotor (which we already have) is a Newport 8301-V. Both stage and picomotor are vacuum compatible (rated at 1e-6 Torr) and tested down to 1e-8 Torr by Steve.

Here's the jig integrated in the full system: