The second version of the gravity powered motor that I made last week was good enough that I made a second one and used them to move a platform back and forth. Below is a photo and a couple of videos showing this prototype. I haven't yet drawn a diagram of how the motors cause the platform to move, but you can see a very similar mechanism here: Y-bed motion. Replace the fixed point on the right of the diagram with a second motor.

A problem with the mechanism is that any small difference in size between the two pulleys that drive the bed will gradually accumulate over time and lead to position errors. I'm not going to worry about this for the time being: I can probably make pulleys with diameters within 1% of each other. I could get better agreement by making lots, comparing them all, and choosing two that closely match. Using a toothed belt or similar (rather than cotton thread) would eliminate the problem, but would add complexity.