This is probably a good place for me to start, because I tried it a few months ago.
Archimedes_Screw_Trough.jpg
The problem with this model is that the ball can still be at an indeterminate place and there's no real simple way to say "yes, a ball is in position X".
I moved on to a dual auger system that allowed me to move balls up and down a very closely sized channel.
I started by printing a set of test pieces to check for fit:
trough_fit_test1.jpg
Once I was happy with that I made the first, hand powered prototype:
trough_concept_1_1.jpg
trough_concept_1_2.jpg
For the next revision I decided to change to a better form factor, include a stepper motor, and have some type of feeder designed to feed the track.
First up was a test of the new form factor:
trough_fit_test2.jpg
Then I moved on to making the parts for the next real test, a stepper powered auger with the capability to also back up balls to the trough. The balls are detected in positions on the track, the augers are intended to go the entire length of the track to the exit, I just haven't printed them yet.
I'm working on the PC for Spaceballs right now, but once I've got that up, my next piece to work on is getting this newest design turning through software.
infinity_trough_new_version_parts.jpg
infinity_trough_new_version.jpg
infinity_trough_closeup.jpg