It may take some time till I can get to this as we are in the middle of mid-terms exam, but this looks like an interesting problem that I want to pursue.
If someone else doesn't respond with solution and this goes several days - be sure to Bump it back to the top so that it isn't overlooked.
I think the solution is really a quite simple 2D contact and 3D contacts as the respective bottles stay in the same plane, but why is the rod 1931 not long enough to contact all bottles? Isn't that what pushes the bottles?
There are some parts missing from your assembly attachment.
I am not concerned with the missing fasteners, but what is part 1908 and part TEST PIN?
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