I took a careful look at your model and I am not surprised it went a bit nuts. I'm going to recommend you simplify the references. The 1st few links do not have a pin connection. One is cylinder and the next is a general connection. The following are the appropriate pins and slots that I could see. However, the pin constraint is mated to cylindrical surfaces which seem to take on a "tangent" situation rather than axial (we've discussed this before in assembly mates). This is a -HUGE- overhead on the model. Each pin mate needs to be made to the axis rather than the surface. Also, simplify the translations of the pin constraints. Make the center plane coincident with the ground feature's center plane in all cases for the pin constraint translations. Also add the rotation limit to the pin constraints. +/- 90 degree from parallel center planes of each link will keep the link from flipping.
Due to the relatively busy part geometry, you might also consider using Component Interfaces to make assembling the links less cumbersome. I've been looking for a way to make the slot curves a single feature (or recognized loop) but I can't seem to make that work ...yet.
The picture below shows what I am seeing with the misaligned links due to the surface mates:
I don't know how much power your computer needs to run this full chain. That's a lot of computation.
However, you can move this chain with the motor I described earlier. You would use rotating motor like my example video, but you would put it on a linear motion motor that would drive between the two ends. It will be an interesting exercise but I would certainly make a simplified model to test the motor aspects before deploying it in this model