I got a lot of ideas from the American Textile History Museum in Lowell, unfortunately now defunct. Once the weave shed supervisor Michael Christian took me around, answered all of my questions, and ran some of the machines for me.
This is a braiding machine, based at one at the ATHM. Three spindles travel around a figure-8 track, pushed by two chain link loops. To make the spindles move in a figure 8, rather than just one loop or the other, an arm at the end of the center path pushes the spindle to the other side on each pass. The gearing to make the arm swing in time with the chain link conveyors required calculation to have exactly the right timing. The spindles held yarn, the figure 8 movement caused the yarn to braid. The worm gear driven wheels to the right draw off the braided yarn. Let's see if I can dig up a video too.
Here's a view of the bottom, with the carefully designed gears to the left
The big project, which I have attempted a few times, is the loom. I built this one to be controlled by the RCX, with just 2 motors (not steppers) and 2 touch sensors. Instead of a flying shuttle, this version passes the shuttle back and forth using a long arm... the arm passes the shuttle to a catcher on the other side, releases it to the catcher, and retracts to allow the heddles to reverse. Then the arm goes back though the shed to retrieve the shuttle. Touch sensors determine when the arm has moved far enough in each direction. A second motor handles the heddles, beater bar, and the winch for completed cloth. This version wove yarn, with 14 strands of warp.