Connecting a piston to a shaft

We’re looking at building a transmission. We know to build the gear train, but how do we move a shaft with pnuematics?

You don’t move the shaft, you move the gears. Look at the VCAT worlds reveal.#303

Here’s a concept you could use to move a rotating shaft using a pneumatic cylinder. Screw the hex nuts tightly to clamp the flat bearing to the cylinder rod (you’ll need to sand off the bearing nubs), and leave the shaft collars on the moving/rotating shaft a little loose, maybe even slip in some teflon washers.
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You can move the shaft. We had a working transmission this year that did exactly that. More than one way to skin a cat…(what a lovely saying)

I would also look at the Green Egg Robotics transmission reveal. It’s like the VCAT transmission only it is shown in more detail.
https://www.youtube.com/watch?v=eC_tcbsR2HU
However, I would highly advise you to use a direct driven drive instead of a transmission. The loss in power you get from a transmission is not usually worth the ability to switch gears. Last year I built a robot that transferred power from an 8 motor turbo drive to an 8 motor flywheel, but the weight and added friction of the transmission made the drive burn out after about 2 minutes. The problem was partially because of the build quality, but the transmission added about 6 or 7 pounds.

I know that our C team has had a thing for transmissions, you can briefly see a pretty effective one right here:

Might help you in designing the transmission.