[Pault]

I think I have a suggestion that could give you what you want without the turning problem.

Last year my team had a 4WD long bot (actually, it could also move sideways, but I am going to leave out the features that let it do that because they don't matter right now). We used AM 8in Pneumatic Wheels, so as you can imagine turning was horrible. However, we also had 4 stingers in the corners of our robot. And each one had a slippery foot* to touch the ground. So what we discovered was that, because they were motor powered rather than pneumatic, we could lower 2 of them just enough so that our back wheels were lifted off of the floor. Now, we could turn really well while still being able to convert into high traction mode.

So, anyways, my suggestion to you would be to do the something similar. Have a little slippery foot on the back of your robot that can be lowers by pneumatics. Your turning problem is gone without sacrificing anything.

This could also be an excuse to not use transmissions. You claim that you want power and agility, but you don't say anything about speed. So if you don't mind having an average speed drivetrain, than transmissions is definitely an idea I would trash. It would be a big task to produce them, and 4 is a lot of weight to add on. If you do want speed, maybe try a 4 CIM, 4 miniCIM drivetrain. Or buy 2 COTS shifters and just use chain (this is what I would do, but I dislike direct drive, so I might be a little biased).

*Made from UHMWPE