I get it now. It took a couple hours of staring at Ether's equations and drawing diagrams, but it makes sense. Here was the breakthrough for me: a mecanum drive is identical to a 4-wheel omni killough/kiwi drive with smaller wheels. I've always considered this drive in terms of force, but never thought through it in terms of speed.
Compared to a traditional drive with equal-diameter wheels, an omni drive can only exert between 71% of the force going forward, due to the force vectors cancelling. But it goes 41% faster, since the wheels are moving forward while the rollers slide. The power remains the same. Make the wheels 41% smaller, and now it goes the same speed and has the same force as a traditional drive.
I get it.
Now time to get some sleep - since I don't have a robot to build like the rest of y'all.
