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Both had two driven wheels, and two casters. One bot had two andymark kit wheels per axle for driving, and the other had only one per axle. I added extra weight on each in order to make the weight exactly equal. I wired both motors to a single battery and switch. Then I put the two drive bases head to head, set so they would drive directly into each other, on an area of FIRST carpet. When turned on, the base with 2 wheels per axle could overcome and outpush the other base every time. We did this 12 times, each time changing to a new, fresh battery.
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After a bit more thinking, I do not have the answer to why this particular scenario happened. However, I do have a solution for how to avoid it -- move to 4 driven wheels on your drive train.
Since friction for a given wheel is dependant upon the force of gravity exerted by the mass of your robot, a 2-driven-wheel system needs most, if not all, of its weight centered directly over the drive axle for maximum traction.
Changing to a 4-driven-wheel system spreads out where in your system the center of mass can be and still practically contribute to all of the wheels' traction. Come to think of it, I think someone previously in this thread mentioned it, and most of the drive train guides I've come across have the underlying assumption Center of Mass rests somewhere in between 4 driven wheels.