This is based on my understanding from freshman physics at Cal. One of the reasons explanations tend to be so poor is that (a) this is not generally taught in an engineers' physics classes and (b) physicists are just now beginning to fully understand the nano-scale forces at work between two bodies in contact.
The basic f = uN model only applies to very hard/rigid materials on an incompressible surface. Therefore, the model is broken by treads/tires in several ways.
(1) Treads are rubber, and thus compress (changing the orientation of their atoms and thus the coefficient of friction.
(2) Carpet is also "squishy" to a degree, and thus violates the model in the same manner as above.
(3) The pile (fibers) of a carpet form a small layer ABOVE the "main" surface of the carpet, meaning that your static friction is dependent on how far you have "sunk" into the carpet and thus how long you've been sitting there.
(4) The rubber that tread is made from is molecularly composed of VERY long hydrocarbon chains (think sort of like a pearl necklace). These chains then are tangled into an "elastomer" structure, like a big bundle of cords thats bunched up (and stretches out, thus the strechiness). Because of the girth of these molecules and the uneven electron density between the carbon "backbone" of the chain and its outer hydrogens, significant Van der Waals (VdW) forces occur between the molecules of the two surfaces at contact point, essentially forming very weak "temporary bonds" which require energy to break (i.e. move). See "dispersive adhesion" on Wikipedia.
We now understand VdW forces to be the reason that geckos can stick to surfaces so well. Their feet are coated in thousands of hairs (which then split into even smaller sub-hairs like a tree) called setae. These hairs' miniscule size and sheer number allow the gecko to exploit the same polar/VdW forces mentioned above, and stick to a surface using even just one toe.
I know this is long, but I hope it helped. I guess the moral is that friction is one thing but "sticktion" is another. Gratefully, this years' playing surfaces are essentially ideal as far as the "standard model" for friction goes, so you won't have to worry about making gecko wheels for your bot until next season
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