I think 234 tried to do a study a few years back and got fairly inconclusive results.
I will say, my perspective in the great traction debate is:
The most important factor is weight and wheel material. This will give you a fairly accurate, but not perfect model.
After those, the likely most critical factor is typically referred to as "static" vs. "dynamic". Often two co-efficients of friction are called out, one for static or rolling, and the other for slip. This is a better model, but still not 100% accurate for compliant materials (like say rubber or carpet).
Typically, more compliant compounds (softer) are "stickier" and provide more grip, but also usually wear out quicker.
To get more accurate for a rubber based wheel, you might want to use a slip ratio model:
https://www.hondata.com/help/tractio..._operation.htm
Note, it this chart it shows that 8% slip is where traction tends to peak. With two static elements, this might look like the tiny amount of deformation you see as you pull on an object, right before it starts sliding.
While this model works well for traction control, it can yield some very weird results with launch control as it effectively says at 0% slip, you have a CoF of 0 as well... Back to using that static value for the static launch condition. The decrease with % slip can help explain why with the same wheels, having a super low ratio vs. just a low ratio can help you push better. IE, with a typical FRC drivetrain with a 2 speed, you have the ability to spin your wheels if top speed is geared below about 6-7 FPS for most drivetrains.
With the static and dynamic CoF model, having a lower ratio below being able to break the tires loose would be silly and ineffective. In practice, given the same weight, same wheels, a 4 FPS drivetrain still tends to outpush a 6FPS drivetrain. This may be because the lower speed drivetrain is operating at a lower slip ratio....
Lastly, a little bit on surface area. It Matters in compliant models, just maybe not as much or how you might think.
There are 3 schools of thought (I am aware of) for treaded tires in soft soil:
Keep them narrow enough to "dig in". This tends to work in really soupy material.
Keep them wide and big to maximize surface area: This tends to be large "floatation" style tires or tracks. The idea there is to interface with so much material that you are not shearing the ground below you. For 4 wheel drive trucks driving in sand, it may be recommended to drop tire pressure to help with this.
Add flaps/cleats to throw the mud which creates thrust. This is usually very specialized for sand rail and mud dragsters. Some amazing amounts of acceleration can be generated this way.
On a hard surface with a compliant tire, a few different effects can happen relative to surface area. The first is some rubber compounds can actually be sticky. If they are sticky, then more surface area = more stick = more traction. The second is that when slip occurs, the rubber will get warmer. Typically as it gets warmer it gets stickier. However if it gets too warm, the rubber can begin to break down and it will get slicker then. Sometimes really slick which racer call "getting greasy". A wider tire will tend to heat up slower which means it will take a bit longer to "get greasy". Also when it is slightly heated, it has the extra grip from the stickiness perspective.
You can feel the wheels get hot with robots in a pushing match, and if you use a thermal camera, you can sometimes see trails of hot carpet following the robots being pushed. Sometimes the wheels and carpet get hot enough to melt.
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Back to what this has to do with robots:
If pushing is important, use as much weight as you can.
Pick the grippiest tires or tire combinations.
Understand a little slip can be increase traction, but a lot of slip reduces traction. Having the right ratio and driver control can make a big difference here.
More surface area is the last parameter, but can be add that extra bit, especially if it has mechanical interlocking features to the carpet.
Anyway, that is my $0.02...