Contact Area and its Relation to Friction?

Ok, I’ve searched around and have found many posts on CD dealing with this question… but no one has given a concrete answer.

As many of you know… a normal force * coefficient of friction = friction force… at least in everybody’s text book.

Obviously more contact area increases traction, or else race cars would have little skinny tires, not big fat wide tires. So… where does contact area factor in to friction force? I think it might have something to do with the non-homogeneous nature of tires and carpet or tires and asphalt or whatever the contact situation may be in most real world tire/track/other land propulsion applications… but I don’t have an equation to give me friction force with non-homogeneous surfaces in that case.

I’m hoping i don’t have to break down and dervie a formula from empirical data, but if thats what has to happen so be it.

Does anyone have a concrete (preferrably mathematical) answer to this vexing (or first-ing :wink: ) problem?

Thanks CD’ers/FIRST’ers,


Andy Baker or Raul Olivera will have a good handle on this question. You may want to PM them or use the email option directly in case they haven’t seen this thread.

This isn’t exactly a mathematical answer, but a quick search on google gives this link:

Remember that most physics books deal in an ideal world where most things are point masses and all surfaces are completely uniform and flat.

Consider too that since Force=Pressure*Area, that as area goes up, pressure can go down resulting in tires that can be made of softer materials or in the case of pnuematic tires, thinner walls because they need to contain lower air pressure.

(but I’m a Computer Engineer not a Mechanical one, and could be completely wrong too)

At their operating temperature, race car tires are actually sticky, so they get traction related to contact area, regardless of the weight. They’re also made of soft and relatively weak rubber, so they need to be wide in order to spread out the force and keep from tearing under acceleration.

Dragster tires are extremely wide because dynamic friction does depend on contact area, and they want to maximize the acceleration even when the tires slip.

In 2003, we TechnoKats did a bunch of friction tests to compare the coefficient of friction between various types and sizes of rubber on the HDPE ramp. We also compared the forces required to push a robot on carpet depending on the type of drive base it had (wheels vs treads).

We found that there is a very, very slight (under 5%) advantage to having a larger surface area on the plastic. This could have been due to a number of things… but each time we pulled on a larger area sample, it took a bit more force to move our object.

We also found this same slight advantage (about 5%) when comparing treads to wheels on carpet. A treaded robot was a bit harder to move.*

In both cases, there is not an ideal flat-to-flat surface interaction. I believe that the mechanical interaction between one surface and the other creates this slight advantage for larger surfaces. It’s easy to see between wheels and carpet, and it’s at a smaller scale between rubber and HDPE.

    • This test was enough to prove to me that treads on a FIRST robot were NOT worth the effort. We did treads in 99, 01, 02, 03, but not ever since. We were very surprised at the results and did not see the 5% advantage justify the effort to do treads. Even though there is this 5% increase in the friction coefficient, a track system has more efficiency losses (maybe as high as 5% more, to offset the friction advantages) and uses more hardware to make the system weigh more.

I’ll contend that a wheeled FRC robot will push with as much force (within 3%) as a tank tread robot, as long as some conditions are met:

  1. The robots weigh the same
  2. The robots have similar Cg locations
  3. The tread material for the wheels and treads are the same
  4. There is a PID control system for the wheels (and the treads, to make it fair) so they don’t slip

All of these friction comparisons above were between STATIC situations. During a static comparison (when the wheels or treads are not spinning or moving), then the friction is very close. I believe that in FIRST, a wheeled robot with good treads on the wheels will hold it’s ground very well until the wheels start spinning and DYNAMIC traction starts to come into play. Possibly the reason why treaded robots push around wheeled robots at times in FIRST is because wheeled robots get into dynamic friction situations by not having a traction limiting program, like PID control.

This is definitely an area where more testing is needed. I am sure that others out there have opinions on this, and I am eager to see what people say.

Andy B.

Friction is weird, so yeah, you probably do have to derive a formula from empirical data.

An example of friction being weird: Most material interfaces have a higher coefficient of static friction, than of dynamic friction. But aluminum to aluminum has a higher coefficient of dynamic friction than static friction.

And when you consider automobile tires, think about what might be happening when the situation in my avatar occurs…

Good luck!

Possibly the reason why treaded robots push around wheeled robots at times in FIRST is because wheeled robots get into dynamic friction situations by not having a traction limiting program, like PID control.

PID Traction Control – unless each wheel on a given side is indepedently driven from the transmission, what reliable way exists to tell the difference between wheel speed and robot speed? Being able to tell the difference in traction loss vs. the robot turning seems even more difficult to grasp.

Another impact of contact area is in turning itself. We all know that the bots turn different on a hard floor like concrete than on the competition carpet.

<speculation>The carpet under the wheel locally acts like a spring. We know that things sitting on carpet sink in some amount (especially obvious when moving furnature) and the the area of carpet engaged affects how far into the carpet the item sinks. Therefore by increasing the size of the wheels contact patch (relative to the other wheels) we can affect the amount the wheel sinks into the carpet. <edit> …by making the larger carpet patch behave like a stiffer spring</edit></speculation>

How does this impact the friction? By moving changing the normal forces. It won’t change the overall pushing capacity of the bot, but it will allow us to change which wheels have the majority of the traction. This is really what we are doing when we drop the center wheel of a 6x6 after all isn’t it. So by widening the center wheel we can move the normal forces to the center axle which lessens the normal loads at the corners… reducing skidding forces… making it easier to turn… with less wheel drop (and rock)… without giving up pushing force.

Attached is a crude spreadsheet attempting to explain my point.

p.s. I am working on getting hard data to back-up my point, but until then feel free to roast me.

Local Normal Forces - 20071031.xls (30 KB)

Local Normal Forces - 20071031.xls (30 KB)

I also don’t understand the idea of PID traction control. I’ve heard of PID velocity control using encoders, and I’m planning to implement that on our robot this year, but I don’t understand the idea of PID traction control. How would you do that?

Jesse, there are some ways. One way is to have a passive wheel on an independent axis in the center of your robot, and encode it. It will only move if the robot is moving. You could also use mouse sensors or trackballs to accomplish similar things, I think.

This thread is pretty well covered. However, the math and actually what happens on the FIRST field regarding friction are two different things. Mark I like the spreadsheet some interesting data there nice work. My suggestion regarding this is test,test and test. In addition to that remember the basics for instance you want to be heavy as possible without being overweight especially with a game like 07’s. watch your CG and keep it as low as possible and finally keep your tread in keep in good condition at all times. Besides this experiment with different materials and tread widths, design is an on going process. Finally, you don’t want to spin your wheels in a pushing match, thats how you get pushed. The easiest way too figure this out is to put the bot pushing against the wall and record the numbers from dashboard when the wheels slip then set them up in programming as limits. Obviously do this while the bot is on a practice field were real carpet or as close to real carpet is present to get the most accurate results. Just a few things my team is doing regarding this issue

my two cents,

I find this concept very confusing. So it takes more force to continue to move aluminum against aluminum then it does to start moving it :confused: I read this and I thought to myself… well that just can’t be right, so I looked elsewhere to verify it, and go figure it’s true. I just don’t see how that works.

PID traction control is pretty simple, just not anything that’s typically done in FIRST. The whole point is to keep your wheels from slipping. Wheels slip when applied force exceeds the static friction force. Applied force is proportional to applied torque which is (mostly) proportional to motor torque which is proportional to motor current. So your goal would be to PID control the current being supplied to (or sourced from) the motors. Current-mode motor drivers and amplifiers are awesome for this, but we don’t have any, sooo the idea would be to use a solid state current sensor on your motor leads, and PID control this.

Now, I’m not sure our available loop rates are really adequate for good stable control of this current, but you could certainly easily implement a simple controller to back-off on commanded PWM signals to keep the current in an acceptable bound that you know won’t slip.

The above means I’m slightly skeptical about programming hard limits into your code based on what PWM values caused slipping at a stand still against a wall. The PWM values control (to a 1st order approximation) the voltage that you’re applying to the motor, not the current. As a motor spins, it creates its own source of voltage (back EMF) proportional to the motor speed that cancels most of this out, with the left over voltage differential driving the current flow and thus creating torque. What this boils down to is that, if your robot is moving forward while you’re commanding a constant voltage, you’re theoretically applying less torque than you could get away with. Annoying, but not really a problem. However, if you’re applying your constant voltage and still getting pushed backwards without slipping, this actually increases the torque you’re putting out. Which means that if someone pushes you back fast enough (and it doesn’t have to be a lot if you’re cutting things close) you’ll suddenly put out too much torque, spin your wheels, and very rapidly lose the pushing match. Plus, you’d be pointlessly limiting your top speed when you’re not pushing someone, because you’ve simply put in a limit to how much voltage you’ll put out and, thus, what your top speed is. Now, there are ways to compensate for this using velocity feedback and such, but they’re not going to be as accurate and reliable as a current sensor feedback.

I apologize for continuing off on a tangent, but I feel we’re on a roll with the tangent and it’s pertinent to the original topic to an extent. The biggest advantage I see for traction control is the ability to climb rough terrain (ramps) without too much driver input.

We too use the encoders for PID velocity control in order to keep the robot driving straight at high velocity.

Hmm, after a bit more thinking the mouse sensors seem easy enough to do if you have 1 mouse sensor on each side – even though the PID control, for perfection and theory, would slightly change during a turn (higher I value) than in a straight (higher P value). I’ll have to bring this up to the drive train design team tonight to see if we can focus a bit of time experimenting with it.

This might be a good start for general traction control as I know exactly what you’re talking about. We definitely need some data before we can come up with anything concrete for limiting values however, and be able to test many scenarios to make sure it dynamically understands turns vs straights. We’ll also have to review the rules on custom circuits since this sensor would be inlined with the motor leads. Bah, such a great idea so little time!

Similar sensors were legal and included in the Kit a few years ago. In fact, R62 and R63 from last year make it clear that these are legal. And yes, making the control adapt properly to the dynamic nature of the FIRST field would be challenging. I think true traction control would be decidedly difficult, and would basically end up monitor wheel velocity and motor load and maybe a few other factors to decide if the wheels actually are slipping.

Grippy non-linear materials like natural rubber, when on relatively smooth surfaces, can have a higher effective coefficient of friction at lower pressures–so for a fixed robot weight, larger contact patches can give higher friction becuase the rubber of the contact patch is under less pressure.

For an experiment showing the non-linear coefficient of friction of rubber (higher coefficient with lower load on the interface) see particularly graph 1 and graph 3. See also toward the bottom of the page, where you find the statement:

“The confusion here comes from the fact that rubber has a very unusual property. The more lightly it is loaded, the higher its apparent coefficient of friction.”

Of course carpet can change everything, so you need to experiment for yourself.

So… the consensus is that contact area has little or nothing to do with friction? :confused: But it does? :confused: :confused:

Ok… i’ll go get empirical data sometime…


QBranch (aka Alex) wants a definitive answer and he will get it right now.

Static friction force does not depend upon surface area. Static friction force does not depend upon surface area. Static friction force does not depend upon surface area.

This assumes one major thing: The surface pressure between the two items is low enough to not cause material failure at either surface (wheel/tread or carpet).

You must make your wheel width wide enough to not rip up the carpet and not yield your rubber (at least, too much). You should design your wheel width to not fail either material. Once you have done that, the width doesn’t mean squat.

Alan is correct about dynamic friction: surface area plays a bigger role.

I have posted numerous times on this and the width does not matter.

I will not argue with any of you about this. I am as certain as can be on this issue … believe me.

Word. Thanks, Paul. Definitive answers are a good thing. (**emphasis **mine)

You can change this slightly and avoid certain potential counterarguments:

Static friction force depends only on the coefficient of friction and the “normal force” (weight, for horizontal surfaces).

There are boundary conditions for some combinations of materials where the coefficient of friction can change based on pressure, and since pressure depends on area and force, changing the area can affect the friction force, but the static friction force still depends only on the coefficient of friction and the normal force.

So if width doesnt mean squat quick question. in st louis us (1625) had to face 217 and 148 at times 217 had 6 wheel drive with im guessing 1inch wide tires and 148 had 6 wheel drive with 2 inch wide ifi traction wheels. both with what i believe to be identical tread. yet we could push 217 easily and 148 we tied head on. we had a 3 speed 4 wheel swerve drive with 1.75" wheels covered in lower cof blue nitrile roughtop from mcmaster. any explanation? my next years plans already inclue 2.5"wide wheels at the moment