# How to calculate pressure exerted on the wheels of a robotic car?

I need some help in designing my robotic car. So its going to have 4 wheels, each driven by a 12-volt motor. It occurs to me that the weight of the chassis itself will exert some pressure on the wheels and squeeze them against the ground, right? Thereby increasing friction. So is there a way to calculate the pressure on the wheels, that way I can design the body accordingly.

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What's to calculate? Just take all the components and weigh them. – Mike Dunlavey Jun 1 '13 at 15:37
If you inflate the wheels well enough, the contact is friction is not going to be your problem. – Mathusalem Jun 1 '13 at 15:55

You have to know where the center of gravity is. If $a$ is the % distance along the wheelbase for the center of gravity (50% = center, 0% = front, 100%=back), and $b$ the % distance along the track for the center of gravity (50% = center, 0% = left, 100% right) then the weight fractions for each wheel are:

$$(\mbox{front-left}) = (\mbox{Weight}) \frac{3-2a-2b}{4}$$ $$(\mbox{front-right}) = (\mbox{Weight}) \frac{1-2a+2b}{4}$$ $$(\mbox{rear-left}) = (\mbox{Weight}) \frac{1+2a-2b}{4}$$ $$(\mbox{rear-right}) = (\mbox{Weight}) \frac{2a+2b-1}{4}$$

There equations come from the balance of moments in two planes. Here is a top view of the balance.

## Example:

I the front back-balance is $a=0.4$ and the left-right balance is $b=0.55$, with a chassis weight of $0.5\,{\rm kg}$ then the corner weights are:

$$(\mbox{front-left}) = (0.5) \frac{3-2*0.4-2*0.55}{4} = 0.1365 {\rm kg}$$ $$(\mbox{front-right}) = (0.5) \frac{1-2*0.4+2*0.55}{4} = 0.1625 {\rm kg}$$ $$(\mbox{rear-left}) = (0.5) \frac{1+2*0.4-2*0.55}{4} =0.0865 {\rm kg}$$ $$(\mbox{rear-right}) = (0.5) \frac{2*0.4+2*0.55-1}{4} =0.1125 {\rm kg}$$

## Results Check

• Total weight on left wheels = $0.225 {\rm kg}$
• Total weight on right wheels = $0.275 {\rm kg}$
• Left-right balance = $b=0.275/0.5 = 0.55$
• Total weight on front wheels = $0.300 {\rm kg}$
• Total weight on rear wheels = $0.200 {\rm kg}$
• Front-back balance = $a=0.200/0.5 = 0.4$
• Total weight = $0.500 {\rm kg}$
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There are two separate sources of friction you need to worry about there. First is just the bearing friction -- the force due to the bearing used on your axles. This is so complicated that there's no way to really calculate it; you should just measure it. And it's quite possible that it will change as the bearings wear or lose lubrication.

Second is rolling resistance, which is what I suspect you were referring to. This is extremely complicated, and depends on the details of the surface the car will be rolling over; whether there are any obstacles; the details of the size, inflation, and flexibility of the tires; etc. But, if you have a rigid tire of diameter $d$, and can estimate the "sinkage depth" $z$ that the tire sinks into the surface, the effective coefficient of friction is $\sqrt{z/d}$. So to get the force the motors will need to supply to overcome this, just multiply the weight of your robot by $\sqrt{z/d}$. Of course, $z$ depends on the weight of the robot and the surface area of the wheel contact, so this might not be useful a priori.

Alternatively, if your surface is rigid but your wheels are soft, you can get an estimate by thinking of the "sinkage depth" of the surface into the wheels -- which is basically the amount of squish by which the wheels deform under weight. But, as Mathusalem says, the solution there is just to inflate the wheels.

(It might also be useful to note that if your surface is a little uneven and your chassis is rigid and your wheels don't have suspension, there will be situations where only three wheels are in contact with the ground. So if you divide up the force, it might be better to divide it by three rather than four. And, as ja72 points out, it may be necessary to consider the distribution of weight, so that you're not overloading one wheel, for example.)

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