How small is the smallest circle a car can drive?

Question

Lets say we have a model of a car with two fixed back wheels and two wheels in front that steer in the same angle:

• The wheelbase $w$ is the fixed distance of the two wheel axes.
• $\alpha_m$ is the maximum angle we can steer

What is the radius $r$ of the smallest left-circle we can drive with wheelbase $w$ and maximum left angle $\alpha_m$?

Clarification: You can suppose that the car has only one frontwheel and one backwheel.

(Does anybody know if there are books about "car physics" that deal with such questions? I think there has to be plenty of material about it, because this might be important for every car racing game. I was just thinking about how the position and orientation of a car changes when it drives for $t$ seconds with initial orientation $\alpha$ and position $(x,y)$, steering $\beta$ and velocity $v$. But I even can't answer the question above at the moment :-/ )

Answer 1

Getting an exact answer will be tricky, because it will involve issues like how the wheels slip on the road, how the differential(s) work, etc. But I suspect you could get a rough estimate, and a sufficiently interesting problem, by considering a "car" with only one front wheel and one rear wheel, and finding the circle such that both wheels are tangent to the circle at their centers.

Edit: I realize, looking at the first diagram in the paper J.W. Perry linked to, that my estimate is fairly terrible. A much better one allows the front and rear wheel to be on different circles. It should be fairly straightforward to see how to do that. Page 382 gives a two-wheel simplified diagram that should help.

Answer 2

But this is not how cars work! Obviously the radius of the inner circle is smaller than the radius of the outer circle, so the front wheels can't be parallel. If you insist that they be parallel, in defiance of real-world auto design, then the wheels will inevitably slip, and the answer is impossible to calculate using mathematics alone.