# How to calculate the radius of a curve based on vehicle wheel speeds of a car

I have data of the 4 differen wheelspeeds of a vehicle during driving time. Based on that I want to calculate the radius of a curve it is taking. The following are my current thoughts that seem to be errorneuos:

1. Each curve or at least each part of a curve can be approximated by a circle, so:

B = Width of the vehicle
r = Radius of any Curve
UI = Circumference of inner curve the vehicle takes
UO = Circumference of outer curve the vehicle takes
UD = Differences in circumferences of the curves

UD = 2 * pi * (r + B) - 2 * pi * r = 2 * pi * B

Now I have the differences in distance the wheels of the car take in a clean circle.

1. With this Distance I want to calculate the speed difference:

VI = Inner Speed
VO = Outer Speed
t = Time
VD = Speed Difference

VI = UI/t
VO = (UI + UD)/t = (UI + 2 * pi * B)/t
VD = VO - VI = UD/t = (2 * pi * B)/t

Now the problem is that I lost every circle parameter (besides B) on my calculation. The interpretation would be that no matter the curve, the Speed difference is always 2 * pi * B per time. This is kinda contraintuitive as I thought, that the speed difference of wheels should be way higher for smaller radiuses.

Can someone help me to understand where I am wrong?

Regards

• <This is kinda contraintuitive as I thought, that the speed difference of wheels should be way higher for smaller radiuses.> the speed is angular velocity times the radii and if angular velocity is same for all parts of the vehicle, why you think so? Aug 13, 2018 at 17:18
• Hm , so a difference in linear Speed of two wheels means a curve, right? And the higher the difference the sharper the curve. So I thought I could go the other way and calculate the difference in velocity based on the curve. What confuses me is, how the radius can get dropped out of this calculation as it should be an indicator for the speed difference. (Sorry if this is basic ^^) Aug 13, 2018 at 17:26
• @Gring-the explicit appearance of the radius is not there as you have written B= difference of radiuses and w the ang. vel. is very much there Aug 13, 2018 at 17:31
• B is the width of the car, so it won't change for different curve sizes. So lets say I have V1 in m/s as the value of the left front wheel and V2 for the reight front wheel. How can I describe the curves the car takes based on the difference of V1 and V2? Aug 13, 2018 at 17:45
• Strongly reminiscent of this classic puzzle: en.wikipedia.org/wiki/String_girdling_Earth Aug 13, 2018 at 21:14

The two wheels roll around different circles in the same time. The difference in speed is proportional to the difference in circumference, which is always $2\pi$ times the difference in radius, which is the width $B$ of the vehicle : $$C=2\pi R$$ $$\Delta C=2\pi \Delta R=2\pi B$$ The difference in circumference, and therefore also the difference in the speed of the wheels, is independent of the radius $R$. So it is impossible to find radius from difference in speeds.
However, you can determine the radius $R$ of the turning circle from the ratio of speeds of the two wheels, as follows :
Suppose that the car turns through a small angle $\theta$ on a circle of radius $R$ as measured from the centre of the vehicle, which has width $2b$. Then the wheels trace out arc lengths $s_1=(R-b)\theta, s_2=(R+b)\theta$. Their speeds are in the ratio $$\frac{v_2}{v_1}=\frac{s_2}{s_1}=\frac{R+b}{R-b}$$ The radius of the turning circle is therefore $$R=\frac{v_2+v_1}{v_2-v_1} b$$