# Looking for the curve traced by a moving bicycle when its steering wheel is fully rotated

I am looking for a curve traced by a moving bicycle when its steering wheel is fully rotated either clockwise or anti-clockwise.

How to model it mathematically?

Is the curve a circle?

My attempt is as follows:

Let $\vec{r}_1(t)$ and $\vec{r}_2(t)$ be the position vectors for the tangent points (between road and tires) on the rear and front tires, respectively. I know that $|\vec{r}_2(t)-\vec{r}_1(t)|$, $|\dot{\vec{r}}_1(t)|=|\dot{\vec{r}}_2(t)|$ and $\dot{\vec{r}}_1(t)\cdot\dot{\vec{r}}_2(t)$ are constants. $\dot{\vec{r}}_1(t)$ is in the direction of $\vec{r}_2(t)-\vec{r}_1(t)$.

Assuming the tires rolls without slipping then their linear velocity is the same.

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I see that $|\vec{r}_2-\vec{r}_1|$ is the fixed distance between the front and rear tires, but why does $|\dot{\vec{r}}_2|=|\dot{\vec{r}}_1|$? – robjohn Jul 30 '12 at 17:43
@robjohn: rolling without slipping. – kiss my armpit Jul 30 '12 at 17:50
However, if $|\vec{r}_1|\not=|\vec{r}_2|$, they can have the same angular velocity, but different velocities, or am I missing your point? – robjohn Jul 30 '12 at 17:56
@robjohn: If the rear tires rotate and travel $\Delta s$ unit on the road then its center moves $\Delta s$ and the front tire's center also moves $\Delta s$. – kiss my armpit Jul 30 '12 at 18:20
The only way that both could follow the same circle is if both tires were at the same angle with respect to the body. Each tire moves perpendicular to its axis. – robjohn Jul 30 '12 at 19:50

It almost has to be a circle, as if the steering wheel isn't moving then the curvature of the track has to be constant.

One way to model it is to imagine just two wheels (like a bicycle). If you draw line for the direction of travel of each wheel you can find a circle that is tangent to both lines. This would allow you to get a relation between the angle of the front wheels and the radius of curvature. The next step would be to handle the width of the car, but then you need the wheels to slip or to rotate different amounts.

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It depends on the amount of detail you'd like to have in your model. A simple dynamic model, like the single track car model allready reveals that the curvature is a function of forward velocity and tyre sideslip angle. This is due to oversteer/understeer properties of a car

you can take a look at

http://www.unibw.de/lrt1/gerdts/lehre/praktikum-optimale-steuerung/einspurmodell.pdf

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The usual model I have seen has the front and rear tires moving perpendicular to their respective axles, and the rear axle is perpendicular to the body of the vehicle (bicycle, car, or bus). This means the centers of the front and rear tire(s) trace circles as pictured below:

$\hspace{4.5cm}$

Note that both the red path, rear tire(s), and green path, front tire(s), are perpendicular to their respective axle. The fixed structure of the vehicle between the centers of the front and rear tire(s) is the black line and is tangent to the path of the rear tire(s). The front end of the black line is not tangent to the path of the front tire(s) because the axle is not perpendicular to the structure when the vehicle is turning.

The triangle formed by the centers of the curvature of the paths of the front and back tire(s) and the middles of the front and back tire(s) forms a right triangle with the right angle at the rear tire(s). Let $R_f$ and $R_r$ be the radii of the circles traced by the front and rear tires, and $L$ be the distance between the centers of the front and rear axles. Then, we get $$R_f^2=R_r^2+L^2$$

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