Ackermann Steering and the Bicycle Model

A car steers by turning its front wheels. On a curve the inner wheel travels a smaller circle than the outer one, so it must turn more sharply. Ackermann steering geometry makes all four wheels roll around the same center point, so none of them skid.

Steering angle Wheelbase Track width Speed

The bicycle model

Merge the two front wheels into one and the two rear wheels into one, and the car becomes a bicycle with wheelbase $L$. With steering angle $\delta$, the rear axle moves on a circle of radius

$$R = \frac{L}{\tan\delta} \qquad \dot\theta = \frac{v}{L}\tan\delta$$

Ackermann geometry

For no skidding, every wheel's axis must point at the same center. With track width $w$, the inner and outer front wheels turn by

$$\tan\delta_{\text{in}} = \frac{L}{R - w/2} \qquad \tan\delta_{\text{out}} = \frac{L}{R + w/2}$$

What to notice

The dashed lines through each wheel's axle all meet at one point on the line of the rear axle. The inner wheel always turns more than the outer one. A longer wheelbase makes a wider turning circle for the same steering angle. Car-like robots, from self-driving cars to farm robots, cannot turn in place, so their planners must respect this minimum radius.