Wheel Slip and Tire Forces

A driven wheel always slips a little: its rim moves slightly faster than the ground under it. The grip force rises with slip up to a peak, then falls as the wheel starts to spin. Where that peak is, and how high, depends on the surface.

Surface Motor torque Traction control

Slip and grip

With wheel speed $\omega$, radius $r$, and vehicle speed $v$, the slip ratio and the grip from a simplified Pacejka "magic formula" curve are

$$s = \frac{\omega r - v}{\max(\omega r,\ v)} \qquad F_x = \mu_{\text{peak}}\,N\,\sin\!\left(C \arctan(B\,s)\right)$$

What to notice

On dry asphalt the peak grip is high and comes at about 10 percent slip. On ice the whole curve is far lower. With too much torque, the wheel passes the peak, slip shoots up, grip drops, and the wheel spins while the vehicle barely moves. Traction control watches the slip and cuts torque to hold it near the peak, which gives more push than full power does.

The same curve limits braking and cornering. For robots on grass, gravel, or wet floors, wheel slip also ruins odometry, since the wheels turn without the robot moving.