Quarter-Car ABS Braking

What is modelled

One wheel of radius r and inertia I carries the normal force N = m g. Its longitudinal slip is s = (v − ωr)/v and the road gives a friction force μ(s) N. The Pacejka magic formula describes μ(s): it rises to a peak at a slip of about 10 to 20 %, then falls toward the locked-wheel value at s = 1. A locked wheel therefore brakes worse than a wheel held near the peak, and it cannot steer. ABS releases and re-applies brake torque to keep the slip near the peak.

Equations

• μ(s) = D sin( C arctan( B s − E (B s − arctan(B s)) ) ).
• m dv/dt = −μ(s) N, I dω/dt = μ(s) N r − Tb.
• Locked wheel: d = v0² / (2 μ(1) g). Constant slip s*: d = v0² / (2 μ(s*) g).
• Peak slip s_p: dμ/ds = 0.

Checks run in the page

• Simulated locked-wheel stop against v²/(2 μ(1) g).
• Peak slip found by golden-section search against a brute-force scan.
• Simulated stop at constant slip against v²/(2 μ(s*) g).
• Closed-loop ABS stop lies between the ideal-slip distance and the locked-wheel distance on dry road.

Honest simplifications

Pure longitudinal slip, a single wheel with fixed load (no weight transfer), a rate-limited bang-bang controller instead of a production hydraulic modulator, and a speed-independent friction curve. Real ABS tuning also handles lateral force, road changes and sensor noise.

• Choose Locked wheel on dry road, then Wet road with ABS, and compare the stopping distance read-outs.
• Choose Ice with ABS and see the long stop that even ideal slip control cannot avoid.
• Move the ABS target slip away from the peak and watch the distance grow.