Anti-Lock Braking with Wheel-Slip Control

The problem

Locked wheels lose lateral grip and usually lengthen stopping distance on most surfaces. ABS keeps wheel slip near the value that gives maximum longitudinal friction while preserving steering. In EVs the motor can also contribute fast torque modulation, so understanding slip control is the base for electric ABS and traction control.

Objectives

• Build a quarter-vehicle model with wheel rotational dynamics and a slip-dependent friction curve.
• Implement bang-bang and PI slip controllers with realistic brake actuator dynamics.
• Compare stopping distance and slip behaviour against locked-wheel braking on several surfaces.
• Study the effect of actuator delay, wheel-speed noise and surface changes.
• Report a tuned controller with justified parameters.

What you need

Key concepts

Longitudinal slip

In braking, λ = (v − ω·R)/v. Zero means free rolling and one means a locked wheel.

Friction–slip curve

Friction coefficient rises with slip to a peak, often around 0.1–0.2 on dry asphalt, then falls towards the locked-wheel value. Models such as Burckhardt or the Pacejka Magic Formula describe this shape.

Wheel dynamics

J·dω/dt = μ·Fz·R − T_brake. When brake torque exceeds what friction can return, the wheel decelerates rapidly towards lock.

Slip control

The controller measures wheel and vehicle speed, computes slip and modulates brake torque to hold slip near a target close to the friction peak.

Step-by-step guide

1. 1

   Define the model and surfaces

   Use an example quarter-vehicle of about 350 kg (a quarter of a compact car), wheel inertia, rolling radius and maximum brake torque. Define friction curves for dry asphalt, wet asphalt and gravel with the Burckhardt model μ(λ) = c1·(1 − e^(−c2·λ)) − c3·λ using published coefficient sets. Plot the three curves and note the slip at peak friction for each.

   Checkpoint: Three friction curves with their peak values and peak-slip positions tabulated.

2. 2

   Build the quarter-vehicle model

   Model vehicle speed with m·dv/dt = −μ·Fz and wheel speed with J·dω/dt = μ·Fz·R − T_brake. Compute slip with a guard against division by zero at low speed, for example by stopping the simulation below 1 m/s. Implement μ(λ) as a function or 1-D Lookup Table selected by surface. Use a fixed-step solver at 0.1 ms or a variable-step solver with a small maximum step, since wheel dynamics are fast.

   Checkpoint: With zero brake torque the wheel rolls freely with near-zero slip; with excessive torque the wheel locks within a fraction of a second.

3. 3

   Baseline: locked-wheel braking

   Apply a brake torque step large enough to lock the wheel from 80 km/h on each surface and record stopping distance and time. Compare with the ideal stopping distance from v²/(2·μ_peak·g). The locked-wheel case uses the sliding friction value, so it is usually longer on asphalt. Use the same initial speed, normal load and brake torque command for every surface. Record wheel and vehicle speed traces so the moment of lock-up is visible, and compute stopping distance by integrating vehicle speed.

   Checkpoint: Locked-wheel stopping distances longer than the ideal peak-friction distance on asphalt surfaces.

4. 4

   Model the brake actuator

   Add a hydraulic actuator model as a first-order lag with a time constant such as 10–30 ms plus a transport delay of a few milliseconds and a rate limit on torque. Add wheel-speed sensor noise and a vehicle speed estimate. These realistic effects are what make ABS tuning hard and must be present before controller design. Keep these parameters fixed during controller design.

   Checkpoint: A step command produces a delayed, rate-limited torque response with the chosen time constant.

5. 5

   Bang-bang slip controller

   Implement a simple controller that increases brake torque when slip is below the target and releases it when above, with hysteresis. Set the target near the dry-asphalt peak. Run emergency stops on each surface and record stopping distance and slip oscillation amplitude and frequency. Tune the hysteresis band and torque rate to balance stopping distance against oscillation. Record the cycling frequency and check that slip never stays near one for more than a brief moment.

   Checkpoint: Slip cycles around the target and stopping distance is shorter than the locked-wheel case on dry asphalt.

6. 6

   PI slip controller

   Design a PI controller on slip error that commands brake torque, with anti-windup and output limits. Linearise the wheel dynamics around the target slip to choose initial gains, then tune in simulation. Compare oscillation amplitude and stopping distance with the bang-bang controller on all surfaces. Because the wheel dynamics change sharply near the friction peak, check the controller at the target slip on each surface rather than only on dry asphalt, and add gain scheduling with surface if needed.

   Checkpoint: Smoother slip regulation than bang-bang with equal or shorter stopping distance on each surface.

7. 7

   Robustness tests

   Test a split in time from dry to wet surface during the stop, increased actuator delay and higher sensor noise. Use a scripted loop with sim() to run all combinations and record stopping distance and maximum slip excursion. Note any case where the wheel briefly locks. Include a case with friction dropping mid-stop, such as dry to wet, and a case with larger sensor noise. Present stopping distance and maximum slip for each controller and disturbance in one table.

   Checkpoint: A robustness table showing performance degradation for each disturbance, with no sustained lock-up.

8. 8

   Steerability margin and target choice

   Lateral friction falls as longitudinal slip rises, so a target slightly below the longitudinal peak keeps more steering capability. Use a combined-slip approximation, such as a friction-ellipse factor, to estimate the lateral grip left at each target slip. Choose a target that balances stopping distance and lateral grip and justify it. Present the trade-off as a plot and explain why production systems avoid holding slip exactly at the longitudinal peak.

   Checkpoint: A plot of stopping distance and remaining lateral grip against target slip with the chosen value marked.

9. 9

   Report

   Present the models, baseline, both controllers, robustness and target-choice results. State the limitations: quarter-vehicle only, no load transfer, a known surface curve and an idealised speed estimate. Discuss how an EV motor could add fast torque modulation on the driven axle as an extension. Include stopping distances for every surface and controller in one table, the tuned parameters and the robustness results. Explain any case where ABS did not shorten stopping distance, such as gravel.

   Checkpoint: A report with stopping distances on every surface for every controller and a justified final design.

Testing & validation

• Free-rolling and lock-up behaviour of the model.
• Locked-wheel stopping distance compared with the ideal value.
• Slip regulation and stopping distance for both controllers on three surfaces.
• Robustness to surface change, delay and noise.

Expected results

A quarter-vehicle ABS simulation with tuned bang-bang and PI slip controllers that shorten stopping distance versus locked wheels and stay robust to actuator delay, noise and surface changes.

Take it further

• Use the traction motor for fast slip modulation alongside hydraulic brakes.
• Add an online friction-peak estimator that adapts the slip target.
• Extend to a half-car with load transfer.
• Generate code for the controller and test it in processor-in-the-loop.

Viva and review questions

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