abs-system

name	abs-system
description	Anti-lock braking system (ABS) — wheel slip control, slip ratio, hydraulic modulator, ECU logic, brake force distribution, stability, SAE J2246, vehicle dynamics.
metadata	{"priority":7,"promptSignals":{"phrases":["ABS system","anti-lock braking","wheel slip","brake force distribution","ABS controller","brake slip ratio"],"minScore":3}}

Anti-Lock Braking System (ABS) — Complete Skill

Fundamentals of Wheel Slip

Slip ratio λ: λ = (V_vehicle - V_wheel) / V_vehicle [dimensionless; 0 = free rolling; 1 = locked wheel]

V_vehicle = longitudinal vehicle speed [m/s] V_wheel = ω_wheel × R_wheel [m/s; ω in rad/s; R = effective rolling radius]

Peak friction coefficient μ vs. slip: μ peaks at λ_opt ≈ 0.10–0.20 (road-dependent) μ_peak (dry asphalt) ≈ 0.8–1.0 at λ ≈ 0.15 Above λ_opt: friction decreases → locked wheel (λ = 1) → μ_slide ≈ 0.5–0.7 ABS target: maintain λ near λ_opt for maximum braking force

Brake force: F_brake = μ(λ) × F_normal [N] F_normal = m_wheel × g + load transfer [N; depends on CG height and wheelbase]

Wheel Dynamics

Wheel angular equation of motion: I_wheel × α_wheel = T_brake_input - R_wheel × F_brake I_wheel = wheel + tire rotational inertia [kg·m²; typically 1–4 kg·m² for passenger car] T_brake = brake torque applied by caliper [N·m]

Wheel deceleration (slip threshold): If α_wheel < -A_threshold → slip increasing → reduce brake pressure If α_wheel > +A_threshold → wheel recovering → increase pressure Typical A_threshold: -20 to -40 rad/s²

ABS Control Phases (Hydraulic)

Pressure build: pump pressurizes caliper; solenoid valve OPEN (inlet), solenoid CLOSED (outlet) Pressure hold: both solenoids closed; pressure maintained constant Pressure dump: outlet valve opens to reservoir; caliper pressure released; wheel recovers Re-apply: cycle repeats (typical frequency: 5–15 Hz)

ECU deceleration/acceleration thresholds: Phase transition logic (simplified Bosch algorithm):

1. If dω/dt < -a₁ → Hold
2. If dω/dt < -a₂ (more severe) → Dump
3. If ω recovering and dω/dt > +a₃ → Build
4. Maintain until λ returns to λ_opt range

Sensor Types

Wheel speed sensor: Passive (variable reluctance): tone wheel + coil; no external power; output: sine wave proportional to speed; minimum speed ~5 km/h Active (Hall effect): square wave output; works from near 0 km/h; required for EBD, TCS Tone wheel teeth: 48–96; resolution: Δω = 2π/(N_teeth) per pulse

Reference speed estimation: V_ref = max(V_FL, V_FR, V_RL, V_RR) → select highest wheel speed as vehicle reference During braking: V_ref decelerates at estimated max −1 g

Brake Force Distribution

Static rear/front ratio: Ideal brake force distribution = dynamic weight transfer line F_rear / F_total = m_r / m_total at static; shifts forward under braking

Electronic Brake force Distribution (EBD): Controls rear brake pressure to avoid rear lock-up before ABS activates Rear bias maintained at ≤ 70% of ideal for stability margin

Brake Assist (BA): Detects panic braking (fast pedal travel rate); builds maximum pressure regardless of pedal force Reduces stopping distance 15–25% (driver hesitation compensation)

Vehicle Stability Effects

Yaw stability with ABS: Without ABS: locked front wheels → no steering response; locked rear → spin With ABS: both axles maintain slip at μ_peak → shortest stop + steering maintained

Stopping distance comparison (60–0 km/h, dry): ABS vehicle: ~36–40 m Locked wheels: ~42–50 m Benefit: 10–20% shorter stop; more important on wet/ice

Hydraulic Modulator

Components:

• Inlet solenoid valves (one per wheel, normally open)
• Outlet solenoid valves (one per wheel, normally closed)
• Low-pressure accumulator (dump fluid storage)
• Return pump (EC pump; returns fluid to master cylinder)
• Motor: 12V DC; 150–300 W; cycled at 5–15 Hz during ABS event

Accumulator sizing: V_accum ≥ V_caliper × n_dump_cycles [cm³] V_caliper = piston displacement at max pressure ≈ 0.5–2 cm³ per caliper

ABS + Stability Control Integration

ESC (Electronic Stability Control) = ABS + TCS + YAW control:

• ABS: prevents wheel lock on braking
• TCS: prevents wheel spin on acceleration
• ESC: individual wheel braking to correct understeer/oversteer

Yaw moment control: M_yaw = ΔF_brake × track_width / 2 [N·m] ESC applies brake to outside wheel (understeer) or inside wheel (oversteer) to correct yaw

Performance Metrics and Testing (SAE J2246)

ABS effectiveness metric: η_ABS = (stopping distance with ABS / ideal minimum stopping distance) η ≥ 0.75 on split-μ; η ≥ 0.95 on uniform μ

Test conditions (FMVSS 135, ECE R13H):

• Dry asphalt: μ ≈ 0.9; initial speed 80–100 km/h
• Wet asphalt: μ ≈ 0.5; initial speed 80 km/h
• Split-μ (left dry/right wet): checks yaw behavior
• Low-μ (ice/snow): μ ≈ 0.1–0.3; ABS reduces distance vs. locked

Stopping distance requirement (FMVSS 135): From 100 km/h: ≤ 57.2 m (car); ≤ 70.1 m (light truck)

Standards

Standard	Scope
SAE J2246	ABS performance test procedures
FMVSS 135	US Federal; passenger car brake requirements
ECE R13H	European; passenger car braking
ISO 21994	Stopping distance test method

Output

Provide: slip ratio target λ_opt at μ_peak condition, ABS cycle frequency [Hz], sensor type recommendation (passive/active), hydraulic modulator configuration (3-channel/4-channel), stopping distance improvement [%] vs. locked wheel at given μ, EBD rear bias ratio, ESC yaw moment authority [N·m], test standard (FMVSS 135/ECE R13H), and ABS effectiveness metric η_ABS.