Clean MATLAB implementation of the Pacejka simplified four-parameter Magic Formula tyre model — lateral and longitudinal — with load-sensitive peak coefficient and a friction-envelope view of combined slip.
Originally implemented during the Vehicle Dynamics Modelling and Simulation course at Hochschule Trier; this repository is a focused standalone extraction of the tyre model used inside both the vehicle dynamics models toolkit (where it sits inside the 14-DoF full-vehicle model) and — planned — the v06 release of the GT3 Lap Time Simulator, where it will replace the simplified μ(F_z) surrogate currently used for QSS calibration.
Lateral force F_y vs slip angle α for four vertical loads. The classic Pacejka shape: linear at zero slip, peaks at a slip-angle around 5–10°, and degrades beyond that. Higher load gives more peak force in absolute terms but lower peak coefficient F_y / F_z — the degressive friction behaviour every race engineer designs setups around.
Longitudinal force F_x vs slip ratio κ for the same loads. Peak occurs at lower slip than lateral (typically κ ≈ 0.10–0.15) and is the basis of traction control and brake balance design.
Real race tyres are tuned per-axle. The 14-DoF model uses different Magic Formula coefficient sets front and rear (Cf=1.2947, Bf=0.0813, Ef=-8.3966 vs Cr=1.2617, Br=0.0925, Er=-8.7012); this plot at a single representative load shows the resulting asymmetry. A symmetric coefficient set hides the understeer / oversteer balance behaviour the suspension has to deal with.
The Kamm-ellipse approximation of combined-slip behaviour: peak F_x and peak F_y (computed from pure-slip Magic Formula) define an elliptical envelope inside which the tyre can deliver any combination of longitudinal and lateral force. Operating outside the envelope requires more grip than the tyre has at that load.
The simplified four-parameter Pacejka Magic Formula:
F = D * sin( C * atan( B*s - E*(B*s - atan(B*s)) ) )
with parameters:
| Symbol | Role | Effect |
|---|---|---|
B |
Stiffness factor | Slope at zero slip — higher B = stiffer tyre |
C |
Shape factor | Peak vs tail shape — typical 1.2–1.6 lateral, 1.4–1.7 longitudinal |
D |
Peak force | Maximum force the tyre can produce |
E |
Curvature factor | Sharpness around peak — typically negative |
The peak coefficient D is load-sensitive:
D = (a1 * F_z + a2) * F_z
which gives a peak friction coefficient μ_peak = D / F_z = a1·F_z + a2 that decreases linearly with vertical load — the textbook degressive friction of pneumatic tyres at high F_z.
The same formula form drives both axes (lateral with slip angle, longitudinal with slip ratio) using different coefficient sets. Combined slip is approximated by the Kamm ellipse — a textbook simplification that captures the dominant force-trade-off without the inter-axis interaction terms of the full Pacejka 2002 combined-slip model.
| Axis | B | C | E | a1 | a2 |
|---|---|---|---|---|---|
| Lateral (single-axle baseline) | 0.1453 | 1.5511 | −2.6166 | −1×10⁻⁵ | 1.25 |
| Longitudinal (driven axle) | 5.7507 | 1.2421 | −6.8251 | −1×10⁻⁵ | 1.25 |
| Lateral — front axle (14-DoF) | 0.0813 | 1.2947 | −8.3966 | −1×10⁻⁵ | 0.95 |
| Lateral — rear axle (14-DoF) | 0.0925 | 1.2617 | −8.7012 | −1×10⁻⁵ | 0.95 |
Lifted directly from the parameter files of the 14-DoF model in the VD models repo.
Every chassis decision that moves vertical load between tyres (anti-roll bar split, ride height, aero balance, weight distribution) interacts with the degressive friction term. Adding ARB stiffness on one axle redistributes load to the outer tyre on that axle, but the gained outer-tyre force is less than the lost inner-tyre force because the outer tyre's μ_peak is lower at higher F_z. The net axle force drops — and that's why ARB tuning matters for handling balance even though it doesn't change total grip across the car. The same physics underpins lateral-load-transfer modelling in the VD models 14-DoF and the lap simulator's lateral-transfer pass.
pacejka-tyre-model/
├── parameters.m # default coefficients (3 sets)
├── src/
│ ├── pacejka_magic_formula.m # the F = D sin(C atan(B s ...)) function
│ └── pacejka_peak_value.m # D = (a1 Fz + a2) Fz
└── figures/
├── plot_lateral_curves.m
├── plot_longitudinal_curves.m
├── plot_axle_comparison.m
└── plot_friction_envelope.m
Originally implemented during the Vehicle Dynamics Modelling and Simulation course of the M.Sc. Interdisciplinary Engineering (Automotive) programme at Hochschule Trier. Course is in progress — final assignment outstanding, certificate not yet issued.
MIT. See LICENSE.