Pacejka Magic Formula fitting from TTC/FSAE data. Automated coefficient optimisation with thermal and pressure sensitivity.
Shows Fy vs. slip angle at 0.5×, 1×, and 1.5× nominal Fz
The Pacejka Magic Formula (MF 5.2/6.1) parameterises tyre force generation as a function of slip ratio κ, slip angle α, vertical load Fz, camber γ, and inflation pressure.
F = D · sin(C · arctan(B·x − E·(B·x − arctan(B·x)))) + Sv(1)Where B (stiffness), C (shape), D (peak), E (curvature) are each load-dependent polynomials. The fitting process minimises the sum of squared residuals between measured and predicted forces across all load/slip/camber conditions.
D = (pDy1 + pDy2·dfz) · Fz · λμy(2)Kyα = pKy1 · Fz₀ · sin(pKy4 · arctan(Fz / (pKy2 · Fz₀))) · λKyα(3)Combined slip uses weighting functions Gxa(α) and Gyk(κ) to couple lateral and longitudinal force generation under simultaneous braking/cornering.
function [coeffs, rmse] = fit_pacejka(data, tyre_params)
% Automated Pacejka MF5.2 coefficient fitting
% Uses Levenberg-Marquardt with analytical Jacobian
%
% Inputs: data - struct: SA, SR, FZ, FY, FX, MZ
% tyre_params - initial guess struct
%
% Outputs: coeffs - optimised coefficient vector
% rmse - fit quality metrics
% Extract test conditions
alpha = data.SA; % slip angle (deg)
kappa = data.SR; % slip ratio (-)
Fz = data.FZ; % vertical load (N)
Fy_meas = data.FY;
% Initial coefficient vector
x0 = pack_coeffs(tyre_params);
% Levenberg-Marquardt (no toolbox)
lambda = 1e-3;
for iter = 1:200
[F, J] = pacejka_residual(x0, alpha, kappa, Fz, Fy_meas);
H = J'*J + lambda*eye(length(x0));
dx = -H \ (J'*F);
x_new = x0 + dx;
% ... convergence check
end
TyreParam.m — nominal load, tyre radius, and which coefficients to fit (pure lat, pure long, combined, aligning).example_ttc_fit.m. The Levenberg-Marquardt loop converges in ~50-100 iterations. No Optimization Toolbox required.Full source code, documentation, and email support. Unlimited use within your FS team.
Get StartedFull source + 1hr setup call + priority support. Custom vehicle config and tyre fitting included.
Contact UsNo toolboxes required. All optimisation uses custom solvers — no Optimization Toolbox, no Curve Fitting Toolbox. Works on base MATLAB R2020a+ / Python 3.8+.