Hi @Panagiotis
Have you tried using your own "Magic Formula"?
You can create one by taking the derivative of the product of a linear function and a relatively fast saturating sigmoidal curve. However, the inverse tangent, 
, is 'too slow' because it takes a very large value to saturate at 
.
, is 'too slow' because it takes a very large value to saturate at .
x = linspace(-1, 1, 2001);
k1 = 3; % parameter 1
k2 = 5; % parameter 2
y1 = k1*x; % Linear function, where k1 > 0
y2 = erf((sqrt(pi)/2)*k2*x); % Gaussian Integral function (aka Error function)
y3 = (2/pi)*atan(pi/2*k2*x); % Inverse tangent
y = y1.*y2;
dy = gradient(y)./gradient(x);
figure
plot(x, [y1; y2; y3; y]), grid on
xlabel('x'), ylabel('y')
title('Find Your Own Magic Formula')
legend('Linear fcn (y_{1})', 'Sigmoid (y_{2})', 'ArcTan (y_{3})', 'y_{1}·y_{2}', 'location', 'best')
figure
plot(x, dy, 'linewidth', 2, 'color', '#265EF5'), grid on
xlabel('Slip ratio'), ylabel('Longitudinal Force')
title('Your Tire Model')
