void Tire::findSigmaHatAlphaHat(btScalar load, btScalar & output_sigmahat, btScalar & output_alphahat, int iterations)
{
btScalar x, y, ymax, junk;
ymax = 0;
for (x = -2; x < 2; x += 4.0/iterations)
{
y = PacejkaFx(x, load, 1.0, junk);
if (y > ymax)
{
output_sigmahat = x;
ymax = y;
}
}
ymax = 0;
for (x = -20; x < 20; x += 40.0/iterations)
{
y = PacejkaFy(x, load, 0.0, 1.0, junk);
if (y > ymax)
{
output_alphahat = x;
ymax = y;
}
}
}
void Tire::findSigmaHatAlphaHat(
btScalar load,
btScalar & output_sigmahat,
btScalar & output_alphahat,
int iterations)
{
btScalar Fz = load;
btScalar Fz0 = nominal_load;
btScalar dFz = (Fz - Fz0) / Fz0;
btScalar camber = 0.0;
btScalar mu = 1.0;
btScalar Dy, BCy, Shf; // unused
btScalar Fxmax = 0.0;
btScalar smax = 2.0;
for (btScalar s = -smax; s < smax; s += 2 * smax / iterations)
{
btScalar Fx = PacejkaFx(s, Fz, dFz, mu);
if (Fx > Fxmax)
{
output_sigmahat = btFabs(s);
Fxmax = Fx;
}
}
btScalar Fymax = 0.0;
btScalar amax = 30.0 * (M_PI / 180.0);
for (btScalar a = -amax; a < amax; a += 2 * amax / iterations)
{
btScalar Fy = PacejkaFy(a, camber, Fz, dFz, mu, Dy, BCy, Shf);
if (Fy > Fymax)
{
output_alphahat = btFabs(a);
Fymax = Fy;
}
}
}
btVector3 Tire::getForce(
btScalar normal_force,
btScalar friction_coeff,
btScalar inclination,
btScalar ang_velocity,
btScalar lon_velocity,
btScalar lat_velocity)
{
if (normal_force < 1E-3 || friction_coeff < 1E-3)
{
slide = slip = 0;
ideal_slide = ideal_slip = 1;
fx = fy = fz = mz = 0;
vx = vy = 0;
return btVector3(0, 0, 0);
}
// pacejka in kN
normal_force = normal_force * 1E-3;
// limit input
btClamp(normal_force, btScalar(0), btScalar(max_load));
btClamp(inclination, -max_camber, max_camber);
// get ideal slip ratio
btScalar sigma_hat(0);
btScalar alpha_hat(0);
getSigmaHatAlphaHat(normal_force, sigma_hat, alpha_hat);
btScalar Fz = normal_force;
btScalar gamma = inclination; // positive when tire top tilts to the right, viewed from rear
btScalar denom = btMax(btFabs(lon_velocity), btScalar(1E-3));
btScalar lon_slip_velocity = lon_velocity - ang_velocity;
btScalar sigma = -lon_slip_velocity / denom; // longitudinal slip: negative in braking, positive in traction
btScalar tan_alpha = lat_velocity / denom;
btScalar alpha = -atan(tan_alpha) * 180.0 / M_PI; // sideslip angle: positive in a right turn(opposite to SAE tire coords)
btScalar max_Fx(0), max_Fy(0), max_Mz(0);
// combining method 1: beckman method for pre-combining longitudinal and lateral forces
// seems to be a variant of Bakker 1987:
// refined Kamm Circle for non-isotropic tire characteristics
// and slip normalization to guarantee simultaneous sliding
btScalar s = sigma / sigma_hat;
btScalar a = alpha / alpha_hat;
btScalar rho = btMax(btSqrt(s * s + a * a), btScalar(1E-4)); // normalized slip
btScalar Fx = (s / rho) * PacejkaFx(rho * sigma_hat, Fz, friction_coeff, max_Fx);
btScalar Fy = (a / rho) * PacejkaFy(rho * alpha_hat, Fz, gamma, friction_coeff, max_Fy);
// Bakker (with unknown factor e(rho) to get the correct direction for large slip):
// btScalar Fx = -s * PacejkaFx(rho * sigma_hat, Fz, friction_coeff, max_Fx);
// btScalar Fy = -a * e(rho) * PacejkaFy(rho * alpha_hat, Fz, gamma, friction_coeff, max_Fy);
/*
// combining method 2: orangutan
btScalar Fx = PacejkaFx(sigma, Fz, friction_coeff, max_Fx);
btScalar Fy = PacejkaFy(alpha, Fz, gamma, friction_coeff, max_Fy);
btScalar one = (sigma < 0.0) ? -1.0 : 1.0;
btScalar pure_sigma = sigma / (one + sigma);
btScalar pure_alpha = -tan_alpha / (one + sigma);
btScalar pure_combined = sqrt(pure_sigma * pure_sigma + pure_alpha * pure_alpha);
btScalar kappa_combined = pure_combined / (1.0 - pure_combined);
btScalar alpha_combined = atan((one + sigma) * pure_combined);
btScalar Flimitx = PacejkaFx(kappa_combined, Fz, friction_coeff, max_Fx);
btScalar Flimity = PacejkaFy(alpha_combined * 180.0 / M_PI, Fz, gamma, friction_coeff, max_Fy);
btScalar x = fabs(Fx) / (fabs(Fx) + fabs(Fy));
btScalar y = 1.0 - x;
btScalar Flimit = (fabs(x * Flimitx) + fabs(y * Flimity));
btScalar Fmag = sqrt(Fx * Fx + Fy * Fy);
if (Fmag > Flimit)
{
btScalar scale = Flimit / Fmag;
Fx *= scale;
Fy *= scale;
}
*/
/*
// combining method 3: traction circle
btScalar Fx = PacejkaFx(sigma, Fz, friction_coeff, max_Fx);
btScalar Fy = PacejkaFy(alpha, Fz, gamma, friction_coeff, max_Fy);
// determine to what extent the tires are long (x) gripping vs lat (y) gripping
// 1.0 when Fy is zero, 0.0 when Fx is zero
btScalar longfactor = 1.0;
btScalar combforce = btFabs(Fx) + btFabs(Fy);
if (combforce > 1) longfactor = btFabs(Fx) / combforce;
// determine the maximum force for this amount of long vs lat grip by linear interpolation
btScalar maxforce = btFabs(max_Fx) * longfactor + (1.0 - longfactor) * btFabs(max_Fy);
// scale down forces to fit into the maximum
if (combforce > maxforce)
{
Fx *= maxforce / combforce;
Fy *= maxforce / combforce;
}
*/
/*
// combining method 4: traction ellipse (prioritize Fx)
// issue: assumes that adhesion limits are fully reached for combined forces
btScalar Fx = PacejkaFx(sigma, Fz, friction_coeff, max_Fx);
btScalar Fy = PacejkaFy(alpha, Fz, gamma, friction_coeff, max_Fy);
if (Fx < max_Fx)
{
Fy = Fy * sqrt(1.0 - (Fx / max_Fx) * (Fx / max_Fx));
}
else
{
Fx = max_Fx;
Fy = 0;
}
*/
/*
// combining method 5: traction ellipse (prioritize Fy)
// issue: assumes that adhesion limits are fully reached for combined forces
btScalar Fx = PacejkaFx(sigma, Fz, friction_coeff, max_Fx);
btScalar Fy = PacejkaFy(alpha, Fz, gamma, friction_coeff, max_Fy);
if (Fy < max_Fy)
{
Fx = Fx * sqrt(1.0 - (Fy / max_Fy) * (Fy / max_Fy));
}
else
{
Fy = max_Fy;
Fx = 0;
}
*/
/*
// no combining
btScalar Fx = PacejkaFx(sigma, Fz, friction_coeff, max_Fx);
btScalar Fy = PacejkaFy(alpha, Fz, gamma, friction_coeff, max_Fy);
*/
btScalar Mz = PacejkaMz(sigma, alpha, Fz, gamma, friction_coeff, max_Mz);
camber = inclination;
slide = sigma;
slip = alpha;
ideal_slide = sigma_hat;
ideal_slip = alpha_hat;
fx = Fx;
fy = Fy;
fz = Fz;
mz = Mz;
vx = lon_slip_velocity;
vy = lat_velocity;
// Fx positive during traction, Fy positive in a right turn, Mz positive in a left turn
return btVector3(Fx, Fy, Mz);
}
btVector3 Tire::getForce(
btScalar normal_load,
btScalar friction_coeff,
btScalar camber,
btScalar rot_velocity,
btScalar lon_velocity,
btScalar lat_velocity)
{
if (normal_load * friction_coeff < btScalar(1E-6))
{
slip = slip_angle = 0;
ideal_slip = ideal_slip_angle = 1;
fx = fy = fz = mz = 0;
vx = vy = 0;
return btVector3(0, 0, 0);
}
// limit input
normal_load = Clamp(normal_load, btScalar(0), btScalar(max_load));
camber = Clamp(camber, -max_camber, max_camber);
// sigma and alpha
btScalar denom = Max(btFabs(lon_velocity), btScalar(1E-3));
btScalar lon_slip_velocity = lon_velocity - rot_velocity;
btScalar sigma = -lon_slip_velocity / denom;
btScalar alpha = btAtan(lat_velocity / denom);
// force parameters
btScalar Fz = normal_load;
btScalar Fz0 = nominal_load;
btScalar dFz = (Fz - Fz0) / Fz0;
// pure slip
btScalar Dy, BCy, Shf;
btScalar Fx0 = PacejkaFx(sigma, Fz, dFz, friction_coeff);
btScalar Fy0 = PacejkaFy(alpha, camber, Fz, dFz, friction_coeff, Dy, BCy, Shf);
btScalar Mz0 = PacejkaMz(alpha, camber, Fz, dFz, friction_coeff, Fy0, BCy, Shf);
// combined slip
btScalar Gx = PacejkaGx(sigma, alpha);
btScalar Gy = PacejkaGy(sigma, alpha);
btScalar Svy = PacejkaSvy(sigma, alpha, camber, dFz, Dy);
btScalar Fx = Gx * Fx0;
btScalar Fy = Gy * Fy0 + Svy;
// ideal slip and angle
btScalar sigma_hat(0), alpha_hat(0);
getSigmaHatAlphaHat(normal_load, sigma_hat, alpha_hat);
slip = sigma;
slip_angle = alpha;
ideal_slip = sigma_hat;
ideal_slip_angle = alpha_hat;
fx = Fx;
fy = Fy;
fz = Fz;
mz = Mz0;
vx = lon_slip_velocity;
vy = lat_velocity;
return btVector3(Fx, Fy, Mz0);
}