Home

Pacejka's Magic Formula is used for expressing tire forces.

Pacejka's Magic Formula is used to estimate tire forces due to slip angle and ratio. For more information on the basic formula(e), see the Pacejka Magic Formula page here. This page deals only with the implementation in Racer.

Racer uses only Fx, Fy and Mz curves (longitudinal, lateral and aligning moment; note that Racer uses a different coordinate system: the Fx notation uses the SAE coordinate system). Also, even though the model indicates Pacejka96, there seem to be a lot of Pacejka variations out there with varying properties & sign conventions. So, examining the source code is your best if you want to know what happens exactly.

Sign conventions

Racer's Pacejka implementation has the following properties:

• Slip angle - this is in degrees and positive for a left hand turn. A positive slip angle will result in a positive lateral force (Fy in SAE). Note that this is the reverse of what Racer displays in the Ctrl-1 screen; slipAngle is shown as negative in left-hand turn. The input to Pacejka, though, is negated, so the Pacejka curves receive a positive slipAngle when Racer's slipAngle is negative.
• Slip ratio - this is positive for accelerating. A positive slip ratio will results in a positive (tractive) force (Fx in SAE). Notice that the input slip parameter for Pacejka is a percentage, so the input to the Pacejka functions below is 100 times what you see in the Ctrl-1 debug screen in Racer.
• Fz - positive normally, and in kN (not just Newtons). So a car weighing 8000N will receive approximately Fz=2kN as input at all 4 wheels (assuming the car is perfectly balanced).
• Camber angle - this is in degrees. A negative camber angle will make tires lean in towards the car body.

The source code - longitudinal tire force

// Fx - longitudinal
rfloat RPacejka::CalcFx96()
// Pacejka96 model
// Calculates longitudinal force (Fx)
// From G. Genta's book, page 63
// Note that the units are inconsistent:
//   Fz is in kN
//   slipRatio is in percentage (=slipRatio*100=slipPercentage)
//   camber and slipAngle are in degrees
// Resulting forces are better defined:
//   Fx, Fy are in N
//   Mz     is in Nm
{
   rfloat B,C,D,E;
   rfloat Fx;
   rfloat Sh,Sv;
   rfloat uP;
   rfloat FzSquared;
   
   // Calculate derived coefficients
   FzSquared=Fz*Fz;
   C=b0;
   uP=b1*Fz+b2;
   D=uP*Fz;
   
   // Avoid div by 0
   if((C>-D3_EPSILON&&C<D3_EPSILON) || (D>-D3_EPSILON&&D<D3_EPSILON))
   {
     B=99999.0f;
   } else
   {
     B=((b3*FzSquared+b4*Fz)*expf(-b5*Fz))/(C*D);
   }
   
   E=b6*FzSquared+b7*Fz+b8;
   Sh=b9*Fz+b10;
   Sv=b11*Fz+b12;
   
   // Notice that product BCD is the longitudinal tire stiffness
   longStiffness=B*C*D; // *RR_RAD2DEG;    // RR_RAD2DEG == *360/2PI
   
   // Remember the max longitudinal force (where sin(...)==1)
   maxForce.x=D+Sv;
   
   // Calculate result force
   Fx=D*sinf(C*atanf(B*(1.0f-E)*(slipPercentage+Sh)+E*atanf(B*(slipPercentage+Sh))))+Sv;
 
  return Fx;
}
 

Lateral force

// Fy - lateral
rfloat RPacejka::CalcFy96()
// Pacejka 96
// Calculates lateral force
// Note that B*C*D is the cornering stiffness, and
// Sh and Sv account for ply steer and conicity forces
{
   rfloat B,C,D,E;
   rfloat Fy;
   rfloat Sh,Sv;
   rfloat uP;
   
   // Calculate derived coefficients
   C=a0;
   uP=a1*Fz+a2;
   D=uP*Fz;
   E=a6*Fz+a7;
   
   // Avoid div by 0
   if((C>-D3_EPSILON&&C<D3_EPSILON) || (D>-D3_EPSILON&&D<D3_EPSILON))
   {
     B=99999.0f;
   } else
   {
     if(a4>-D3_EPSILON&&a4<D3_EPSILON)
     {
       B=99999.0f;
     } else
     {
       // Notice that product BCD is the lateral stiffness (=cornering)
       latStiffness=a3*sinf(2*atanf(Fz/a4))*(1.0f-a5*fabs(camber));
       B=latStiffness/(C*D);
     }
   }
   
   Sh=a8*camber+a9*Fz+a10;
   Sv=(a111*Fz+a112)*camber*Fz+a12*Fz+a13;
   
   // Remember maximum lateral force
   maxForce.y=D+Sv;
   
   // Calculate result force
   Fy=D*sinf(C*atanf(B*(1.0f-E)*(sideSlip+Sh)+
   E*atanf(B*(sideSlip+Sh))))+Sv;

   return Fy;
}
 

(last updated January 14, 2013 )