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

WHAT IS PACEJKA'S MAGIC FORMULA?

Pacejka curves are used in Racer as a big part of the tire models. They model the forces that are generated by the tire as a result of the tire not following the road precisely. Steer the tire a little, and you get a slip angle, and this is input into the Pacejka Fy formula, giving a sideways force. Press the throttle, and the wheel starts spinning a bit; this gives a different ratio of wheel spin speed vs. ground speed, and this gives a forward (longitudinal) force.

Pacejka's Magic Formula is a standard in a lot of today's racing simulations applications (not necessarily games; it's mostly professional applications that use it). Multiple versions exist - Racer mostly uses a 1989 version of Pacejka it seems; the formula was originally taken from Giancarlo Genta's book 'Motor Vehicle Dynamics', which is from 1997. The book also contains a few Pacejka example sets, not all too accurate but useful nevertheless.

The default Pacejka formula in Racer uses coefficients a<n> for lateral forces, b<n> for longitudinal and c<n> for aligning moment (force feedback/Mz). Racer will soon support MF5.2, which stands for Magic Formula 5.2, which is used more often these days. Although newer, the real crux is to get tire data, which is scarce and heavily protected intellectual property just about all the time. However, sometimes you may get your hands on approximate data, which just hides subtleties which are not too important anyway.

MF 5.2 uses newly named variables, such as PCx1, PDy1 etc. These are the base fitting coefficients to get the mathematical curves to match the empirical (measured) tire data. Then, lambda values are used in various places to be able to tweak the tire in more real-life terms.

The text below focuses on the default model, Pacejka89. Note that the text speaks of '96', which I though the version was until recently. The future may see more use of MF5.2 Pacejka sets, since that is what I've seen in the field more often the last couple of years.

Magic Formula 5.2 usage

Currently implemented fully in (v0.8.8), the MF5.2 set can be activated by setting 'model=1' in your car.ini pacejka data. An example extract:

pacejka_mf52
{
  ; Pacejka89 (0) or MF5.2 (1)?
  model=1
  pcx1=1.3
  pdx1=1.2
  pkx1=30
  ; Lateral
  pcy1=1.34
  pdy1=1.04
  pky1=-25.0
  pky2=2.71
  pky3=0.39
} 

wheel0
{
...
  pacejka~pacejka_mf52
  {
  }

}

The source code from the v0.8.9 version:

void RPacejka::CalcMF52()
// Pacejka MF5.2 (~2006)
{
   rfloat Fx0,Dx,Cx,Ex,Bx,dfz,Fz0,Fz0_der;             // Nominal load, adapted nominal load
   rfloat kx,k,mux,sign_kx;
   rfloat Shx,Svx,Kx,gamma,gamma_x;
   rfloat Fzn;                     // Fz in Newtons
   rfloat tmp;

   //
   // Global
   //
   k=slipPercentage*0.01f;   // Slip ratio
   Fzn=Fz*1000.0f;
   Fz0=fz0; //4500;
   Fz0_der=lfz0*Fz0;
   dfz=(Fzn-Fz0_der)/Fz0_der;
   gamma=camber/RR_RAD2DEG;
   //
   // Fx (pure slip)
   //
   Shx=(phx1+phx2*dfz)*lhx;
   Svx=Fzn*(pvx1+pvx2*dfz)*lvx*lmux;
   kx=k+Shx;
   if(kx<0)sign_kx=-1.0f;
   else    sign_kx=1.0f;

   gamma_x=gamma*lgax;

   Cx=pcx1*lcx;
   mux=(pdx1+pdx2*dfz)*(1.0f-pdx3*gamma_x*gamma_x)*lmux;     // Different in Pac2006
   Ex=(pex1+pex2*dfz+pex3*dfz*dfz)*(1.0f-pex4*sign_kx)*lex;
   // Limiter on Ex (eq 23)
   if(Ex>1.0f)Ex=1.0f;
   Dx=mux*Fzn;
   Kx=Fzn*(pkx1+pkx2*dfz)*expf(pkx3*dfz)*lkx;                // K=BCD (=stiffness)
   // Low velocity trouble
   tmp=Cx*Dx;
   if(fabs(tmp)<0.001f)
   Bx=Kx/(tmp+0.001f);
   else
   Bx=Kx/tmp;

   tmp=Bx*kx;
   Fx0=Dx*sinf(Cx*atanf(tmp-Ex*(tmp-atanf(tmp))))+Svx;

   //
   // Fy
   //
   rfloat Fy0,Dy,Cy,Ey,By;
   rfloat localMuy;
   rfloat Shy,Svy,Ky;
   rfloat say,sign_alpha_y;
   rfloat gamma_y;
   rfloat alpha;

   alpha=sideSlip/RR_RAD2DEG;
   gamma_y=gamma*lgay;

   Cy=pcy1*lcy;
   localMuy=(pdy1+pdy2*dfz)*(1.0f-pdy3*gamma_y*gamma_y)*lmuy;
   Dy=localMuy*Fzn;
   Ky=pky1*Fz0_der*sinf(2.0f*atanf(Fzn/(pky2*Fz0_der)))*(1.0f-pky3*fabs(gamma_y))*lfz0*lky;    // =BCD=stiffness at slipangle 0

   tmp=Cy*Dy;
   if(fabs(tmp)<0.001f)
     By=Ky/(tmp+0.001f);
   else
     By=Ky/tmp;

   Shy=(phy1+phy2*dfz)*lhy+phy3*gamma_y;
   Svy=Fzn*((pvy1+pvy2*dfz)*lvy+(pvy3+pvy4*dfz)*gamma_y)*lmuy;
   say=alpha+Shy;

   if(say<0)sign_alpha_y=-1.0f;
   else     sign_alpha_y=1.0f;
   Ey=(pey1+pey2*dfz)*(1.0f-(pey3+pey4*gamma_y)*sign_alpha_y)*ley;
   // Limiter on Ey (eq 35)
   if(Ey>1.0f)Ey=1.0f;

   // Lateral base force
   tmp=By*say;
   Fy0=Dy*sinf(Cy*atanf(tmp-Ey*(tmp-atan(tmp))))+Svy;

   //
   // Mz
   //
   rfloat Mzr;
   rfloat t0,Dt,Ct,Bt,alpha_t,Et,gamma_z;
   rfloat Sht,Shf;
   rfloat Br,R0;
   rfloat alpha_r,Dr;
   const float lr=1.0f;        // Not found in paper at lambda section

   R0=r0;
   gamma_z=gamma*lgaz;

   Sht=qhz1+qhz2*dfz+(qhz3+qhz4*dfz)*gamma_z;
   alpha_t=alpha+Sht;

   // Avoid division by zero
   if(fabs(Ky)<0.001f)
   if(Ky<0)Ky=-0.001f;
   else    Ky=0.001f;

   Shf=Shy+Svy/Ky;
   alpha_r=alpha+Shf;
   Bt=(qbz1+qbz2*dfz+qbz3*dfz*dfz)*(1.0f+qbz4*gamma_z+qbz5*fabs(gamma_y))*lky/lmuy;    // Pac2006 adds gamma_y^2 dependence (qbz6)
   Ct=qcz1;
   Dt=Fzn*(qdz1+qdz2*dfz)*(1.0f+qdz3*gamma_z+qdz4*gamma_z*gamma_z)*(R0/Fz0_der)*lt;        // lt=lamba_t?

   Et=(qez1+qez2*dfz+qez3*dfz*dfz)*(1.0f+(qez4+qez5*gamma_z)*(2.0f/3.14159265f)*atanf(Bt*Ct*alpha_t));   // <=1
   // Clamp Et (eq 51)
   if(Et>1.0f)Et=1.0f;

   Br=qbz9*lky/lmuy+qbz10*By*Cy;      // Preferred qbz9=0
   tmp=Bt*alpha_t;
   t0=Dt*cosf(Ct*atanf(tmp-Et*(tmp-atan(tmp))))*cosf(alpha);           // t_alpha_t in the paper
   Dr=Fzn*((qdz6+qdz7*dfz)*lr+(qdz8+qdz9*dfz)*gamma_z)*R0*lmuy;        // *cosf(alpha_t) for Pac2006 (in MF52 this is still in Mzr=... below)
   Mzr=Dr*cosf(Ct*atanf(Br*alpha_r))*cos(alpha);                       // last cos(alpha) is cos(alpha_t) in Pac2006

   // No combined aligning moment
   Mz=-t0*Fy0+Mzr;

   //
   // Combined slip
   //
   // Combine (page 30+)
   // Longitudinal
   float Shxa,Exa,Cxa,Bxa,alpha_s,Gxa0,Gxa;
   Shxa=rhx1;
   Exa=rex1+rex2*dfz;
   Cxa=rcx1;
   Bxa=rbx1*cosf(atan(rbx2*k))*lxal;   // +rbx3*gammaStar*gammaStar) (Pac2006)
   alpha_s=alpha+Shxa;
   Gxa0=cos(Cxa*atan(Bxa*Shxa-Exa*(Bxa*Shxa-atan(Bxa*Shxa))));
   if(fabs(Gxa0)>D3_EPSILON)
   Gxa=cos(Cxa*atan(Bxa*alpha_s-Exa*(Bxa*alpha_s-atan(Bxa*alpha_s))))/Gxa0;
   else
   Gxa=0;        // Or 1 for super grip?
   Fx=Gxa*Fx0;

   // Lateral
   float Dvyk,Svyk,Shyk,Eyk,Cyk,Byk,ks,Gyk0,Gyk;

   Dvyk=muy*Fz*(rvy1+rvy2*dfz+rvy3*gamma)*cosf(atanf(rvy4*alpha));
   Svyk=Dvyk*sin(rvy5*atan(rvy6*k))*lvyka;
   Shyk=rhy1+rhy2*dfz;
   Eyk=rey1+rey2*dfz;
   Cyk=rcy1;
   Byk=rby1*cosf(atanf(rby2*(alpha-rby3)))*lyka;
   ks=k+Shyk;
   Gyk0=cosf(Cyk*atanf(Byk*Shyk-Eyk*(Byk*Shyk-atanf(Byk*Shyk))));
   Gyk=cosf(Cyk*atanf(Byk*ks-Eyk*(Byk*ks-atanf(Byk*ks))))/Gyk0;

   Fy=Gyk*Fy0+Svyk;

   // Aligning torque
   float alpha_r_eq,alpha_t_eq,Mzr,Fy_der;
   float sign_alpha_t,sign_alpha_r;
   float kk,s,t;

   if(alpha_t>=0)sign_alpha_t=1.0f;
   else          sign_alpha_t=-1.0f;
   if(alpha_r>=0)sign_alpha_r=1.0f;
   else          sign_alpha_r=-1.0f;
   kk=Kx/Ky;
   kk=(kk*kk*k*k);     // kk^2*k^2
   alpha_r_eq=sqrtf(alpha_r*alpha_r+kk)*sign_alpha_r;
   alpha_t_eq=sqrtf(alpha_t*alpha_t+kk)*sign_alpha_t;
   s=(ssz1+ssz2*(Fy/Fz0_der)+(ssz3+ssz4*dfz)*gamma)*R0*ls;
   Mzr=Dr*cosf(atanf(Br*alpha_r_eq))*cosf(alpha);
   Fy_der=Fy-Svyk;
   // New pneumatic trail
   tmp=Bt*alpha_t_eq;
   t=Dt*cosf(Ct*atanf(tmp-Et*(tmp-atanf(tmp))))*cosf(alpha);

   // Add all aligning forces
   Mz=-t*Fy_der+Mzr+s*Fx;

   // Postprocess; negate for Racer?!
   Fy=-Fy;
   Mz=-Mz;

   // Static results
   latStiffness=-By*Cy*Dy;     // There's that '-' again for lateral force (Fy)
   longStiffness=Bx*Cx*Dx;
}
 

WHAT DO THE CURVES LOOK LIKE?

Here you can see a typical bunch of Pacejka curves for longitudinal (forward) force (Fx), lateral (sideways) force (Fy) and aligning moment (Mz, which you feel at the steering wheel, the force-feedback for consumer FF wheels). Horizontally you see the inputs, as used for the 3 different curves. Vertically you see resulting forces (Fx/Fy) and torque (Mz). SR means slip ratio, and is defined as wheelSpinVelocity/groundVelocity-1. SA means slip angle and is the angle (in degrees) between the wheel heading and the wheel's actual velocity. Last, SA for Mz is the slip angle as used for the aligning moment (Mz). Note that Mz even crosses the horizontal line, which means that once you steer TOO MUCH, you even get negative feedback, making your wheel want to turn even MORE. Take a look at the black Fx curve: Fx peaks and then lowers back down again; this curve is a bit exaggerated perhaps; it should go down to about 75% peak force perhaps. Looking at the lateral curve, you can say that Fy peaks at about 4 degrees. This also is a bit rough; you would normally be more in the 6-15 degree range for your peak.

Another example of a Pacejka curve from a different program is this curve, as used in EA's F1-2002 (data from the tire.tbc file). Used as a base tire curve, this is tweaked according to the situation (tire temperature for example). Notice that it stays at the peak very long and only gradually drops down: this tire is very forgiving.

SO WHAT DOES THE CURVE DO PHYSICS-WISE?

Longitudinal Pacejka (Fx) for example gives the amount of force that the tire generates. That is the force that moves the body forwards.

Input (horizontally) is slip ratio (SR); this is the ratio of wheel velocity vs. ground velocity. So if the wheel spins at twice the velocity of the ground, you get: SR=wheelVel/groundVel-1 (the -1 is there to make 0 the null situation) => SR=1. So if the wheel spins faster than the ground below, you get a forward force.

However, as you can see in the curve, you have a peak. Once your tire is really spinning a lot faster than the ground, you actually LOSE grip, which is why the graph drops again. The coefficients are just there to tweak the curve. High peak = high possible force, high line after the peak = still a lot of grip once you get a lot of wheelspin. Notice that because the curve dives down again, it's also harder to regain grip, since not much force is generated by the tire to reduce the spin back to normal again.

With lateral Pacejka it's the same idea, only based on slip angle (wheel direction vs. ground direction). The peak location is important here: F1 tires may peak around 6 degrees (displayed horizontally), road cars at perhaps 10-15 degrees. Again, if the graph drops too much after the peak, steering too much will cause more severe understeer effects. If the peak force slip angle is low (6 degrees) you'll have a fast reacting tire. If it's higher (some say Grand Prix Legend's peak slipangles are artificially high) you get smoother steering.

A big chunk in tire modelling is the way longitudinal and lateral forces are generated. Another big chunk is how they are combined, since (if you've heard of the friction circle) a tire cannot generate more than some maximum force, and this has to be split for lateral and longitudinal directions. So in other words, a maximum accelerating car couldn't keep this up while turning.

An editor for the Pacejka curves in Racer is part of the package and there is documentation for the Pacejka Player too.

Racer uses the Pacejka Magic Formula model for the tire forces.

Pros:

• A standard model which a lot of people already understand.
• Common in the high-end racing simulation world.

Cons:

• Not too much non-proprietary data available.
• The standard coefficients don't all have a direct physically explanation (but there are ways around that, although you won't read about that here ;) ).

Still, the Genta book contains 5 sets on which to build. But that means you have to understand a little of them. Unfortunately, the data is a bit lacking; some coefficients (for load sensitivity) are missing here & there. Still, it's good that they're there.

Longitudinal forces

The Pacejka96 formula calculates Fx (X is the SAE axis for the forward vehicle axis) as follows (in pseudo-code style, the actual formula is normally shown a little cleaner). The static input is the b<n> coefficients, and the dynamic input is load (Fz) and slipRatio (slipPercentage). Don't know where camber has gone. ;-)

The official Pacejka-96 longitudinal formula goes like this:

y(x) = D sin[ C arctan { Bx - E ( Bx - arctan(Bx))}]

where Y(X) = y(x) + Sv, and x = X + Sh.

The resulting force, Fx, is equal to Y(X). Actually, the same formulation is used for Fy and Mz (lateral force and aligning moment), only with different variations of calculating the BCDE variables. Y here represents output (Fx/Fy/Mz), X represents the input (slip ratio, slip angle).

A lot of things can be said on these variables. The basics are:

• D is the peak value of the curve (except for the Sv addition/shift)
• C is the shape factor (determines the shape of the peak)
• Sh and Sv are shifting values; they shift the curve horizontally and vertically (hence the 'h' and 'v'), and if I recall correctly they were added in a later stage. Sh represents conicity forces, which appear because the tire may look a bit like a cone, and Sv represents plysteer forces (which appear because of the direction and method with which the plies are manufactured into the tire).

The B, C, D and E variables are functions of the wheel load, slip angle, slip ratio and camber. The influence of these inputs are given by the b0-b10 constants. Calculating these goes as follows (in pseudo-code format):

• C=b0
• D=(b1*Fz+b2)*Fz
• B=((b3*Fz^2+B4*Fz)*exp(-b5*Fz))/(C*D)
• E=b6*Fz^2+b7*Fz+b8
• Sh=b9*Fz+b10
• Sv=0
• at the end, Fx=D*sin(C*atan(B*(1-E)*(slipPercentage+Sh)+E*atan(B*(slipPercentage+Sh))))+Sv

Here, Fz is the normal force, the load in English. The result is Fx, the longitudinal force. The input for Racer is the b0-b10 coefficients, which define the curve's peak and flow. Some interpretations for the Pacejka formula are:

• D+Sv is the maximum force the tire can generate, at its peak performance
• B*C*D is the longitudinal stiffness of the tire (I don't know the units though)
• arctan(B*C*D) gives the angle of the curve where it passes through the origin (apart from Sh, Sv). This is a stiffness indicator as well.
• The curve usually has the following shape: linear from the origin, peaking (at height D+Sv), then gradually coming down again.
• The asymptotic height to which the curve tends to go at big inputs is called Ys and is equal to D*sin(0.5*PI*C). This is what you're left with at high slip ratios or slip angles.
• The load sensitivity is b1, while b2 is the constant friction coefficient (the offset of the friction is b1*Fz).
• The b2 value is like the friction coefficient only multiplied by 1000, so b2=1688 means a peak friction coefficient of 1.688 (apart from Sv). An F1 tire may go beyond that, to over 2.0, especially when include load sensitivity.
• All the sin() and atan() functions just create a nice curve which has a peak at which D+Sv is reached, and then moves down again, after a reasonably sharp fall.

There are 10 coefficients, b0-b10. With help of Brian Beckman chapter 21 (the example values are Genta's Ferrari), these are:

A note about the example values; there are a lot of zeroes in here, and for example 'load sensitivity' (b1) is also 0 which makes me believe these numbers are not too realistic, since every tire seems to have load sensitivity. But well, we'll have to make do.

Lateral forces

More on this later...

Sign conventions

The following sign conventions apply to Racer's Pacejka implementation:

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.

Combined Pacejka formulae

The Fx and Fy formulae are really only applicable when there is only 1 of them; so Fx is valid if Fy=0, and Fy is valid if Fx=0. If both longitudinal and lateral forces are applied, things are different. A tire can only generate so much total force. This is where the term 'friction circle' comes from: when combining Fx and Fy, you'll never get an added result (as Pythagoras, sqrt(Fx^2+Fy^2)) that goes beyond a maximum radius. This radius is the maximum total force that the tire can generate.

Actually, the friction circle really looks more like an ellipse, and for most tires adding just a little lateral force generates more total force than just the maximum longitudinal force (Fx) alone.

A simple method to combine Fx and Fy (when both are non-zero) is an elliptical approach:

• Calculate Fx and Fy separately (as usual)
• Cut down Fy so that the vector (Fx,Fy) doesn't exceed the maximum magnitude:
  Fy=Fy0*sqrt(1-(Fx/Fx0)^2)
  Where:
  - Fy is the resulting combined slip lateral force
  - Fy0 is the lateral force as calculated using the normal Fy formula
  - Fx is the longitudinal force as calculated using the normal Fx formula
  - Fx0 is the MAXIMUM longitudinal force possible (calculated as D+Sv in the Pacejka Fx formula).

This method favors longitudinal forces over lateral ones (cuts down the lateral force and leaves Fx intact).

The magic formula was extended around 1996 for combined slip operation. Note that in Racer v0.4.9, this method is not used, but instead there are methods from Gregor Veble and Brian Beckman, which seem to do very ok. The more formal Pacejka combined slip goes as follows (note this text may be off a bit, as I am interpreting some things on my own):

• Fx = Fx0 * Gxa(a,k,Fz)

Where Fx is the longitudinal force, Fx0 is the original Pacejka Fx formula described above, Gxa is a weighting function. We move on:

• Fx = Dxa*cos [ Cxa * arctan{ Bxa * ( a + SHxa ) } ]

with coefficients:

• Bxa = r_Bx1 * cos [ arctan ( r_Bx2*k ) ] * lamba_xa
• Cxa = r_Cx1
• Dxa = Fx0 / cos [ Cxa * arctan ( Bxa * Shxa ) ]
• SHxa = r_Hx1

and the weighting function is defined as:

• Gxa = cos [ Cxa * arctan { Bxa * (a + SHxa ) } ] / cos [ Cxa * arctan ( Bxa * Shxa ) } ]

Notice that Shxa and SHxa are 2 different variables. The parameters for combined slip are:

• rBx1, rBx2; stiffness factor B
• rCx1; shape factor C
• rHx1; horizontal shift H

The problem here is more coefficients to worry about. What are realistic 'r' values? Actually, this scheme is part of a rewrite of the Pacejka coefficients, and I haven't yet investigated too much into how the old a*, b* and c* coefficient names relate to the above equations.

Aligning moment (Mz) - Pacejka96

The following code is from the source, so it details the exact Mz calculation.

rfloat   RPacejka::CalcMz96()
// Calculates aligning moment
{
   rfloat Mz;
   rfloat B,C,D,E,Sh,Sv;
   rfloat FzSquared;
     
   // Calculate derived coefficients
   FzSquared=Fz*Fz;
   C=c0;
   D=c1*FzSquared+c2*Fz;
   E=(c7*FzSquared+c8*Fz+c9)*(1.0f-c10*fabs(camber));
   if((C>-D3_EPSILON&&C<D3_EPSILON)||
      (D>-D3_EPSILON&&D<D3_EPSILON))
   {
     B=99999.0f;
   }   else
   {
     B=((c3*FzSquared+c4*Fz)*(1-c6*fabs(camber))*expf(-c5*Fz))/(C*D);
   }
   Sh=c11*camber+c12*Fz+c13;
   Sv=(c14*FzSquared+c15*Fz)*camber+c16*Fz+c17;
     
   Mz=D*sinf(C*atanf(B*(1.0f-E)*(sideSlip+Sh)+
         E*atanf(B*(sideSlip+Sh))))+Sv;
   return Mz;
}

Pacejka's Magic Formulae in Racer

Racer uses most of the above, except for the combined slip (where you have longitudinal AND lateral forces at the same time, which is just about all the time when racing). The combined slip is much more complex, using the optimal_slipratio and optimal_slipangle values from car.ini files to normalize tire force use. It then modifies inputs & outputs from the Pacejka formulae to get a combined version.

Note that the MF5.2 (Magic Formula 5.2) Pacejka formulation already contains combined forces and no such trickery is used; instead, the empirical values are used and the optimal_* values are ignored.

For the source code to Racer's Pacejka implementation (just the Pacejka formula, the combined force calculation is spread through a few modules and is not included), see the page on Racer's Pacejka source code.

Generic tire notation

You may have encountered tire names like 225/60HR-16. Here, the coding is (thanks to Alpine for the explanations on the forum):

(last updated May 28, 2010 )