approximations to the magic formula

International Journal of Automotive Technology, Vol. 11, No. 2, pp. 155−166 (2010)

DOI 10.1007/s12239−010−0021−5

Copyright © 2010 KSAE

1229−9138/2010/051−03

155

APPROXIMATIONS TO THE MAGIC FORMULA

A. LÓPEZ*, P. VÉLEZ and C. MORIANO

Industrial Engineering Department, Universidad Antonio de Nebrija, C/Pirineos 55, Madrid 28040, Spain

(Recevied 5 December 2008; Revised 8 June 2009)

ABSTRACT−Pacejka’s tire model is widely used and well-known by the automotive engineering community. The magic

formula describes the brake force, side force and self-aligning torque in terms of the longitudinal slip and slip angle, plus

several corrections. This paper uses approximation theory to obtain different types of approximations to the magic formula:

rational functions (RA) resulting from the Remez algorithm, expansions in a series of Chebyshev polynomials (ACh), a series

of Chebyshev rational polynomials (ARChPs), a series of rational orthogonal functions (ORF) and a series of ARChPs that

result from grade-1 ORFs. The last expansion shows the fastest convergence and most effective computation. Jacobi rational

polynomials can also be obtained to complement this expansion and facilitate fine-tuning in specific areas of the error curve.

This work is complemented by obtaining the original rational approximations to the inverse tangent function, which take

advantage of the curve symmetry to reduce the computation load and provide models that include the influence of the vertical

load. The convergence properties of the development in series and the error values resulting from numeric examples for the

three types of stress are shown. The proposed final ARChP expressions show very low error (1%) compared to the original

magic formula. They can be computed 20 times faster; they can be evaluated, derived and integrated analytically easily; and

their coefficients can be obtained from tests using common least-squares algorithms.

KEY WORDS : Magic formula, Tire model, Approximation theory

1. INTRODUCTION

This paper searches for a Chebyshev series expansion of

Pacejka’s tire model in order to obtain a more efficient

mathematical expression with enhanced analytical proper-

ties that can be integrated in the series expansion of the

equations that describe vehicular dynamics. The final aim

is to advance toward analytic solutions of those equations

using symbolic computing.

López et al. (2006) has provided an example of expan-

sion in the power series of a simple longitudinal dynamics

of a vehicle. The resulting polynomial expressions facili-

tate very fast computation of the dynamic equations in real

time. Moreover, pre-computation of answers dependent on

the model entries can be achieved simply with the use of

symbolic computation tools (MAPLE). The need to obtain

simple and accurate formulations for the tire model to be

integrated into the previous dynamic model led to the

publication (López et al., 2007) of a first paper, which in

turn led to the development of RA, ACh and ARChP

approximations for longitudinal stress and bivariate ap-

proximations to the magic formula. Now, that report has

been completed and further expanded with the addition of

rational orthogonal functions theory and ORF and ARChP

expansions stemming from ORFs. Expansions in Jacobi

polynomials are also added for exact shift adjustments at

the origin and to obtain the error in different sections of the

curve. This article covers three types of stress in addition to

simplifications based on curve symmetry.

A new, efficient bivariate expansion is presented. Coeffi-

cients are easily obtained from the tests using standard least

squares algorithms.

2. REVIEW OF THEORETICAL BASIS

2.1. Approximation of a Function in a Chebyshev Series

(ACh)

Chebyshev polynomials (Fox and Parker, 1968) of the first

kind are defined by

and are orthogonal to the function w(x)=(1−x2)−1/2 on the

interval [−1, 1].

To work in different [a, b] intervals, shifted polynomials

must be used:

Their general expression (Abramowitz and Stegun, 1972)

is the following:

n=1, 2, 3...; T0(x)=1

Tnx( )=cos[n arccos x( )]

t=1/2 b a–( )x+a+b[ ]

Tnx( )=

n

2---

m 0=

n/2

∑ 1–( )m n m– 1–( )!

m! n 2m–( )!---------------------------- 2x( )

n 2m–

;

*Corresponding author. e-mail: [email protected]

156 A. LÓPEZ, P. VÉLEZ and C. MORIANO

where is the largest whole number less than or equal

to n/2.

They fulfill the following recursive property:

n=1, 2,... (1)

Chebyshev polynomials can be computed and manipulated

using the MAPLE Orthopoly library.

The expansion of a function in a Chebyshev series

(ACh) has the following form:

,

The single comma in the summation indicates that the first

term must be divided by 2.

This expansion usually converges faster than the power

series, and the coefficients are described by the following:

If we truncate the series at N degrees, we get an approxi-

mation to the function: the accuracy of the approximation

improves as N increases. Because of the properties of

Chebyshev polynomials, truncating the function at N-1

degrees is the best N-1-degree polynomial approximation

to the function with N degrees.

The coefficients an can be assessed with direct integ-

ration in some functions, but, in general, this calculation is

not possible, and the previous integral must be approximat-

ed by some other quadrature formula. MAPLE uses quad-

rature algorithms that first analyze the singularities and

then use Clenshaw-Curtis quadrature (Clenshaw and Curtis,

1960; Waldvogel, 2006); if the result is not satisfactory,

Newton-Cotes adaptive formulae are used. All of these

algorithms are carried out in the Chebpade function from

the MAPLE Numapprox library of approximation of func-

tions.

2.2. Approximation Using Rational Functions (RA)

RA approximations are more efficient when the function

varies rapidly in some areas but not in others, which occurs

in tire behavior, especially when longitudinal stress is con-

sidered.

The Padé approximation provides rational expressions

with their numerators and denominators developed in power

series. They are processed efficiently as a continuous frac-

tion. Chebyshev-Padé developments generate more com-

pact and accurate rational expressions with Chebyshev

polynomials in their numerator and denominator. The

MAPLE Numapprox library also implements the rational

approximations. Its Chebpade function turns the initial

Chebyshev function into a power series, carries out a Padé

approximation and turns the resulting numerator and de-

nominator into Chebyshev series again.

Chebyshev-Padé functions obtain good approximations,

but not those of minimum-maximum error (known as

minimax). To find the latter, the second Remez algorithm

(Remez, 1934) is used, which is a modified Chebyshev-

Padé approximation; it fine-tunes the result with numeric

iterations and converges to an improved minimax approxi-

mation.

The second Remez algorithm produces optimal results

that approximate both rational and polynomial functions.

This function allows the minimum error of any given func-

tion f (t) weighted with any weight term w(t) to be calcu-

lated. If w(t)=1/| f (t)| is used, the minimum relative error is

obtained. The minimax approximation with n-degree poly-

nomials in the numerator and m-degree polynomials in the

denominator requires (n+m) additions and (n+m) multipli-

cations for its evaluation, which are indicated as a minimax

approximation [n,m].

These methods are described in many books on approxi-

mation theory (Powel, 1981).

In MAPLE, the Remez algorithm is implemented by the

minimax function that is included in the Numapprox library

of function approximations.

2.3. Approximation to a Function in a Series of Chebyshev

Rational Polynomials (ARChP)

More recent works (Guo et al., 2002; Wynn, 2006) show

the suitability of using rational polynomials, which accele-

rate convergence when the functions to approximate have

singularities or quick variation areas. These inherit the pro-

perties of Chebyshev polynomials and have the form:

;

where

The development of a function in a series of Chebyshev

rational polynomials is:

for

Chebyshev polynomials are orthogonal on the interval [−1,

1], but our independent variables (slip K and lateral slip)

vary between 0 and 100 and between −15o and 15o, respec-

tively (because the formula is the same, we will generically

call both of them x, and their initial and end points xin and

xfin, respectively). Thus, the Chebyshev expansion in series

on the original variable x cannot be performed because its

domain exceeds the orthogonality of Chebyshev polynomials.

Therefore, shifted polynomials at v (Fox and Parker, 1968,

p. 49), , with the following variable change

must be used:

n/2

Tn 1+ x( )=2xTn x( )−Tn 1– x( );

f x( )=

n 0=

∞

∑ ′anTn x( )

an =2

π---

1–

1

∫ 1 x2

–( )1/2–

f x( )Tn x( )dx

Rn x( )=Tn

x 1–

x 1+------------⎝ ⎠⎛ ⎞=Tn v( ); 0 x ∞≤ ≤

v=x 1–

x 1+----------; x=

v 1+

v 1–----------–

f x( )=

n 0=

∞

∑ ′βnRn x( )=

n 0=

∞

∑ ′βnTn

x 1–

x 1+------------⎝ ⎠⎛ ⎞=

=

n 0=

∞

∑ ′βnTn v( ); 1– v 1≤ ≤

xin x xfin vin v vfin≤ ≤⇒≤ ≤

vin=xin 1–

xin 1+-------------; vfin=

xfin 1–

xfin 1+--------------

Tn

dv( )=Tn u( )

APPROXIMATIONS TO THE MAGIC FORMULA 157

where u shifts between −1 and 1. Therefore, the final

development is:

We can see that a double transformation, from the x to y

domains and from the v to u domains, was required. The

function in the u domain is approximated by a Chebyshev

development in series, and in the resulting approximate

function, the two previous transformations are undone to

obtain the approximate function in the original domain x

(slip or lateral slip).

2.4. Rational Orthogonal Functions (ORF) Theory

2.4.1. Introduction

According to Bultheel et al. (1999), if A={α1, α2..}, is a

sequence of real numbers other than zero, the linear vector

space of n-degree rational functions with poles at {α1 , …,

αn..} is defined by the space of functions L={b0, b1…, bn},

where the base functions are defined by:

If we orthonormalize and assume an interval on the real

line that excludes every pole, then these functions meet the

recurrence relation.

Van Deun et al. (2004) has obtained the coefficients of the

recurrence relation (En, Fn) for the case of Chebyshev ORF

functions with Chebyshev weight functions. A Chebyshev

weight function is a Jacobi weight function of the type:

where δ y γ = ±1/2

When the Joukowski transformation is introduced,

Van Deun gets the coefficients:

If n=1, we get:

;

where

Examples and application to the magic formula are shown

in section 3.6.

2.4.2. Expansion in ORFs series

The best least-square approximation obtained after truncat-

ing the expansion in a series of orthogonal functions F(x)

(of any type) of a function f (x) is (Burden and Douglas,

1998);

(2)

where the coefficients are:

The weight function w(x) defines the importance of the

approximation of different sections of the interval [x1, x2].

For example, the Chebyshev weight function:

has very little influence in the center of the interval and

more influence at its ends.

In this particular case, Fj(x) are ORFs. The value of rj

represents the norm of the ORF function, and for ORF

functions with Chebyshev weight functions, it takes the

constant value rj=π. We recall that, in the case of Cheby-

shev polynomials Fj(x)=T( j, x), this value was rj=π/2.

MAPLE does not support any function related to ORFs,

neither the generation nor expansion of functions. Expan-

sion of the magic formula in ORF is presented in section

3.6.

2.4.3. Expansion in a series of Jacobi polynomials

Within the families of classic orthogonal polynomials gene-

rated from the Sturm-Liouville differential equation, from

which Chebyshev polynomials are also derived, we consi-

der Jacobi polynomials (Totik, 2005). The weight function

v=1/2 vfin vin–( )u+ vin vfin+( )[ ]; ⇒

u⇒ =2v vin vfin+( )–

vfin vin–( )-------------------------------

f x( )=

n 0=

∞

∑ ′βnTn

sv( )=

n 0=

∞

∑ ′βnTn

2

vfin vin–---------------- v

vin vfin+( )vfin vin–

---------------------–⋅⎝ ⎠⎛ ⎞

=

n 0=

∞

∑ ′βnTn u( )

bk x( )=x. 1 x.αk–( ).bk 1– x( ), b0 x( )=1; αk=1αk

-----

ϕn x( )=En.x. 1 x.αn–( )+Fn 1 x.αn 1––( ). 1 x.αn–( ).ϕn 1– x( )

−En

En 1–

---------1 x.αn 2––( )1 x.αn–

--------------------------.ϕn 2– x( )

w x( )=1 x–( ) δ

1 x+( )γ-----------------

w x( )=

1 x2

–( )1/2–

, case a)

1 x2

–( )1/2

, case b)

1 x–

1 x+----------⎝ ⎠⎛ ⎞ case c)

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

α =1

2--- β z+( ); α=J β( )

β2−2αβ+1=0; β =α± α

2

1– ; β =J1–

α( )

En=2

1 βn 1–

2

–( ) 1 βn

2

–( ) 1 βn 1– β n

–( )

1+βn 1–

2

( ) 1+βn

2

( )--------------------------------------------------------------------------

Fn=−

1 βn

2

–

1 βn 1–

2

–

------------------1 β

n 1–

2

–( ) βnβ

n 2–+( )+2βn 1– 1 β

n– β

n 2–( )

1+βn

2

( ) 1+βn 1– β n 2–( )

-----------------------------------------------------------------------------------------------

E1

a= 2c; E1

b=2c; E1

c=2c

F1

a=− 2β1c; F1

b=−β1c F1

c= 1 β1–( )c

c=1 β 1

2

–

1 β 1

2

+-----------------

f x( )

n 0=

∞

∑≈ ak.Fk x( )

aj=1

rj

--- x1

x2

∫ w x( ).f x( ).Fj x( )dx

rj= x1

x2

∫ w x( ). Fj x( )[ ]2

dx

w x( )=1

1 x2

–

---------------


in this type of polynomial

is controlled by two parameters, δ and γ, which allow the

area of a best approximation in the orthogonality interval to

be chosen. In practice, this is very interesting because it

allows us to improve the error adjustment in any area of the

longitudinal stress, lateral stress, or self-aligning torque

curves, depending on the application in which the approxi-

mation is used: for instance, looking either for a more

reduced error in slip values close to zero or for values close

to the maximum stress or the maximum slip point (100%).

The norm rj in Jacobi polynomials is not constant, and it

is a function of δ, γ and the degree of the n-polynomial.

The recurrence relation seen for the Chebyshev polyno-

mials (1) in section 2.1 is made more general in the case of

Jacobi polynomials:

where the recurrence coefficients are now:

Jacobi polynomials can also be computed and manipulated

using the MAPLE Orthopoly library.

The expansion of a function in a series of Jacobi poly-

nomials uses the same expression (2) as in section 2.4.2,

but with a Jacobi weight function. The integral must be

programmed in MAPLE. A library for expansions of func-

tions in Jacobi series is not available.

2.5. Magic Formula

The well-known model proposed by Bakker et al. (1987,

1989) and Pacejka (2002), is a semi-empirical tire model

based on the “magic” formula:

Y=D.sin[C.arctan(BX–E.[BX-arctan(BX)])]

The shape of the curve is controlled by four parameters: B,

C, D and E. The equation can calculate the following:

• Lateral forces in a tire, Fy, as a function of the slip angle

of the tire, α, (in degrees)

• Braking force, Fx, as a function of longitudinal slip K (%)

• Self-aligning torque, Mz, as a function of the slip angle α.

B, C, D and E are constants that describe the inclination of

the curve at the origin (BCD), the peak value (D), the

curvature (E) and the basic form (C) for each case (lateral,

braking or self-aligning torque). In addition, the curve can

have vertical (Sv) or horizontal (Sh) shifts at the origin.

The full expression is:

Y=D.sin[C.arctan(B(X+Sh)–E.[B(X+Sh)

−arctan(B(X+Sh))])]+Sv

Coefficients B, D and E are functions of the vertical load in

the tire, Fz:

BCD2=a3.sin(a4(arctan(a5.F)));

BCD1 is valid for the longitudinal force and the self-

aligning torque with C=1.65 and C=2.4, respectively.

BCD2 is valid for the lateral force with C=1.3.

The Camber angle γ in the wheel modifies the shifts Sh

and Sv and the stiffness BCD:

;

E1 is the E value modified by the camber angle in the self-

aligning torque calculation.

The aforementioned authors published the following

values of the coefficients a1...a13 for a given tire:

3. APPROXIMATIONS TO THE MAGIC

FORMULA

3.1. Rational Approximations (RA) to the Functions Arctan(x)

and Sin(x)

3.1.1. Arctan(x)

We approximate the function arctan(x) that appears in the

expressions included in the magic formula. We find values

of x between −20 and 30 rad approximately, with a function

value which is maximum at its asymptote and equal to π/2.

By using RA minimax [2, 2], deleting the independent

term in the numerator, adjusting and rounding up or down

the coefficients, we obtain the following:

arctan(x)≈

which is valid , anti-symmetrical, has a very low ab-

solute maximum error ε |ε | < 0.0025, is far more accurate

than other pseudoarctan(x) formulations presented in the

literature , and has “nice” coefficients.

In continuous fraction form, it can be expressed as:

arctan(x)≈sign(x).

The function requires four additions and two divisions

plus the sign. Even lower errors can be obtained by ap-

proximations [2, 3] or higher.

w x( )=1 x–( )δ

1 x+( )γ----------------

rj=2

δ γ 1+ + Γ n δ 1+ +( ).Γ n γ 1+ +( )n!. 2n δ γ 1+ + +( ).Γ n δ γ 1+ + +( )----------------------------------------------------------------------------

Pn 1+

δ,γ( )x( )= an bb+( ).Pn

δ,γ( )x( )−cn.Pn 1–

δ,γ( )x( ); n=1,2,...

an=

2n 1 δ γ+ + +( ) 2n 2 δ γ+ + +( )

2 n 1+( ) n 1 δ γ+ + +( )----------------------------------------------------------------------

bn=

δ2

γ2–( ) 2n 1 δ γ+ + +( )

2 n 1+( ) 2n δ γ+ +( ) n 1 δ γ+ + +( )------------------------------------------------------------------------------

an=

n+δ( ) n+γ( ) 2n 2 δ γ+ + +( )

n 1+( ) n 1+δ γ+ +( ) 2n δ γ+ +( )--------------------------------------------------------------------------

d=a1.Fz

2+a2.Fz; B=BCD/ C.d( ); E=a6.Fz

2+a7.Fz+a8;

BCD1=a3Fz

2

a4Fz+

ea5.F

z

-------------------------;

∆Sh=a9.γ ; ∆Sv= a10Fz

2+a11Fz( ).γ ; ∆B=−a12 γ .B

E1=E0

1 a13. γ–---------------------

x. 4.66 8. x+( )5 6. x 5.1.x

2+ +

-----------------------------------

x∀

1.5686270.931719

x 0.4741–1.762934

x 1.650574+-------------------------------+

--------------------------------------------------------------–

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞


3.1.2. Sin(x)

The function sin(x) also appears with −4.1 < x < 4.1 rad. If

we proceed in the same way as with arctan(x), we get:

|ε | < 0.018, 9 Op (5 Mul, 1Div and 3 Add)+2.abs(x)

There are more efficient approximations [2,1] at the

longitudinal and lateral stress ranks:

(−2.3 < x < 2.3)

|ε | < 0.014, 6 Op (3 Mul, 1Div 2 Add)

3.2. Direct Approximations ACh to the Magic Formula

In this article, we consider relative error to be the absolute

error divided by the maximum absolute value of the func-

tion. This approach is convenient because approximations

with a minimax classic relative error (divided by the

modulus in each x value) give good results in low force

sections of the curve close to 0 (the least interesting

section), but very poor results in the rest of the curve (the

most interesting part). Our definition allows us to compare

errors for different vertical loads easily.

The expansion in the Chebyshev series of the magic

formula does not allow the use of low degree polynomials;

the following table shows the polynomial degree and the

relative error (as defined in the previous paragraph), with a

vertical weight Fz=8 kN.

The high values of the normal weight are those that need

a higher polynomial degree.

Regarding lateral force, the direct ACh of the magic

formula requires n ≥ 5 polynomials at the rank 0 < x < 15o

to cover all the weight values in our sample tire (from 0 to

10 kN). For the rank −15 < x < 15o, we need at least n ≥ 20

degree polynomials: low normal weight values require a

higher polynomial degree.

For the self-aligning torque we need n ≥ 13 for the half

interval and n ≥ 45 for the complete interval 15 < x < 15o.

Therefore, using direct expansions in Chebyshev series

in the magic formula is not a good idea, because conver-

gence is not fast enough.

3.3. Rational Approximations (RA) to the Magic Formula

3.3.1. Longitudinal force

The minimax [2,2] adjustments carried out with the MAPLE

Numapprox library related to our sample tire give relative

error values that increase with the vertical weight Fz from

Rel.Error=0.9% for Fz=1 kN, to 1.36% for Fz=8 kN.

Using minimax approximations [2,3], we get Relative

Error values that fluctuate between 0.36% for Fz=1 kN and

0.79% for Fz=8 kN:

Figures 1 and 2 show both the adjustment of the approxi-

mation [2,3] and the error. For the case of [2,2], the curve

has a similar shape but a slightly higher error.

If we accept an acceptable maximum relative error

criterion of less than 1%, we must work with minimax

approximations [2,3].

If we delete the independent term from the numerator in

the previous approximation [2,3], we get a slightly higher

error; however, this error is zero at the origin.

3.3.2. Lateral force and self-aligning torque

If we proceed in the same way, we can see Error < 1% in

minimax adjustments [5,3] for self-aligning torque, where

−15 < x < 15o. The figures show the lateral stress curve and

the error curve with Fz=8 kN.

We will show how to take advantage of the symmetry

using the results of the approximation between 0 and 15o in

section 3.9.

Regarding the self-aligning torque, if we want to ap-

sin x( )x. 0.96 0.306– . x( )

1.025 0.357– . x +0.1121.x2

----------------------------------------------------------------≈ ;

sin x( )x. 1.3 0.45– . x( )

1.177– +0.154. x----------------------------------------≈

Fx3.1564.10

6

. x 13.7157+( ). x 0.01928–( )

x 592.4785+( ). x2

3.695.x 22.7886+ +( )--------------------------------------------------------------------------------------------≈

Figure 1. Rational approximation [2,3] of braking force as

a function of slip.

Figure 2. Absolute error (N) at the rational approximation

[2,3] of the braking force, as a function of the slip.


proximate the whole rank between −15 and 15 with a Rel

Error < 1%, we must use an approximation [5,5].

We check the resulting curves for Fz=3 kN.

If we focus on the rank 0..15, the approximations [2,3]

produce results with a Relative Error < 1%.

3.4. Approximations ARChP to the Magic Formula with

Constant Fz

Applying the expansion described in section 2.3, if we

include the suggested double transformation, we can see

that the ARChPs of the form:

give results with a Relative Error < 1% with polynomials:

n ≥ 8 for Fx with 0 ≤ x ≤ 100, Fy with 0 ≤ α ≤ 15 and

0 ≤ Fz ≤ 8 kN

n ≥ 12 for Mz with 0 ≤ α ≤ 15, and 0 ≤ Fz ≤ 8 kN

It is evident that convergence is faster than in the case of

ACh direct expansions; however, it is still not satisfactory

because the polynomials have at least eight degrees.

3.5. ARChP Approximations from ORFs

The convergence speed can be improved if we expand in a

Chebyshev series of rational functions of the type:

The optimal factor b in each case varies with Fz.

f x( )=

n 0=

∞

∑ ′βnT

n

s x 1–

x 1+-----------

⎝ ⎠⎛ ⎞ f x( )=

n 0=

∞

∑ ′φnT

n

s x

x b+-----------

⎝ ⎠⎛ ⎞

Figure 3. Rational approximation [5,3] of the lateral force

as a function of the lateral slip.

Figure 4. Absolute error (N) at RA [5,3] of the lateral force

versus lateral slip.

Figure 5. Rational approximation [5,5] of the self-aligning

torque.

Figure 6. Absolute error (N.m) at the rational approxi-

mation [5,5] of the self-aligning torque.


We obtain Relative Error < 1% in the following expan-

sions:

n ≥ 4 with b=4 for Fx with 0 ≤ x ≤ 100 and 0 ≤ Fz ≤ 8 kN

n ≥ 4 with b=4 for Fy with 0 ≤ α ≤ 15 and 0 ≤ Fz ≤ 8 kN

n ≥ 9 with b=3.5 for Mz with 0 ≤ α ≤ 15 and 0 ≤ Fz ≤ 8 kN

The specified values of b guarantee a RelError < 1% for

0 ≤ Fz ≤ 8 kN. However, we can improve the error for

every value of Fz by modifying b slightly.

In Fx, for 4 ≤ Fz ≤ 7 kN, the result is n ≥ 3 with Error < 1%.

This expansion calculation in MAPLE is performed with

the Minimax-Remez function, which produces more accu-

rate results than the Chebpade function, as has already been

stated.

These results are excellent: for example,

Fz=6 kN (constant)

C=1.65; D=6097.2; B=0.2064; E=0.606; BCD=2076.600

a1=-21.3; a2=1144; a3=49.6; a4:=226; a5=0.69e-1;

a6=-0.6e-2; a7=0.56e-1; a8=.486

xin=0; xfin=100; vin=0; vfin=100/104; b=4

Original equation (Magic formula)

Fx=6097.2.sin(1.65.arctan(0.0813 x+0.606.arctan(0.2064 x)))

Approximation

Fxap=3557.1888.T s(0,v)+2587.3377.T s(1,v)−

1536.3996.T s(2,v)−515.9311.T s(3,v)

being: T s(n,v)=T(n, .v−1)

In addition, according to the Horner normal form:

Fxap=−50.61+(8506.27+(13491.33−18571.27.v).v).v

where

If we execute expansions in x for different values of Fz,

look for the optimal b in each case and calculate the re-

gression of b=f(Fz), we increase the convergence speed; in

the case of our sample tire, we can get:

n ≥ 3 with b=5.4629-0.2829.Fz

for Fx with 0 ≤ x ≤ 100 and 0 ≤ Fz ≤ 8 kN.

3.6. Monopole ORF Approximations to the Magic Formula

When applying the results given in section 2.4, we can get

the base of the monopole ORF functions from the values of

b.

As an example, the three ORFs and the approximation

for the braking force are shown for the same tire data and

maximum force (6 kN).

.

In this case, the error curve is very similar to that shown in

Figure 8, although it fluctuates between −60 N and +60 N.

This approximation is less accurate and requires more

computation than the previous one because the previous

approximation was a minimax (this can be seen when

checking the maximum local error leveled in Figure 8), and

this approximation is a minimum squared expansion,

which is less precise.

3.7. Bipole ORF Approximations to the Magic Formula

Using the same tire data as in the previous example, start-

ing from pole 4, the third pole increases and the initial

bipole decreases until the minimum error is observed:

Fxap=

n 0=

∞

∑ ′βnTn

2

vfin vin–------------------.v

vin vfin+

vfin vin–------------------–

⎝ ⎠⎛ ⎞

52

25------

v=x

x 4+----------

Figure 7. Braking Force (N) versus Longitudinal slip in

Chebyshev series of the rational function x/(x+b).

Figure 8. Absolute Error (N) versus Longitudinal slip in a

Chebyshev series of the rational function x/(x+b).


In this case, the error is better than the previous ones, and

because it is similar to Figure 8, it fluctuates between −40

N and +40 N.

This bipolar ORF approximation has the same error as

RA minimax [2,3], although the minimax has complex poles.

Similar ORF expansions can be performed for lateral force

and self-aligning torque.

3.8. Approximations in a Series of Jacobi Rational Poly-

nomials

If we use the expressions in section 2.4.4., we can start

from the values δ =−1/2 and γ =1/2, which correspond to

the Chebyshev weight function (which, in turn, is a parti-

cular case of Jacobi polynomial). Increasing both values

reduces the error in the central area of the curve and

increases it at the ends. The error can also be adjusted to

zero at the ends by keeping one of the two parameters fixed

and changing the other, while keeping the maximum error

values constant for the whole curve. Thus, for instance, the

value of shifts from the origin to the values Sv and Sh

indicated in section 2.5 can be adjusted.

For example, for the same tire with Fz=6 kN, an adjust-

ment with a null error is shown at the origin (Sv=Sh=0)

with the values δ =−1/2 and γ =−0.4685/2. The resulting

approximation (with b=3.85) is the following:

Fxap=(7369.26+(15916.3-19867.52.v).v).v

where

The error curve looks very similar to that shown in Figure

8, which is also between ±50 N, but the current error curve

has a null error at the origin.

We can adjust the null error at the end of the curve or at

its maximum using this method.

We can also adjust the slope at the origin (the value BCD

in the original Pacejka formula) to obtain an exact value or,

with a moderate error, to achieve global maximum error

values around 1%. For example, the following approxi-

mation (δ =−0.16 and γ =−0.68, b=3.85):

Fxap=−62.86+(7904.7+(14744.57−19130.30.v).v).v

being

has a slope at the origin of BCDap=2053.

(BCD original=2076). (Error=1.1%)

The approximate peak value is Dap=6135

(D original=6097) (Error=0.6%), with a maximum global

error of 67 N (Error=1.09%)

In this type of approximation:

Fap=A0+(A1+(A2+A3.v).v).v

The value of A1/b is the derivative at the origin (BCD).

3.9. Use of Symmetry

As we have expanded the lateral force Fy and the self-

aligning torque in the current semi-axis, we must look for

valid formulations for the entire real line to take advantage

of the performed computation with a lower degree poly-

nomial.

To calculate the approximate expression we do the

following:

(1) Calculate the approximation to the original function

without shifts with a null error at the origin in the

interval 0..vfin using Jacobi polynomials.

(2) Calculate the valid expression for the interval -xfin..xfin,

which passes through the origin.

(3) Apply Sv and Sh shifts to the approximate expression.

The following is an example for the lateral force Fy:

a1=−22.1; a2=1011; a3=1078; a4=1.82; a5=0.208; a6=0;

a7=−0.354; a8=0.707;

D=5270.4; BCD=1076.149; B=0.1571: E=−1.417

Sh:=−0.126:Sv:=−181:Fz=6 kN.

(1) Approximation using a series of Jacobi rational poly-

nomials of the function with Sv=Sh=0 and a null error

at the origin (calculated with α =−1/2 and β =0.71).

v=x

x+3.85----------------

v=x

x+3.85----------------

v=x

x b+----------

Figure 9. Lateral force (N) versus slip angle in a 3-degree

symmetric Jacobi approximation, with shifts Sv and Sh.


The maximum absolute error is 68 N:

Fxap=(5166.26+(24265.833-30220.43.v).v).v

(2) A 3-degree expression valid for -xfin ≤ x ≤ xfin

F1xap=sign(x).(5166.26+(24265.833-30220.43.v1).v1).v1

(3) Shifted final approximation

F2xap=F1xap(x+Sx)+Sv

The resulting curves are shown in Figures 9 and 10.

4. INFLUENCE OF THE VERTICAL LOAD

For the longitudinal force, we approximate the influence of

the vertical load from the curve Fap1 obtained for 1 kN by

adding the peak value factor D(Fz) that coincides with that

of the original formula (now it gives the approximate peak

value) and a second shape factor Ff. Both factors are func-

tions of the vertical load Fz. We associate the linear coeffi-

cient A’1 with stiffness at the origin. We calculate the regre-

ssion with optimal values of Fs and A’1 for each value of

Fz.

Fxap=D.(

We show an example of the braking force with three degrees

and a maximum relative error of 1.1% for 1 kN ≤ Fz ≤ 8 kN:

From the Jacobi approximation with Fz=1;

δ =−1/2; γ =−1.98/2;

b=5.5; A1=1249; A2=3191; A3=−3828;

In addition, using the original peak value factor,

D=a1. Fz2+a2Fz; a1=−21.3; a2=1144; and

D1=D(Fz =1)=1122.7 N

The approximate shape factor giving minimum error is:

Fs=5.956−0.5181.Fz+0.0255.Fz2

The stiffness factor with minimum error is:

A’1=−0.00102906.Fz2+0.0092337.Fz+1.104

The slope at the origin is D.A’1/FS, where D the approxi-

mate peak value.

Figures 11 and 12 show the error in this approximation.

If we use the original BCD value to calculate A’1, where

A’1=Fs.BCD/D, we can use existing data; however, the

error is three times greater.

Clearly, we can integrate the factor D into the A’i, coeffi-

cients by finding the products D.A’2 and D.A’3, and then

reducing the product D.A’1 to two degrees using Minimax-

Remez [2,0]. We can also consider A’2 and A’3 to vary with

Fz for a longer period of time to obtain more accurate

expressions: shorter expressions of D.A’1(Fz) have a larger

error. There are many possibilities. One of the simplest is

the following:

Fxap=Fz.(B1+(B2+B3.v).v).v

with a maximum RelError=3.2% and the following values

of Bi:

and working with the same shape factor:

Fs=5.956−0.5181.Fz+0.0255.Fz2

For the lateral force Fy, the shape of the curve changes

v=x

x+6.5-------------

v1=x

x 6.5+------------------; sign x( )=

x

x-----

A1′+ A2′ A3′.v+( ).v).v

v=x

x Fs+-------------; A2′=

A2

D1

------; A3′=A3

D1

------

A2′=3191

1122.7----------------=2.842; and A3′=

3828–

1122.7----------------=−3.409.

v=x

x Fs+-------------; Bi=const.

B1=930; B2=3910; B3=−4250

Figure 10. Absolute error (N) versus slip angle in a degree

3 symmetrical Jacobi approximation, with shifts Sv, Sh.

Figure 11. Braking force versus longitudinal slip for Vari-

able vertical load (1 kN ≤ Fz ≤ 8 kN).


more with Fz; thus, we must use variable coefficients A2

and A3 with Fz. The expression is the same:

Fyap=D.

and has the following optimum values:

=1.26−0.231.Fz+0.0251.Fz2

=1.91+0.495.Fz

=−2.51−0.129.Fz−0.068.Fz2

FS=3.3−0.46.Fz+0.156.Fz2

We also use the original peak value D for Fy:

D=a1. Fz2+a2Fz; a1=−22.1; a2=1011;

Relative Error curves are similar to those in Figure 12, with

a maximum error of 2.6% for a 3-degree expansion with 1

kN < Fz < 8 kN.

Once again we can integrate the products D.A’i=f(Fz),

with different results in terms of complexity and error.

In all these examples, for both Fx and Fy, the error is 0 at

the origin and we can add original shifts Sv and Sx, as seen

in section 3.9, using existing data.

5. COMPUTATIONAL EFFICIENCY

In order to compare the computational efficiency, we consi-

der the expressions shown in section 3.5:

Original equation (Magic formula)

Fx=6097.2.sin(1.65.arctan(0.0813 x+0.606.

.arctan (0.2064 x)))

Approximation

Fxap=−50.61+(8506.27+(13491.33−18571.27.v).v).v

where

Our proposed approximated formula computes 20 times

faster than the original Magic formula.

(C-Compiler: MinGW (c); Intel Core2 CPU. T5600 at

1.83 GHz; 987 MHz, 1,99 GB RAM).

If we compare the expressions shown in section 3.9,

which are valid for both the positive and negative sides:

F1xap=sign(x).(5166.26+(24265.833−30220.43.v1).v1).v1

F2xap=F1xap(x+Sx)+Sv

with the full magic formula, including shifts:

Y=D.sin[C.arctan(B(X+Sh)–E.[B(X+Sh)

−arctan(B(X+Sh))])]+Sv

Then the approximated expression runs eleven times faster.

In addition, integrating the model into the Chebyshev

series expansion of the equations that describe the vehi-

cular dynamics is also possible. Thus, analytic solutions with

a very high computational efficiency can be determined.

6. DETERMINING THE COEFFICIENTS FROM

TESTS

The previous equation:

Fxap=A0+(A1+(A2+A3.v).v).v

is a polynomial; thus, it is easy to obtain the coefficients Ai

from tests using the Least-Squares standard algorithms.

Previously, we had to transform data from the original vari-

able (for example, slip) to the transformed variable, v=x/

(x+b). For example:

The slip data vector (%) (22 values, simulated example) is

the following:

[0,1,2,3,4,5,6,7,9,13,17,21,25,31,37,43,49,58,68,78,88,100]

The braking force data vector Fx (22 values, simulated ex-

ample) is the following:

(A1′+ A2′ A3′.v+( ).v).v

v=x

x+Fs------------;

A1′

A2′

A3′

x

x+4---------

v1=x

x 6.5+----------------; sign x( )=

x

x----;

Figure 12. Relative error for variable vertical load.

Figure 13. Transformation of slip values; v=x/(x+b).


[−55.6, 1264.3, 2239.6, 2937.4, 3464.3, 3838.4, 3980.0,

4095.6, 4278.3, 4176.5, 4125.3, 3915.4, 3824.9, 3686.0,

3492.6, 3446.6, 3357.7, 3213.7, 3199.7, 3031.9, 3043.7,

2878.3]

The optimal b is 4.195, and the transformed slip data vector

(v=x/(x+4,195)) is the following:

[0, 0.1925, 0.3228, 0.4170, 0.4881, 0.5438, 0.5885,

0.6253, 0.6821, 0.7560, 0.8021, 0.8335, 0.8563,

0.8808, 0.8982, 0.9111, 0.9211, 0.9326, 0.9419,

0.9490, 0.9545, 0.9597]

The Least-Squares curve 3 in Figure 14 shows (0 ≤ v ≤

0.9597).

Fxap=−47.9487+(4917.92+(-13916.4+11471.01.v).v).v

If we take v=x/(x+4.195), we obtain the final approximated

curve 4 in Figure 14 (0 ≤ x ≤ 100).

The optimal value b used in the transformation is un-

known, but at the end of section 3.5, we were able to obtain

the optimal values of b for a given tire, which can be

expressed with a linear expression in terms of Fz:

b=5.46−0.28.Fz

The optimal value of the coefficient b for this tire, can take

values from 3.2 to 5.2 when the normal load Fz changes

from 1 to 8 kN.

If b takes values higher or lower than the optimal, the

addition of quadratic deviations from the test values (curve

4 in Figure 14) is always bigger. The minimum addition of

quadratic deviations is found when b is optimal. We only

need to program a loop to calculate the following for every

step:

− The transformed points (Curve 2 at Figure 13) for the

given b value.

− Curve 3 of Figure 14 using a common least-squares

algorithm with three degrees.

− Curve 4, which undoes the transformation from v to x.

− The addition of quadratic deviations from curve 4 to test

values.

In the loop, we vary the value of b in a wide range (from 1

to 20 for example), to find the optimal value of b for a

given normal load Fz. Corresponding Ai values for the

optimal b are also optimal.

Computing this loop takes 3 or 4 seconds. The full pro-

cess is automatic.

Determining coefficients for variable Fz is accomplished

as described in Section 4 from the set of constant Fz curves.

7. CONCLUSION

From the analysis of the different types of approximations

to the magic formula with constant Fz, we propose an

efficient and accurate calculation using the following type

of expression:

being

which results from expansion in a series of shifted Jacobi

rational polynomials as they converge at relative errors

around 1% with low degrees (3 or 4); they can be adjusted

in specific curve areas, especially at the origin, and they

allow the use of the same expression for both sides of the

symmetric curve for Fy and Mz, as seen in section 3.9.

Additionally, we can obtain analytic derivatives and inte-

grals of this expression easily. The latter is not possible in

the original magic formula. The slope at the origin is also

calculated easily (A1/b), and if we use 3-degree polyno-

mials as proposed, we can calculate the abscissa of the

maximum value analytically.

If maximum accuracy is the main goal for a constant

value of Fz, the use of Minimax-Remez rational approxi-

mations, such as those seen in section 3.3, is recommended.

The described techniques use state of the art theories of

function approximation together with the symbolic compu-

tation programs (MAPLE) and show the advantages of

handling the equations analytically, especially for Fx and

Fy, for which we can use 3-degree polynomials with very

low error. Self aligning torque requires higher degrees.

We can obtain different expressions of Fx and Fy de-

pending on the normal load Fz, as in Pacejka’s original

formulation, and use the same peak value factor D, to take

advantage of the already existing data.

The proposed expressions can be computed much faster

(20 times faster) than the original magic formula. In addi-

tion, integration in the Chebyshev series expansion of the

equations that describe vehicular dynamics is also possible

with high computational efficiency.

Obtaining the coefficients from test samples is also easy

because the proposed expressions are polynomials and auto-

matic Least-Squares algorithms can be used.

ACKNOWLEDGEMENTS−This report has been financed by

the Dirección General de Universidades (Comunidad de Madrid)

and the Instituto Madrileño de Desarrollo IMADE.

Fap= n 1=

N

∑ Ai.vn

; v=x

x b+----------Figure 14. Transformation of the Least-Squares curve.


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approximations to the magic formula

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