Application of the Pacejka Magic Formula Tyre Model on a Study of a Hydraulic Anti-Lock Braking...

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Application of the Pacejka MagicFormula Tyre Model on a Studyof a Hydraulic Anti-Lock BrakingSystem for a Light MotorcycleChen-Yuan Lu & Ming-Chang Shih

Version of record first published: 09 Aug 2010.

To cite this article: Chen-Yuan Lu & Ming-Chang Shih (2004): Application of the PacejkaMagic Formula Tyre Model on a Study of a Hydraulic Anti-Lock Braking System for a LightMotorcycle, Vehicle System Dynamics: International Journal of Vehicle Mechanics andMobility, 41:6, 431-448

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Vehicle System Dynamics2004, Vol. 41, No. 6, pp. 431–448

Application of the Pacejka Magic Formula Tyre

Model on a Study of a Hydraulic Anti-Lock

Braking System for a Light Motorcycle

CHEN-YUAN LU1 AND MING-CHANG SHIH1,2

SUMMARY

The object of the study is to apply the Pacejka magic formula tyre model on a study of a hydraulic anti-lock

braking system, especially applied to a light motorcycle. A sliding mode PWM controller is designed and

tested. Both simulation and experimental studies of an anti-lock braking system are undertaken. The paper

presents an analytical approach for estimating the longitudinal adhesive coefficient between a tyre and the

road through the magic formula tyre model, the parameters of which are identified by a genetic algorithm. A

dynamic analysis of a light motorcycle is carried out in detail. The experimental results show that the anti-

lock braking system designed in the study is effective to prevent wheels locking during emergency braking.

The proposed simulation results match experimental data well.

1. INTRODUCTION

In the past few decades, anti-lock braking systems and several different types of

controllers have been mounted in many vehicles. In general, there are two major

advantages of an anti-lock braking system over conventional braking: (1) shorter

stopping distances on most road surfaces, and (2) enhancement of steering control

during hard braking maneuvers. However, both of these phenomena relate strongly to

slip ratio. The braking performance index slip ratio, S, is defined as S ¼ Vv �Vtyre=Vv, where Vv is the velocity of the vehicle and Vtyre is the velocity of the tyre.

In Figure 1, the relationship between the longitudinal coefficient (related to the

braking force), the lateral coefficient (related to steering control) and the slip ratio are

clearly described. As the slip ratio increases from zero, the longitudinal adhesive

coefficient typically reaches a maximum and subsequently tends to a horizontal

1Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, ROC.2Address correspondence to: Professor Ming-Chang Shih, Department of Mechanical Engineering,

National Cheng Kung University, 700, No. 1, Dashiue Rd., Tainan City, Taiwan, ROC. E-mail:

[email protected]

10.1080/00423110512331383848$22.00 # Taylor & Francis Ltd.

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asymptote. Stopping distance can be shortened effectively if the slip ratio is kept

between 8% and 30% [1]. At the same time, there is also good steering controllability

because the lateral adhesive coefficient is still high.

The tyre model illustrates the relationship between the tyre adhesive coefficient and

parameters such as slip, normal force and vehicle velocity. In the study, the magic

formula tyre model [2] is adopted because of its accurate description of measured

steady-state tyre behavior and good predictability of vehicle performance [3]. The

parameters of the magic formula represent typifying quantities of the tyre characteristic.

The assessment of the magic formula parameter values has been treated by Van Oosten

et al. [4]. The relationship between the vehicle velocity and the magic formula

parameters is discussed in the present study. Genetic algorithm identification is used to

determine the magic formula parameters. Road braking tests were run on both high

adhesive road surfaces and low adhesive road surfaces to provide the data needed to

calculate the magic formula parameters.

Most anti-lock braking systems employ an additional hydraulic pump and valves to

regulate the braking pressure during the anti-lock operation. But the volume and the cost

of these are both too high to install them on a light motorcycle, and a general defect of

such anti-lock braking systems is a shock of braking pressure generated by non-smooth

mechanical behavior of hydraulic valves. The latter can be avoided by design of a

hydraulic anti-lock braking modulator instead of the hydraulic valves, and a smooth

power source, an electric motor, is used to drive the modulator in this study. A pulse

width modulation sliding mode controller [5] is adopted to control the electric motor due

to its robustness [6].

Fig. 1. Adhesive coefficient versus slip ratio curve [1].

432 C.-Y. LU AND M.-C. SHIH

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2. DESIGN AND ANALYSIS OF A HYDRAULIC

ANTI-LOCK BRAKING SYSTEM

The diagram of the hydraulic anti-lock braking system designed in the study is shown

in Figure 2. A plate-cam is designed and driven by an electric motor, and a hydraulic

anti-lock braking modulator is also designed and driven by the cam.

2.1. Basic Shape of the Cam

The displacement of the cam in the study is decided by the required decrease of

braking pressure, as shown in Figure 3. The shape of the cam can be composed of the

displacement of the driven member.

2.2. Torque Required to Drive the Cam

As shown in Figure 3, total force, Fc, acted on the driven member is:

Fc ¼ mc � ac þ cc � vc þ mc � g þ Fe; ð1Þ

where mc is the mass of driven member, cc is the damping ratio, g is the acceleration of

gravity, Fe is the external load on driven member, vc is the velocity of the driven

member and ac is the acceleration of the driven member. Here, Fe ¼ Ps � Ah, Ps is the

Fig. 2. Scheme of the anti-lock braking system designed in the study.

HYDRAULIC ANTI-LOCK BRAKING SYSTEM 433

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pressure of braking disc and Ah is the area of the spool. The pressure angle � of the

cam is shown below:

¼ tan�1

�vc

ðrb þ hÞ!c

�; ð2Þ

where rb is the radius of the basic circle in the cam, !c is the angular velocity of the

cam and h is the displacement of cam. From Figure 3 and Equations (1) and (2), the

torque required to drive the cam Tc is shown below:

Tc ¼ Fcðrb þ hÞ tan ¼ Fc � vc

!c

ð3Þ

From Equation (3), the torque required to drive the cam can be calculated and a

suitable electric motor can be selected to drive the cam. As Figure 4a shows, the

Fig. 3. Design and analysis scheme of the cam.

Fig. 4. Operating state of the ABS modulator designed in the paper.


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modulator is not acted (initial state), and the pressure of the braking pump is about

equal to the pressure of the braking disc. In Figure 4b, the modulator is in the

decompressed process. In Figure 4c, the modulator is in the compressed process.

2.3. Vehicle Model

In the study, a two-axle vehicle model is evolved to analyze the motion of a light

motorcycle, as shown in Figure 5. In the model, the body of the motorcycle is sim-

plified as a mass-block. The mass and the forces of the motorcycle are all assumed in

the symmetrically longitudinal plane. The stiffness and the damping ratio of the

suspensions are ignored, which generates an instantaneous load transfer between

the wheels. The effect of the rider’s posture is ignored. The force of the side wind, the

camber angle and the gradient resistant force are assumed to be zero. The model is

simplified as the parameters related to this model are easier to measure. It is more

practical than other complex models.

In the study, the subscripts f and r represent the front and the rear axle, respec-

tively. The normal forces of the front and the rear wheels are shown as follows:

Nf ¼ ðM � g � Lb þ Fint � HG � Fw � HwÞ=L

Nr ¼ ðM � g � La � Fint � HG þ Fw � HwÞ=L;

�ð4Þ

where M is the total mass of the motorcycle and the rider, L is the distance between the

front and the rear wheel, La and Lb are the distance from the center of gravity of the

vehicle to the front and the rear axle respectively, HG is the height of the center of

Fig. 5. The force condition upon the vehicle and the wheels.


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gravity, Fw is the longitudinal wind load, Hw is the height of the longitudinal wind

load and Fint is the inertial load. Fw can be obtained as Fw ¼ 12� � � cw � Aw � ðVv þ u0Þ,

where � is the density of air, cw is the air resistant coefficient, Aw is the area of the wind

load acting on the motorcycle and u0 is the opposite absolute velocity of the wind. In

the study, u0 is assumed to be zero. The corresponding specifications are shown in

Table 1, then, we obtain Fw ¼ 0:15 � V2v . Fint can be obtained as Fint ¼ M � a, where a

is the acceleration of the motorcycle. The normal forces are evaluated measuring

the longitudinal acceleration and velocity.

Iw is the wheel moment of inertia, ! is the angular velocity of the wheel, R is

the radius of the wheel, Tb is the braking torque acted from the braking calipers to the

wheel, Tb ¼ Kb � Ps, Kb is a torque gain and Tt is the adhesive torque acted from the

road to the tyre, Tt ¼ Fx � R, where Fx is the longitudinal adhesive force acted from

the road to the tyre. The relationship of it is Iw � _!! ¼ Tt � Tb and is shown in Figure 5.

The inertia force can be shown as Fint ¼ M � a ¼ Fw þ Fxf þ Fxr then, the

acceleration of the vehicle can be obtained as a ¼ Fw þ Fxf þ Fxr=M: After the

acceleration is integrated twice, the stopping distance, Dx, is obtained.

In the identification procedure, the mathematical model is used to evaluate

the longitudinal force and the normal forces:

Rotational equilibrium: Nf ¼ f ða;VÞ, Nr ¼ f ða;VÞLongitudinal equilibrium: Fxf ¼ f ða;VÞ

The purpose is the evaluation of the longitudinal adhesive coefficient as a function

of the slip ratio.

In the simulation, the mathematical model is used to evaluate the dynamic

behavior of the vehicle:

Longitudinal acceleration: a ¼ �F=M

Wheel acceleration: _!! ¼ ðTt � TbÞ=Iw

The purpose is the evaluation of the longitudinal acceleration and velocity of the

vehicle and the slip ratio of the front wheel.

Table 1. The specifications of the test motorcycle.

Mass (including rider), M 160 kg

Air density, � 1.1774 kg/m3

Air resistance coefficient, Cw 0.4789

Area of the wind load, Aw 0.55 m2

Radius of wheel, R 0.215 m

Height of C.G., HG 0.58 m

Wheel base, L 1.2 m

Distance between C.G. and rear axle, Lb 0.5 m


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2.4. Longitudinal Adhesive Force

The adhesive force is the force between the tyre and the road surface. In this study, the

magic formula tyre model is adopted to describe the relationship. The longitudinal

adhesive force, Fx, defined positive if acting backward from the road to the tyre equals

the longitudinal adhesive coefficient �x multiplied by the normal force N:

Fx ¼ N � �x; ð6Þ

where N can be determined by Equation (4). The vehicle acceleration, a, and the

wheel angular velocity, !, can be obtained from a road braking test described in

Section 4. The original state curve of the longitudinal adhesive coefficient of magic

formula tyre model and the slip ratio is shown in Figure 6, which typically reaches a

maximum and subsequently tends to a horizontal asymptote. Figure 6 also illustrates

the meaning of some of the parameters. The fitting formula is the Pacejka Magic

Formula as follow:

�x ¼ D � sinðC � tan�1ðB � s � EðB � s � tan�1ðB � sÞÞÞÞ; ð7Þ

where s¼ Sþ SH, SH ¼ f=cs is a horizontal shift, cs ¼ BCD and f is the rolling

resistance coefficient assessed by experimentation. The values of f, D, C, B and E with

parameters of vehicle velocity and adhesive torque can be obtained from the

following:

f ¼ c1 þ c2 � Vv

D ¼ �x;max þ c3 � Vv

C ¼ 2

sin�1

��x;s

D

�þ c4 � ln

�1 þ Vv

c5

�

Fig. 6. A typical original state of tyre characteristic indicating the meaning of some of the parameters of

the magic formula.


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B ¼ cs

CDþ c6 � Vv

E ¼B � sm � tan

�

2C

�B � sm þ tan�1ðB � smÞ

þ c7 � Vv;

where �x;max is the maximum value of �x, sm ¼ Sm þ SH , sm is the shifted slip ratio

where the peak �x;max occurs, �x;s is the asymptotic level of �x at large slip value and

Vv is defined in m/s. The parameters of Equation (7), c1� c7, can be identified by a

genetic algorithm described in Section 3.1. The values of �x;max, �x;s and Sm can be

obtained after the braking tests described in the Section 4.

3. IDENTIFICATION AND CONTROLLER DESIGN

3.1. Genetic Algorithm Identification

The method discussed has been historically named genetic algorithm because it

follows some procedures which appear in Darwin’s doctrine of evolution [7]. The

problem of identification consists in the determination of a set of values of the tyre

model parameters, which minimizes a pre-fixed error function between the model and

the experimental data available. The target of the genetic algorithm is to find an

individual x�, which minimizes the error characterized by

Error ¼Xk

i¼1

ð�kx � �xðGk; sk;Vk

v ÞÞ2; ð8Þ

where ð�kx;G

k; sk;Vkv Þ are the k experimental values available. Each individual is

characterized by a set of seven genes.

3.2. Sliding Mode PWM Slip-Ratio Controller

In the study, sliding mode control theory is adopted to design the controllers because

of its robustness. The sliding-mode index Sslide is defined as

Sslide ¼ Eslide þ l � _EEslide; ð9Þ

where Eslide ¼ S� � S, S� is the target slip and l is a strictly positive constant. _EEslide,

which can be shown as

_EEslide ¼dEslide

dt Eslideðk þ 1Þ � EslideðkÞ

Ts

; ð10Þ

where Ts is the sampling period.


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In Figure 7, the sliding surface Sslide ¼ 0 separates the phase plane into two semi-

planes: one is Sslide > 0 and the other is Sslide < 0. Figure 8 illustrates the decision

algorithms of the sliding mode PWM control theorems, where 1 means that the

electric motor is maintained in the compressed process, as shown in Figure 4c, while

� 1 means that the electric-motor is maintained in the decompressed process, as

shown in Figure 4b. If control signal U0 is between 1 and � 1, it means that the

rotation of the electric motor is not continuous, the respective rotation time being

proportional to the absolute value of signal U0.

To compare with the result of sliding mode PWM controller, the P-R conditions

presented by Guntur and Ouwerkerk [8] in the 1970s is adopted in the study. So called

P-R conditions consist of four prediction and eight reselection conditions for releasing

or applying brakes to prevent wheel lock-up. Most of the control laws of existing ABS

Fig. 7. Phase plane of sliding mode control.

Fig. 8. PWM control decision algorithm.


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are basically the combinations of these conditions. If P condition is satisfied, which

means that the wheel tends to be locked, ABS will release the braking pressure in

order to increase the wheel angular velocity. While the R condition is satisfied, i.e. the

danger of lock-up is averted, the braking pressure will be applied again. The P-R

criteria selected in the study are described as follows:

P1: if � _!!R > k1, then braking pressure is decreased,

R4: if €!! < 0, then braking pressure is increased.

4. ROAD BRAKING TESTS

Figure 9 is the photograph of a road test motorcycle built in the study. Two rotary

encoders are installed on the front and the rear wheel respectively to measure the

angular velocity through the frequency of its signals. An accelerometer was used to

measure the deceleration of the test motorcycle while braking. Only the front wheel is

braked and measured while testing. Two pressure transmitters are installed on the

braking pump and braking disc respectively. A microcomputer is used to record data

measured from the sensors and to control the electric motor through an AD/DA card.

After a modification of the effect of the orientation change of the accelerometer

while braking, the vehicle velocity can be accurately calculated by the integration of

deceleration. The experiments were run on both an indoor dry slippery road surface

(low adhesive) and an outdoor dry asphalt road surface (high adhesive). Although an

indoor dry slippery surface is not a standard road surface, it is easier to keep the

Fig. 9. Photograph of the road test motorcycle.


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environmental parameters unchanged than on other outdoor, low adhesive road

surfaces at many times of tests in the study. To calculate the parameters of the magic

formula tyre model, the tyres locked and released at different velocities are needed.

Two worn tyres with an average groove depth of 1.8 mm and an inflation pressure of

2.2 bar (in both) were used for the road test motorcycle.

Table 2 shows the values of the magic formula parameters identified by the genetic

algorithm described in Section 3.1. The curves of the longitudinal adhesive coefficient

versus the slip ratio of the fitting values of the magic formula and experimental results

with velocity varied at steps of 10 km/h from 65 km/h to 5 km/h on the dry asphalt

road surface are shown in Figure 10. Figure 11 shows the same curves with velocity

varied at steps of 10 km/h from 45 km/h to 5 km/h on the dry slippery road surface.

The degree of deviation between the test value and the fitting value is very small

(in the intended range of operation), so it is suitable to express the relation between

Table 2. The values of magic formula parameters.

Dry asphalt road surface Dry slippery road surface

C1 0.008476 0.007385

C2 0.00071 0.00038

C3 � 0.001736 � 0.002

C4 0.242983 0.15537

C5 5.605701 2.793245

C6 0.453304 0.425224

C7 � 0.0075 � 0.009061

Fig. 10. Longitudinal adhesive coefficient of experiments and the magic formula versus slip curve of the

front wheel varied with steps of 10 km/h from 65 km/h to 5 km/h on the dry asphalt road surface.


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�x and S with magic formula. Table 3 shows the stopping distances between

experiment and estimation in the study without ABS. Although the maximum

velocity of the data of experiments is 65 km/h and 45 km/h, respectively, the

Fig. 11. Longitudinal adhesive coefficient of experiments and the magic formula versus slip curve of the

front wheel varied with steps of 10 km/h from 45 km/h to 5 km/h on the dry slippery road surface.

Table 3. The stopping distance of the experiments and the magic formula without ABS.

Initial velocity (in km/h) Stopping distance (in m)

Dry asphalt road surface 65 Experiment 13.44

(moderate initial

braking pressure)

Magic Formula 13.45

65 Experiment 14.85

(high initial

braking pressure)

Magic Formula 14.89

70 Experiment 15.78

(moderate initial

braking pressure)

Magic Formula 15.70

Dry slippery road surface 45 Experiment 13.13

(moderate initial

braking pressure)

Magic Formula 13.02

45 Experiment 15.63

(high initial

braking pressure)

Magic Formula 15.83

50 Experiment 15.29

(moderate initial

braking pressure)

Magic Formula 15.16


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parameters of the magic formula are still applicable when the velocity is higher than

the maximum experimental velocity, as shown in Table 3. Figures 12 and 13 show the

curves of the longitudinal adhesive coefficient versus the slip ratio of the magic

formula tyre model at the velocity varied at the steps of 10 km/h from 95 km/h to

5 km/h on the dry asphalt and the dry slippery adhesive road, respectively. One may

determine the longitudinal adhesive force Fx to calculate the stopping distances

through the vehicle model described in Section 2.3.

Fig. 13. Longitudinal adhesive coefficient of magic formula versus slip curve of the front wheel varied each

10 km/h from 95 km/h to 5 km/h on the dry slippery road.

Fig. 12. Longitudinal adhesive coefficient of magic formula versus slip curve of the front wheel varied each

10 km/h from 95 km/h to 5 km/h on the dry asphalt road.


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5. SIMULATION AND EXPERIMENTAL RESULTS

Figure 14 shows a block diagram of the anti-lock braking control system. The analysis

of the anti-lock braking system described in Section 2 and the sliding mode PWM

controller described in Section 3.2 are used in the computer simulation. Further, the

simulation designs were built and used in experiments on the road test motorcycle.

Both during the experiments and simulations, only the front wheel is braked. The

brake handle is gripped at around 65 km/h on the dry asphalt road surface and at

Fig. 14. Block diagram of the anti-lock braking control system.

Table 4. The results of experiments and simulations.

Control type Braking

time

(s)

Stopping

distance

(m)

Average

deceleration

(m/s2)

Without ABS Experiment (Locked wheel) Simulation (dry asphalt road) 1.55 14.04 11.61

Simulation (dry slippery road) 2.06 12.9 6.08

Without ABS Experiment (dry asphalt road) 1.74 15.71 10.37

(Low initial braking pressure) Simulation (dry asphalt road) 1.74 15.75 10.35

Experiment (dry slippery road) 2.36 14.75 5.30


P1R4 Experiment (dry asphalt road) 1.49 13.45 12.12

(Moderate initial braking pressure) Simulation (dry asphalt road) 1.50 13.54 12.03



Sliding mode PWM Experiment (dry asphalt road) 1.47 13.27 12.27

(Moderate initial braking pressure) Simulation (dry asphalt road) 1.48 13.36 12.22




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around 45 km/h on the dry slippery road surface, respectively. All tests are run on a

10 ms sampling period. Detailed experimental data about braking times and stopping

distances are shown in Table 4. One of these is arbitrarily chosen and compared with

Table 5. The stopping distance of the 50 times experimental result using sliding mode PWM control on both

dry slippery road surface and dry asphalt road surface.

Dry asphalt road Dry slippery road

Stopping distance Times Stopping distance Times

Under 13 m 0 Under 11.5 m 1

13–13.5 m 48 11.5–12 m 24

Over 13.5 m 2 12–12.5 m 18

12.5–13 m 5

Over 13 m 2

Fig. 15. One of the simulation results of front wheel using sliding mode PWM controller on the dry

slippery road surface with a moderate initial braking pressure.


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the mathematical simulation. The target slip S� is set to be 0.2. Although the target

slip is set as 0.2 for simplification, in fact, optimal slip is time-varying and difficult to

determine during the braking process, but the steering controllability while the slip

ratio is around 0.2 is expected to be acceptable judging from Figure 1. The ABS is set

not to control the braking when the vehicle velocity is lower than 5 km/h. Table 5

shows the stopping distance of the 50 times experimental results at the initial velocity

around 65 km/h on the high adhesive road surface and around 45 km/h on the low

adhesive road surface, respectively to show the reliability of the anti-lock braking

system designed in the study. The braking conditions, such as initial velocity and

initial braking pressure, at each of the road experiments are difficult to keep the same.

Obviously, deviations have an influence on the stopping distance. Figure 15 shows

Fig. 16. One of the experimental results of front wheel using sliding mode PWM controller on the dry

slippery road surface with a moderate initial braking pressure.


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one of the simulation results using sliding mode PWM controller on the dry slippery

road surface, with a moderate initial braking pressure. The evaluation of normal

forces of simulation is shown in Figure 5 and Equation (4). Figure 16 shows one of the

experimental results using sliding mode PWM controller on the dry slippery road

surface, with a moderate initial braking pressure.

6. CONCLUSION

This paper presents a procedure to determine the longitudinal adhesive coefficient

between a tyre and the road through the magic formula tyre model, which aims at an

accurate description of measured steady-state tyre slip behavior. A good selection of

starting range of genetic algorithm for the calculation of the magic formula

parameters is not difficult, but, without concealing, it’s often a matter of experience.

The results of longitudinal adhesive force presented by the magic formula tyre model

can be very satisfactory as shown in the paper.

The hydraulic modulator designed in the paper can be manufactured easily. The

safe-circuit in the modulator does not need to be added, because the brake fluid can be

flowed from the braking pump to the braking disc through the orifice of the check

valve, as shown in Figure 3. As shown in Table 4, the efficiency of the ABS modulator

designed in the study is good to control the braking behavior. The variation of the

braking pressure is small, as shown in Figure 16, by the design of a hydraulic anti-lock

braking modulator and a smooth power source, electric motor, which is used to drive

the modulator; therefore the rider will not feel violent shocks on the hand while the

ABS modulator is operating. Both simulation and experimental results show that

effective regulation of wheel slip during emergency braking can be achieved by

implementation of sliding mode PWM control.

REFERENCES

1. van Zanten, A., Erhardt, R. and Lutz, A.: Measurement and Simulation of Transients in Longitudinal and

Lateral Tire Forces. SAE Paper 900210, 1990.

2. Pacejka, H.B. and Bakker, E.: The Magic Formula Tyre Model. Veh. Syst. Dyn. 21 (1993), pp. 1–18.

3. Lidner, L.: Experience With the Magic Formula Tyre Model. Veh. Syst. Dyn. 21 (1993), pp. 30–46.

4. van Oosten, J.J.M. and Bakker, E.: Determination of Magic Formula Tyre Model Parameters. Veh. Syst.

Dyn. 21 (1993), pp. 19–29.

5. Wu, M.-C. and Shih, M.-C.: Hydraulic Anti-Lock Braking Control Using the Hybrid Sliding-Mode Pulse

Width Modulation Pressure Control Method. In: Proceedings of the Institution of Mechanical Engineers,

Part I: J. Syst. Cont. Eng. 215 (2) (2001), pp. 177–187.

6. Tan, H.S. and Tomizuka, M.: Discrete-Time Controller Design for Robust Vehicle Traction. IEEE Cont.

Syst. Mag. (April 1990), pp. 107–113.


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7. Vetturi, D., Gadola, M., Manzo, L. and Faglia, R.: Genetic Algorithm for Tyre Model Identification in

Automotive Dynamics Studies. In: Proceedings of the 29th International Symposium on Automotive

Technology and Automation (ISATA), Florence, Italy, June 4, 1996.

8. Guntur, R.R. and Ouwerkerk, H.: Adaptive Brake Control System. In: Proceedings of the Institution of

Mechanical Engineering, 186 (68/72) (1972), pp. 855–880.


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Application of the Pacejka Magic Formula Tyre Model on a Study of a Hydraulic Anti-Lock Braking...

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Transcript of Application of the Pacejka Magic Formula Tyre Model on a Study of a Hydraulic Anti-Lock Braking...