This paper endeavors to examine a friction estimation algorithm which is used to measure tire-road friction coefficient. The data collected can be used to make systems that can improve vehicle comfortability and stability. We would examine the Pacekja Magical Formula in calculating the tire-road friction coefficient.

Vehicle comfortability and stability in all terrain is a crucial requirement for today’s client. This has led to use of technology to improve the vehicle’s comfortability and stability. Increased competition in the motor industry has also led to the business development of new technologies to give added advantage to specific manufacturers(Muller, Uchanski and Hedrick (2003).

Although development of technology has increased significantly, one cannot override the physical traction limit as a result of friction between the tires and the road. This means that for all vehicle control systems, the tire-road friction is a crucial element to consider when accounting for comfortability and stability.

Tire surface area, texture, inflation pressure, temperature and weight are the parameters that affect car traction, grip and friction generated(Villagra 2011). Larger surface area increase the contact patch between the tire and ground which may create better traction. Texture of the tire determines deformation of the elements in the adhesive and sliding regions of the tire during contact(Rajamani 2012).

Several algorithms have been developed to calculate the tire-road friction coefficient. For this paper we will adopt the Pacejka Magic Formula to determine the tyre-road friction coefficient. Longitudinal and lateral forces can be estimated from the formula by fitting coefficients in pure and combined slip conditions. The magic formula is as follows:

y(x) = D.sin[C.atan{B.x - E(B.x - atan(B.x))}]

with: Y(x) = y(x) + Sv and x=X +Sh

Output variable Y(x) represents the braking and traction force Fx, input variable X represents the slip ratio. The equation can be used to calculate the longitudinal or lateral forces depending on variable X and equation coefficients D, C, B, E, Sy and Sh. Tyre tests have to be conducted to find the coefficients to complete the equation. The below table indicates the coefficient descriptions where subindexes x and y have been used to indicate longitudinal and lateral parameters respectively.

Longitudinal force (Fx) would be as follows:

Dx= μx· Fz

μx= (PDX1+ PDX2· dfz) · λμx

dfz=(FzFz0) /Fz0

Cx= PCX1

BxCxDx= Fz· (PKX1+ PKX2· dfz) · ePKXdfz

Bx=BxCxDx /CxDx

Ex=(PEX1+PEX2dfz+PEX3dfz2)(1−PEX4sign(s+Shx))

Shx= PHX1+ PHX2· dfz

Svx= (PVX1+ PVX2· dfz) · Fz

Type of the road relates to parameter λμxwith values ranging from [0-1]. The slip ratio X = s defined ass=1−(ωR / vx)whereωis the angular tyre velocity, R is tyre radius and Vx is the vehicle’s longitudinal speed.

Lateral force (Fy) would be determined as follows:

Dy= μy· Fz

μy= (PDY1+ PDY2· dfz) · (1 − PDY3· γ2) · λμy

Cy= PCY1

ByCyDy=Fz0PKY1sin(2atan(Fz /PKY2Fz0))(1−PKY3|γ|)

By=ByCyDy /CyDy

Ey= (PEY1+ PEY2· dfz) · (1 − (PEY3+ PEY4· γ) · sign(α + Shy))

Shy= PHY1+ PHY2· dfz

Svy= (PVY1+ PVY2· dfz) · Fz

Where x = is the slip angle and the camber angle is y.

Tire-road friction information can be very useful in building vehicle control systems like traction control system, anti-lock brake system, electronic stability control, automatic cruise control and collision avoidance systems (Shi 2018). The information can also be used to improve road build structure and traffic control systems. Braking and cornering of the car is also dependent on the longitudinal and lateral tire forces. The stiffness of braking or cornering can be computed by linear derivation of the pure slip friction curve (Singh and Taheri 2015).

Tire-road friction coefficient is necessary for development of stable, safe and comfortable cars. A lot of systems in car functioning is dependent on the coefficient and its application in building the system. Technological changes may render the coefficient useless if the hovering car concept is fully developed.

### Reference List

1. Cabrera, J. A., Castillo, J. J., Pérez, J., Velasco, J. M., Guerra, A. J., & Hernández, P. (2018). A Procedure for Determining Tire-Road Friction Characteristics Using a Modification of the Magic Formula Based on Experimental Results. Sensors (Basel, Switzerland), 18(3), 896. http://doi.org/10.3390/s18030896
2. Muller, S., Uchanski, M. and Hedrick, K., (2003). Estimation of the maximum tire-road friction coefficient. Journal of dynamic systems, measurement, and control125(4), pp.607-617.
3. Rajamani, R., Phanomchoeng, G., Piyabongkarn, D. and Lew, J.Y., (2012). Algorithms for real-time estimation of individual wheel tire-road friction coefficients. IEEE/ASME Transactions on Mechatronics17(6), pp.1183-1195.
4. Shi, Y., Li, B., Luo, J. and Yu, F., (2018). A practical identifier design of road variations for anti-lock brake system. Vehicle System Dynamics, pp.1-33.
5. Singh, K.B. and Taheri, S., (2015). Estimation of tire–road friction coefficient and its application in chassis control systems. Systems Science & Control Engineering, 3(1), pp.39-61.
6. Villagra, J., D’Andréa-Novel, B., Fliess, M. and Mounier, H., (2011). A diagnosis-based approach for tire–road forces and maximum friction estimation. Control engineering practice19(2), pp.174-184.
7. Yi, K., Hedrick, K. and Lee, S.C., (1999). Estimation of tire-road friction using observer based identifiers. Vehicle System Dynamics31(4), pp.233-261.