 Abstract— Tire model is required in order to study vehicle dynamic behavior for designing control system such as electronic stability control. In this study, a Magic Formula tire model was implemented using Matlab Simulink block diagram. Tire modeling is the first step to investigate vehicle handling stability. The model was developed based on a set of mathematical equations. The model was developed based on pure slip and combine slip condition. The feasibility of Magic Formula block diagram is validated via simulation software which is Carsim. For the validation procedure, Double Lane Change (DLC) test was used. The output forces and moments from Carsim are compared with the development model. The validation results are discussed. Once the model was validated, the Magic Formula model will be use as a subsystem for vehicle stability control. Keywords— Carsim, Magic Formula, Simulink, Tire Slip I. I NTRODUCTION OMMONLY, tire forces and moments are generated when there is friction between tire and road surface. The forces and moments produced are crucial parameters that influence vehicle handling. Effective tire model for handling simulation should comprise of two basic elements; lateral tire force and longitudinal tire force which depend on slip angle and slip ratio respectively. Aligning moment is computed by multiplying the lateral force with the pneumatic trail produced by the deformation of rubber tire [1]. Mohammad Safwan Burhaumudin, is with the Department of Automotive Engineering, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (corresponding author phone: +60137147460; e-mail: safwan_burhaumudin@yahoo.com). Pakharuddin Mohd Samin, is with the Department of Automotive Engineering, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail: pakhar@fkm.utm.my). Hishamuddin Jamaluddin, is with the Department of System Dynamics and Control, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail: hishamj@fkm.utm.my). Roslan Abd Rahman, is with the Department of System Dynamics and Control, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail: roslan@fkm.utm.my). Syabillah Sulaiman, is with the Department of Automotive Engineering, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail: syabillahsulaiman@gmail.com). Combination slip conditions are represented in Magic Formula mathematical formulae. Magic Formula tire is able to handle combine slip condition. This model provides an accurate behavior of tire mechanics based on real experimental test data. Vertical wheel load is one of the input variables that influence the production of tire forces and moments. Vertical wheel load will contribute to different contact patch area and tire deformation [2]. In this study, vertical wheel load input are kept constant but the vehicle slip ratio and slip angle are varied. The objective of this paper is to implement a tire model based on Magic Formula mathematical equation [3], [4]. The equations consist of several coefficients to be tuned so that the output trend is similar to the experimental test data. The details of the equations for Magic Formula are shown in section II. While in the section III, the modeling block is introduced and validated with Carsim. For the purpose of tire forces and moment analysis, section IV discusses the results obtained from the proposed model. Some researchers are working on several tire models such as UniTire Model [2], Dugoff’s Tire Model [4], and others. Different tire model used different approach and equations and of course different model parameters . This tire model will be used for vehicle dynamic analysis and control. The forces and moments that are generated by the tire are monitored in order to enhance vehicle stability. A control algorithm required to control the tire forces and moments to be maintained at its adhesion limit to ensure vehicle handing stability. II. MATHEMATICAL MODELING A. Slip Ratio and Slip Angle Slip ratio, κ defines the ratio between slip velocity and vehicle velocity. The expression represents the slip ratio is v vv vehicle vehiclewheel    . (1) Slip ratio exist when vehicle is under braking and accelerating condition. Slip angle, α, defines the angle between the directions of tire to the velocity vector of the vehicle [5]. The value is Modeling and Validation of Magic Formula Tire Model Mohammad Safwan Burhaumudin, Pakharuddin Mohd Samin, Hishamuddin Jamaluddin, Roslan Abd Rahman, Syabillah Sulaiman C International Conference on Automotive, Mechanical and Materials Engineering (ICAMME'2012) Penang (Malaysia) May 19-20, 2012 113 typically small under steady cornering. However, the value suddenly changes in critical driving condition. Fig. 1 illustrates a tire moving along the velocity vector, v, at a sideslip angle, α. The tire is steered by the angle δ. If the angle between the velocity vector, v, and the vehicle x-axis is shown by β, then slip angle, α, is defined as     . (2) In the Matlab Simulink simulation the slip angle generated by Carsim software was used. The slip angle produce by Carsim yielded from the movement of vehicle under Double Lane Change (DLC) maneuver. Fig. 1 Tire slip angle. B. Magic Formula Magic Formula [3], [4] developed by Pacejka has been widely used to calculate steady-state tire force and moment characteristics. The semi-empirical tire model combined slip situation from a physical view point. The pneumatic trail is introduced as a basis to calculate this moment about the vertical axis [6]. The formulae consist of coefficient B, C, D, and E that determine the trend of force and moment similar to the actual test data. To avoid symmetrical asymptote about the origin, the Magic Formula introduce coefficient S v and S h . This offset from the origin occurs due to the presence of camber angle. Table 1 shows the coefficients that govern the Magic Formula equation. The general form of graph produced by Magic Formula is shown in Fig. 2. TABLE I C OEFFICIENT IN MAGIC FORMULA Symbol Quantity B stiffness factor C shape factor D peak value E curvature factor S h horizontal shift S v vertical shift Fig. 2 Original sine version of the Magic Formula graph. The full set of equation of Magic Formula [6] that represents pure and combine slip condition are defined in equation (3) to (16). In this study, 3 outputs are considered which are longitudinal force, lateral force and aligning moment. Longitudinal force for pure slip, F xo , consists of coefficients B, C, D, E and S v . The subscript x represents condition along x-axis. Slip ratio, κ, is the input of F xo as given by       . arctanarctansin S Vx x B xx B x E xx B x C x D x F xo    (3) Lateral force for pure slip, F yo consists of coefficients B, C, D, E and S v . The subscript y represents condition along y-axis. Slip angle, α, is the input to F yo as given by         . arctanarctansin S Vy y B y y B y E y y B y C y D y F yo    (4) Aligning moment for pure slip, M zo , is the product of pneumatic trail, t o with lateral force, F yo , as given by   . M zro F yo t o M zo     (5) The second term on the right side of equation (5) represents moment occurs due to camber angle. Pneumatic trail, t o consist of coefficients B, C, D and E as given by      .arctanarctancos  t B tt B t E tt B t C t D t t o  (6) Aligning moment due to camber angle, M zro , as given by     .arctancos  r B r C r D r M zro  (7) Longitudinal force for combined slip, F x , is the product of factor G α with pure longitudinal force, F xo , as given by . F xo G x F x    (8) Factor G α represents increment or decrement factor when slip angle is introduced in the presence of slip ratio as given by International Conference on Automotive, Mechanical and Materials Engineering (ICAMME'2012) Penang (Malaysia) May 19-20, 2012 114  G ox S B x S B x E x S B x C x G x           / arctan arctancos                (9) and                 S Hx B x S Hx B x E x S Hx B x C x G ox          arctan arctancos (10) Lateral force for combined slip, F y , is the product of factor G κ with the pure lateral force, F yo , as given by . S Vy F yo G y F y   (11) Factor G κ represents increment or decrement factor when slip ratio is introduced in presence of slip angle as given by  G oy S B yS B y E yS B y C y G y           / arctan arctancos                (12) and                   S Hy B y S Hy B y E y S Hy B y C y G oy          arctan arctancos (13) Aligning moment for combined slip, M z , as shown by        . , arctancos ' F x s eqr B r C r D r F y t M z   (14) The aligning moment, M z , is the summation of moments occurs due to lateral force, camber angle and longitudinal force. The offset location where the force acting in y-axis and x-axis is given by          eqt B t eqt B t E t eqt B t C t D t t , arctan ,, arctancos  (15) and   R o s 1.0 (16) III. S IMULATION AND VALIDATION A. Simulation All the equations in section II are converted into Matlab Simulink block diagram. The final simplified model is shown in Fig. 3. The inputs to the subsystem are vertical load, F z , road friction coefficient, μ, slip ratio, κ, and slip angle, α. Camber angle is set to zero degree for simplicity. Fig. 3 Modeling of Magic Formula in Matlab Simulink. The slip ratio and slip angle that are generated under Double Lane Change (DLC) maneuver at 120 km/h are shown in Fig. 4. The values of slip ratio and slip angle generated by Carsim are fed into Matlab Simulink as inputs variable. Fig. 4 Slip ratio and slip angle during Double Lane Change (DLC) Maneuver. B. Validation For validation purpose, Carsim software was used to simulate a vehicle moving in Double Lane Change (DLC) maneuver. The default build-in testing module was used for the Carsim simulation. The command window of Carsim testing simulation is illustrated in Fig. 5 can be divided into 3 sections. In the command window, vehicle type and testing procedure were selected from section 1. The vehicle specification chosen was E-Class, Sedan and the testing procedure was Double Lane Change (DLC) at 120 km/h maneuver. Once the vehicle type and testing procedure have been chosen, the solver was selected in section 2. The results and animation was chosen section 3. Vehicle maneuver for Double Lane Change (DLC) is shown in Fig. 6. Fig. 5 Carsim command window for simulate vehicle maneuver. The left image in Fig. 6 shows the vehicle about to change lane while the right image shows the vehicle at the end of the maneuver. Sec ti o n 1 Sec ti o n 3 Sec ti o n 2 Fz μ α к Fxo, Fx Fyo, Fy Mzo, Mz International Conference on Automotive, Mechanical and Materials Engineering (ICAMME'2012) Penang (Malaysia) May 19-20, 2012 115 Fig. 6 Double Lane Change Maneuver. IV. RESULTS AND DISCUSSIONS During vehicle travelling on the designated course, the vehicle response will generate slip angle and slip ratio. Those slip generated by Carsim is fed into Matlab Simulink Magic Formula block diagram. The trends of the graph are compared and the results are shown in Figs.7 to 9. Fig. 7 shows the longitudinal force generated at the tire. The trend of the graph produced by Matlab Simulink is quite similar to Carsim except when the gradient is increasing and decreasing. These happen due to some factors that are neglected by Magic Formula tire model but in Carsim are considered. Aerodynamic effect, gear shifting effect and kinematics of vehicle suspension are some factors that are neglected by Magic Formula tire model. Additional vehicle model is required to compensate the situation, which is will considered in future work. Fig. 7 Longitudinal force computed from Matlab Simulink and Carsim. The lateral force shown in Fig. 8 has 2 peaks and a trough. The range of lateral force generated is from -4000 N to 3000 N. Maximum lateral force was generated due to increasing of slip angle during the maneuver. Fig. 8 Lateral force computed from Matlab Simulink and Carsim. Aligning moment shown in Fig. 9 has inverse trend to lateral force. This moment exists due to the deformation of tire prevailing at Double Lane Change (DLC) maneuver. Tire deformation will create concentrated point at the tire contact patch. This point is generally not at the center of the contact patch. Distance offset from the origin of the tire creates pneumatic trail that contribute to the generation of aligning moment . Fig. 9 Aligning moment computed from Matlab Simulink and Carsim. IV. CONCLUSION Modeling of Magic Formula tire model in Matlab Simulink was developed in order to initiate a future project in vehicle stability control. Tire model is a paramount subsystem affecting vehicle dynamic behavior. The developed tire model was validated using Carsim software. The Magic Formula tire was validated based on Double Lane Change (DLC) testing method maneuver at 120 km/h using Carsim. During maneuver, tire slip ratio and slip angle were generated. The slip ratio and slip angle yielded from that maneuver were used as the input variable to the Magic Formula tire developed using Matlab Simulink. The longitudinal force, lateral force and aligning moment produced during that course maneuver were compared. International Conference on Automotive, Mechanical and Materials Engineering (ICAMME'2012) Penang (Malaysia) May 19-20, 2012 116 From the validation results, the trends between Matlab Simulink and Carsim are similar with some difference in the magnitude. The difference arises due to aerodynamic effect, gear shifting effect and kinematics of vehicle suspension effect being ignored in the model. The validation result shows the Magic Formula tire can be used to represent actual tire dynamic behavior under any maneuver. A CKNOWLEDGMENT The author wish to thank the Ministry of Higher Education (MOHE) and Universiti Teknologi Malaysia (UTM) for providing the research facilities and support, especially all staff’s of Department of Automotive, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia. R EFERENCES [1] C. Long, H. Chen, “Comparative Study between the Magic Formula and the Neural Network Tire Model Based on Genetic Algorithm,” Third International Symposium on Intelligent Information Technology and Security Informatics, pp. 280-284, 2 April 2010. [2] N. Xu, D. Lu, S. Ran, “A Predicted Tire Model for Combined Tire Cornering and Braking Shear Forces Based on the Slip Condition,” International Conference on Electronic & Mechanical Engineering and Information Technology, pp. 2073-2080, 12 August 2011. [3] E. Bakker, L. Nyborg, and H.B. Pacejka, “Tyre Modelling for Use in Vehicle Dynamics Studies,” SAE Paper 870421, pp. 1-15, 1987. [4] R. Rajamani, Vehicle Dynamics and Control, New York: Springer, 2006, ch. 13. [5] Jazar, G. Nakhaie, Vehicle Dynamic Theory and Application, New York: Springer, 2008, pp. 600-605. [6] H.B. Pacejka, Tire and Vehicle Dynamics, SAE International and Elsevier, 2005, ch. 4. International Conference on Automotive, Mechanical and Materials Engineering (ICAMME'2012) Penang (Malaysia) May 19-20, 2012 117