Magic Formula Model Research Articles

An enhanced motorcycle tyre model characterised through experimental riding data

Tyre-road interaction governs motorcycle dynamics; however, the most widespread tyre model formulations must be characterised through a dedicated test bench on the lab or road, unavailable to many interested subjects. This article proposed a new tyre model formulation, conceived to be characterised through riding data using standard instrumentation. Albeit its coefficients are identified through quasi-static, uncombined slip manoeuvres, the model addresses transient, combined manoeuvres and is adaptive to road friction levels and static weight through statistical relationships from the literature. A pre-existing formulation was improved and expanded. The model’s behaviour in demanding conditions was investigated through a high-fidelity simulation environment, using a Magic Formula tyre model as the reference. Next, the characterisation procedure was carried out using actual riding data. The model’s accuracy is shown by reproducing numerically one of the manoeuvres and through comparison with the results of a bench test. The proposed model could correctly reproduce the primary behaviour of a Magic Formula model, also concerning tyre moments and steering torque. Characterising the tyre model through real riding data proved feasible, and its robust formulation limited the propagation of estimation errors. The proposed tyre model formulation and characterisation procedure should interest, among others, those subjects that lack access to a tyre testing machine.

A physical tire model for real-time simulations

We present a novel tire brush model with carcass flexibility, primarily designed for real-time applications while preserving the physical significance of its parameters. The tire’s geometry is discretized and represented using a series of ribs, each intersecting the local road surface independently, on the other hand carcass in-plane fore-aft displacement, in-plane deflection, and out-of-plane sidewall torsion are approximated through a second-order polynomial. Forces and torques originating from the tire–road contact area are described using brush mechanics, where bristles’ kinematics is directly linked to carcass deformation. Extensive effort is dedicated to understanding the possible sources of instability, as well as describing and implementing a robust numerical scheme to effectively solve the nonlinear system of equations arising from the modeling. Timing performance results are provided to demonstrate the suitability of the presented model for demanding hard real-time simulations. Finally, validation of the presented tire model is carried out by fitting experimental data and comparing it with the state-of-the-art Magic Formula model, proving its reliability and accuracy in reproducing tire behavior.

Measurement of the lateral characteristics and identification of the Magic Formula parameters of city and cargo bicycle tyres

The paper presents measurements of the lateral force and self-aligning torque from cargo and city bicycle tyres. Based on the experimental data, we have determined the parameters for the Magic Formula model, for lateral force and self-aligning torque. We performed tests with VeTyT, an indoor test rig specific for bicycle tyres, under different vertical loads (ranging from 343 N to 526 N), camber angles (−5, 0, 5) deg and inflation pressure (from 300 kPa to 500 kPa). For each condition, we evaluated the cornering stiffness and found that it generally decreases with the increase in inflation pressure for the tour/city bicycle tyres. However, the cargo tyre tested showed an opposite trend, with an increase in the cornering stiffness as the inflation pressure increased from 300 kPa to 400 kPa.

Testing, Modelling, and Performance Evaluation of a Rotary Eddy-Current Damper with Inherent Nonlinear Damping Characteristics for Structural Vibration Control

This paper investigates the inherent nonlinear damping characteristics of a rotary eddy-current damper (RECD) and evaluates its vibration control performance on a single-degree-of-freedom (SDOF) system subjected to harmonic and seismic excitations. First, a RECD prototype was manufactured to experimentally identify its intrinsic nonlinear eddy-current damping characteristics. A magic formula model is then proposed to characterize the relationship between the eddy-current damping force and the velocity of the RECD, and its applicability in depicting the nonlinear damping characteristics of the RECD is evaluated by comparing with the electromagnetic finite-element (FE) model and the commonly used Wouterse’s model. Moreover, by comparing the experimental and analytical frequency–domain responses of an SDOF structure equipped with a RECD, the applicability of the magic formula model in evaluating the structural vibration control performance of the RECD is further verified. Finally, the control performance of the RECD for an SDOF structure subjected to earthquake ground motions is numerically evaluated and compared. Results show that the eddy-current damping force of the RECD presents obvious nonlinear characteristics with increasing velocity, and the proposed magic formula model can adequately depict the nonlinear eddy-current damping characteristics of the RECD. Moreover, the RECD exhibits superior structural control performance under harmonic and seismic excitations.

Research on Direct Braking Force Estimation and Control Strategy Using Tire Inverse Model

With the rapid development of computer control and vehicle intelligence technology, speed and safety of vehicles have been greatly improved, and the requirements for vehicle control performance are getting higher and higher. For the direct braking force control, in the process of deceleration, a fast braking response can be obtained, which improves the braking performance and vehicle safety. This paper concentrates on direct braking force estimation and control strategy using a tire inverse model based on the antilock braking system, and to solve the problem of the existing ABS system is mainly antilock braking function, no direct braking force control function. Taking magic formula model for reference inverse model, the critical parameters under different road surfaces are obtained according to experience data. Then, the desired slip ratio corresponding to braking force can be obtained via fast tire inverse model look-up table method. The tyre friction self-adjustment decision making is obtained using the tire inverse model method. A direct braking force antilock braking system (DBF-ABS) controller is built using the nonsingular fast terminal sliding mode method. The simulation results indicated that the control strategy has adaptability and stability to the change of road conditions.

Tire multiphysical modeling for the analysis of thermal and wear sensitivity on vehicle objective dynamics and racing performances

The handling behavior of a vehicle is one of its most important properties because of its relation to performance and safety and to its deep link with concepts such as “over-steer” or “under-steer”. Tire–road interaction models play a fundamental role in the vehicle system modeling, since tires are responsible for the generation of forces arising within contact patches, fundamental for both handling and ride/comfort. Among the models used to reproduce such forces, Pacejka’s Magic Formula (MF) is undoubtedly one of the most used ones in real-time automotive simulation environments because of its ability to fit quite easily a large amount of experimental data, but its original formulation did not take into account of the tire thermodynamics and wear conditions, which clearly affect tire and vehicle dynamics and are not negligible, especially for high level applications, such as motorsport competitions. Exploiting a multiphysical tire model, which consists in an evolved version of the standard MF model (MF-evo), and a vehicle model properly validated throughout experimental data acquired in outdoor testing sessions carried out with an industrial partner, the current work presents a study on vehicle behavior variation induced by thermodynamic and wear parameters, defining a series of metrics to analyze and show results. One of the elements of interest on which the focus is placed is the possibility to highlight how under-over-steering behavior of a car changes according to different thermodynamic states of tires; to do this, a commercial software VI CarRealTime has been used to perform a series of objective steady-state maneuvers and long runs, exploiting the logic of a lap time optimizer.

Dynamic modeling of MR dampers based on quasi–static model and Magic Formula hysteresis multiplier

In this research, a novel parametric dynamic model based on quasi-static (QS) model and a Magic Formula hysteresis multiplier, called QSMF model, is investigated to predict accurately and more meaningfully explain the hysteresis behavior of magnetorheological (MR) dampers. The proposed hysteresis model consists of a quasi–static (QS) component inherited from the QS model and a hysteresis multiplier improved from the Magic Formula (MF) operator [1]. The QS mechanical parameters reflect the physical nature while the MF ones characterize the practical nonlinear dynamic responses of MR dampers. This combination enables the model to be flexibly applicable for both design based on quasi-static model and dynamic modeling of MR dampers. In the MF hysteresis component, the original MF model is modified in order to describe MR dampers with high asymmetric hysteresis behavior since each hysteresis branch can be simulated separately. Besides the physical meanings of the parameters, the favorable approach is another advantage of the QSMF model as it contains no differential equation. In addition, the QS parameters can be identified independently via QS experiments. To broaden applicability, the proposed model is formulated for two general operating modes of MR dampers, i.e., shear mode and flow mode. Performances of the model under different excitation conditions are then analyzed and compared with those of the Spencer’s phenomenological model [2] and Pan’s model [3].

Enhancing brush tyre model accuracy through friction measurements

Accurate tyre models are important to ensure valid and reliable simulation of vehicle behaviour. To this purpose, the Magic Formula (MF) became the de facto standard for vehicle dynamics simulations, despite requiring many empirically derived coefficients. This paper shows how the accuracy of a simple physical-based brush-type tyre model can be enhanced to simulate tyre behaviour that closely matches MF results. To do so, the real, highly complex rubber friction characteristics are incorporated into the brush model. The friction characteristics were obtained from friction measurements with a tread rubber block. The developed model is validated against experimental tyre data obtained on a flat-track test rig and the corresponding MF model. Results show that the inclusion of friction characteristics allow accurate simulation of longitudinal and lateral slip conditions over a wide range of normal loads with the simple brush model. At extreme loading scenarios, the simulation accuracy deteriorates because of the significant influence of highly nonlinear deformation behaviour of the tyre that is not accounted for in the brush model formulation.

On-Board Road Friction Estimation Technique for Autonomous Driving Vehicle-Following Maneuvers

In recent years, autonomous vehicles and advanced driver assistance systems have drawn a great deal of attention from both research and industry, because of their demonstrated benefit in reducing the rate of accidents or, at least, their severity. The main flaw of this system is related to the poor performances in adverse environmental conditions, due to the reduction of friction, which is mainly related to the state of the road. In this paper, a new model-based technique is proposed for real-time road friction estimation in different environmental conditions. The proposed technique is based on both bicycle model to evaluate the state of the vehicle and a tire Magic Formula model based on a slip-slope approach to evaluate the potential friction. The results, in terms of the maximum achievable grip value, have been involved in autonomous driving vehicle-following maneuvers, as well as the operating condition of the vehicle at which such grip value can be reached. The effectiveness of the proposed approach is disclosed via an extensive numerical analysis covering a wide range of environmental, traffic, and vehicle kinematic conditions. Results confirm the ability of the approach to properly automatically adapting the inter-vehicle space gap and to avoiding collisions also in adverse road conditions (e.g., ice, heavy rain).

Analyzing quarter car model with Magneto-Rheological (MR) damper using equivalent damping and Magic formula models

Mathematical modelling of Magneto-Rheological (MR) damper has been an intriguing field of research ever since the invention of the device itself. An accurate model of MR damper results in development of an efficient controller for a semi-active system with MR damper. Hence, a number of models have been put forward to accurately predict the MR damper behavior. One of these models is Magic formula model. Based on the famous Magic formula used in tire force calculation, this model can be used for representing the peak damper force vs damper piston velocity amplitude graph. This model was later modified to capture the force displacement diagram of MR damper. The former model is denoted as Magic Formula Model-1 (MFM-1) and the latter one is denoted as Magic Formula Model-2 (MFM-2) here onwards. In the current study a commercial MR damper has been tested for various piston velocities and currents. The equivalent damping coefficient is then calculated for the tested conditions. The equivalent damping coefficients are used for analyzing a quarter car model. Two quarter car models with MR damper are simulated, one uses MFM-1 for MR damper and the other uses MFM-2. All the quarter car models are subjected to single pulse input and the sprung mass response is measured in terms of displacement. The RMS error between the response of quarter car model with equivalent damping and quarter car models with MR damper is used to determine the performance of each mathematical model. The study revealed that MFM-1 represents the MR damper behavior more accurate than that of MFM-2.

Analysis and Prediction of Tire Cornering Properties for Different Inflation Pressures Based on Deflection Control

ABSTRACT The variation of inflation pressure has an important effect on the mechanical characteristics of tires that can affect vehicle performance, so the influence of inflation pressure should be taken into account in high-precision tire models. Much research has focused on tire mechanical characteristics under different inflation pressures and modeling methods, such as the Improved Magic Formula/Swift Tire model developed by Schmeitz et al. that takes inflation pressure into account. Based on a large number of tire testing data under different inflation pressures, an empirical model can be obtained, but numerous tire tests are expensive and cannot reveal the mechanism. In previous studies, most tire tests were based on load control, wherein the load is constant while the footprint, carcass, and belt stiffness can change with the variation of inflation pressure, making the cornering characteristics much more complicated. Herein, a method of deflection control is used and the contact patch is well maintained. The effect of tire inflation pressure is simplified to be the change of structure stiffness and averaged contact pressure. The effect of inflation pressure on cornering properties under deflection control is systematically observed under different tire deflection values. In addition, a prediction model extended from the Magic Formula model is proposed. The model of Schmeitz et al. [1,2] needs test data from at least three inflation pressures; the model proposed herein requires tests at only two inflation pressures. The validation shows that this prediction method has good accuracy, almost the same as that of the model of Schmeitz et al. The error index of the lateral force is at most 2%, and the error index of the self-aligning torque is 4% at the maximum.

Study on improvement of LuGre dynamical model and its application in vehicle handling dynamics

The accurate description of contact characteristics between tire and road surface may significantly affect the vehicle motion simulation and control analysis. Based on the LuGre tire model, the pressure distribution function of loading is modified as a non-uniform distribution pattern through new method of recursion formula. The mathematical expression of modified LuGre tire model is achieved through introducing the Bouc-Wen model to reflect tire hysteresis behavior, which takes into consideration the process of dynamic effect of tire vertical loading and tire footprint length. The effectiveness of model is verified through comparison with classic magic formula (MF) and LuGre tire model. Furthermore, the stability of solution for the modified LuGre tire model in the prediction of tire force and alignment torque is demonstrated in frequency domain. Finally, a nonlinear four-DOF vehicle handling dynamics model is built. The front wheel steering angle and the rotational angular velocities of the six tire from a calibrated virtual vehicle is entered into the vehicle model system with modified LuGre tire model as input data. The responses of longitudinal velocity, lateral velocity, yaw rate and roll angle are obtained under two kinds of limit conditions. The comparison results demonstrate the correctness and effectiveness of the built model, which further prove that the modified LuGre tire model is applicable for handling performance analysis of vehicle system.

Neural Network Identification of a Racing Car Tire Model

In order to meet the demands of small race car dynamics simulation, a new method of parameter identification in the Magic Formula tire model is presented in this work, based on an analysis of the Magic Formula tire model structure. A high-precision tire model used for vehicle dynamics simulation is established via this method. It is difficult for students to build a high-precision tire model because of the complexity of widely used tire models such as Magic Formula and UniTire. At a pure side slip condition, building a lateral force model is an example, which illustrate the utilization of a multilayer feed-forward neural network to build an intelligent tire model conveniently. In order to fully understand the difference between the two models, a two-degrees-of-freedom (2 DOF) vehicle model is established. The advantages, disadvantages, and applicable scope of the two tire models are discussed after comparing the simulation results of the 2 DOF model with the Magic Formula and intelligent tire model.

A comprehensive active-steering control method for improvement of vehicle handling performance

Non-linear system control has always been a difficult point for vehicle stabilization. To improve the vehicle handling performance, a comprehensive active-steering control method is proposed and derived. Different from traditional strategy, this new controller is based on a piecewise tyre modelling ideology combined with feedback linearization controlling method. In the linear region of wheel–terrain contact, vehicle dynamic system turns to be a linear system, an optimal control is designed for the sake of rapid response in tracking desired values. In the non-linear region, where the controlling difficulty always lies in, the tyre lateral force is described by a new polynomial formula model, which is simpler than magic formula model and more accurate than linear model. This new tyre modelling ideology ensures the feasibility of feedback linearization method in non-linear system control. To verify the proposed controller, a numerical seven-degrees-of-freedom vehicle model is built and validated by standard input simulation. Then, simulation under limit conditions, including high friction case and low friction case, are conducted and results are presented and discussed. Compared with optimal controller and free-control method, comprehensive controller has a much more desirable applicability in both cases and greatly improves the vehicle handling performance.

Envelope Control for Four-Wheel Independently Actuated Autonomous Ground Vehicle Through AFS/DYC Integrated Control

This paper proposes an envelope control framework for four-wheel independently actuated autonomous ground vehicle (AGV) to regulate it on desired path and simultaneously control it to the driving limits. The envelope control framework is achieved based on the integrated control of active front-wheel steer and direct yaw-moment control. In a speed controller, the G-G diagram is used to describe the driving limits on each path segment. The desired traction and braking force is calculated according to the G-G diagram and desired path. In a path-following controller, a feedforward–feedback lateral controller is designed to calculate the desired steering angle to follow the desired path. In a yaw-moment controller, the β–r phase portraits are utilized to describe the handling limits. The yaw-moment controller aims at keeping the AGV from losing stability in limit driving, which is calculated through a sliding mode controller and provided by the independent motors actuation. Through an independent driving technique, the tyre cornering stiffness is estimated online based on the predefined Magic Formula model to improve the controller's robustness. An autonomous Formula Student racecar developed by the authors is used as testbed. The autonomous driving experiments on racetrack validate the efficiency of the proposed controller.

A lateral dynamics of a wheelchair: identification and analysis of tire parameters

In vehicle dynamics studies, the tire behaviour plays an important role in planar motion of the vehicle. Therefore, a correct representation of tire is a necessity. This paper describes a mathematical model for wheelchair tire based on the Magic Formula model. This model is widely used to represent forces and moments between the tire and the ground; however some experimental parameters must be determined. The purpose of this work is to identify the tire parameters for the wheelchair tire model, implementing them in a dynamic model of the wheelchair. For this, we developed an experimental test rig to measure the tires parameters for the lateral dynamics of a wheelchair. This dynamic model was made using a multi-body software and the wheelchair behaviour was analysed and discussed according to the tire parameters. The result of this work is one step further towards the understanding of wheelchair dynamics.

Development of a Parameter Identification Method for MF-Tyre/MF-Swift Applied to Parking and Low Speed Manoeuvres

A vehicle parking manoeuvre is characterized by low or zero speed, small turning radius and large yaw velocity of the steered wheels. To predict the forces and moments generated by a wheel under these conditions, the Pacejka Magic Formula model has been extended to incorporate the effect of spin (turn slip model) in the past years. The extensions have been further developed and incorporated in the MFTyre/MF-Swift 6.2 model. This paper describes the development of a method for the identification of the turn slip parameters. Based on the operating conditions of a typical parking manoeuvre, the dominant parameters of the turn slip model are firstly defined. At an indoor test facility, the response of a tyre under the identified operating conditions is measured. An algorithm is developed to identify the dominant turn slip parameters from the measured responses. Wherever possible, the algorithm is based on direct analytical relationships between the turn slip model parameters and the measured signals, otherwise an iterative method is applied. By means of comparing vehicle simulation results to instrumented vehicle measurements, the turn slip model, including the identified parameters, is validated. The results show the effectiveness of the developed method.

Definition and Modeling of the Power Loss of Tires

ABSTRACT Tires represent a crucial leverage for reducing CO2 emissions: in the future, the importance of analyzing power losses under real-vehicle-driving conditions and conflicts between optimizing power loss, longitudinal dynamics, lateral dynamics, and comfort will increase. The objective of the present study is to develop both the set of physical characteristic values (CV-set) and the genetic algorithm used to generate Magic Formula parameter sets that have the desired physical characteristics. To this purpose, improvements concerning the genetic algorithm are presented and a definition for the power loss of tires is derived and validated through measurements on an outer drum test bench. Then, four physical characteristic values describing the power loss are added to the existing CV-set. This CV-set, in combination with a genetic algorithm, allows us to generate virtual tires (e.g., Magic Formula model) with defined physical characteristics of longitudinal force, lateral force, aligning moment, and power loss.

Stochastic vehicle handling prediction using a polynomial chaos approach

One fundamental difficulty in understanding the physics of the off-road traction and in predicting vehicle performance is the indeterminacy of certain important parameters on the interface between tyre and terrain/road surface, for instance, the slip ratio, the slip angle, the normal forces, and the friction coefficients. It is not possible to accurately capture the effect of such uncertainties on the tyre behaviour (resultant force and moments) using a deterministic model. In addition, current measuring techniques have certain limitations and sometimes non-negligible measurement errors could be a source of relatively rough approximations in estimating some important parameters involved in vehicle dynamics simulations and control algorithms. In this study, mathematical tools to quantify the impact of uncertainties in tyre-terrain interface on vehicle handling were employed. We treated the uncertainty in key parameters associated with the tyre-road interface using a polynomial chaos approach. Two of the most popular semi-empirical models for predicting the pneumatic tyres performance under steady-state and transient conditions, the Friction Ellipse Model (FEM) and the Magic Formula Model (MFM), were selected to be extended from the deterministic to a stochastic workframe, to account for the uncertainties in the tyre-terrain friction coefficient, the slip ratio, the slip angle, and the normal forces in the contact patch. The modelling approach presented in this paper was able to capture the stochastic nature of parameters of interest and to predict the response of the system under those uncertainties, in an effort to provide a better understanding and a more realistic prediction of the tyre-terrain interaction than a deterministic formulation. To illustrate the application of the stochastic tyre-road models on vehicle dynamics, they were simulated in conjunction with a bicycle-car model.

Study on steering wheel shimmy with clearance of kingpin

The non-linear factors of tyre lateral force and contact-impact force of kingpin clearance are considered in this paper. They are brought in the shimmy model by the tyre magic formula and dynamic model of a two-state clearance mechanism. The non-linear mathematical model of steering wheel shimmy with non-independent suspension is established by using Lagrange equation method. And the influences of different initial incentives and parameters are numerically investigated. The simulation results indicate that the clearance of kingpin can increase the amplitude of steering wheel shimmy, the speed range become larger, and the non-sensitive parameters may have an impact on the steering wheel shimmy.