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On the Implementation of an Innovative Temperature-Sensitive Version of Pacejka’s MF in Vehicle Dynamics Simulations
Abstract
The characterization and reproduction of tire behaviour for vehicle modelling is a topic of particular interest both for real-time driving simulations and for offline performance optimization algorithms. In such contexts, the Pacejka’s Magic Formula (MF) tire model [1] represents a standard that gained in the last 25 years a role of high relevance due to its low computational request and attitude to allow an efficient parameterization for a wide range of tires working conditions. Nevertheless, the original MF formulation was conceived with the aim to provide tire/road interaction forces and moments as a function of vertical load, longitudinal and lateral slip, and inclination (or camber) angle; such variables are fundamental but not totally satisfying in the description of the complex multi-physical phenomena occurring at tire/road interface [2]. In particular, the relationship between interaction forces and the cited input variables is highly influenced by further effects, linked to tire temperature, tread wear, compound viscoelastic characteristics and road roughness. Among these, the influence that the thermal conditions of the different layers constituting the global thickness of tires have on the friction and on their stiffness characteristics, is highly significant and definitely not negligible in case a full reliability of the vehicle dynamics simulations is required, especially in motorsport applications [4]. The paper illustrates the basic concepts linked to the development of a novel version of a MF-based formulation, able to take into account uncommon factors affecting tire/road interaction. Once described the structure and the parameters identification process [4], some results obtained with the MF-evo employed in a simulative loop with a thermal model are reported.
... However, the accuracy of the Magic Formula decreases when thermodynamic and wear influences on tire grip and cornering stiffness can not be neglected, since this model only accounts for kinematic and dynamic variables. To properly consider these variables, the authors in [11,12] developed a multiphysical tire model integrating the Magic Formula with additional polynomial dependencies on the tire thermodynamic and wear states. ...
... It is worth noting that Eqs. (7)(8)(9)(10)(11)(12) contain several approximations regarding the model kinematics. In par-ticular, the rim velocities defined in the adopted reference frame are calculated starting from the axle velocities through linearization. ...
... The forces (F cx , F cy ) and torques (M cx , M cy , M cz ), linked to the tangential interaction are determined coherently with [8,11], starting from kinematic slips both in steady-state and transient conditions. ...
In the development of physical tire models, the complexity of the composite structure and the multiphysical variables require strongly nonlinear mathematical formulations to guarantee a desired degree of accuracy. The aim of the current work is to extend the applicability of the multiphysical magic formula-based tire model, already developed and presented by the authors, within a wider frequency range, interposing a rigid ring body between the contact patch and the wheel hub. The contact patch, varying in terms of size, shape, and relative position, is evaluated using instantaneous cams to define the effective plane. Here the advanced slip model, taking into account thermodynamic and wear effects, is then integrated. The adopted formulations have been mathematically and physically justified. They have been analytically compared to formulations related to the rigid-ring implementation available in the literature. Specific experimental activities concerning both the tire’s vertical kinematics and dynamics have been conducted to demonstrate the model’s improved physical consistency on small wavelength unevennesses.
... Later, the original model, designed for vehicle handling applications, has been reformulated to include the internal pressure effect [21,22] and to extend the applicability in dynamic scenarios with higher frequency [23]. The MF model has been further enhanced in [24], where the authors have proposed an advanced multiphysical MF-based (MF-evo) realtime tyre model with the aim to extend the Pacejka's Magic Formula tyre model in the whole range of the tyre operating conditions, taking into account its internal temperature distribution [25,26], inflation pressure [22], tread wear [27,28], compound viscoelastic characteristics and road roughness [29,30]. The potential risks, related with the employment of empirical models, are linked with their parametrisation and the quality of data, since the adoption of numeric data-based techniques makes it possible to completely misinterpret the tyre behaviour even in case of a good fitting towards experimental results. ...
... To overcome the above modelling limits, the authors have proposed in [24] an advanced methodology making use of the additional polynomial formulations for the analytical description of the macro-and micro-parameters: ...
To cite this article: Aleksandr Sakhnevych (2021): Multiphysical MF-based tyre modelling and parametrisation for vehicle setup and control strategies optimisation, Vehicle System Dynamics, ABSTRACT Starting from the earliest phases of design of the vehicle and its control systems, the understanding of tyres is of fundamental importance to govern the overall vehicle dynamics. A properly charac-terised tyre-road interaction model is essential to achieve a reliable vehicle dynamics model on which more design variations can be studied directly in simulation environment optimising both cost and time. The possibility to count on computationally efficient and reliable formulations represents nowadays a great advantage, and the multiphysical Pacejka's Magic Formula (MF-evo) tyre model presented is one of the best trade-off solutions to meet the strict real-time requirements and to reproduce multiphysical variations of the tyre dynamic behaviour towards temperature, pressure and wear effects. A specific methodology has been developed to characterise and to identify the MF-evo parameters with a high grade of accuracy and reliability directly from experimental data. The proposed technique is based on a pre-processing procedure to remove non-physical outliers and to cluster the data, which allows to optimise the multidimensional parameterisation process. To the purpose of validation of the parametrisation routine, data from a motorsport case, exceptionally difficult to reproduce in simulation due particularly significant variations of the tyre dynamics during a single test, have been employed demonstrating the MF-evo model potential and robustness. ARTICLE HISTORY
... Similar correlations are available in [26] when road roughness is concerned. To include the effect of the tyre surface temperature on friction curves, the variation of the Pacejka's parameters can be studied as well, see [27]. In real-time simulation environments, moreover, a tyre thermal model (see e.g., [17,28,29]) is frequently employed, to account for temperature variations during the operation of the tyre. ...
... A full vehicle model contains an important number of bodies connected not only by kinetic joints, but also by nonlinear springs, shock absorbers, silent-blocks (bushings), elastic bodies (anti-roll bars, etc.) and highly nonlinear tires [25]. In addition, tire behavior is affected by variables such as temperature, internal pressure and wear, with a significant impact on the vehicle performance [26,27]. ...
A new methodology for constructing stability maps (phase-plane analysis) is presented and validated for application to complex multibody vehicle models implemented in Multibody Dynamics simulation software (Adams®). Traditional methodologies are developed to be applied to explicit mathematical models. Given the complexity of some special multibody systems, particularly in vehicle dynamics, simplifications are needed to apply this stability analysis technique. The main limitation when using simplified models is the need to neglect components which could have a significant influence on the dynamic behavior of the system and therefore on its stability. In the proposed methodology it is not necessary to have access to explicit mathematical models of multibody systems. Thus, the stability map of a vehicle model can be constructed by considering highly nonlinear dynamic elements, such as tires and silent-blocks components, modeled using the nonlinear finite element technique.
... As seen in Calabrese's work [23], the two main effects that the tyre temperature has on the tyre characteristics are on grip and stiffness; grip experiences a considerable change. However, in high-fidelity models such as MFevo, the grip and stiffness changes are more precisely represented as functions of different internal tyre layer temperatures and also pressure, as shown in [13,26]. In myTyre, the grip and slip stiffness effect were included using the K µ (T s ) and K k (T s ) functions by scaling D x and B x , respectively. ...
Vehicle dynamics can be deeply affected by various tyre operating conditions, including thermodynamic and wear effects. Indeed, tyre temperature plays a fundamental role in high performance applications due to the dependencies of the cornering stiffness and potential grip in such conditions. This work is focused on the evaluation of a potentially improved control strategy’s performance when the control model is fed by instantaneously varying tyre parameters, taking into account the continuously evolving external surface temperature and the vehicle boundary conditions. To this end, a simplified tyre thermal model has been integrated into a model predictive control strategy in order to exploit the thermal dynamics’ dependents within a proposed advanced ABS control system. We evaluate its performance in terms of the resulting braking distance. In particular, a non-linear model predictive control (NMPC) based ABS controller with tyre thermal knowledge has been integrated. The chosen topic can possibly lay a foundation for future research into autonomous control where the detailing of decision-making of the controllers will reach the level of multi-physical phenomena concerning the tyre–road interaction.
... On the other hand the carcass temperature reacts more slowly making it more difficult to manage-this is particularly true in cases where carcass-temperature reductions are required. In order to recognise the long-run impact of tyre degradation, the grip is modelled as a function of both tyre wear [67] and the tread temperature [68]. Simulations on the Circuit de Catalunya show that with new tyres the first lap tends to be slow reflecting the combined influences of a low operating temperature and tyres that have not yet been 'rubbed in'. ...
The paper begins with a survey of advances in state-of-the-art minimum-time simulation for road vehicles. The techniques covered include both quasi-steady-state and transient vehicle models, which are combined with trajectories that are either pre-assigned or free to be optimised. The fundamentals of nonlinear optimal control are summarised. These fundamentals are the basis of most of the vehicular optimal control methodologies and solution procedures reported in the literature. The key features of three-dimensional road modelling, vehicle positioning and vehicle modelling are also summarised with a focus on recent developments. Both cars and motorcycles are considered.
In recent years the increasing needs of reducing the costs of car development expressed by the automotive market have determined a rapid development of virtual driver prototyping tools that aims at reproducing vehicle behaviors. Nevertheless, these advanced tools are still not designed to exploit the entire vehicle dynamics potential, preferring to assure the minimum requirements in the worst possible operating conditions instead. Furthermore, their calibration is typically performed in a pre-defined strict range of operating conditions, established by specific regulations or OEM routines. For this reason, their performance can considerably decrease in particularly crucial safetycritical situations, where the environmental conditions (rain, snow, ice), the road singularities (oil stains, puddles, holes), and the tyre thermal and ageing phenomena can deeply affect the adherence potential. The objective of the work is to investigate the possibility of the physical model-based control to take into account the variations in terms of the dynamic behavior of the systems and of the boundary conditions. Different scenarios with specific tyre thermal and wear conditions have been tested on diverse road surfaces validating the designed model predictive control algorithm in a hardware-in-the-loop real-time environment and demonstrating the augmented reliability of an advanced virtual driver aware of available information concerning the tyre dynamic limits. The multidisciplinary proposal will provide a paradigm shift in the development of strategies and a solid breakthrough towards enhanced development of the driving automatization systems, unleashing the potential of physical modeling to the next level of vehicle control, able to exploit and to take into account the multi-physical tyre variations.
In the paper a new physical tyre thermal model is presented. The model, called Thermo Racing Tyre (TRT) was developed in collaboration between the Department of Industrial Engineering of the University of Naples Federico II and a top ranking motorsport team.
The model is three-dimensional and takes into account all the heat flows and the generative terms occurring in a tyre. The cooling to the track and to external air and the heat flows inside the system are modelled.
Regarding the generative terms, in addition to the friction energy developed in the contact patch, the strain
energy loss is evaluated. The model inputs come out from telemetry data, while its thermodynamic parameters come either from literature or from dedicated experimental tests.
The model gives in output the temperature circumferential distribution in the different tyre layers (surface,
bulk, inner liner), as well as all the heat flows.
These information have been used also in interaction models in order to estimate local grip value.
A flat track tire testing machine developed by the IMMa group is described. It permits the simulation and study of the dynamic behavior of a great variety of tires under controllable and repetitive highly dynamic realistic working conditions in the laboratory for a diversity of vehicles, from motorcycles to light trucks. The machine incorporates: – a hydraulically operated tire support and loading system with wide operating ranges; – a computer controlled brake system to simulate braking maneuvers with ABS systems; – a complete sensorial system; – a data acquisition and control system continually monitoring and acting on the experimental variables, i.e., tire and belt speed, longitudinal slip, slip and camber angles, tire pressure, tire normal force, etc. As an application example, results are presented that adjust the parameter of the magic formula for a standard 175/70 R14 passenger vehicle tire. Accurate mathematical tire models are recognized as essential for the prediction of vehicle dynamic performances using simulation tools.
To improve and better understand the tire wet grip mechanism, it is essential to perform test bench measurements under wet conditions. On both public streets and the track surfaces of the internal drum test bench of the Karlsruhe Institute of Technology (KIT), a drop in the friction level of the track surface can be observed with an increasing number of wet measurements. For this purpose, a new measuring device was developed and built: the continuous friction monitoring system (CFM-System). The measuring principle is a continuously braked standard test tire, whereby a longitudinal friction coefficient is determined. To establish a suitable measuring method using the CFM-System, a basic understanding of the longitudinal behavior of this standard test tire was achieved. For this purpose, l-slip characteristics were determined to investigate the friction behavior of this tire under different slip conditions. Furthermore, the influence of wheel load and driving speed were examined and analyzed. Based on these findings, an adequate test method was derived. The investigations presented in this article show a good relation between the results of the CFM-System and the results obtained from measurements using the standard reference test tire (SRTT). The results show that this new CFM-System is suitable for monitoring the grip level of the track surface on the internal drum test bench when performing wet grip measurements. It is assumed that the results can be directly transferred to the real road or to proving grounds because real track surfaces are used on the test bench.
Vehicle performances, especially in motorsport, are deeply affected by tire behavior and in particular by tire compound proper working conditions. In this research activity, a series of innovations have been introduced on the Thermo Racing Tire (a physical-analytical tire thermal model, based on Fourier’s law of heat transfer applied to a three-dimensional domain) in order to take into account all the main aspects actively involved in the thermal behavior of the tire, as the presence of exhausted gases eventually impacting at the rear axle and the inhomogeneous distribution of local variables (pressure, stress and sliding velocity) within the contact patch, caused in example by the tire camber angle. The new model developed considers the presence of the sidewalls, actively involved in the convective heat exchanges, respectively, with the external airflow and the inner gas fluid, located inside the inflation chamber. The aim of the new version of the tire thermal model is a better physical comprehension of all the phenomena concerning the contact with the asphalt and the prediction of the link between the thermal state and the frictional performance, crucial for the definition of an optimal wheel and vehicle setup.
Tires are the interface between vehicle and road. The forces and moments generated by the tires determine the motion of a road vehicle. Dedicated tire tests provide insight in these forces and moments and their dependency on slip, inclination angle and vertical force. The brush tire model can explain the measured characteristics qualitatively, the Magic Formula is a semi-empirical tire model to quantitatively describe them. To account for tire dynamic behavior, relaxation effects are discussed and modeled.
Tyres play a key role in ground vehicles' dynamics because they are responsible for traction, braking and cornering. A proper tyre-road interaction model is essential for a useful and reliable vehicle dynamics model. In the last two decades Pacejka's Magic Formula (MF) has become a standard in simulation field.
This paper presents a Tool, called TRIP-ID (Tyre Road Interaction Parameters IDentification), developed to characterize and to identify with a high grade of accuracy and reliability MF micro-parameters from experimental data deriving from telemetry or from test rig. The tool guides interactively the user through the identification process on the basis of strong diagnostic considerations about the experimental data made evident by the tool itself. A motorsport application of the tool is shown as a case study.
The tire and vehicle setup definition, able to optimise grip performance and thermal working conditions, can make the real difference as for motorsport racing teams, used to deal with relevant wear and degradation phenomena, as for tire makers, requesting for design solutions aimed to obtain enduring and stable tread characteristics, as finally for the development of safety systems, conceived in order to maximise road friction, both for worn and unworn tires. The activity discussed in the paper deals with the analysis of the effects that tire wear induces in vehicle performance, in particular as concerns the consequences that tread removal has on thermal and frictional tire behaviour. The physical modelling of complex tire–road interaction phenomena and the employment of specific simulation tools developed by the Vehicle Dynamics UniNa research group allow to predict the tire temperature local distribution by means of TRT model and the adhesive and hysteretic components of friction, thanks to GrETA model. The cooperation between the cited instruments enables the user to study the modifications that a reduced tread thickness, and consequently a decreased SEL (Strain Energy Loss) and dissipative tread volume, cause on the overall vehicle dynamic performance.
For the lightweight design of the wheel rim of motorcycles, the knowledge of the way in which contact forces are transmitted by the tire is of crucial importance. In this paper, an analytical model of the tire is developed and explicit formulae giving the distribution of the radial and axial forces acting on the wheel rim for a given vertical load are derived. The analytical model is validated by means of a finite element method (FEM) model and experimental tests. The proposed analytical model is able to predict the radial deflection of both a front and a rear tire for a racing motorbike with very good accuracy over a wide range of inflating pressures and vertical loads. The force distributions are in very good agreement with the results of the FEM model. Experimental tests show that the force distribution at the interface between the tire and rim can be used to predict the stress distribution in the rim with a good accuracy.
The minimum lap time optimal control problem has been solved for a go-kart model. The symbolic algebra software Maple has been used to derive equations of motion and an indirect method has been adopted to solve the optimal control problem. Simulation has been successfully performed on a full track lap with a multibody model endowed with seven degrees of freedom. Geometrical and mechanical characteristics of a real kart have been measured by a lab test to feed the mathematical model. Telemetry recorded in an entire lap by a professional driver has been compared to simulation results in order to validate the model. After the reliability of the optimal control model was proved, the simulation has been used to study the peculiar dynamics of go-karts and focus to tyre slippage dynamics, which is highly affected by the lack of differential.
The most powerful engine, the most sophisticated aerodynamic devices or the most complex control systems will not improve vehicle performances if the forces exchanged with the road are not optimized by proper employment and knowledge of tires. The vehicle interface with the ground is constituted by the sum of small surfaces, wide about as one of our palms, in which tire/road interaction forces are exchanged. From this it is clear to see how the optimization of tire behavior represents a key-factor in the definition of the best setup of the whole vehicle.Nowadays, people and companies playing a role in automotive sector are looking for the optimal solution to model and understand tire's behavior both in experimental and simulation environments. The studies carried out and the tool developed herein demonstrate a new approach in tire characterization and in vehicle simulation procedures. This enables the reproduction of the dynamic response of a tire through the use of specific track sessions, carried out with the aim to employ the vehicle as a moving lab.The final product, named TRICK tool (Tire/Road Interaction Characterization and Knowledge), comprises of a vehicle model which processes experimental signals acquired from vehicle CAN bus and from sideslip angle estimation additional instrumentation. The output of the tool is several extra "virtual telemetry" channels, based on the time history of the acquired signals and containing force and slip estimations, useful to provide tire interaction characteristics. TRICK results can be integrated with the physical models developed by the Vehicle Dynamics UniNa research group, providing a multitude of working solutions and constituting an ideal instrument for the prediction and the simulation of the real tire dynamics.
This paper deals with the frictional behaviour of a tyre tread elementary volume in sliding contact with road asperities. Friction is supposed as composed by two main components: adhesion and deforming hysteresis. The target, fixed in collaboration with a motorsport racing team and with a tyre manufacturing company, is to provide an estimation of local grip for on-line analyses and real time simulations and to evaluate and predict adhesive and hysteretic frictional contributions arising at the interface between tyre tread and road. A way to approximate asperities, based on rugosimetric analyses on macro and micro scale, has been introduced. The adhesive component of friction has been estimated by means of a new approach based on two different models found in literature, whose parameters have been identified thanks to a wide experimental campaign previously carried out. The hysteretic component of friction has been estimated by means of an energy balance taking into account rubber viscoelastic behaviour, characterized by means of proper DMA tests, and internal stress / strain distribution due to indentation with road. Model results are finally shown and discussed and the validation experimental procedure is described. The correct reproduction of friction phenomenology and the model prediction capabilities are highlighted making particular reference to grip variability due to changes in working conditions.
Multi-Physical approach for tire contact and wear mechanisms modelling
- A Sakhnevych
Sakhnevych A., Multi-Physical Approach for Tire Contact and Wear Mechanisms
Modelling. PhD thesis, Università degli Studi di Napoli Federico II, Naples (2017).