An Alternative Method to Determine the Magic Tyre Model Parameters Using Genetic Algorithms

Summary Tyre behavior plays an important role in vehicle dynamics research. Knowledge of tyre properties is necessary to properly design vehicle components and advance control system. For that purpose mathematical models of the tyre are being used in vehicle simulation models. The Magic Formula Tyre Model is a semi-empirical tyre model which describes tyre behavior quite accurately. The Magic Formula Tyre Model needs a set of parameters to describe the tyre properties; the determination of these parameters is dealt with in this paper. A new method based on genetic techniques is used to determine these parameters. The main advantages of the method are its simplicity of implementation and its fast convergence to optimal solution, with no need of deep knowledge of the searching space. So to start the search, it is not necessary to know a set of starting values of the Magic Formula parameters. The comparison between analytical optimization methods and the method proposed is discussed in this paper.     

An Alternative Method to Determine the Magic Tyre Model Parameters Using Genetic Algorithms

Summary Tyre behavior plays an important role in vehicle dynamics research. Knowledge of tyre properties is necessary to properly design vehicle components and advance control system. For that purpose mathematical models of the tyre are being used in vehicle simulation models. The Magic Formula Tyre Model is a semi-empirical tyre model which describes tyre behavior quite accurately. The Magic Formula Tyre Model needs a set of parameters to describe the tyre properties; the determination of these parameters is dealt with in this paper. A new method based on genetic techniques is used to determine these parameters. The main advantages of the method are its simplicity of implementation and its fast convergence to optimal solution, with no need of deep knowledge of the searching space. So to start the search, it is not necessary to know a set of starting values of the Magic Formula parameters. The comparison between analytical optimization methods and the method proposed is discussed in this paper.     

The IMMa optimisation algorithm without control input parameters

The IMMa optimisation algorithm (IOA) consists of a heuristic method based on a differential evolution algorithm for choosing the Magic Formula (MF) tyre model parameters. In a previous paper, we demonstrated that the IOA improved the searching procedure of optimum MF parameters with respect to the starting value optimisation (SVO) methods. But we had to introduce some control input parameters that were fixed during the running process. Now, the new version does not require control input variables to be chosen by the user. That is, we use an algorithm with self-adapting control parameters and it continues being easy to use, robust and fast. Hence, users do not need any kind of knowledge to use the IOA.     

A study of the relationship between Magic Formula coefficients and tyre design attributes through finite element analysis

Pacejka's Magic Formula Tyre Model is widely used to represent force and moment characteristics in vehicle simulation studies meant to improve handling behaviour during steady-state cornering. The experimental technique required to determine this tyre model parameters is fairly involved and highly sophisticated. Also, total test facilities are not available in most countries. As force and moment characteristics are affected by tyre design attributes and tread patterns, manufacturing of separate tyres for each design alternative affects tyre development cycle time and economics significantly. The objective of this work is to identify the interactions among various tyre design attributes-cum-operating conditions and the Magic Formula coefficients. This objective is achieved by eliminating actual prototyping of tyres for various design alternatives as well as total experimentation on each tyre through simulation using finite element analysis. Mixed Lagrangian–Eulerian finite element technique, a specialized technique in ABAQUS, is used to simulate the steady-state cornering behaviour; it is also efficient and cost-effective. Predicted force and moment characteristics are represented as Magic Formula Tyre Model parameters through non-linear least-squares fit using MATLAB. Issues involved in the Magic Formula Tyre Model representation are also discussed. A detailed analysis is made to understand the influence of various design attributes and operating conditions on the Magic Formula parameters. Tread pattern, tread material properties, belt angle, inflation pressure, frictional behaviour at the tyre–road contact interface and their interactions are found to significantly influence vehicle-handling characteristics.     

A comparison of various algorithms to extract Magic Formula tyre model coefficients for vehicle dynamics simulations

Tyre models are a prerequisite for any vehicle dynamics simulation. Tyre models range from the simplest mathematical models that consider only the cornering stiffness to a complex set of formulae. Among all the steady-state tyre models that are in use today, the Magic Formula tyre model is unique and most popular. Though the Magic Formula tyre model is widely used, obtaining the model coefficients from either the experimental or the simulation data is not straightforward due to its nonlinear nature and the presence of a large number of coefficients. A common procedure used for this extraction is the least-squares minimisation that requires considerable experience for initial guesses. Various researchers have tried different algorithms, namely, gradient and Newton-based methods, differential evolution, artificial neural networks, etc. The issues involved in all these algorithms are setting bounds or constraints, sensitivity of the parameters, the features of the input data such as the number of points, noisy data, experimental procedure used such as slip angle sweep or tyre measurement (TIME) procedure, etc. The extracted Magic Formula coefficients are affected by these variants. This paper highlights the issues that are commonly encountered in obtaining these coefficients with different algorithms, namely, least-squares minimisation using trust region algorithms, Nelder–Mead simplex, pattern search, differential evolution, particle swarm optimisation, cuckoo search, etc. A key observation is that not all the algorithms give the same Magic Formula coefficients for a given data. The nature of the input data and the type of the algorithm decide the set of the Magic Formula tyre model coefficients.     

A novel semi-empirical tyre model for combined slips

A new tyre-force model for simultaneous braking and cornering is presented, which is based on combining existing empirical models for pure braking and cornering with brush-model tyre mechanics. The aim is to offer an easy-to-use, accurate model for vehicle-handling simulations. On a working tyre the contact patch between the tyre and the road is, in general, divided into an adhesion region where the rubber is gripping the road and a sliding region where the rubber slides on the road surface. The total force generated by the tyre is then composed of components from these two regions. The brush model describes this in a mechanical framework. The proposed model is based on a new method to extract adhesion and sliding forces from empirical pure-slip tyre models. These forces are then scaled to account for the combined-slip condition. The combined-slip self-aligning torque is also described. A particular feature of the model is the inclusion of velocity dependence, even if this is not explicitly present in the empirical pure-slip model. The approach is quite different from most previous combined-slip models, in that it is based on a rather detailed mechanical model in combination with empirical pure-slip models. The model is computationally sound and efficient and does not rely on any additional parameters that depend on combined-slip data for calibration. It can be used in combination with virtually any empirical pure-slip model and in this work the Magic Formula is used in examples. Results show good correspondence with experimental data.     

Environmental effects on Pacejka’s scaling factors

A set of scaling factors has been introduced by Pacejka [Pacejka, H.B., 2002, Tyre and Vehicle Dynamics (Oxford: Butterworth Heinemann Editions)] into his Magic Formula tyre model to take into account the influence of a number of external overall parameters such as road roughness, weather conditions, suspension characteristics and so on. These scaling factors are important for a correct prediction of tyre–road contact forces, but are not a function of the tyre itself. Changing the point of view, one could say that scaling factors should remain constant for different tyres on the same circuit, with the same weather conditions and with the same car. After characterizing different tyres through indoor tests (that do not consider external overall parameters) and after having identified Pacejka’s coefficients with scaling factors equal to one, several outdoor experimental tests have been carried out to assess the influence of vehicle and road surface conditions on scaling factors. These experimental data allowed us to identify, through a minimization approach, the ‘best’ set of Pacejka’s scaling factors for that vehicle and for that tyre on that track. Scaling factors for equal track and vehicle but different tyres were compared to check whether their values remained constant. To access the validity of scaling factors, a comparison between experimental data, collected on an instrumented passenger car, and MB simulations considering unity and identified scaling factors’ values, were carried out. All experimental data shown in this article come from tests carried out within the VERTEC project, a European founded research project (Task 2.a and 2.b) that puts together knowledge coming from vehicle manufacturers (Volvo, Porsche and Centro Ricerche Fiat CRF), tyre manufacturers (Pirelli and Nokian Tyres), control logic manufacturers (Lucas Varity GmbH), road maintenance experts (Centres d’Études Techniques de l’Équipement CETE), transport research organizations (Transport Research Laboratory TRL, Swedish National Road and Transport Research Institute VTI) and universities (Helsinki University of Technology HUT, Politecnico di Milano and University of Florence UNIFI).     

Magic Formula轮胎模型参数辨识的一种混合优化方法

Magic Formula(MF)轮胎模型能够准确描述轮胎的侧偏特性,广泛应用于车辆动力学的研究。由于MF轮胎模型参数多,且高度非线性,从大量的试验数据中准确辨识这些参数相当困难。提出一种基于遗传算法和数值优化算法的混合优化方法,采用由粗到精的辨识过程,先利用遗传算法得出近似最优解,再利用数值优化算法辨识出精确的参数。利用辨识出的参数计算轮胎的侧偏特性,计算结果与试验数据吻合良好,表明该方法是辨识MF轮胎模型参数的有效手段。     

A study on model fidelity for model predictive control-based obstacle avoidance in high-speed autonomous ground vehicles

This paper investigates the level of model fidelity needed in order for a model predictive control (MPC)-based obstacle avoidance algorithm to be able to safely and quickly avoid obstacles even when the vehicle is close to its dynamic limits. The context of this work is large autonomous ground vehicles that manoeuvre at high speed within unknown, unstructured, flat environments and have significant vehicle dynamics-related constraints. Five different representations of vehicle dynamics models are considered: four variations of the two degrees-of-freedom (DoF) representation as lower fidelity models and a fourteen DoF representation with combined-slip Magic Formula tyre model as a higher fidelity model. It is concluded that the two DoF representation that accounts for tyre nonlinearities and longitudinal load transfer is necessary for the MPC-based obstacle avoidance algorithm in order to operate the vehicle at its limits within an environment that includes large obstacles. For less challenging environments, however, the two DoF representation with linear tyre model and constant axle loads is sufficient.     

轮胎垂直振动动力学模型

本文提出一种自适应轮膜模型参数的简易测定方法，给出轮胎垂直振动动力学方程及相应的求解方法，并进行计算分析与试验测定。将分析结果与通常的点接触式模型相比较，对点接触式模型及本文所建立的自适应模型做了综合评价，结果表明，本模型的计算结果是令人满意的。     

Unscented Kalman filter for vehicle state estimation

Vehicle dynamics control (VDC) systems require information about system variables, which cannot be directly measured, e.g. the wheel slip or the vehicle side-slip angle. This paper presents a new concept for the vehicle state estimation under the assumption that the vehicle is equipped with the standard VDC sensors. It is proposed to utilise an unscented Kalman filter for estimation purposes, since it is based on a numerically efficient nonlinear stochastic estimation technique. A planar two-track model is combined with the empiric Magic Formula in order to describe the vehicle and tyre behaviour. Moreover, an advanced vertical tyre load calculation method is developed that additionally considers the vertical tyre stiffness and increases the estimation accuracy. Experimental tests show good accuracy and robustness of the designed vehicle state estimation concept.     

Analysis of vehicle rollover dynamics using a high-fidelity model

Recent data show that 35% of fatal crashes in sport utility vehicles included vehicle rollover. At the same time, experimental testing to improve safety is expensive and dangerous. Therefore, multi-body simulation is used in this research to improve the understanding of rollover dynamics. The majority of previous work uses low-fidelity models. Here, a complex and highly nonlinear multi-body model with 165 degrees of freedom is correlated to vehicle kinematic and compliance (K&C) measurements. The Magic Formula tyre model is employed. Design of experiment methodology is used to identify tyre properties affecting vehicle rollover. A novel, statistical approach is used to link suspension K&C characteristics with rollover propensity. Research so far reveals that the tyre properties that have the greatest influence on vehicle rollover are friction coefficient, friction variation with load, camber stiffness and tyre vertical stiffness. Key K&C characteristics affecting rollover propensity are front and rear suspension rate, front roll stiffness, front camber gain, front and rear camber compliance and rear jacking force.     

Identification of the characteristics of motorcycle and scooter tyres in the presence of large variations in inflation pressure

Stability and safety of road vehicles are largely affected by tyre properties. Single-track vehicles are characterised by weakly damped modes of vibration (weave and wobble) and therefore this phenomenon is even more important. This article focuses on the study of both steady-state and transient properties of motorcycle and scooter tyres in the presence of very low and very high inflation pressures. The steady-state properties are defined as lateral forces (side-slip and camber forces) and yaw torques (self-aligning and twisting). The transient properties are described in terms of relaxation length, which represents the distance needed to reach a certain percentage of the steady-state value of the tyre force. Experimental tests are carried out on a specific rotating disk machine. Three sets of tyres are analysed. Steady-state properties are measured by increasing step by step the values of camber and side-slip angles. Transient properties are studied carrying out tests with harmonic side-slip excitation and measuring the phase lag between the excitation (input) and the tyre force (output). Experimental results show important variations in tyre properties with inflation pressure with general trends of all the tested tyres and particular features related to the tyre's geometry. After the analysis and discussion of experimental results, the measured data are fitted by means of a specific version of the Magic Formula. The dependence of the Magic Formula's coefficients on inflation pressure is analysed and interpolation curves are given.     

土钉支护结构优化设计中的混合粒子群优化算法

为了在土钉支护设计中寻找一组最佳设计参数,以达到既经济又安全的目的,将混沌搜索的粒子群优化(PSO)算法应用到基坑土钉支护优化设计中,以单位长度土钉墙的土钉材料造价作为优化的目标函数。该方法保持了PSO算法结构简单的特点,改善了PSO算法的全局寻优能力,提高了算法的收敛速度和计算精度。同时,对不活动粒子的处理使算法避免了早熟现象的出现。工程实例计算表明:该方法是进行土钉支护结构优化设计的有效方法。     

Design of tyre force excitation for tyre–road friction estimation

Knowledge of the current tyre–road friction coefficient is essential for future autonomous vehicles. The environmental conditions, and the tyre–road friction in particular, determine both the braking distance and the maximum cornering velocity and thus set the boundaries for the vehicle. Tyre–road friction is difficult to estimate during normal driving due to low levels of tyre force excitation. This problem can be solved by using active tyre force excitation. A torque is added to one or several wheels in the purpose of estimating the tyre–road friction coefficient. Active tyre force excitation provides the opportunity to design the tyre force excitation freely. This study investigates how the tyre force should be applied to minimise the error of the tyre–road friction estimate. The performance of different excitation strategies was found to be dependent on both tyre model choice and noise level. Furthermore, the advantage with using tyre models with more parameters decreased when noise was added to the force and slip ratio.     

轮胎侧偏动力模型的非线性分析

轮胎径向变形与轮胎接地印迹长度呈非线性关系，从而导致轮胎侧偏力和回正力矩与车轮动载荷也是非线性关系。本文用一轮胎模型对这一现象进行了理论分析，并得到了与实验数据基本一致的结果。     

整车多体动力学模型的建立、验证及仿真分析

利用多体动力学方法建立了某轿车的整车非线性多体动力学模型,模型中考虑了前后悬架、转向系统的详细几何结构参数,以及连接处的橡胶衬套、阻尼器的非线性特性,轮胎采用M agic Formu la模型。对所建模型进行了多种试验验证,并分析了该样车的操纵稳定性等相关特性,仿真结果表明所建整车多体模型有较高的精度。     

汽车四轮转向非线性系统的神经网络控制

考虑了轮胎的非线性特性，引入Magic Formula建立汽车非线性力学模型，利用前向BP神经网络来辨识该非线性模型，并利用另一个神经网络进行PID参数的整定，进行离散控制系统和控制算法的设计，有效地提高了汽车四轮转向的稳定性。