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 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.     

利用遗传算法识别汽车轮胎魔术公式参数

汽车轮胎动力学模型的准确度对汽车运动仿真的精度影响很大。汽车轮胎魔术公式是一种精度较高的轮胎模型。利用遗传算法的非线性优化特点,对汽车轮胎纯滑移魔术公式参数进行识别,并讨论算法参数对参数识别的影响。识别结果,纵向力和侧向力的相对残差在5%左右,回正力矩的相对残差在10%左右。     

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。     

XJTU系统在FAT32下的使用

针对XJTU系统在FAT32中使用时出现的问题 ,进行了分析 ,并提出了解决方法。     

Magic Formula减振器数值模型的参数识别与虚拟调校

为研究减振器的调校和匹配,需要建立可以完整描述减振器的外特性曲线理论模型。介绍了一种新型减振器模型——Magic Formula减振器数值模型,同时给出了一种有效、精确地模型参数辨识方法。然后将模型参数化,嵌入到7自由度整车Simulink模型中,以整车的动力学性能为优化目标,联合Isight软件,使用多目标优化算法,在C级路面对减振器的阻尼特性进行虚拟调校。优化结果表明,调校后的减振器的阻尼特性能够较明显改善车辆的动力学性能。     

车辆电子稳定性程序的最优控制

引入轮胎魔术公式,建立了车辆的两自由度非线性动力学模型．以车辆质心侧偏角和横摆角速度为控制变量,基于车辆的线性动力学模型设计了最优控制器,将此控制器应用于非线性动力学模型并进行了仿真．结果表明,车辆电子稳定性程序显著提高了车辆的操纵稳定性,使驾驶员在大侧向加速度、大侧偏角的极限工况下能够对车辆进行正常操纵．