Mathematical Problems in Engineering

Volume 2017, Article ID 6924506, 9 pages

https://doi.org/10.1155/2017/6924506

## Nonlinear Research and Efficient Parameter Identification of Magic Formula Tire Model

College of Engineering, Nanjing Agricultural University, Nanjing 210031, China

Correspondence should be addressed to Zhixiong Lu; moc.qq@5615673001

Received 6 May 2017; Revised 16 August 2017; Accepted 22 August 2017; Published 28 September 2017

Academic Editor: Stefan Balint

Copyright © 2017 Zhun Cheng and Zhixiong Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Magic Formula tire model can describe the mechanical properties of tire accurately and thus is applied in the research field of vehicle dynamics widely. The Magic Formula tire model has the characteristics of a great number of parameters and the high nonlinearity, so it is hard to identify parameters. Researchers generally use different intelligent optimization algorithms for parameter identification. However, in the process of parameter identification, with a few experimental data, parameter identification results generally have the low accuracy, while, in the case of a large number of experimental data, the amount of work done in the experiment will increase and there will be many experimental errors. To solve these problems, this paper researches the longitudinal force of tire and proposes an interpolation method and a method based on the nonlinear research of the tire force. The results of parameter identification experiments on the two kinds of tire data show that both of the two methods can be used for the parameter identification of Magic Formula tire model fast and accurately with only a few experimental data. In addition, this paper proposes a method estimating the maximum longitudinal force and corresponding slip rate.

#### 1. Introduction

Magic Formula tire model is able to well reflect the change rule of the tire force [1–4] and widely used; however, due to the nonlinear property and the required excessive parameters, there are many problems in the current researches on parameter identification of Magic Formula. van Oosten and Bakker [5] proposed that, in order to improve the fitting accuracy, 2000 and 1500 measured data points were needed, respectively, for pure slip the conditions and for combined slip conditions. Cabrera et al. [6, 7] and Ortiz et al. [8] proposed calculating the globally optimal solution based on differential evolution algorithm, but it is still likely to fall into the local minimum. Moreover, for the parameter identification of pure longitudinal force, the number of iterations for genetic algorithm is more than 500, and the elapsed time is 180 s. Wang et al. [9] proposed a new self-adaptive differential evolution algorithm (NSADE) and compared with other two differential evolution algorithms IOA and SSPDE; based on NSADE algorithm, the convergence to the global optimum is high in speed, the number of iterations is about 200, and the elapsed time is about 700 s. Zhang et al. [10] proposed a hybrid optimization method, based on which, firstly, the approximate optimal solution is solved according to the genetic algorithm, and then the accurate parameters are identified by the numerical optimization algorithm; the number of iterations for genetic algorithm is 10000; then the calculation results are well fit with the experimental data.

In conclusion, there are some problems in the parameter identification of Magic Formula. If the measured data points are fewer, then the result of the parameter identification is not unique, and there will be many values [11]; due to a small amount of data from the introduced algorithm, many curves may meet the requirements of the objective function; however, for the same tire in the same case, the characteristic curve of the tire force is unique; moreover, the experimental error cannot show and guarantee that several measured data are the highest in accuracy, and the obtained curve is the actual tire force curve. If the measured points are increased, more experimental errors will be introduced, and the workload will be greatly increased. In addition, the interaction of parameter increases the difficulty in algorithm identification; the researchers usually calculate , respectively, and substitute the values into Magic Formula, so the number of the iterations for the algorithm increases, and a great number of data need to be calculated for iteration each time.

In order to solve these problems, this paper proposes an interpolation method and a method based on the nonlinear research of the tire force. The former is based on 9 real experimental types of data under the same kind of load and uses the Lagrange, Hermite, and Spline interpolation to get more experimental data through simulation and then verifies the validity of experimental data obtained through simulation by calculating the errors which may be produced in simulated experimental data. In this way, we can get more experimental data with fewer experiments and then make parameter identification using the genetic algorithm. The method has the high accuracy and a very fast rate of convergence to the optimal solution. The latter, based on the nonlinear characteristic of Magic Formula tire model and the research on the changing laws for longitudinal force of tire, proposes two parameter identification models of and , respectively, and a method estimating the maximum longitudinal force. The results indicate that under the same kind of load, we can make a fast and accurate parameter identification with a few, sometimes only three, real experimental types of data and get a very accurate estimate of maximum longitudinal force. This paper offers some reference to parameter identifications in other fields.

#### 2. Parameter Identification of Magic Formula Tire Model Based on Interpolation Method

The basic Magic Formula of tire longitudinal force is as follows:where represents the theoretical longitudinal force of tire; represents the real longitudinal force of tire; represents the real slip rate; represents the theoretical slip rate; parameters represent the rigidity factor, the shape factor, the peak factor, the curvature factor, the horizontal displacement, and the vertical displacement, respectively. In addition, are related to the function with the vertical load as the independent variable, so there are secondary parameters [12]. If we can get primary parameters accurately and then we can get secondary parameters easily, and because there are various tire models based on Magic Formula, this paper does not research solving secondary parameters. and are small, so the research in this paper neglects their values.

The parameter identification method of tire model generally adopts the heuristic intelligent optimization algorithm, like the simulated annealing algorithm, the genetic algorithm, the particle swarm optimization algorithm, and so on. People generally use the coefficient of determination as the objective function of iteration of algorithm. The closer the value of to 1 is, the higher the accuracy of parameter identification is. The calculation formula of is as follows:where represents the predicted value of longitudinal force of tire; represents the measured value of longitudinal force of tire; represents the th slip rate; represents the total number of measured values.

The number of experimental data of tire force is the key to parameter identification of Magic Formula tire model. If the number of measured data points is limited, many curves may meet the requirements of objective function, and some curves making the objective function optimal may have strange shapes and thus make the results of parameter identification untrustworthy. Although the tire force has small errors in measurement points, it has considerable errors in some unmeasured points locally; that is, there may be many abnormal points. Substantial data points measured may increase the difficulty and time consumption of experiment and also introduce more experimental errors. With the interpolation method, we can obtain the high-accuracy simulation of experimental data and the proper interpolation method can reduce the error of interpolated data points and the errors of first and second derivatives and then ensure the accuracy of interpolation.

To further ensure the accuracy of interpolation, this paper uses the piecewise interpolation method and each segment is based on 4 measured data points. This is because, according to the changing laws [13], the changes of longitudinal force of tire are generally divided into the following three segments: the first segment approximates to a line; the second segment approximates to a parabola with the slope decreasing first and then increasing; the third segment approximates to a curve with the slope decreasing continuously. Therefore, the parameter identification based on the interpolation method only requires the measurements of 9 types of data under each vertical load.

This paper chooses the measured data of 2 kinds of tire for the research. The first kind of tire’s measured data comes from literature [14]; the second kind of tire’s measured data comes from ADAMS built-in mdi_pac_94 tire data. The numbers of measured data of two kinds of tire are both 101 under each vertical load. According to the previous analysis, we select 9 measured types of data under the same vertical load as the raw data for interpolation method. Figure 1 shows all measured data and selected 9 measured data types of the first kind of tire. Figure 2 shows all measured data of the second kind of tire.