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Vol. 43, Supplement, 2005, 253–266

Experience with the IMMa tyre test bench for the

determination of tyre model parameters using

genetic techniques

J. A. CABRERA*, A. ORTIZ, E. CARABIAS and A. SIMÓN

University of Málaga, Málaga, Spain

The study of tyre behaviour is an interesting area in vehicle dynamics research. There are a great number of researchers working in this area. So knowledge of the tyre properties is necessary to design vehicle components and advance control systems properly. For that purpose, mathematical models of the tyre are used in vehicle simulation models. Semiempirical models can be used for this. These kinds of tyre model use a certain number of parameters that define them. Semiempirical models need to fit measurement data to the tyre properties that they define. The determination of these sets of parameters and the measurement data obtained are 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 an optimal solution, with no need for an extensive knowledge of the searching space. So, to start the search, it is not necessary to know a set of starting values of the parameters that define the tyre model. A set of measurements that define several tyre properties are obtained on the IMMa tyre test bench. These measurement data are used to compare with results from the method proposed, and a new mutation procedure has been developed.

Keywords: Tyre model; Tyre model parameters; Genetic algorithms; Tyre test bench

1. Introduction

Tyre behaviour plays an important role in vehicle dynamics research. Knowledge of the tyre properties is necessary to design vehicle components and advanced control systems properly. For that purpose, mathematical models of the tyre are being used in vehicle simulation models. There are some semiempirical tyre models [1, 2] that describe tyre behaviour quite accurately because the experimental parameters of those semiempirical models are optimized using test data. Of course, these parameters are only useful for a pneumatic tyre tested for a full range of vertical loads.

Research groups are very interested in obtaining the proper values of parameters of these models, but they always have the same problems, which are as follows. What method of optimization should be employed? What are the initial values of the parameter to start the

*Corresponding author. Email: [email protected]

Vehicle System Dynamics

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optimization process? Which optimization method guarantees that you will not stop in a local minimum? Is it necessary to define valid ranges of variation for the values of the parameters in which the method performs the search?

All these questions prevent tyre researchers from planning the use of those semiempirical tyre models, because the cost to obtain the optimal parameters that make the model work properly is a great impediment, although they are easy to implement and apply to a vehicle dynamic simulation process.

The answers to these questions will be dealt with in this work. We use an optimization method [3] that does not require knowledge of the searching space, the initial values of the parameters to be optimized and the ranges of the values of the parameters. A new method based on genetic techniques has been developed by the IMMa group and has been used to determine these parameters. This new method is easy to implement and use and also solves all the previous disadvantages mentioned above; so it is a powerful tool for saving computational effort and time.

The semiempirical tyre models for which the method has been tested have parameters with different ranges, and these ranges depend on units of the test input data. Certain optimization techniques use a limited searching space, because the ranges of initial parameters are prede-fined and must be known [4]. Our method is not influenced by the range of initial parameters and it is able to find the optimum solution although proper starting values are not chosen. This feature has been tested with different parameter ranges and with different tyre characteristics. This is strongly influenced by the optimization techniques used and by the input variables chosen to run the algorithms. It is only necessary to choose the value of six variables in the algorithms:NP, initial population;F, disturbing factor [0, 2]; CP, crossover probability [0, 1]; MP, mutation probability [0, 1]; range, range of variation in the parameter values in the mutation process; itermax, maximum number of iterations of the algorithms. Obviously, this set of values is very easy to choose and not much previous experimental knowledge is necessary.

The optimization algorithm is based on the technique called differential evolution [5], but a new mutation procedure of the parameters to be optimized will be presented in this work. Mutation of a parameter is the random variation in its value. We have verified that this procedure is fundamental to obtaining the optimum when the parameter range values are very different. The mutation procedure developed will be described in this paper and we shall demonstrate the improvements obtained when we use this algorithm. The mutation procedure changes only some of these parameters, allowing us to find the correct optimum and not to stop in a local minimum.

The main advantages of the method are its simplicity of implementation and its fast conver-gence to an optimal solution, with no need for extensive knowledge of the searching space. So, to start the search, it is not necessary to know a set of starting values of the tyre model parameters.

A comparison between different sets of input variables for running the algorithms is discussed in this paper.

The data test used in the process has been obtained on the IMMa tyre test bench for different tyres and for pure manoeuvres. Hence the procedure to perform this work is the following.

(i) Describe the optimization algorithm (crossover, selection and mutation procedure). (ii) Obtain the test data using the IMMa tyre test bench.

(iii) Apply the previously described algorithms to different tyre models to obtain the longitudinal and lateral forces and the autoaligning torque.

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2. Optimization method

Genetic algorithms are different from more normal optimization and search procedures in four ways.

(i) Genetic algorithms work with a coding of the parameter set, and not the parameter themselves.

(ii) Genetic algorithms search with a population of points, and not with a single point. (iii) Genetic algorithms use evaluations of goal functions, and not derivatives or other auxiliary

knowledge.

(iv) Genetic algorithms use probabilistic transition rules, and not deterministic rules.

Altogether, these four differences contribute to a genetic algorithm’s robustness and turn out to be an advantage over other more commonly used techniques.

The optimization problem is given by

min [f (p1(X), p2(X), . . . , pn(X))]

subject to

gj(X)≤0, j =0,1,2, . . . , m,

xi ∈[lii, lsi] ∀xi ∈X,

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wherefis the goal function, where each individualXobtains a value, where its fitnessespi(.) are functions of the properties that show the objectives of the system to be optimized and wheregj(.)are the constraints defining the searching space.

The strategy of evolutionary methods for optimization problems begins with the genera-tion of a starting populagenera-tion. For the problem dealt with, the starting populagenera-tion is a set of parameters of the tyre models (i.e.the ‘magic formula’ or the Bayle et al.model), whose values are randomly generated within the searching space. Each individual (chromosome) of the population is a possible solution to the problem and it is formed by parameters (genes) that set the variables of the problem.

Genes can be schematized in several ways. In the first approach by Holland [6, 7], they are binary chains; so eachxigene is expressed by a binary code of sizen.Another way to express the genes, as is done in this work, is directly as real values. All genes are grouped in a vector that represents a chromosome [5, 8]:

X=[x1, x2, . . . , xn] ∀x∈ . (2)

Next the starting population has to evolve to populations where individuals are a better solution. This task can be reached by natural selection, reproduction, mutation or other genetic operators. In this work, selection and reproduction are carried out sequentially and mutation is used as an independent process.

2.1 Selection

For selection, two individuals are randomly chosen from the population and they form a couple for reproduction. The selection can be based on different probability distributions, such as a uniform distribution or a random selection from a population where the weight of each individual depends on its fitness, so that the best individual has the greatest probability of being chosen.

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In this paper, the best individual and two individuals randomly selected with uniform distri-butions are chosen for reproduction and they make up a disturbing vectorV. The scheme [5], known as differential evolution, yields

Xi:i∈[1, NP],

V=Xbest+F (Xr1−Xr2) ,

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whereXbestis the best individual of a population ofNPindividuals,Xr1andXr2are two indi-viduals randomly selected in the population andFis a real value that controls the disturbance of the best individual.

2.2 Reproduction

Next, for reproduction,Vis crossed with the individualiof the current population to generate the individualiof the next population. This operator is called crossover.

In natural reproduction, parents’genes are interchanged to form the genes of their descendant or descendants. As shown in figure 1, reproduction is approached by a discrete multipoint crossover that can be used to generateXN

i; parentXi provides its descendant with a set of genes randomly chosen from its entire chromosome and parentVprovides the rest.

If the new descendantXNi is better than its antecedentXi, it will replace the antecedent. OtherwiseXiis retained andXNi is rejected. Therefore the population neither increases nor decreases.

Crossover is carried out with a probability defined as CP∈[0, 1].

2.3 Mutation

A new mutation procedure of the parameters to be optimized is developed in this work. Mutation is an operator consisting of random change of a gene during reproduction. We have verified that this procedure is fundamental to obtaining the optimum when the parameter range values are very different. The mutation procedure changes only some of these parameters,

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Figure 2. Differential evolution without the mutation procedure.

allowing us to find the correct optimum and not to stop in a local minimum. This feature is shown in figure 2.

The whole procedure to obtain a new descendant is shown in figure 2. In this case, there are two different parameters (genes) and the optimum has very different values for these two parameters. For this reason, if the mutation procedure does not work properly, it is possible to drop in a local minimum. In figures 2 and 3 the differences between both strategies with and without mutation procedure are shown.

The way to obtain a new descendant of the next population without the mutation procedure is shown in figure 2. In this case theV–Xicouple generates theXNi descendant, but this new chromosome may not reach the global minimum because the absolute values of the genes that compose it are very different, and the selection plus reproduction operations by themselves are not able to make the new descendant overcome the valley of the local minimum.

With the mutation procedure it is possible to solve the problem explained before. The gen-eration of a new descendant using the mutation procedure is shown schematically in figure 3. Here, the value of one or several of the genes of theV–Xicouple is changed in a range defined by the user, when the reproduction is taking place. This fact yields a new descendant XN

i,

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which has a different fitness from theXNi descendant studied in the previous case. This allows the algorithm to look for individuals with better fitness in the next generation.

In this work, mutation is defined as follows: when gene xi mutates, the operator ran-domly chooses a value within the interval of real values (xi, xi±range), which is added

to or subtracted fromxi, depending on the direction of the mutation.

Mutation is carried out with a probability defined as MP∈[0, 1], much lower than CP. Once the genetic operators are described, the optimization algorithm will be explained.

The proposed algorithm starts with the random generation of a starting population withNP individuals. The process to generate a new population ofNPindividuals is the following: the best individualXbestand two random individualsXr1andXr2of the actual population are chosen and the disturbing vectorVis formed as stated in equation (3). Then the disturbing vectorVand theith individualXiare crossed and mutated with a crossover probability CP and a mutation probability MP respectively, yielding a candidateXNi for the subsequent population. The next step consists in evaluating the candidateXNi and theith individualXiin the goal function; the best of these will become theith individual in the new population. This process is repeated to obtain theNP individuals of the new population, in which the new best individualXNbest appears. New populations are generated until one of the two following conditions is satisfied: the number of iterations reaches the maximum number of iterations or the evaluation of the best ofNpopulations is lower than the minimum error.

To evaluate the goal function we need to obtain test data for different tyre characteristics. For this reason, in the next section, the IMMa tyre test bench [9] will be described and a set of test data for different tyre characteristic will be obtained.

3. Test data

The IMMa tyre test bench [9] has been used to obtain test data. The tyre bench basically consists of a high-stiffness structure (handling platform) and a flexible closed-loop steel belt between two drums that present a flat rolling surface to the tyre. The structure has the shape of a bridge with a three-legged support on a height control system and is formed by a central mechanism set to place the tyre on the steel belt and to control its orientation (slip and camber angles, and vertical displacement) and the force applied in the vertical direction. In fact, the mechanism set acts on a plate where a sensor measures forces and torques at the three Cartesian axes. Also, an axle box enables the tyre to turn freely. A commercial disc brake (disc, pads and caliper) is installed at the tyre axle box and this allows the tyre to stop when an appropriate braking force is introduced.

The lateral force of the tyre is introduced by pushing the support of the drums that is built in another high-stiffness structure and that is able to move along a guide.

The steel belt turns round two drums owing to the frictional force between the two surfaces; one of the drums is a driver and the other adjusts the lateral displacement of the belt to prevent it from running off the drum’s lateral boundary.

The traction force of the tyre is caused by an electrical motor that moves the driver drum. Owing to this, the bench is able to apply a set of variable forces to the tyre.

In this work, to perform the test on the proposed algorithm, we have made measurements on the tyre test bench of the pure longitudinal and lateral forces and the combined lateral force. The set of measured data used in this analysis is given for the following tyre type: INSA TURBO 185/60 R14. These sets of measurements are shown in table 1 (lateral forces at pure slip), table 2 (autoaligning torques at pure slip), table 3 (longitudinal forces at pure slip) and table 4 (lateral forces at combined slip and camber angles).

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Table 1. The lateral forceFyversus the slip angleαfor different vertical loads.

Fy(N) for the followingα

Fz(N) 0◦ 0.5◦ 1◦ 1.5◦ 2◦ 2.5◦ 3◦ 4◦ 6◦ 8◦ 10◦

1650 0 310 646 950 1129 1323 1464 1704 1982 2139 2060 2100 0 439 792 1215 1456 1653 1852 2101 2451 2630 2636 2500 0 463 845 1308 1624 1868 2081 2462 2859 2970 3011

Table 2. The autoaligning torqueMzversus the slip angleαfor different vertical loads.

Mz(N m) for the followingα

Fz(N) 0◦ 0.5◦ 1◦ 1.5◦ 2◦ 2.5◦ 3◦ 4◦ 6◦ 8◦ 10◦

1650 0 7.48 15.3 18.43 17.77 18.68 16.26 13.99 11.46 8.88 8.74 2100 0 13.13 22.22 27.15 28.61 27.59 26.62 24.38 19.15 14.04 12.69 2500 0 17.41 28.97 37.06 38.81 39.01 38.33 34.43 25.70 18.31 15.49

Table 3. The longitudinal forceFxversus the longitudinal slip ratio for different vertical loads.

Fx(N) for the followingκ

Fz(N) 0 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1

1100 0 900.5 1030 1070 1088 1097 1106 1088 1071 1060 1051 1650 0 645 1083 1332 1472 1555 1695 1715 1696 1676 1659 2000 0 1207 1858 1946 1969 1979 1990 1994 1994 1995 1995

Table 4. The lateral forceFyversus the slip angleαfor different camber anglesγ(Fz=2000 N).

γ(deg) −10 −8 −6 −4 −3 −2.5 −2 −1.5 −1 −0.5 0

−6 −2818 −2593 −2408 −2075 −1848 −1610 −1384 −1162 −772 −330 −180

0 −2226 −2180 −2117 −1913 −1691 −1586 −1304 −1008 −750 −427 −0 6 −2138 −2188 −2073 −1850 −1629 −1559 −1371 −1155 −848 −515 −116

γ(deg) 0.5 1 1.5 2 2.5 3 4 6 8 10

−6 −181 411 800 1022 1248 1487 1714 2047 2232 2457

0 427 750 1008 1304 1586 1691 1913 2117 2180 2226

6 282 616 922 1138 1326 1396 1617 1840 1955 1905

These sets of data will be used to perform a test on the proposed algorithm in the next section.

4. Results

The proposed algorithm is used to fit the parameters of the tyre model so that the tyre char-acteristics resemble the test data. Basically we have a tyre model that possesses certain tyre properties and this tyre model has several parameters that define it. The goal of the algorithm is to search for the best set of parameters that make the tyre model fit the measurement data more adequately; so a goal function has to be developed in every case.

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Normally the goal function evaluates the squared error between measurement data and tyre model data; so the objective of the searching algorithm is to try to reduce this error.

In this work, five cases are studied.

4.1 Case 1: pure lateral force

The problem is defined as follows.

(i) The tyre model used to define the pure lateral force has been proposed by Bakkeret al.[10]. This model needs nine parameters to define it without the camber angle having an influence and it is well known that they have different ranges and meanings. (ii) The optimization process of the parameters uses the test data in table 1. Only the

mea-surements that correspond to those obtained when the vertical load Fz=1650 N and

Fz=2100 N were used during the training process.

(iii) The parameters of the algorithm are as follows: number of iterations, 1000;F=0.6; CP=0.4;NP=100; MP=0.1; mutation range, 0.01.

(iv) The values of the initial population occur randomly in the range [0,1].

Once the problem has been defined, the algorithm is used to solve and search for the best parameters to fit the curves defined in table 1. In this case, the best parameters that were calculated with the vertical load and the measurement data on the lateral force in kilonewtons and the slip angle in degrees are given in table 5.

There is an important parameter in the previous problem, namely the mutation range. We have checked that, if this parameter is zero or the mutation procedure is not used in the optimization process, it is possible to drop in a minimum local; so this procedure is used when the values of the objective parameters are very different.

The results are shown in figure 4(b). Note how the best parameters obtained fit theFyvalues properly, although the algorithm has not been trained when the vertical load isFz=2500 N. The evolution of the goal function along the iterations is shown in figure 4(a). The final error is 0.045 650 kN2_.

4.2 Case 2: pure autoaligning torque

(i) The tyre model used to define the pure autoaligning torque has been proposed by Bakkeret al.[10]. This model needs nine parameters to define it without the camber angle having an influence and it is well known that they have different ranges and meanings. (ii) The optimization process of the parameters uses the test data in table 2. Only the

mea-surements that correspond to those obtained when the vertical load Fz=1650 N and

(iii) The parameters of the algorithm are as follows: number of iterations, 1000;F=0.6; CP=0.4;NP=100; MP=0.1; mutation range, 0.000 625.

Table 5. Parameters of the tyre model for the lateral forceFycharacteristic.

a0=C a1 a2 a3 a4 a5 a6 a7 a8

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Figure 4. (a) Sum-squared error evolution during the training of the pure lateral force; (b) the lateral force versus the slip angle at three vertical loads.

Once the problem has been defined, the algorithm is used to solve and search for the best parameters to fit the curves defined in table 2. In this case, the best parameters that were calculated with the vertical load in kilonewtons and the measurement data on the autoaligning torque in newton metres and the slip angle in degrees are given in table 6.

The results are shown in figure 5(b). Note how the best parameters obtained fit theMzvalues properly, although the algorithm has not been trained when the vertical load isFz=2500 N. The evolution of the goal function along the iterations is shown in figure 5(a). The final error is 13.60 N2_m2_.

Table 6. Parameters of the tyre model for the autoaligning torqueMzcharacteristic.

a0=C a1 a2 a3 a4 a5 a6 a7 a8

1.965 5.0734 2.841 −0.006 71 11.3267 −0.085 12 −0.241 74 0.709 54 −0.341 33

Figure 5. (a) Sum-squared error evolution during the training of the pure autoaligning torque; (b) the autoaligning torque versus the slip angle for three vertical loads.

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4.3 Case 3: pure longitudinal force

(i) The tyre model used to define the pure longitudinal force has been proposed by Pacejka and Besselink [11]. This model needs ten parameters to define it without the camber angle having an influence (pHx1,pHx2,pVx1andpVx2are neglected) and it is well known that they have different ranges and meanings.

(ii) The optimization process of the parameters uses the test data in table 3. Only the mea-surements that correspond to those obtained when the vertical load Fz=1100 N and

(iii) Parameters of the algorithm are as follows: number of iterations, 1000; F=0.6; CP=0.4;NP=100; MP=0.1; mutation range, 0.001.

Once the problem has been defined, the algorithm is used to solve and search for the best parameters to fit the curves defined in table 3. In this case, the best parameters that were calculated with the vertical load and the measurement data on the longitudinal force in Newtons and the longitudinal slip ratio in percentages per unit are given in table 7.

The results are shown in figure 6(b). Note how the best parameters obtained fit the

Fx values properly, although the algorithm has not been trained when the vertical load is

Fz=1650 N.

The evolution of the goal function along the iterations is shown in figure 6(a). The final error is 6537.9386 N2_.

Table 7. Parameters of the tyre model for the longitudinal forceFxcharacteristic (the Pacejka–Besselink model).

pCx1 pDx1 pDx2 pKx1 pKx2 pKx3 pEx1 pEx2 pEx3 pEx4

0.888 1.0421 0.046 14 6.8139 −1.088 1.6444 −10.413 12.483 0.9556 0.010 56

Figure 6. (a) Sum-squared error evolution during the training of the pure longitudinal force; (b) the longitudinal force versus the longitudinal slip ratio for three vertical loads (the Pacejka–Besselink model).

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4.4 Case 4: pure longitudinal force

(i) The tyre model used to define the longitudinal force has been proposed by Burckhardt [12]. This model needs three parameters to define it without the camber angle having an influence.

(ii) The optimization process of the parameters uses the test data in table 3. Only the mea-surements that correspond to those obtained when the vertical load Fz=1100 N and

(iii) The parameters of the algorithm are as follows: number of iterations, 100; F=0.6; CP=0.4;NP=100; MP=0.1; mutation range, 1.

Once the problem has been defined, the algorithm is used to solve and search for the best parameters to fit the curves defined in table 3. In this case, the best parameters that were calculated with the vertical load and the measurement data on the longitudinal force in Newtons and the longitudinal slip ratio in percentages per unit are given in table 8.

The results are shown in figure 7(b). Note how the best parameters obtained fit theFxvalues properly, although the algorithm has not been trained when the vertical load isFz=1650 N. The evolution of the goal function along the iterations is shown in figure 7(a). The final error is 55 913.5 N2_.

Table 8. Parameters of the tyre model for the longitudinal forceFxcharacteristic (the Burckhardt

model).

C1 C2 C3

1.0053 55.8164 0.020 07

Figure 7. (a) Sum-squared error evolution during the training of the pure longitudinal force; (b) the longitudinal force versus the longitudinal slip ratio for three vertical loads (the Burckhardt model).

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4.5 Case 5: lateral force due to combined slip and camber angles

(i) The tyre model used to define the combined lateral force has been proposed by Pacejka and Besselink [11]. This model needs 18 parameters to define it and it is well known that they have different ranges and meanings.

(ii) The optimization process of the parameters uses the test data in table 4. Only the mea-surements that correspond to those obtained when the camber angle isγ =0◦andγ =6◦ and the vertical loadFz=2875 N were used during the training process.

(iii) The parameters of the algorithm are as follows: number of iterations, 1000;F=0.6; CP=0.4;NP=100; MP=0.1; mutation range, 0.001.

Once the problem has been defined, the algorithm is used to solve and search for the best parameters to fit the curves defined in table 4. In this case, the best parameters that were calculated with the vertical load and the measurement data on the lateral force in Newtons and the slip and camber angles in degrees are given in table 9.

The results are shown in figure 8(b). Note how the best parameters obtained fit theFyvalues properly, although the algorithm has not been trained when the camber angle isγ = −6◦.

The evolution of the goal function along the iterations is shown in figure 8(a). The final error is 139 626.083 N2_.

Table 9. Parameters of the tyre model for the longitudinal forceFycharacteristic

(the Pacejka–Besselink model).

pCy1 pDy1 pDy2 pDy3 pKy1 pKy2 pKy3 pEy1 pEy2

0.6659 1.7198 0.7435 0.0018 0.5404 0.1094 −0.0525 0.2426 0.5314

pEy3 pEy4 pHy1 pHy2 pHy3 pVy1 pVy2 pVy3 pVy4

1.838 −0.1222 0.2117 0.3668 0.0012 0.2196 0.5659 0.3016 0.7758

Figure 8. (a) Sum-squared error evolution during the training of the combined lateral force; (b) the lateral force versus the slip angle for three camber angles with a vertical loadFz=2875 N.

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5. Discussion and conclusions

The first two solved cases, cases 1 and 2, deal with the pure lateral force and the autoaligning torque respectively using the ‘magic formula’ published by Bakkeret al.[10]. These two cases employ a small number of parameters to define the model. These first two cases show the goal function evolution with iterations (see figure 4(a) and figure 5(a)), meeting an optimum with little error. It was verified that, using the Simplex search algorithm given by Nelder and Mead [13], if initial values of the parameters were far from the optimum, the final error was larger than the error made before and, if the initial values were near the optimum, the final error was of the same magnitude as the error made by the proposed method but continued to be larger; so the disadvantage of this analytical optimization method is finding a suitable initial starting condition. However, the proposed method here does not need to know the physics of the parameters to be optimized; therefore the initial values of the parameters are irrelevant. For instance, in this paper the values are in the range [0,1].

Cases 3 and 4 deal with the pure longitudinal force. Case 3 uses the ‘magic formula’ model with ten parameters and it was published by Pacejka and Besselink [11] and case 4 uses the Burckhardt longitudinal model with only three parameters and it was published by Burckhardt [12]. In this case, the optimum set of parameters is reached with 50 iterations as against the 500 iterations that case 3 needs. However, the Burckhardt model is not able to reproduce traction situations; so the ‘magic formula’ model shows a wider range of use, but with a higher computational cost than the Burckhardt model.

Case 5 deals with the combined lateral force with 18 parameters to be optimized and the methodology proposed here also works properly, but some differences are observed. The first is that the optimum is reached at a higher number of iterations than in the previous cases. Another is that the mutation range is the key to obtaining an optimum set of parameters. In this case a set of mutation ranges were tested and the result was 0.001. This result makes the algorithm fit the test data more closely.

In this paper a set of measurements obtained on the IMMa tyre test bench have been used to compare how the algorithm based on differential evolution works with different tyre models and with different numbers of optimized parameters.

All the cases studied were fitted to test data in a suitable way. The simplicity of the algorithm implementation and the option of starting the optimization process with any initial values for the parameters were the main advantages of the proposed method; so the experience required for the method users is reduced significantly and substantially.

One large area of study to improve this algorithm is to develop a procedure to modify the mutation range. This would allow the optimum to be reached during the running time using the information obtained in the previous iteration. The same principle can be used to modify the disturbing factorF.

References

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[13] Nelder, J.A. and Mead, R., 1965, A simplex method for function minimization.Computer Journal,7, 308–313. [14] Cabrera, J.A., Ortiz, A., Carabias, E. and Simon, A., 2004, An alternative method to determine the magic tyre