A comparison of various algorithms to extract Magic ...

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

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A. Vijay Alagappan , K.V. Narasimha Rao & R. Krishna Kumar a

Department of Engineering Design, Indian Institute of Technology Madras, Chennai, India

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RPS CoE for Tyre and Vehicle Mechanics, J.K. Tyre and Industries Ltd., IIT Madras, Chennai, India Published online: 06 Dec 2014.

To cite this article: A. Vijay Alagappan, K.V. Narasimha Rao & R. Krishna Kumar (2015) A comparison of various algorithms to extract Magic Formula tyre model coefficients for vehicle dynamics simulations, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 53:2, 154-178, DOI: 10.1080/00423114.2014.984727 To link to this article: http://dx.doi.org/10.1080/00423114.2014.984727

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Vehicle System Dynamics, 2015 Vol. 53, No. 2, 154–178, http://dx.doi.org/10.1080/00423114.2014.984727


A comparison of various algorithms to extract Magic Formula tyre model coefficients for vehicle dynamics simulations A. Vijay Alagappana , K.V. Narasimha Raob and R. Krishna Kumara∗ of Engineering Design, Indian Institute of Technology Madras, Chennai, India; b RPS CoE for Tyre and Vehicle Mechanics, J.K. Tyre and Industries Ltd., IIT Madras, Chennai, India

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(Received 29 March 2014; accepted 2 November 2014 ) 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. Keywords: tyre models; Magic Formula; fitting; optimisation; trust region algorithms; Nelder–Mead simplex; pattern search; differential evolution; particle swarm optimisation; cuckoo search

Introduction Virtual prototyping has become indispensable in the automotive industry due to the benefits it offers in terms of reduced cost and development cycle times. These can be used to evaluate various performance measures related to vehicle dynamics such as braking, traction, handling, ride comfort, etc. Tyre models are an integral part of such vehicle simulations since the tyre road interface is one of the primary sources of all dynamic forces, the other being the aerodynamic forces. Modelling the dynamics of a vehicle requires a tyre model to translate the forces and moments generated at the tyre–road interface to the wheel hub. Numerous tyre models have been proposed in the past by researchers, which are broadly categorised as semi-empirical, simple physical and complex models, based on the modelling technique. The Magic Formula tyre model proposed by Bakker, Nyborg and Pacejka is a semi-empirical *Corresponding author. Email: [email protected] c 2014 Taylor & Francis


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model which is one of the most widely used tyre models in vehicle dynamics simulations.[1– 3] This model is unique in nature because of its ability to model accurately the tyre force and moment data. Some of the parameters have a physical significance. The parameters of this model can be used as indicators of tyre force and moment characteristics and used in the tyre design and development process. Rao et al. [4] have studied the relationship between tyre design attributes and Magic Formula coefficients using finite element analysis instead of experiments to find out which design attributes can improve handling characteristics. While finite element analysis is one of the options for acquiring force and moment data of a tyre, generally the data are obtained from experiments conducted on track with a test trailer or in laboratory with a force and moment testing machine. Olsson et al. [5] explain the differences in the results between these two types of methods. Some of these testing procedures have been standardised. Van Oosten et al. [6] describe a standard tyre interface and the TIME project aiming at a standard tyre testing procedure that is reliable and consistent with realistic driving conditions. Hüsemann and Wöhrmann [7] present a review of required tyre measurements for different tyre simulation models. They discuss the impact of the difference in tyre measurement results caused by using different test rigs and the effect of different measurement procedures on the tyre modelling performance and vehicle dynamics output. The Magic Formula model has gained wide acceptance for representing tyre force and moment data in both the academia and industry. There have been parallel developments in the fitting procedure for the determination of the coefficients. Since the model is nonlinear in nature and involves a wide range of parameters, the fitting procedure needs knowledge of the search space, physical significance of the coefficients and numerical optimisation techniques for least-squares minimisation. Van Oosten and Bakker [8] have used the NAG subroutine E04FDF for least-squares minimisation in the fitting process and also mentioned some of the demands that the measurement data need to fulfil in order to get a good fit. For example, they suggest that the peak of each tyre force characteristic should be covered in case the peak factor C is not fixed. Schuring et al. [9] have presented the BNPS (Bakker, Nyborg, Pacejka and Smithers Scientific Services Inc.) model, which is an automated implementation of the Magic Formula tyre model. The effect of the primary parameters such as C and E on the shape of the curve has been explored and reported in their work. The fitting process requires guesses for the initial values and the results are dependent on the starting points of the minimisation algorithm. Guessing initial values requires some experience in the fitting process and some knowledge of the search space. Hopkins [10] used a systematic procedure for calculating initial guesses from the curve features for the lateral force case. This is not possible for combined slip cases where the parameters apparently do not have any physical basis. Cabrera et al. [11] suggest using the differential evolution algorithm (a variant of genetic algorithm) to minimise the sum of square of errors (SSEs). This technique has the advantage of simplicity of implementation and also does not require initial values as it starts with random values between 0 and 1. Since it uses a global optimisation technique, there is also a lower probability of getting trapped in local minima. In further developments, they proposed a new mutation scheme,[12] compared their algorithm with a starting values optimisation technique used by TNO Automotive [13] and modified it so that there is no need for a user to enter any of the algorithm control parameters.[14] Palkovics and El-Gindy [15] propose fitting tyre force and moment data with neural networks consisting of multi-level perceptrons using backpropagation, while Boada et al. [16] propose the same using radial basis functions, but these techniques do not use Magic Formula. Though all the above techniques can be used for fitting the tyre force and moment data, each method has specific advantages and disadvantages. Though these techniques minimise the SSEs between the predicted and measured values, none of them guarantees physically


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meaningful solutions. This is because parameters can take multiple sets of values, that is, non-unique values, to produce the same curve. For example, if the peak of the curves is not reached, there is a lot of interplay between the primary coefficients C, D and E. Similarly, absence of camber data or insufficient number of loads and camber can also result in such problems. Each technique predicts different coefficients since each algorithm may find a different minimum. There is also the possibility of a difference between the coefficients predicted by the constrained and unconstrained optimisation. These differences could be due to multiple minima with some falling outside the bounds and others inside. In this paper, such problems related to fitting have been highlighted. Ultimately, the fitted parameters are required for modelling the tyre in vehicle dynamics simulations. The need of the hour is to find techniques that can make this process of identifying parameters simpler, faster and also give accurate results. This can only be done by comparing the performance of different algorithms and also the conditions under which these work. This work has been done and described in this paper. A few sets of test data have been fitted and the different algorithms for fitting have been compared. It is hoped that this work can serve as a guideline for choosing minimisation algorithms for fitting Magic Formula parameters and also enable further research in this direction.

Curve fitting methods and procedure – a review The Magic Formula tyre model The Magic Formula tyre model gives a relationship between the slip quantities and the forces at different normal loads and inclination angles. The slip quantities are the slip ratio in the case of longitudinal force and slip angles in the case of lateral force and self-aligning torque. In this paper, the 2004 version (commercially known as Magic Formula 6.0) is used. The basic form of the Magic Formula is given as follows [17]: y(x) = D sin[C arctan{Bx − E(Bx − arctan(Bx))}], Y (X ) = y(x) + SV , x = X + SH ,

(1) (2) (3)

where Y is the longitudinal force, lateral force or self-aligning torque and X is the slip angle α or the slip ratio κ. The coefficients B, C, D, E, S H and S V are known as the stiffness factor, shape factor, peak value, curvature factor, horizontal shift and vertical shift, respectively. These are the primary coefficients and they are expressed in terms of the secondary coefficients using a set of formulae. The secondary coefficients are used to express the variation of each of the primary coefficients with respect to the load, load squared, camber, camber squared, etc. In the pure slip cases, there are totally 14 coefficients for the longitudinal force, 22 coefficients for the lateral force and 27 coefficients for the self-aligning torque [17]. The process of fitting or the estimation of parameters, which describe the Magic Formula, depends on many factors. They include the number of independent and useful points, intervals at which the measurements are taken, range of independent variables covered, noise in the data, constraints in the range of parameters and so on. Least square technique is the most commonly used technique for curve fitting, the other approaches being maximum likelihood, minimum absolute deviation and method of moments. In the few papers that have been published related to fitting of Magic Formula,[8,11–14] the least square minimisation has been used. In this technique, the SSEs


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(given by the following expression) is minimised to obtain the best fit: n

{FMF (xi ) − Fmeasured (xi )}2 ,

(4)


i=1

where F MF is the force value calculated using the predicted Magic Formula coefficients and F measured is the measured force value for each data point. The objective function here is nonlinear in nature and the problem can be solved using minimisation procedures that can find the lowest sum of squares in an m parameter dimensional search space. Nonlinear least-squares minimisation can be performed using either unconstrained or constrained minimisation techniques. Constraints can be simple bounds for each of the parameter or certain conditions such as C x > 0, Dx > 0, K xκ = cornering stiffness (slope at origin), Ex ≤ 1, etc. as mentioned in [17]. The minimisation can be carried out using either gradient-based algorithms or derivativefree algorithms. While gradient-based algorithms can only be used when information about the slope or derivatives is available, the derivative-free algorithms need only the function values. The gradient-based algorithms are suitable only for problems with smooth objective function and constraints, while the derivative-free algorithms can handle noise and are robust. Many of the derivative-free algorithms are better at finding the global minimum than the gradient-based ones (Nelder–Mead simplex is an exception; it is not a global optimisation algorithm) and usually do not require start points unlike the latter (both Nelder–Mead simplex and pattern search are exceptions, they require an initial guess). Algorithms used for minimising the SSEs In this work, different numerical optimisation algorithms have been used to minimise the SSEs. The algorithm codes used here are from MATLAB’s Optimization Toolbox or modified and customised versions of the codes posted by the developers who proposed or implemented the algorithms.[18–21] Many algorithms were tried out, and finally only six of them have been used to solve the minimisation problem since the other algorithms were too slow, had complicated parameters to be chosen or performed poorly. Table 1 summarises important aspects of the algorithms used here. Since the performance and methodology of most gradient-based algorithms was found to be similar, only one among them – the trust region reflective algorithm, was considered. The rest of the algorithms used in this work are all derivative-free minimisation algorithms. All the methods, except trust region reflective and Nelder–Mead simplex, are global optimisation techniques that can search for the global solution, compared to the previous ones which converge to a minimum which may not be a global minimum. The algorithms used here are easy to implement and understand and also give good fits in a reasonable amount of time. These algorithms have been used in this work to fit tyre force and moment data and have been compared for their performance. Procedure In this study, data sets for pure slip cases have been fitted to the Magic Formula 2004 version, its equivalent commercial version being called MF 6.0. This version was selected since it had significantly more number of parameters compared to the previous version MF 5.2. Version 6.1 had some more parameters, but took into account also the forces at different inflation pressure that required significantly more amount of data without a corresponding increase in

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the number of parameters. So, it was decided to stick with the MF 6.0 version. Three sets of data – finite element simulation, experimental and mathematically constructed data, have been used for fitting the lateral force and self-aligning torque curves. Six different algorithms,

Table 1. Algorithms used for minimisation to fit the Magic Formula model in this paper.


S.No. 1

2

3

4

Algorithm

Description

Trust region reflective A gradient-based optimisation technique (TRR) in which the region around the start point is approximated by a quadratic function and the size of this region is altered based on the quality of the approximation.[22] Using wellestablished minimisation techniques for quadratic functions, the minimum in the region is found and this is taken as the next start point for subsequent iterations. The system of equations cannot be underdetermined in this case Nelder–Mead simplex Solves unconstrained problems without (NMS) using any derivative information using an extension of the simplex method proposed by Nelder and Mead.[23] It can often handle discontinuity, particularly if it does not occur near the solution. But, it may only give local solutions and may also converge to a non-stationary point Pattern search (PTS) This is a direct search technique to find a set of points near the current point that have objective function values lesser than that of the current point. This is multiplied by a vector to generate a pattern of iterates with non-increasing objective function values. The solution is ultimately found using searching and polling methods.[24] It does not require gradient information at all

Differential evolution (DE)

Control parameters/options/settings (a) Start point (b) User supplied Jacobian or step factors for approximation using finite differences, finite difference type – central, forward or backward, preconjugate gradient parameters, etc. (c) Tolerances and stopping criteria: termination tolerance on function value and parameters, maximum number of function evaluations and iterations (a) Start point (b) Tolerances and stopping criteria: termination tolerance on function value and parameters, maximum number of function evaluations and iterations (c) No other extra parameters or options

(a) Start point (b) Poll options – poll method (three different algorithms – GPS, GGS and MADS each having a basis 2N or N + 1), complete poll, polling order; search options – complete search, search method; mesh options – initial size, max. size, accelerator, rotate, scale, expansion factor, contraction factor; constraint parameters (c) Tolerances and stopping criteria: termination tolerance on mesh, function value, parameters and constraints, max. number of function evaluations, iterations and time limit An evolutionary optimisation technique (a) Range of starting values or seed that solves the optimisation problem values (these can also be used as based on the natural processes of bounds if needed) selection, combination and mutation (b) Number of population members, considering the vector differences DE step size and crossover probbetween members of the population. ability constant It is a variant of the genetic algorithm (c) Strategy or version of DE proposed by Storn and Price [25] (d) Tolerances and stopping criteria: and has been used for solving many function value to reach, maxinumerical optimisation problems. This mum number of iterations (gentechnique has been used by Cabrera et erations) al. and reported as successfully being able to fit the force and moment data with the Magic Formula [11] (continued).


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Table 1. Continued.


S.No.

Algorithm

Description

5

Particle swarm This is a swarm intelligence optimisation (PSO) technique introduced by Kennedy and Eberhart [26] where candidate solutions move around the search space to locate the global minimum. Each candidate solution is an agent that has its own velocity and position which are altered in subsequent iterations by the best solution using acceleration constants and inertia weights

6

Cuckoo search (CS)

Control parameters/options/settings

(a) Bounds (compulsory) and seed values (optional) (b) Population size, acceleration constants for local and global best influence, initial and final inertia weights, epochs (iterations) to reach final inertia weight (c) Type of PSO (d) Tolerances and stopping criteria: function value to reach or the error goal, maximum number of iterations or epochs, minimum error gradient and number of iterations to wait for termination if SSE does not change This algorithm inspired by the cuckoo (a) Range of starting values (these species’s brood parasitism was can also be used as bounds if developed by Yang and Deb [27] needed) for solving numerical optimisation (b) Number of nests, discovery rate problems. It detects and rejects of alien eggs/solutions, options bad solutions analogous to a host for Levy flight parameters (but bird spotting and throwing away the author says only the first two cuckoo’s eggs from its nest. It parameters need to be tuned) uses Levy flights instead of simple (c) Tolerances and stopping criteria: random walks to change the maximum number of iterations, solutions in each iteration to reach value to reach or the error goal a better solution. This algorithm has been successful in giving better solutions to spring design and welded beam problems than existing solutions in the literature

which have been highlighted in Table 1, were used for minimising the SSEs and fitting the data sets. The following sections describe the algorithm parameters, the initial guesses and bounds, the input data used for fitting and the way of evaluation of results. Algorithm parameters The performance of all the algorithms depends on the algorithmic parameters and other settings. These include tolerances and stopping criteria, parameters such as population, rate of change of the candidate solutions, etc. In order to preserve uniformity, all the data sets have been fitted with the same set of algorithm control parameters. Hence, algorithms that perform well for all type of input data sets, without requiring modification of the algorithm control parameters, can be identified. A good algorithm should be able to give solutions for all cases without user intervention and tweaking of algorithm control parameters. Table 2 summarises the parameters and other settings for each of the minimisation algorithms used here: Initial values and bounds In order to solve the fitting problem, all the algorithms either require starting values and/or bounds. Some of the ways using which the initial values and bounds were set are given below.

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Table 2. Algorithm parameters used for fitting [18,19,20,21]. Algorithm (abbreviation) Trust-region reflective (TRR)


Nelder–Mead simplex (NMS)

Pattern search (PTS)

Differential evolution (DE)

Particle swarm optimisation (PSO)

Cuckoo search (CS)

Important parameters and settings Both finite and infinite bounds Uses finite difference approximation for gradients Maximum number of function evaluations = 1e6 Maximum number of iterations = 1e6 Termination tolerance on function value = 1e − 6 Termination tolerance on parameters = 1e − 6 Only finite bounds Maximum number of function evaluations = 1e6 Maximum number of iterations = 1e6 Termination tolerance on function value = 1e − 4 Termination tolerance on parameters = 1e − 4 Both finite and infinite bounds Mesh contraction factor = 0.5 Mesh expansion factor = 2.0 Maximum number of function evaluations = 1e6 Maximum number of iterations = 1e6 Termination tolerance on function value = 1e − 6 Termination tolerance on parameters = 1e − 6 Mesh tolerance = 1e − 6 Both finite and infinite bounds Range of starting values = [ − 1,1] Number of population members = 100 DE step size = 0.6 Crossover probability constant = 0.4 Value to reach = 0.1 Maximum number of iterations = 1e4 Strategy or type of DE = DE/local-to-best/1 Only finite bounds Range of starting values = Bounds Maximum number of iterations (epochs) to train = 1e5 Population size = 60 Acceleration Constant 1 (local best influence = 2) Acceleration Constant 2 (global best influence = 2) Initial inertia weight = 0.9 Final inertia weight = 0.4 Epoch when inertial weight at final value = 25000 Minimum global error gradient = 1e − 25 Epochs before error gradient criterion terminates run = 2500 Error goal = 0.1 Type of PSO = Common PSO with inertia Both finite and infinite bounds Number of nests = 75 Discovery rate of alien eggs/solutions = 0.25 Value to reach = 0.1 Maximum number of iterations = 1e4

Extraction of curve features. Most of the initial values or starting points can be set by extracting features of the curves. The peaks are close to the value of the primary parameter D, the slope at the origin to the value of K, and the difference of the coordinates of the peak to the S H and S V values. In this way, the initial values for the primary parameters can be guessed. The guesses for secondary parameters can be obtained by solving a set of equations formed by substituting the primary guesses for each curve into the expressions relating the primary and the corresponding secondary parameters. For example, let us take the case of the primary parameter Dy . This primary parameter changes with respect to the load and camber angles and is therefore given by an expression relating it to these dependent variables with three secondary parameters – pDy1 , pDy2, and pDy3. If there are five curves for five different



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normal loads at 0 camber angle, then one can get an initial value guess for each of the 5 Dy s and thus five different equations. These five equations in two variables pDy1 and pDy2 (the third variable pDy3 is dependent only on camber which is zero) can then be solved to get the estimates for the secondary parameters. Similarly, once pDy1 and pDy2 are estimated, pDy3 can be solved for using the equations for Dy with curves at different camber angles. In this manner, each of the secondary parameter initial values can be guessed from the primary parameter initial values using the equations relating the primary to the secondary parameters. This approach has been described in detail by Hopkins [10] who used it for setting initial values for the lateral force case. Some of the bounds can also be set using this approach. In the case of lateral and longitudinal forces, this approach can be used. On the other hand, for selfaligning torque and combined slip, the parameters are given by composite nonlinear functions obtained by multiplying and summing two or more of the basic nonlinear expressions. The extraction in this case becomes difficult. In order to guess C and E, rather than using fixed values (e.g. − 1.3 as C for lateral case as given by Bakker et al. [1]), there are a few formulae given by Pacejka based on the difference in the height of the peak and asymptotes.[17] The use of physical meaning. There are some values that have a physical meaning such as the longitudinal friction pDx1, variation of friction with load pDx2 , etc. In such cases, one can make use of that knowledge to set bounds and initial guesses. For example, friction on most road surfaces is between 0.7 and 1.3, while on wet road and ice it may be lower, whereas it may reach up to 2.0 for high-performance motorcycle and racing tyres.[17] So, a range of 0–2.5 can safely be set as lower and upper bounds. Also, friction usually does not drop by more than 10–20% in most cases. So, initial guesses and bounds for this parameter can be set safely as − 0.1 and ( − 0.5, 0), respectively. Study of shape – size sensitivity. The sensitivity of the parameters can be studied by varying each of the parameters on an existing model or mathematically fabricated model. For example, by studying the sensitivity of the horizontal shift parameters pHy1 , we understand that using a small value such as 0.1 for pHy1, one can shift the curve horizontally by 0.1 radians or 5.73°, which seems improbable from past observations of lateral force curves. Hence, the values can be safely expected to be below 0.1. Study of the SSE plots. While the previous techniques focus mainly on the model parameters, this can give an idea about the actual minimisation problem and the search space. While it is impossible to visualise plots of the SSEs with respect to all the variation of all the parameters (since the number of dimensions is usually too high to represent in a plot), it is possible to plot the variation of SSE with a range of one or two parameters. These plots are obtained by fixing all parameters at some feasible values, except one or two parameters that need to be studied. But, selecting the parameters to be studied and region for plots has to be done by trial and error. Most of the initial values and bounds have been obtained by the above-mentioned procedure. When none of the methods could clearly indicate the range, they are obtained by trial and error. Though the software can dynamically calculate the initial guess for each specific data set by using the features of the curve, fixed values for initial guesses have been used in this case. It must be noted that these values are used only by algorithms that require start points or bounds. The other algorithms start from random points in the range [ − 1, 1] or in the range within the bounds. While the bounded versions of the algorithms (marked with B in brackets) have parameters that are bounded by finite values, the unbounded algorithms (marked with UB in brackets) have ( − ∞, ∞) as the bounds for the parameters.

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Input data for fitting In this work, three types of data (classified based on their source) have been used as input data set for the fitting procedure – finite element analysis data, experimental data and mathematically constructed data. The reason for using all these three types of data is that each of these types has distinct characteristics. For example, mathematically constructed data are perfect, devoid of any noise and can give zero error when fitted. This data type could serve as a basic test for any algorithm. If an algorithm cannot fit such smooth, noiseless data, it most probably cannot fit any other data. The second input data type – finite element data, also has almost zero noise, but follows closely the experimental behaviour than the mathematically constructed data since it is based on the tyre mechanical properties and physics. Hence, one may not get zero error since the obtained curve shape could be possibly slightly different from the curve given by the Magic Formula. Also, tight design deadlines mean that the ability to extract Magic Formula parameters early in the design analysis stage is important. This requires that the fitting method be evaluated for finite element data too. The third type of data – experimental data, always has noise in addition to the curve behaviour possibly being slightly different from a curve given by the formula. This type of data is the most realistic data. Thus, there are three different cases ranging from idealistic zero error data to noisy real world data. Hence, the algorithms must be able to handle all these cases equally well. All the input data sets discussed here are available only for the pure slip cases. The combined slip cases are not discussed. All these characteristics of the input data types apply equally well to longitudinal, lateral, self-aligning torque and combined-slip cases. The characteristics of each of these input data types and the source for the corresponding input data sets is explained below. Finite element analysis data. The data obtained from finite element analysis is generally smooth and devoid of noise unlike experimental data. The curves appear smoother if the number of data points is more. As the force and moment characteristics in such cases are obtained from finite element analysis, the algorithms must be tested on such data. Since the primary focus is on longitudinal force characteristics, data from the steady-state brakingtraction finite element analysis of a 205/60R16 tyre have been used. The slip ratio has been varied from approximately − 0.6 to 0.6 for each of the five load cases at 60%, 80%, 100%, 120% and 140% of the nominal load (4210 N). Since the slip ratio at which the force is calculated depends on the convergence of the analysis, neither are all the points equally spaced nor do all the loads have the same number of points. Experimental data with slip sweep. This is the most common type of input data for the determination of Magic Formula coefficients of a tyre. The development of the Magic Formula and other empirical tyre models was itself based on this type of data obtained from tyre steady-state force and moment testing machines. These data consist of the forces obtained at different loads and camber angles by sweeping through a range of slip angles or slip ratios depending on whether the force is longitudinal or lateral. As it is experimental, the data are noisy. But the data can be spaced evenly and all loads can have the same number of points because the slip ratio or angles at which the readings are taken are controllable. Since, the lateral force and self-aligning torque are among the most important tyre force and moment characteristics affecting cornering, and as the longitudinal force has already been illustrated with finite element analysis data, the lateral force and self-aligning torque have been discussed using this data type. For these input data sets, a total of 85 points (17 slip angles at 5 different loads) have been used. The load at the centre data set (4196 N) has been taken as the nominal load in each of these cases. These data have been obtained from experiments


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conducted on a 185/70R14 using TMSI’s OnLEVELTM Tire Test System at JK Tyre & Industries Ltd. The effect of the camber angle has not been considered and the camber angle has been set as 0o . As a result, only the non-camber-related coefficients have been extracted in this case. Mathematically constructed data. This data set has been constructed purely using the mathematical formulae in the Magic Formula tyre model. The coefficients of a 205/60R15 tyre, given in [17], were used to construct the lateral force vs. slip angle curves at different loads. The nominal load for this tyre is 4000 N. This data set has the same slip angle range, load range and number of points as the data set from slip angle sweep experiments mentioned above. The points in the curves are used as input to get back the coefficients. The purpose of using such a mathematically constructed data was to evaluate the performance of the algorithms when the data are purely mathematical without any noise unlike in the above case. As a result, zero error after fitting is feasible and this case can be used to verify if the original set of coefficients is obtained from the fitting process. Hence, these data can serve as a basic test for minimisation algorithms used to fit tyre force and moment data. The data sets have been organised as follows: (1) longitudinal force (pure slip) (a) finite element analysis data (2) lateral force (pure slip) (a) experimental slip angle sweep data (b) mathematically constructed data (3) self-aligning torque (pure slip) with experimental data (a) experimental slip angle sweep data (b) mathematically constructed data Evaluation of fits There are many ways to evaluate the resulting fits. The quality of the fit can be ascertained by the difference between the predicted and the observed values. Usually, for purposes of evaluation and comparison, the use of a single number to represent the efficiency of the fit is desirable. This can be achieved by calculating the SSEs between the predicted and the observed values, which is the function being minimised. The root-mean-square error (MSE) can also be used and this measure is preferred as it is on the same scale as the data. Other measures such as the coefficient of determination, known as the R-squares (RSQ), bring out the fraction of variance in the data which is explained by the model. An Rsquare value close to one indicates a very good fit. The formulas used are summarised as follows [28]: SSE =

n

{FMF (xi ) − Fmeasured (xi )}2 ,

(5)

i=1

n i=1

{FMF (xi ) − Fmeasured (xi )}2 , n 2 i=1 {Fmeasured (xi )} n 2 i=1 {FMF (xi ) − Fmeasured (xi )} RSQ = 1 − n . n 2 i=1 {FMF (xi ) − i=1 Fmeasured (xi ) /n}

MSE =

(6) (7)

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Results and discussions Comparison of fits The fits given by the different algorithms on the input data sets for the longitudinal force, lateral force and self-aligning torque are discussed in the following sections.


Fitting of longitudinal force (pure slip) data Finite element analysis data. The longitudinal force vs. slip ratio curves obtained from the finite element analysis of a 205/60R16 tyre were fitted to the Magic Formula model using all the six mentioned algorithms. The extracted coefficients are shown in Table 3. The best and the worst fits are shown in Figure 1. It can be seen from the table that all the algorithms return RSQ values greater than 0.95, but the worst fit, which has an RSQ of 0.956, is not acceptable as observed in Figure 1(b). Hence, it is not wise to rely only on the RSQ for evaluating the absolute fit. The plots must also be checked before concluding. A low RSQ definitely means a bad fit, but a high RSQ need not mean a good fit. But, the RSQ can be used to compare fits. In this particular case, the RSQ does not indicate the difference between the fits of the algorithms as the difference is not visible within the first three decimal places. Hence for this problem, the SSE and MSE are better for comparing the fits than the RSQ. The unbounded cuckoo search fails to fit these data and gives a very high SSE. The bounded cuckoo search, bounded differential evolution and the bounded particle swarm optimisation give the lowest errors. It can also be seen from Table 3 that all these three algorithms not only give the same SSE, but also give the same set of coefficients. Hence, this set of coefficients given by all these three algorithms can safely be assumed to be a good parameter set. All the algorithms producing good fits give the same values for the pDx , pCx , pVx and pHx coefficients, while the values for pEx and pKx coefficients vary with the algorithm. This could be due to crosscorrelation among the latter set of parameters that are interrelated by nonlinear expressions, while the former set may have a unique set of values because of the linear nature of the expressions relating them. The set of equations relating the secondary parameters can be found in [17].

Fitting of lateral force (pure slip) data Experimental slip angle sweep data. The coefficients obtained from the various algorithms for the experimental data of the 185/70R14 tyre along with the goodness-of-fit measures are presented in Table 4. The best fit with the lowest error is given by the unbounded cuckoo search, followed by the unbounded differential evolution and unbounded trust region reflective algorithms. Many of the other algorithms – bounded trust region reflective, bounded pattern search, bounded differential evolution, bounded particle swarm optimisation and bounded cuckoo search – give very good fits with the same SSE and the same sets of coefficients. Only the Nelder–Mead simplex and the unbounded pattern search method give large errors. But, all the fits are acceptable as can be observed from RSQ of Table 4. The fitted coefficients in the worst case are also acceptable. The coefficients returned by all the algorithms giving good fits are mostly the same with the exception of pKy2 and pKy4 . This may again be due to cross-correlation and nonlinear nature of the expressions relating these coefficients, which cause local minima. An important characteristic of this data set is the noise inherent in experimental data.

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Figure 1. Best and worst case fits for a 205/60R16 tyre longitudinal force data obtained from finite element analysis. Only three of the five load cases have been shown here for clarity. (a) Best case fit: obtained using bounded differential evolution. (b) Worst case fit: obtained using unbounded cuckoo search.

Coefficients TRR(UB) TRR(B) NMS(UB) PTS(UB) PTS(B) DE(UB) DE(B) PSO(B) CS(UB) CS(B)

pCx1

pDx1

1.121 0.907 1.255 0.923 1.193 0.917 − 2.240 0.914 1.944 0.921 1.116 0.907 1.116 0.907 1.116 0.907 1.856 − 1.467 1.116 0.907

pDx2 − 0.003 − 0.001 − 0.006 0.000 − 0.015 − 0.002 − 0.002 − 0.002 − 0.005 − 0.002

pEx1 − 3.246 0.017 − 1.089 1.126 1.000 − 3.107 − 3.107 − 3.107 − 15.023 − 3.107

pEx2

pEx3

− 0.553 0.204 1.284 − 0.006 0.025 0.000 − 0.099 − 0.099 1.092 − 0.099

2.813 − 0.502 − 0.280 − 0.020 0.121 0.000 0.201 0.201 − 0.399 0.201

pEx4

pKx1

pKx2

pKx3

pHx1

− 0.226 15.725 0.008 0.121 − 0.001 1.000 19.780 − 0.878 0.165 0.000 − 0.038 18.593 − 0.098 − 0.003 − 0.001 0.023 18.944 0.073 0.095 − 0.001 0.007 19.372 − 0.006 0.123 0.000 − 0.243 15.878 0.000 0.114 − 0.001 − 0.243 15.888 0.000 0.110 − 0.001 − 0.243 15.888 0.000 0.110 − 0.001 − 1.074 − 72.784 8.381 1.009 − 0.672 − 0.243 15.888 0.000 0.110 − 0.001

pHx2

pVx1

pVx2

SSE

0.001 0.001 0.003 0.001 0.001 0.001 0.001 0.001 0.179 0.001

0.000 0.002 179,283 0.002 0.002 951,263 0.006 − 0.022 777,397 0.007 0.002 504,178 0.004 0.002 751,910 0.000 0.002 166,750 0.000 0.002 166,708 0.000 0.002 166,708 0.453 − 0.039 57,606,611 0.000 0.002 166,708

MSE

RSQ

0.012 0.027 0.024 0.019 0.024 0.011 0.011 0.011 0.208 0.011

1.000 0.999 0.999 1.000 0.999 1.000 1.000 1.000 0.956 1.000



166 Table 3. Longitudinal force coefficients and goodness-of-fit measures with different minimisation algorithms on longitudinal force data obtained from finite element analysis of a 205/60R16 tyre.

Coefficients

pCy1

TRR(UB) TRR(B) NMS(UB) PTS(UB) PTS(B) DE(UB) DE(B) PSO(B) CS(UB) CS(B)

1.535 1.535 1.707 1.568 1.535 − 1.535 1.535 1.535 1.598 1.535

pDy1

pDy2

− 1.093 0.395 − 1.093 0.396 − 1.085 0.240 − 1.096 0.418 − 1.093 0.396 − 1.093 0.395 − 1.093 0.396 − 1.093 0.396 1.091 − 0.377 − 1.093 0.396

pEy1 0.172 0.171 0.455 0.272 0.171 0.172 0.171 0.171 0.279 0.171

pEy2

pEy3

pKy1

− 5.185 0.045 − 11.985 − 5.191 0.045 − 11.984 − 0.046 − 0.068 − 15.832 − 4.847 0.191 − 12.089 − 5.190 0.045 − 11.984 − 5.186 0.045 − 11.984 − 5.190 0.045 − 11.984 − 5.190 0.045 − 11.984 − 4.256 0.025 − 12.189 − 5.190 0.045 − 11.984

pKy2

pKy4

259.367 5.000 3.060 13.367 5.000 − 11,099,787 5.000 5.000 − 0.779 5.000

370.925 7.230 2.668 18.587 7.230 − 15,874,868 7.231 7.231 5.046 7.231

pHy1

pHy2

pVy1

pVy2

− 0.002 0.001 − 0.001 − 0.002 − 0.002 0.001 − 0.001 − 0.002 − 0.003 0.001 − 0.004 0.000 − 0.001 − 0.006 0.010 − 0.080 − 0.002 0.001 − 0.001 − 0.002 − 0.002 0.001 − 0.001 − 0.002 − 0.002 0.001 − 0.001 − 0.002 − 0.002 0.001 − 0.001 − 0.002 − 0.002 0.002 − 0.002 0.010 − 0.002 0.001 − 0.001 − 0.002

SSE

MSE

RSQ

497,090 498,877 714,281 544,573 498,877 497,089 498,877 498,877 431,402 498,877

0.022 0.022 0.026 0.023 0.022 0.022 0.022 0.022 0.020 0.022

1.000 1.000 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000



Table 4. Lateral force coefficients and goodness-of-fit measures with different minimisation algorithms for experimental slip angle sweep data of a 185/70R14 tyre.

167


168


Mathematically constructed data. In order to check whether the coefficients obtained by the fitting process are indeed the ones that are correct, a second set of data that were mathematically constructed was used for fitting. These data were constructed in such a way that the load and slip angle intervals are the same as the first data set. If the coefficients given by the fitting process are the same as that used to construct the curves, then it indicates that the correct coefficients have been obtained. The results for this second data set are shown in Table 5. The coefficients used to construct the data are given in the first row labelled as the original set. It is important to note that the data are mathematically constructed, there is no noise and the better algorithms almost reach zero error giving back the original coefficients. The unbounded and bounded versions of trust region reflective, differential evolution and the bounded cuckoo search methods give the best fits for this data set. The pattern search method, the Nelder–Mead simplex method and the unbounded cuckoo search method give larger errors, but these errors do not affect the fit, except for the lateral force at lower loads where the fitted curve does not match the measured curve, as can be seen in Figure 2. It is clear by comparing the SSEs and MSEs of the experimental and constructed data from Tables 4 and 5 that the constructed data, with zero noise, give much lower SSEs and MSEs and this is expected. Also, unlike in the case of experimental data, in this case, the coefficients pKy2 and pKy4 for all the good fits do not deviate much from the original set of coefficients. This analysis of the fits given by the constructed data can give a good idea to choose the better performing algorithms. An algorithm that cannot fit such perfectly smooth data will definitely not be able to fit noisy experimental data. This can act as an important test to identify good algorithms and eliminate the bad ones.

Fitting of self-aligning torque (pure slip) data Experimental slip angle sweep data. The same algorithms without changing any of the algorithm parameters were also tried on the self-aligning torque data. Unlike the lateral force case, where all the algorithms could fit the experimental data reasonably well, in this case the performance of the algorithms varied substantially. The errors corresponding to the unbounded pattern search, unbounded cuckoo search and bounded particle swarm optimisation were much higher compared to the rest of the algorithms. Table 6 shows the set of coefficients obtained using each of the algorithms. The unbounded differential evolution algorithm outperforms all the algorithms with the lowest errors, while the unbounded pattern search gives the worst fit. Figure 3 shows the best and the worst fits for the data. It can be seen clearly that the worst case fit here is totally unacceptable. Such large errors in some of the algorithms could be an indication of the solvers trapped in a local minimum. In this case, the variability among the fitted coefficients is also visible unlike in the lateral force case when most of the algorithms gave the same coefficients with the exception of pKy2 and pKy4 . In this case, most of the coefficients with the exception of a few like qBz1 , qBz2 and qCz1 take different values for each algorithm. This could be due to multiple local minima caused by cross-correlation among the parameters, most of which are related by nonlinear expressions. There is more complexity in self-aligning torque as the final value is given by the sum of residual torque and the product of lateral force and pneumatic trail. These three individual quantities are given by a sine or cosine version of the basic Magic Formula equation. Mathematically constructed data. The fits for the constructed data, as expected, are relatively better. Following the trend of the experimental data, the worst fits for this data set too are unacceptable unlike those obtained with the lateral force data. The coefficients obtained with each of the minimisation algorithms used are shown in Table 7. The unbounded

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Figure 2. Best and worst case fits for a 205/60R15 tyre lateral force data obtained from mathematically constructed data set. Only three out of the five load cases have been shown here for clarity. (a) Best case fit: obtained using unbounded trust region reflective algorithm. (b) Worst case fit: obtained using particle swarm optimisation.

Coefficients

pCy1

pDy1

pDy2

pEy1

pEy2

pEy3

pKy1

pKy2

pKy4

pHy1

pHy2

pVy1

pVy2

SSE

MSE

RSQ

Original Set TRR(UB) TRR(B) NMS(UB) PTS(UB) PTS(B) DE(UB) DE(B) PSO(B) CS(UB) CS(B)

1.193 1.193 1.193 1.437 2.889 1.812 − 1.193 1.193 0.532 2.887 1.193

− 0.990 − 0.990 − 0.990 − 0.991 − 0.988 − 1.043 0.990 − 0.990 − 2.500 0.988 − 0.990

0.145 0.145 0.145 0.137 0.144 0.138 − 0.145 0.145 0.378 − 0.144 0.145

− 1.003 − 1.003 − 1.003 − 0.101 1.901 1.000 − 1.003 − 1.003 − 41.403 1.899 − 1.003

− 0.537 − 0.537 − 0.537 − 0.215 − 0.185 0.074 − 0.537 − 0.537 − 3.225 − 0.188 − 0.537

− 0.083 − 0.083 − 0.083 − 0.863 0.034 − 0.504 − 0.083 − 0.083 − 1.296 0.035 − 0.083

− 14.950 − 14.950 − 14.950 − 14.520 − 16.081 − 16.675 15.001 − 15.052 − 4.382 15.725 − 14.858

2.130 2.130 2.130 31.249 2.298 2.332 − 2.099 2.068 3.553 2.553 2.194

2.000 2.000 2.000 30.342 2.006 2.000 1.966 1.931 3.120 − 2.274 2.070

0.003 0.003 0.003 0.003 0.003 − 0.001 0.003 0.003 0.100 0.003 0.003

− 0.001 − 0.001 − 0.001 − 0.001 − 0.001 − 0.003 − 0.001 − 0.001 0.010 − 0.001 − 0.001

0.045 0.045 0.045 0.045 0.045 − 0.005 0.045 0.045 0.824 0.045 0.045

− 0.024 − 0.024 − 0.024 − 0.021 − 0.026 − 0.029 − 0.024 − 0.024 − 0.147 − 0.026 − 0.024

– 0.000 0.000 40,947 15,174 70,657 0.084 0.096 1,209,511 15,172 0.095

– 0.000 0.000 0.007 0.004 0.009 0.000 0.000 0.038 0.004 0.000

– 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 1.000 1.000



170

Table 5. Lateral force coefficients and goodness-of-fit measures with different minimisation algorithms for mathematically constructed data of a 205/60R15 tyre.

Coefficients

qBz1

qBz2

TRR(UB) TRR(B) NMS(UB) PTS(UB) PTS(B) DE(UB) DE(B) PSO(B) CS(UB) CS(B)

10.254 11.125 − 11.227 17.500 10.562 6.171 10.865 0.638 74.117 10.823

− 5.530 − 10.000 0.014 − 193.000 − 10.000 − 1.747 − 10.000 − 0.238 − 780.856 − 10.000

qBz3

qBz9

qBz10

qCz1

qDz1

qDz2

56.377 108.110 10.565 1.159 0.128 0.061 − 0.115 0.927 1.063 1.163 0.131 0.000 0.019 0.000 0.012 − 1.213 0.132 − 0.004 366.750 0.000 0.000 1.250 0.000 0.750 0.000 − 50.000 − 5.385 1.146 0.129 0.000 10.247 − 5550.22 − 509.621 2.032 0.128 0.068 0.000 0.000 0.000 1.156 0.130 0.000 0.000 12.495 − 0.415 5.000 0.062 0.000 265.883 − 125.021 − 16,500.0 32.913 0.237 − 0.276 0.000 − 50.000 − 5.212 1.155 0.130 0.000

qDz6

qDz7

qEz1

− 0.001 0.009 − 0.731 − 0.001 0.007 − 0.465 − 0.002 − 0.016 − 0.052 0.000 0.000 − 0.500 − 0.001 0.008 − 0.751 − 0.001 0.011 1.154 − 0.001 0.006 − 0.565 − 0.003 0.012 − 10.000 − 2.796 17.026 − 4.278 − 0.001 0.007 − 0.591

qEz2

qEz3

qEz4

qHz1

− 0.640 16.352 0.275 0.000 − 1.012 − 1.540 0.323 0.001 0.039 − 0.086 0.008 0.001 − 1392.75 22,653.0 0.000 0.000 − 1.634 − 5.135 0.244 0.000 0.019 1.171 − 0.034 0.003 − 1.373 − 4.225 0.221 0.000 − 0.502 − 10.000 − 10.000 − 0.796 174.896 − 874.505 − 56.875 − 10.758 − 1.405 − 4.369 0.265 0.000

qHz2

SSE

MSE

RSQ

− 0.009 325 0.037 0.999 − 0.015 482 0.045 0.998 − 0.061 1876 0.090 0.992 0.000 94,599 0.637 0.591 − 0.012 464 0.045 0.998 − 0.015 273 0.034 0.999 − 0.009 458 0.044 0.998 − 0.240 45,318 0.441 0.804 80.786 72,816 0.559 0.685 − 0.012 460 0.044 0.998



Table 6. Self-aligning torque coefficients and goodness-of-fit measures with different minimisation algorithms for experimental slip angle sweep data of a 185/70R14 tyre.

171



172

Figure 3. Best and worst case fits for a 185/70R14 tyre self-aligning torque data obtained from slip angle sweep experiments. Only three out of the five load cases have been shown here for clarity. (a) Best case fit: obtained using unbounded differential evolution algorithm. (b) Worst case fit: obtained using unbounded pattern search method.



173

and bounded trust region algorithms give the best fit with the lowest errors, closely followed by the cuckoo search technique with bounds. Other algorithms such as bounded and unbounded differential evolution and bounded pattern search also give good fits. Once again, the unbounded pattern search, unbounded cuckoo search and particle swarm optimisation give very large errors and bad fits. Figure 4 shows the best and the worst fits for the constructed self-aligning torque data. It can be observed that the worst case with the highest errors among all the algorithms is certainly not able to represent the data well. The large errors in these cases may be caused because of reasons such as getting trapped in local minima or lack of convergence. As mentioned earlier in the case of the lateral force constructed data, constructed data for self-aligning torque can help in identification of the better algorithm. A comparison of the fits for all the data sets can provide a clearer picture of which is the best alternative. The SSE cannot be used for comparing different data sets since it depends on the absolute values. Hence, we use only the MSE and RSQ values for comparing different data sets. It is observed that the MSE values for the longitudinal force finite element data range between 0.011 and 0.208 and the RSQ values range between 1.0 and 0.956, from the best to the worst fit, respectively. For the lateral force, in the case of the experimental data, the MSE ranges from 0.020 to 0.026 and the RSQ from 1.0 to 0.999, while for the mathematically constructed data, the MSE ranges from 0.000 to 0.038 and RSQ from 1.0 to 0.999. In the case of the self-aligning torque, while the MSE values are in the 0.034–0.637 range and RSQ values in the 0.999–0.591 range for experimental data, the MSE values are in the 0.001– 0.609 range and RSQ values in the 1.0 to 0.602 range for mathematically constructed data. Hence, it can be seen that experimental data generally gives more errors than finite element and mathematically constructed data. It is also observed that the self-aligning torque is harder to fit compared to longitudinal and lateral forces. The comparison with respect to each of the six algorithms for all the data sets is also important and is given here. The MSE for TRR(UB) ranges from 0 in lateral force mathematically constructed data to 0.037 in self-aligning torque experimental data, while the MSE for TRR(B) ranges from 0 in lateral force mathematically constructed data to 0.045 in selfaligning torque experimental data. The MSE for NMS(UB) ranges from 0.007 in lateral force mathematically constructed data to 0.09 in self-aligning torque experimental data. For the PTS(UB) algorithm, the MSE ranges from 0.004 in lateral force mathematically constructed data to 0.637 in self-aligning torque experimental data, while for the PTS(B) algorithm, the MSE ranges from 0.009 in lateral force mathematically constructed data to 0.045 in selfaligning torque experimental data. The MSE for DE(UB) ranges from 0 in lateral force mathematically constructed data to 0.034 in self-aligning torque experimental data, while the MSE for DE(B) ranges from 0 in lateral force mathematically constructed data to 0.044 in self-aligning torque experimental data. For the PSO(B) algorithm, the MSE ranges from 0.011 in longitudinal force mathematically constructed data (thereby breaking the trend of lateral force mathematically constructed data having the lowest MSEs) to 0.441 in self-aligning torque experimental data. While in the case of CS(UB) algorithm, the MSE ranges from 0.004 in lateral force mathematically constructed data to 0.559 in self-aligning torque experimental data, in the case of CS(B) algorithm, the MSE ranges from 0 in lateral force mathematically constructed data to 0.044 in self-aligning torque experimental data. Thus, it is observed that in the case of all algorithms (except PSO), the lateral force mathematically constructed data have the least error and the experimental self-aligning torque has the maximum error. One must choose an algorithm that can give the least errors in all cases, from the most basic case to the worst case. The inferences based on these comparisons are given in the following section.

Coefficients

qBz1

Original Set 8.964 TRR(UB) 8.976 TRR(B) 8.976 NMS(UB) 10.430 PTS(UB) 15.500 PTS(B) 9.275 DE(UB) 5.568 DE(B) 6.755 PSO(B) 8.425 CS(UB) − 31.082 CS(B) 8.992

qBz2 − 1.106 − 1.098 − 1.098 0.056 − 362.664 − 1.112 − 0.386 − 0.472 0.000 0.533 − 1.005

qBz3

qBz9

qBz10

− 0.842 18.470 0.000 − 0.852 20.766 − 0.211 − 0.852 20.766 − 0.211 − 0.119 − 0.144 − 0.175 1904.95 − 5.141 0.000 − 0.327 47.14 − 6.186 − 0.586 72.942 − 3.657 0.000 16.162 − 1.784 − 2.759 − 50.000 − 50.0 4.388 30.023 − 2.665 − 1.123 31.490 − 1.108

qCz1

qDz1

qDz2

1.180 0.100 − 0.001 1.185 0.101 − 0.001 1.185 0.101 − 0.001 1.263 0.106 0.003 6.766 − 0.016 0.672 1.203 0.101 0.000 2.243 0.104 0.001 1.952 0.105 0.000 5.000 0.125 0.000 − 1.139 0.095 − 0.001 1.186 0.101 − 0.001

qDz6

qDz7

qEz1

qEz2

− 0.008 0.000 − 1.609 − 0.359 − 0.008 0.000 − 1.514 − 0.340 − 0.008 0.000 − 1.514 − 0.340 − 0.006 − 0.003 − 0.225 0.225 − 0.008 − 0.008 − 41.000 624.063 − 0.008 0.000 − 1.136 − 0.275 − 0.008 0.000 1.247 − 0.010 − 0.008 0.000 1.000 − 0.018 − 0.008 0.003 − 10.000 0.200 − 0.007 0.000 1.133 − 0.058 − 0.008 0.000 − 1.498 − 0.285

qEz3 0.000 0.002 0.002 0.242 − 2374.13 0.322 0.002 0.333 − 10.000 0.048 − 0.168

qEz4

qHz1

0.174 0.007 0.184 0.007 0.184 0.007 0.126 0.003 − 1991.23 0.000 0.177 0.010 − 0.033 0.011 0.020 0.003 − 10.000 0.023 − 0.641 − 0.047 0.186 0.007

qHz2

SSE

MSE

RSQ

− 0.002 – – − 0.002 0.084 0.001 − 0.002 0.084 0.001 − 0.008 270 0.046 − 0.063 47,456 0.609 − 0.008 12.395 0.010 − 0.003 17.891 0.012 0.001 68.492 0.023 0.000 5643 0.210 − 0.004 1052.32 0.091 − 0.001 0.100 0.001

– 1.000 1.000 0.998 0.602 1.000 1.000 0.999 0.953 0.991 1.000



174

Table 7. Self-aligning torque coefficients and goodness-of-fit measures with different minimisation algorithms for mathematically constructed data of a 205/60R15 tyre.

175



Figure 4. Best and worst case fits for a 205/60R15 tyre self-aligning torque data obtained from mathematically constructed data set. Only three out of the five load cases have been shown here for clarity. (a) Best case fit: obtained using unbounded trust region reflective algorithm. (b) Worst case fit: obtained using unbounded pattern search method.

176



Inferences From the above results, it can be inferred that the trust region reflective algorithm is one of the best options for obtaining the coefficients. It consistently gives lower errors compared to the rest of the algorithms. This algorithm is the fastest to converge among all the algorithms used. But, it requires good start points. The other algorithms, such as differential evolution, cuckoo search or particle swarm optimisation, do not require starting points, yet they fare reasonably well. The algorithms are not allowed to run till they reach an absolute minimum, but are terminated according to stopping criteria, which may be a fixed number of iterations, error goal or relative tolerance (given in Table 2). This helps in weeding out the slowest algorithms, as they are stopped before reaching a converged value if they are too slow and also if they get stuck. It must be noted that the coefficients obtained by the fitting process are for a particular nominal load. The coefficients will change if a different nominal load is taken for normalising the coefficients. All the algorithms except particle swarm optimisation that is always bounded, and Nelder– Mead simplex that is always unbounded, have been tried with the bounded and unbounded versions and compared. It is observed from the given tables that the performance is not affected by the bounding of variables for the trust region reflective and differential evolution algorithms. On the other hand, the cuckoo search and pattern search are affected by the bounding of variables. It is seen that in general both the unbounded and bounded versions of trust region reflective and differential evolution methods and the bounded version of the cuckoo search give better fits than other algorithms. The pattern search, bounded particle swarm optimisation and unbounded cuckoo search give the largest errors in most of the cases. It is also observed that the type of data fitted, whether lateral force or longitudinal force, noisy or smooth, etc., also affects the performance of the algorithms. The algorithms that perform well and give lower errors regardless of these factors are naturally a better choice for fitting the data. Since the trust region reflective, bounded cuckoo search and differential evolution satisfy these criteria, they can be considered as suitable algorithms for fitting the Magic Formula. From the tyre mechanics point of view, the curves produced by these parameters give the forces at the tyre–road interface under different loading conditions. Different fitting inaccuracies will therefore have different effects on the simulation results. For example, the worst case fit results that have been obtained here, if used in a vehicle dynamics simulation, would completely misrepresent the behaviour of the tyre. These numerical parameters actually represent key tyre mechanics properties such as peak force offered, cornering stiffnesses, saturation force values, plysteer, conicity, etc. The tyre forces and therefore the grip could be, say, grossly underestimated and the simulation could show the vehicle to have a poor handling. Therefore, it is very important that a good fit be obtained and the tyre forces neither be overestimated nor underestimated.

Conclusion The process of fitting tyre force and moment data to the Magic Formula using different optimisation techniques has been described. Different sets of data, each of a unique type such as finite element data, slip angle sweep data and mathematically constructed data, have been used for the purpose of this study. It can be concluded from this study that all algorithms need not perform the same way for all tyre force and moment models. Only the algorithms that can fit all types of data and all types of models such as longitudinal force, lateral force, self-aligning torque, etc. without tweaking any of the parameters for each case separately are



177

suitable for fitting the Magic Formula tyre model. There are three algorithms that give good fits for all the tested cases – trust region reflective (both bounded and unbounded), differential evolution (both bounded and unbounded) and bounded cuckoo search. The rest of the algorithms give good fits only for a few of the cases and not for all the cases. Another conclusion is that even if the algorithms produce good fits, they may not return the same set of parameters due to multiple local minima of similar depth and cross-correlation among parameters. So, one should compare the results of a number of algorithms before taking a decision. If all the results are similar and give the same SSE, it is an indicator that the chances of the obtained parameters set being the most appropriate are higher. One of the major recommendations from this study concerns the initial or starting values. In case the start values are well known or can be guessed, it is always better to use the trust region or other gradient-based methods. In case the bounds are known, but the initial guesses are unknown, bounded DE and bounded CS can be used. In situations, where both initial guesses and bounds are not available, unbounded DE is the best possible option. In future, work on the fitting process can focus on finding the required amount of input data for each case – the number of measurement points must be kept to a minimum to reduce the number of simulations or experiments, but also should not result in overfitting. As the performance of most of these algorithms is sensitive to the algorithm parameters, the tuning of these parameters to deliver optimum performance can also be a good field to explore. Other optimisation algorithms such as Modified Cuckoo Search, Artificial Bee Colony Algorithm, Bat Algorithm, Harmony Search, etc. can also be explored. The various optimisation schemes can also be combined together to take advantage of the individual strengths of the algorithms and compensate for the weakness of each of these algorithms such as lower speed or local convergence. Another very important area that needs to be investigated is that of curves of different ‘character’, for example, curves delivered by the measurement of a stiff and low aspect ratio tyre vs. a soft and high aspect ratio one. Significantly different sets of data with different characteristics should be chosen to challenge the fitting routines.

Acknowledgments The authors acknowledge the management of JK Tyre and Industries for providing the data from the flat track machine.

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