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

Abstract

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.

Circular Path and Linear Momentum Method for Seismic Response Analysis of Vehicles

“…These factors are calculated for pure slip conditions based on the TRR algorithm, which showed the best performance compared with other algorithms. The coefficients used to calculate the MFM parameters are shown in Table 3 (Alagappan et al, 2014). The pressure on the tires is calculated considering the static pressure as well as the effects of the rolling, pitching, and vertical motion of the vehicle (Maruyama and Yamazaki, 2002).…”

“…Figure 4C shows the variation of the self-aligning torque with slip angles for the same cases. We selected an R15 tire, where the nominal load is taken as 6.15 kN for calculation of MFM parameters (Alagappan et al, 2014). The total longitudinal force acting on the tire can be calculated using Eq.…”

“…The forces acting on the tires were calculated using the Magic Formula Model (MFM) (Pacejka, 2006). The MFM coefficients were taken from previously published results (Alagappan et al, 2014) using the trust region reflective (TRR) method algorithm. We analyzed the responses of a car, bus, and truck in the longitudinal and lateral directions for several conditions.…”

On the Vehicle Sideslip Angle Estimation: A Literature Review of Methods, Models, and Innovations

“…However, the training procedure is highly time demanding, and they only tested the network on two scenarios. Also, Acosta et al, Dye et al, Alagappan et al and Huang et al in [107][108][109][110] used a hybrid estimator composed by a neural network and an observer (typically an EKF). The first one has only the task of fitting tyre data, while the second estimates vehicle dynamics states.…”

“…Neural network-based [96][97][98][99][100][101][102][103][104][105][106][107][108][109][110]: This method is specifically used to overcome the need for a vehicle model of any kind and its related complex set of parameters [101]. Artificial neural networks (ANN) are largely considered effective tools for system modelling, as they are suitable to model complex systems using their ability to identify relationships from input-output data pairs.…”