Estimation of the tyre–road maximum friction coefficient and slip slope based on a novel tyre model

In this article, a new approach to estimate the vehicle tyre forces, tyre–road maximum friction coefficient, and slip slope is presented. Contrary to the majority of the previous work on this subject, a new tyre model for the estimation of the tyre–road interface characterisation is proposed. First, the tyre model is built and compared with those of Pacejka, Dugoff, and one other tyre model. Then, based on a vehicle model that uses four degrees of freedom, an extended Kalman filter (EKF) method is designed to estimate the vehicle motion and tyre forces. The shortcomings of force estimation are discussed in this article. Based on the proposed tyre model and the improved force measurements, another EKF is implemented to estimate the tyre model parameters, including the maximum friction coefficient, slip slope, etc. The tyre forces are accurately obtained simultaneously. Finally, very promising results have been achieved for pure acceleration/braking for varying road conditions, both in pure steering and combined manoeuvre simulations.     

Velocity and normal tyre force estimation for heavy trucks based on vehicle dynamic simulation considering the road slope angle

A precise estimation of vehicle velocities can be valuable for improving the performance of the vehicle dynamics control (VDC) system and this estimation relies heavily upon the accuracy of longitudinal and lateral tyre force calculation governed by the prediction of normal tyre forces. This paper presents a computational method based on the unscented Kalman filter (UKF) method to estimate both longitudinal and lateral velocities and develops a novel quasi-stationary method to predict normal tyre forces of heavy trucks on a sloping road. The vehicle dynamic model is constructed with a planar dynamic model combined with the Pacejka tyre model. The novel quasi-stationary method for predicting normal tyre forces is able to characterise the typical chassis configuration of the heavy trucks. The validation is conducted through comparing the predicted results with those simulated by the TruckSim and it has a good agreement between these results without compromising the convergence speed and stability.     

A combined-slip predictive control of vehicle stability with experimental verification

In this paper, a model predictive vehicle stability controller is designed based on a combined-slip LuGre tyre model. Variations in the lateral tyre forces due to changes in tyre slip ratios are considered in the prediction model of the controller. It is observed that the proposed combined-slip controller takes advantage of the more accurate tyre model and can adjust tyre slip ratios based on lateral forces of the front axle. This results in an interesting closed-loop response that challenges the notion of braking only the wheels on one side of the vehicle in differential braking. The performance of the proposed controller is evaluated in software simulations and is compared to a similar pure-slip controller. Furthermore, experimental tests are conducted on a rear-wheel drive electric Chevrolet Equinox equipped with differential brakes to evaluate the closed-loop response of the model predictive control controller.     

An easy procedure to determine Magic Formula parameters: a comparative study between the starting value optimization technique and the IMMa optimization algorithm

In 2004, a new searching algorithm for Magic Formula tyre model parameters was presented. Now, a summary of the results, for pure and combined slip, that this algorithm is able to achieve is presented. The Magic Formula tyre model needs a set of parameters to describe the tyre properties. The determination of these parameters is dealt with in this article. A new method, called IMMa Optimization Algorithm (IOA), based on genetic techniques, is used to determine these parameters. Here, we show the computational cost that has been used to obtain the optimum parameters of every characteristic of the Magic Formula tyre model, called Delft Tyre 96. The main advantages of the method are its simplicity of implementation and its fast convergence to optimal solution, with no need of deep knowledge of the searching space. Hence, to start the search, it is not necessary to know a set of starting values of the Magic Formula parameters (null sensitivity to starting values). The search can be started with a randomly generated set of parameters between [0, 1]. Nowadays, MF-Tool, an application developed by TNO, uses an optimization technique to fit Magic Formula parameters from Matlab toolbox [van Oosten, J.J.M. and Bakker, E., 1993, {Determination of magic tyre model parameters}. Vehicle System Dynamics, 21, 19–29; van Oosten, J.J.M., Savi, C., Augustin, M., Bouhet, O., Sommer, J. and Colinot, J.P., 1999, {Time, tire, measurements, forces and moments, a new standard for steady state cornering tyre testing}. EAEC Conference, Barcelona, 30 June–2 July.]. We refer to that algorithm as the starting value optimization technique. The comparison between the optimization technique employed by TNO and the proposed IOA method is discussed in this article. In order to give a relative idea of adjustment accuracy, the sum-squared error and the mean-squared error, from the curves of the tyre model with the parameters optimized by both applications compared with test data are evaluated.     

Normal and shear forces in the contact patch of a braked racing tyre. Part 1: results from a finite-element model

In this first part of a two-part article, a previously described and validated finite-element model of a racing-car tyre is developed further to yield detailed information on carcass deflections and contact pressure and shear stress distributions for a steady rolling, slipping, and cambered tyre. Variations in running conditions simulated include loads of 1500, 3000 and 4500 N, camber angles of 0° and ?3°, and longitudinal slips from 0% to?20%. Special attention is paid to heavy braking, in which context the aligning moment is of great interest. Results generated are in broad agreement with limited experimental results from the literature and they provide considerable insight into how the tyre deforms and how the contact stresses are distributed as functions of the running conditions. Generally, each rib of the tyre behaves differently from the others, especially when the wheel is cambered. The results form a basis for the development of a simpler physical tyre model, the purpose of which is to retain accuracy over the full operating range while demanding much less computational resource. The physical tyre model is the topic of the second part of the article.     

Slip-Angle Estimation for Vehicle Stability Control

Recently, some direct yaw-moment control systems have been in development. Obviously, such systems need accurate slip-angle information. This paper describes a strategy of vehicle slip angle estimation. The difficulty in slip angle estimation is due to nonlinear characteristics of tyres and influence of relative slant of the road surface. To solve this difficulty, a combined method of model observer and direct integration method is proposed. In this method, two kinds of values of the side forces of the wheels are provided, i.e., direct detected values by the G-sensor and values from a tyre model. Then those values are combined appropriately which results in the combination of model observer and direct integration. A feedback algorithm, redesigned to suppress the influence of tyre model error, is applied in the observer. Considering interference of road surface and its avoidance, road slant angle is estimated and consequently vehicle model is corrected. The estimated value of the road friction coefficient is given by the acceleration, and an adequate bias, depending on yaw-deviation, is added. The calculation method of reference yaw-velocity is improved, in order to avoid interference of road slant and variation of dynamic characteristic of vehicle.     

Slip-Angle Estimation for Vehicle Stability Control

Recently, some direct yaw-moment control systems have been in development. Obviously, such systems need accurate slip-angle information. This paper describes a strategy of vehicle slip angle estimation. The difficulty in slip angle estimation is due to nonlinear characteristics of tyres and influence of relative slant of the road surface. To solve this difficulty, a combined method of model observer and direct integration method is proposed. In this method, two kinds of values of the side forces of the wheels are provided, i.e., direct detected values by the G-sensor and values from a tyre model. Then those values are combined appropriately which results in the combination of model observer and direct integration. A feedback algorithm, redesigned to suppress the influence of tyre model error, is applied in the observer. Considering interference of road surface and its avoidance, road slant angle is estimated and consequently vehicle model is corrected. The estimated value of the road friction coefficient is given by the acceleration, and an adequate bias, depending on yaw-deviation, is added. The calculation method of reference yaw-velocity is improved, in order to avoid interference of road slant and variation of dynamic characteristic of vehicle.     

Unscented Kalman filter for vehicle state estimation

Vehicle dynamics control (VDC) systems require information about system variables, which cannot be directly measured, e.g. the wheel slip or the vehicle side-slip angle. This paper presents a new concept for the vehicle state estimation under the assumption that the vehicle is equipped with the standard VDC sensors. It is proposed to utilise an unscented Kalman filter for estimation purposes, since it is based on a numerically efficient nonlinear stochastic estimation technique. A planar two-track model is combined with the empiric Magic Formula in order to describe the vehicle and tyre behaviour. Moreover, an advanced vertical tyre load calculation method is developed that additionally considers the vertical tyre stiffness and increases the estimation accuracy. Experimental tests show good accuracy and robustness of the designed vehicle state estimation concept.     

Shear forces in the contact patch of a braked-racing tyre

This article identifies tyre modelling features that are fundamental to the accurate simulation of the shear forces in the contact patch of a steady-rolling, slipping and cambered racing tyre. The features investigated include contact patch shape, contact pressure distribution, carcass flexibility, rolling radius (RR) variations and friction coefficient. Using a previously described physical tyre model of modular nature, validated for static conditions, the influence of each feature on the shear forces generated is examined under different running conditions, including normal loads of 1500, 3000 and 4500 N, camber angles of 0° and?3°, and longitudinal slip ratios from 0 to?20%. Special attention is paid to heavy braking, in which context the aligning moment is of great interest in terms of its connection with the limit-handling feel. The results of the simulations reveal that true representations of the contact patch shape, carcass flexibility and lateral RR variation are essential for an accurate prediction of the distribution and the magnitude of the shear forces generated at the tread–road interface of the cambered tyre. Independent of the camber angle, the contact pressure distribution primarily influences the shear force distribution and the slip characteristics around the peak longitudinal force. At low brake-slip ratios, the friction coefficient affects the shear forces in terms of their distribution, while, at medium to high-slip ratios, the force magnitude is significantly affected. On the one hand, these findings help in the creation of efficient yet accurate tyre models. On the other hand, the research results allow improved understanding of how individual tyre components affect the generation of shear forces in the contact patch of a rolling and slipping tyre.     

Shear Force Development by Pneumatic Tyres in Steady State Conditions: A Review of Modelling Aspects

SUMMARY

Modelling of the generation of shear forces by pneumatic tyres under steady state conditions is reviewed. The review is placed in a practical context, through reference to the uses to which models may be put by the vehicle dynamicist and the tyre designer. It will be of interest also to the student of rolling contact problems.

The subject is divided into sections, covering physically founded models which require computation for their solution, physically based models which are sufficiently simplified to allow analytical solutions and formula based, empirical models. The classes are more nearly continuous than this strict division would imply, since approximations in obtaining analytical solutions may be made, empirical correction factors may be applied to analytical results and formula based methods may take into account tyre mechanical principles. Such matters are discussed in the relevant sections. Attention is given to the important matter of choosing model parameters to best represent the behaviour of a particular tyre.

Conclusions relate to the structural and frictional mechanisms present in the shear force generation process, the contributions of carcass and tread elastic properties and of geometrical and frictional factors to the determination of the distributions of force through the contact region, the relationship between accuracy and computational load and the selection of methods for modelling tyre forces in a road vehicle dynamics context. Reference to the most pertinent literature in the field is made and possibilities for the further development of the state of the art are mentioned.     

Eigensystem of Tangential Contact in Tyre Models

Summary In this paper, a simplified model of tangential contact between tyre and rigid surface is investigated. By linearization the eigensystem of the contact equations is obtained and parameter variations are carried out. It is shown, that some vehicle model parameters have great influence on the eigensystem of tangential contact and can determine the highest eigenfrequency of the system vehicle and tyre. Root loci are used to investigate the influence of parameters like vehicle velocity and gridwidth of the discretization. Based on the eigensystem, stability areas of numerical methods in solving the partial differential equations of tangential contact are calculated. Numerical solutions using stiff and nonstiff integrators are compared with respect to the stability areas, computational effort and accuracy. The results are discussed with a view to further development.     

Eigensystem of Tangential Contact in Tyre Models

Summary In this paper, a simplified model of tangential contact between tyre and rigid surface is investigated. By linearization the eigensystem of the contact equations is obtained and parameter variations are carried out. It is shown, that some vehicle model parameters have great influence on the eigensystem of tangential contact and can determine the highest eigenfrequency of the system vehicle and tyre. Root loci are used to investigate the influence of parameters like vehicle velocity and gridwidth of the discretization. Based on the eigensystem, stability areas of numerical methods in solving the partial differential equations of tangential contact are calculated. Numerical solutions using stiff and nonstiff integrators are compared with respect to the stability areas, computational effort and accuracy. The results are discussed with a view to further development.     

Analysis of lateral tyre friction dynamics1

The tyre friction model is a key part of the overall multi-body tyre dynamics model. The LuGre dynamic tyre friction model is analytically linearised for pure cornering conditions. The linearised model parameters are conveniently expressed as functions of static curve slope parameters. The linearised lateral force and self-aligning torque submodels are described by equivalent mechanical systems. The linearised model and equivalent system parameters are analysed for different slip angle and wheel centre speed operating points. An example of the application of linearised tyre friction model to tyre vibration analysis is presented as well.     

Longitudinal wheel slip during ABS braking

Anti-lock braking system (ABS) braking tests with two subcompact passenger cars were performed on dry and wet asphalt, as well as on snow and ice surfaces. The operating conditions of the tyres in terms of wheel slip were evaluated using histograms of the wheel slip data. The results showed different average slip levels for different road surfaces. It was also found that changes in the tyre tread stiffness affected the slip operating range through a modification of the slip value at which the maximum longitudinal force is achieved. Variation of the tyre footprint length through modifications in the inflation pressure affected the slip operating range as well. Differences in the slip distribution between vehicles with different brake controllers were also observed. The changes in slip operating range in turn modified the relative local sliding speeds between the tyre and the road. The results highlight the importance of the ABS controller's ability to adapt to changing slip–force characteristics of tyres and provide estimates of the magnitude of the effects of different tyre and road operating conditions.     

Study of the Pneumatic Tyre Behaviour on Dry and Rigid Road by Finite Element Method

The paper presents a physical tyre model capable of describing the complete pneumatic tyre behaviour during steady and transient states. Given the radial deflection, the longitudinal and lateral slip, the camber angle, the inner pressure and the mechanical parameters describing the tyre structure, the model returns the vertical load, the longitudinal and lateral forces, the self aligning torque. Particular attention has been devoted to the computation (by f.e.m.) of tyre carcass and tread deformations; it is explained how side force increases by moderate braking at constant slip angle. An experimental verification validates the model, although more studies could be needed to improve model effectiveness.