Dynamic Friction Models for Road/Tire Longitudinal Interaction

Summary In this paper we derive a new dynamic friction force model for the longitudinal road/tire interaction for wheeled ground vehicles. The model is based on a dynamic friction model developed previously for contact-point friction problems, called the LuGre model. By assuming a contact patch between the tire and the ground we develop a partial differential equation for the distribution of the friction force along the patch. An ordinary differential equation (the lumped model) for the friction force is developed, based on the patch boundary conditions and the normal force distribution along the contact patch. This lumped model is derived to approximate closely the distributed friction model. Contrary to common static friction/slip maps, it is shown that this new dynamic friction model is able to capture accurately the transient behaviour of the friction force observed during transitions between braking and acceleration. A velocity-dependent, steady-state expression of the friction force versus the slip coefficient is also developed that allows easy tuning of the model parameters by comparison with steady-state experimental data. Experimental results validate the accuracy of the new tire friction model in predicting the friction force during transient vehicle motion. It is expected that this new model will be very helpful for tire friction modeling as well as for anti-lock braking (ABS) and traction control design.     

Dynamic Friction Models for Road/Tire Longitudinal Interaction

Summary In this paper we derive a new dynamic friction force model for the longitudinal road/tire interaction for wheeled ground vehicles. The model is based on a dynamic friction model developed previously for contact-point friction problems, called the LuGre model. By assuming a contact patch between the tire and the ground we develop a partial differential equation for the distribution of the friction force along the patch. An ordinary differential equation (the lumped model) for the friction force is developed, based on the patch boundary conditions and the normal force distribution along the contact patch. This lumped model is derived to approximate closely the distributed friction model. Contrary to common static friction/slip maps, it is shown that this new dynamic friction model is able to capture accurately the transient behaviour of the friction force observed during transitions between braking and acceleration. A velocity-dependent, steady-state expression of the friction force versus the slip coefficient is also developed that allows easy tuning of the model parameters by comparison with steady-state experimental data. Experimental results validate the accuracy of the new tire friction model in predicting the friction force during transient vehicle motion. It is expected that this new model will be very helpful for tire friction modeling as well as for anti-lock braking (ABS) and traction control design.     

Estimation of longitudinal speed robust to road conditions for ground vehicles

This article seeks to develop a longitudinal vehicle velocity estimator robust to road conditions by employing a tyre model at each corner. Combining the lumped LuGre tyre model and the vehicle kinematics, the tyres internal deflection state is used to gain an accurate estimation. Conventional kinematic-based velocity estimators use acceleration measurements, without correction with the tyre forces. However, this results in inaccurate velocity estimation because of sensor uncertainties which should be handled with another measurement such as tyre forces that depend on unknown road friction. The new Kalman-based observer in this paper addresses this issue by considering tyre nonlinearities with a minimum number of required tyre parameters and the road condition as uncertainty. Longitudinal forces obtained by the unscented Kalman filter on the wheel dynamics is employed as an observation for the Kalman-based velocity estimator at each corner. The stability of the proposed time-varying estimator is investigated and its performance is examined experimentally in several tests and on different road surface frictions. Road experiments and simulation results show the accuracy and robustness of the proposed approach in estimating longitudinal speed for ground vehicles.     

A 3D Brush-type Dynamic Tire Friction Model

The use of advanced dynamic friction models can improve the brush-type tire friction models. This paper presents a 3D dynamic brush model based on the LuGre friction model. The model describes the dynamics of longitudinal and lateral tire friction forces, as well as the self aligning torque dynamics. It has been originally derived in a distributed-parameter form, and then transformed to a simpler lumped-parameter form with only three internal states. Both uniform and non-uniform normal pressure distributions are considered. The model has analytical solution for steady-state conditions. The steady-state behavior is validated with respect to “magic” formula static model, which served as an “ideal” benchmark. The lumped model dynamic behavior is validated by comparing its time-responses with original distributed model responses. The model parameterization with respect to normal force and other tire/road parameters is considered as well.     

A 3D Brush-type Dynamic Tire Friction Model

The use of advanced dynamic friction models can improve the brush-type tire friction models. This paper presents a 3D dynamic brush model based on the LuGre friction model. The model describes the dynamics of longitudinal and lateral tire friction forces, as well as the self aligning torque dynamics. It has been originally derived in a distributed-parameter form, and then transformed to a simpler lumped-parameter form with only three internal states. Both uniform and non-uniform normal pressure distributions are considered. The model has analytical solution for steady-state conditions. The steady-state behavior is validated with respect to “magic” formula static model, which served as an “ideal” benchmark. The lumped model dynamic behavior is validated by comparing its time-responses with original distributed model responses. The model parameterization with respect to normal force and other tire/road parameters is considered as well.     

3D brush model to predict longitudinal tyre characteristics

A 3D tyre brush model, which aims to predict the longitudinal tyre characteristic under steady-state conditions by modelling the occurring physical effects in the tyre–road contact patch, is presented. The model includes an analytical method to describe the tyre footprint geometry, the pressure distribution, the slip due to the lateral tyre contour, the slip due to braking or traction and the longitudinal as well as the lateral shear stresses on a flattened tyre. The presented development tool offers a method to investigate different rubber friction data (caused by different tread compounds and/or surface textures) and to analyse its influence on longitudinal tyre characteristics. The tyre design is fixed (same casing, dimension and pattern). The results include the shear stresses as well as the different sliding velocities in the contact patch for different slip conditions. The model was developed for a standard summer pattern design and a standard tyre dimension (205/55R16). It can also be adapted to other tread designs and tyre dimensions. To offer a good comparability between model results and test bench measurements, the surface curvature of an internal test rig is considered.     

A real-time approach to robust identification of tyre–road friction characteristics on mixed-μ roads

ABSTRACT

The interaction between the tyre and the road is crucial for understanding the dynamic behaviour of a vehicle. The road–tyre friction characteristics play a key role in the design of braking, traction and stability control systems. Thus, in order to have a good performance of vehicle dynamic stability control, real-time estimation of the tyre–road friction coefficient is required. This paper presents a new development of an on-line tyre–road friction parameters estimation methodology and its implementation using both LuGre and Burckhardt tyre–road friction models. The proposed method provides the capability to observe the tyre–road friction coefficient directly using measurable signals in real-time. In the first step of our approach, the recursive least squares is employed to identify the linear parameterisation form of the Burckhardt model. The identified parameters provide, through a T–S fuzzy system, the initial values for the LuGre model. Then, a new LuGre model-based nonlinear least squares parameter estimation algorithm using the proposed static form of the LuGre to obtain the parameters of LuGre model based on recursive nonlinear optimisation of the curve fitting errors is presented. The effectiveness and performance of the algorithm are demonstrated through the real-time model simulations with different longitudinal speeds and different kinds of tyres on various road surface conditions in both Matlab/Carsim environments as well as collected data from real experiments on a commercial trailer.     

Effective assessment of tyre–road friction coefficient using a hybrid estimator

Vehicle stability and active safety control depend heavily on tyre forces available on each wheel of a vehicle. Since tyre forces are strongly affected by the tyre–road friction coefficient, it is crucial to optimise the use of the adhesion limits of the tyres. This study presents a hybrid method to identify the road friction limitation; it contributes significantly to active vehicle safety. A hybrid estimator is developed based on the three degrees-of-freedom vehicle model, which considers longitudinal, lateral and yaw motions. The proposed hybrid estimator includes two sub-estimators: one is the vehicle state information estimator using the unscented Kalman filter and another is the integrated road friction estimator. By connecting two sub-estimators simultaneously, the proposed algorithm can effectively estimate the road friction coefficient. The performance of the proposed estimation algorithm is validated in CarSim/Matlab co-simulation environment under three different road conditions (high-μ, low-μ and mixed-μ). Simulation results show that the proposed estimator can assess vehicle states and road friction coefficient with good accuracy.     

Time-optimal control of the race car: influence of a thermodynamic tyre model

The robustness of an existing numerical method for the time-optimal control of the race car is demonstrated through its application to a model of a Formula 1 car equipped with a simplified thermodynamic tyre model. The tyre model includes a temperature- and frequency-dependent model of road/tyre friction. A lumped parameter approach is used to model the thermodynamics of the various parts of the tyre such as the tread, carcass and inflation gas. The influence of tyre, track surface and ambient temperatures on time-optimal manoeuvring is presented.     

Investigation of Features of Tyre Rolling at Non-Small Velocities on the Basis of a Simple Tyre Model with Distributed Mass Periphery

SUMMARY

On the basis of the brush-type tyre model the paper considers the interaction between steady-state rolling deformable wheel and flat road surface as well as corresponding force and moment characteristics of the wheel.

At least two zones of sliding, anisotropic dry friction, sliding friction coefficient speed-dependent and instantaneous leap of the friction coefficient when transition from sliding to adhesion zone occurs, have been taken into account, as well as distributed peripheral mass of tyre, elasticity, pseudo-dry friction and damping properties in radial, tangential and lateral directions of the elements at the wheel periphery, including a visco-elastic belt. Vertical force distribution in the contact area is not supposed to be known in advance and follows from the calculation. As a result, sliding zone lengths, distributed forces in contact area, six components of generalized road reaction reduced to the wheel center, and rolling resistance moment are found as functions of vertical load, movement velocity, longitudinal and side slip, friction in contact area with road, stiffnesses, dry friction and damping in the tyre model elements and of distributed peripheral mass.

A computer program developed in Fortran and results of calculations are of particular interest for qualitative analysis including steady rolling of studded tyre and also racing car and aircraft tyres which peripheral mass shows itself in a special way because of great movement velocities.     

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.     

轮胎垂直振动动力学模型

本文提出一种自适应轮膜模型参数的简易测定方法，给出轮胎垂直振动动力学方程及相应的求解方法，并进行计算分析与试验测定。将分析结果与通常的点接触式模型相比较，对点接触式模型及本文所建立的自适应模型做了综合评价，结果表明，本模型的计算结果是令人满意的。     

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.     

Tyre friction behaviour under abrupt wheel torque transients on slippery road surfaces: experimental analysis and modelling

The paper shows that, during abrupt wheel torque transients for ice surface and low vehicle speeds, the tyre can develop significantly larger longitudinal force than the peak value of the tyre static curve. This so-called dynamic tyre friction potential (DTFP) effect has many influencing factors such as the rate of change of the wheel torque, the vehicle speed, and the tyre dwell time. The paper presents a detailed analysis of the DTFP behaviour based on the experimental data collected by using an in-wheel motor-based tyre test vehicle. The analysis results and an insight into the brush structure of a tyre model lead to the hypothesis that the different influencing factors may be predominantly explained by the bristle dwell time (BDT) effect. Following this hypothesis, the LuGre model of the tyre friction dynamics is extended with a physical BDT sub-model. The experimental validation results show that the proposed model can accurately capture the low-speed tyre–ice friction behaviour during abrupt wheel torque transients.     

An optimal pole-matching observer design for estimating tyre–road friction force

In this paper, considering the dynamical model of tyre–road contacts, we design a nonlinear observer for the on-line estimation of tyre–road friction force using the average lumped LuGre model without any simplification. The design is the extension of a previously offered observer to allow a muchmore realistic estimation by considering the effect of the rolling resistance and a term related to the relative velocity in the observer. Our aim is not to introduce a new friction model, but to present a more accurate nonlinear observer for the assumed model. We derive linear matrix equality conditions to obtain an observer gain with minimum pole mismatch for the desired observer error dynamic system. We prove the convergence of the observer for the non-simplified model. Finally, we compare the performance of the proposed observer with that of the previously mentioned nonlinear observer, which shows significant improvement in the accuracy of estimation.     

Analytical dynamic tire model

Modeling of tire friction is one of the central problems for vehicle control systems design. LuGre-type dynamic tire model has been proposed and well discussed in previous studies, because it offers a compact form of dynamic model that is convenient in advanced control studies. It has been successfully used in tire slip control design and vehicle state estimation problems. In this article, a concept of time-constrained Stribeck effect is introduced to interpret the mechanism of the LuGre friction model in predicting tire friction characteristics. A modified two-dimensional (2D) dynamic LuGre friction model is introduced to make it compatible with the governing theorem in the steady state. An analytical 2D modified LuGre-type dynamic tire model is developed, in which some fundamental limitations of classical LuGre models are eliminated. The main modifications involve a change in the structure of the 2D LuGre friction model, introduction of load-dependent parameters in 1D and 2D tire models, and a changed structure in the distributed parameter model. The proposed model is compared, in the steady state, to both the Magic Formula and the classical LuGre model. It improves model accuracy in the steady state and gives a physically reasonable distribution of the bristle deflection. A first-order lumped parameter (LP) nonlinear model, which has simpler structure than the distributed parameter model and the classical LP LuGre model, is then derived. Numerical simulations show that the proposed LP model has a good estimation for tire transient dynamics. Thus, the proposed model retains the merits of LuGre-type models and improves the agreement with observation and experimental data on friction force distribution along the patch and on the steady-state friction prediction.     

Analytical dynamic tire model

Modeling of tire friction is one of the central problems for vehicle control systems design. LuGre-type dynamic tire model has been proposed and well discussed in previous studies, because it offers a compact form of dynamic model that is convenient in advanced control studies. It has been successfully used in tire slip control design and vehicle state estimation problems. In this article, a concept of time-constrained Stribeck effect is introduced to interpret the mechanism of the LuGre friction model in predicting tire friction characteristics. A modified two-dimensional (2D) dynamic LuGre friction model is introduced to make it compatible with the governing theorem in the steady state. An analytical 2D modified LuGre-type dynamic tire model is developed, in which some fundamental limitations of classical LuGre models are eliminated. The main modifications involve a change in the structure of the 2D LuGre friction model, introduction of load-dependent parameters in 1D and 2D tire models, and a changed structure in the distributed parameter model. The proposed model is compared, in the steady state, to both the Magic Formula and the classical LuGre model. It improves model accuracy in the steady state and gives a physically reasonable distribution of the bristle deflection. A first-order lumped parameter (LP) nonlinear model, which has simpler structure than the distributed parameter model and the classical LP LuGre model, is then derived. Numerical simulations show that the proposed LP model has a good estimation for tire transient dynamics. Thus, the proposed model retains the merits of LuGre-type models and improves the agreement with observation and experimental data on friction force distribution along the patch and on the steady-state friction prediction.     

Total dynamic response of a PSS vehicle negotiating asymmetric road excitations

A planar suspension system (PSS) is a novel automobile suspension system in which an individual spring–damper strut is implemented in both the vertical and longitudinal directions, respectively. The wheels in a vehicle with such a suspension system can move back and forth relative to the chassis. When a PSS vehicle experiences asymmetric road excitations, the relative longitudinal motion of wheels with respect to the chassis in two sides of the same axle are not identical, and thus the two wheels at one axle will not be aligned in the same axis. The total dynamic responses, including those of the bounce, pitch and the roll of the PSS vehicle, to the asymmetric road excitation may exhibit different characteristics from those of a conventional vehicle. This paper presents an investigation into the comprehensive dynamic behaviour of a vehicle with the PSS, in such a road condition, on both the straight and curved roads. The study was carried out using an 18 DOF full-car model incorporating a radial-spring tyre–ground contact model and a 2D tyre–ground dynamic friction model. Results demonstrate that the total dynamic behaviour of a PSS vehicle is generally comparable with that of the conventional vehicle, while PSS exhibits significant improvement in absorbing the impact forces along the longitudinal direction when compared to the conventional suspension system. The PSS vehicle is found to be more stable than the conventional vehicle in terms of the directional performance against the disturbance of the road potholes on a straight line manoeuvre, while exhibiting a very similar handling performance on a curved line.