Dynamic tyre friction models for combined longitudinal and lateral vehicle motion

An extension to the LuGre dynamic friction model from longitudinal to longitudinal/lateral motion is developed in this paper. Application of this model to a tyre yields a pair of partial differential equations that model the tyre-road contact forces and aligning moment. A comparison of the steady-state behaviour of the dynamic model with existing static tyre friction models is presented. This comparison allows one to determine realistic values of the parameters for the new dynamic model. Via the introduction of a set of mean states we reduce the partial differential equations to a lumped model governed by a set of three ordinary differential equations. Such a lumped form describes the aggregate effect of the friction forces and moments and it can be useful for control design and online estimation. A method to incorporate wheel rim rotation is also proposed. The proposed model is evaluated by comparing both its steady-state as well as its dynamic characteristics via numerical simulations. The results of the simulations corroborate steady-state and dynamic/transient tyre characteristics found in the literature.     

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.     

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

SUMMARY

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.     

A theoretical model of speed-dependent steering torque for rolling tyres

ABSTRACT

It is well known that the tyre steering torque is highly dependent on the tyre rolling speed. In limited cases, i.e. parking manoeuvre, the steering torque approaches the maximum. With the increasing tyre speed, the steering torque decreased rapidly. Accurate modelling of the speed-dependent behaviour for the tyre steering torque is a key factor to calibrate the electric power steering (EPS) system and tune the handling performance of vehicles. However, no satisfactory theoretical model can be found in the existing literature to explain this phenomenon. This paper proposes a new theoretical framework to model this important tyre behaviour, which includes three key factors: (1) tyre three-dimensional transient rolling kinematics with turn-slip; (2) dynamical force and moment generation; and (3) the mixed Lagrange–Euler method for contact deformation solving. A nonlinear finite-element code has been developed to implement the proposed approach. It can be found that the main mechanism for the speed-dependent steering torque is due to turn-slip-related kinematics. This paper provides a theory to explain the complex mechanism of the tyre steering torque generation, which helps to understand the speed-dependent tyre steering torque, tyre road feeling and EPS calibration.     

A comprehensive study on the stability analysis of vehicle dynamics with pure/combined-slip tyre models

In this paper, a vehicle's lateral dynamic model is developed based on the pure and the combined-slip LuGre tyre models. Conventional vehicle's lateral dynamic methods derive handling models utilising linear tyres and pure-slip assumptions. The current article proposes a general lateral dynamic model, which takes the linear and nonlinear behaviours of the tyre into account using the pure and combined-slip assumptions separately. The developed methodology also incorporates various normal loads at each corner and provides a proper tyre–vehicle platform for control and estimation applications. Steady-state and transient LuGre models are also used in the model development and their responses are compared in different driving scenarios. Considering the fact that the vehicle dynamics is time-varying, the stability of the suggested time-varying model is investigated using an affine quadratic stability approach, and a novel approach to define the critical longitudinal speed is suggested and compared with that of conventional lateral stability methods. Simulations have been conducted and the results are used to validate the proposed method.     

An Extended Adaptive Kalman Filter for Real-time State Estimation of Vehicle Handling Dynamics

This paper considers a method for estimating vehicle handling dynamic states in real-time, using a reduced sensor set; the information is essential for vehicle handling stability control and is also valuable in chassis design evaluation. An extended (nonlinear) Kalman filter is designed to estimate the rapidly varying handling state vector. This employs a low order (4 DOF) handling model which is augmented to include adaptive states (cornering stiffnesses) to compensate for tyre force nonlinearities. The adaptation is driven by steer-induced variations in the longitudinal vehicle acceleration. The observer is compared with an equivalent linear, model-invariant Kalman filter. Both filters are designed and tested against data from a high order source model which simulates six degrees of freedom for the vehicle body, and employs a combined-slip Pacejka tyre model. A performance comparison is presented, which shows promising results for the extended filter, given a sensor set comprising three accelerometers only. The study also presents an insight into the effect of correlated error sources in this application, and it concludes with a discussion of the new observer's practical viability.     

Noise,vibration, harshness model of a rotating tyre

ABSTRACT

The tyre plays a fundamental role in the generation of acoustically perceptible driving noise and vibrations inside the vehicle. An essential part of these vibrations is induced by the road excitation and transferred via the tyre into the vehicle. There are two basic ways to study noise, vibration, harshness (NVH) behaviour: Simulations in time and frequency domains. Modelling the tyre transfer behaviour in frequency domain requires special attention to the rotation of the tyre. This paper shows the approach taken by the authors to include the transfer behaviour in the frequency range up to 250?Hz from geometric road excitations to resulting spindle forces in frequency domain. This paper validates the derived NVH tyre model by comparison with appropriate transient simulations of the base transient model.     

A thermo-frictional tyre model including the effect of flash temperature

A new tyre model is developed that can predict the influence of both macroscopic and local flash temperature on tyre force generation. The model comprises two heat-transfer solvers. A macroscopic solver calculates the 3D temperature distribution across the tread and sidewall at a resolution of a few millimetres. A separate flash-temperature solver calculates the local hot-spot temperature distribution at the macro-asperity tyre-road contact interface at a resolution of micrometres. The two heat-transfer solvers are coupled with a structural model for the calculation of tyre forces and the sliding speed distribution along the contact patch. The sliding speed distribution feeds into the flash-temperature model and the local coefficient of friction is found as a function of sliding speed, flash temperature, normal pressure, road roughness and the complex modulus of rubber. The proposed tyre model is the first to include the effect of a changing macroscopic temperature distribution on the build-up of the local flash temperature, and to account for road-tread conduction at the macro-asperity contact interface. The model is applicable for identifying the friction envelope and optimum temperature range for tyres on roads with known roughness. This is important in motorsport where knowledge of grip offers a competitive advantage.     

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.     

The Influence of In-Plane Tyre Dynamics on ABS Braking of a Quarter Vehicle Model

For efficient analysis it is important to choose the proper model that fits the problem that needs to be solved. This paper discusses three pragmatic simulation models for longitudinal behaviour of a passenger car tyre (Steady State, Transient and Rigid Ring) that may be used in e.g. an ABS simulation. The characteristics of the simulation models are evaluated using some simple simulations. Simulations with a quarter vehicle model that includes load transfer effects are carried out to determine the deviation in results between the mentioned tyre models for an ABS application. The results show that the Steady State model may only be used below 10 Hz and that the Transient model is valid up to about 30 Hz. The results from the ABS simulation with the Rigid Ring model are most reliable and are clearly different from the Steady State and Transient model, which indicates that ABS simulations should be carried out with the Rigid Ring model. Additionally it is demonstrated that for tyre behaviour on uneven roads the influence of the tyre belt cannot be neglected.     

The Influence of In-Plane Tyre Dynamics on ABS Braking of a Quarter Vehicle Model

For efficient analysis it is important to choose the proper model that fits the problem that needs to be solved. This paper discusses three pragmatic simulation models for longitudinal behaviour of a passenger car tyre (Steady State, Transient and Rigid Ring) that may be used in e.g. an ABS simulation. The characteristics of the simulation models are evaluated using some simple simulations. Simulations with a quarter vehicle model that includes load transfer effects are carried out to determine the deviation in results between the mentioned tyre models for an ABS application. The results show that the Steady State model may only be used below 10 Hz and that the Transient model is valid up to about 30 Hz. The results from the ABS simulation with the Rigid Ring model are most reliable and are clearly different from the Steady State and Transient model, which indicates that ABS simulations should be carried out with the Rigid Ring model. Additionally it is demonstrated that for tyre behaviour on uneven roads the influence of the tyre belt cannot be neglected.     

Pragmatic Tyre Model for Short Wavelength Side Slip Variations

The first-order relaxation length concept is often used to model transient tyre behaviour. It gives a rather good representation of the lateral force response, also at shorter wavelengths. The self aligning moment appears to behave rather differently and its responses cannot be represented by a relaxation length system only. Based on the analytical frequency response functions of the pneumatic trail of the brush type tyre model, a new pragmatic approach is developed. The model consists of a phase leading system in series with the first-order model for the lateral force. The results are compared to the responses of a discrete brush type simulation model. It appears that the aligning moment can be represented by this new pragmatic model very well.     

Pragmatic Tyre Model for Short Wavelength Side Slip Variations

The first-order relaxation length concept is often used to model transient tyre behaviour. It gives a rather good representation of the lateral force response, also at shorter wavelengths. The self aligning moment appears to behave rather differently and its responses cannot be represented by a relaxation length system only. Based on the analytical frequency response functions of the pneumatic trail of the brush type tyre model, a new pragmatic approach is developed. The model consists of a phase leading system in series with the first-order model for the lateral force. The results are compared to the responses of a discrete brush type simulation model. It appears that the aligning moment can be represented by this new pragmatic model very well.     

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.     

Design of tyre force excitation for tyre–road friction estimation

Knowledge of the current tyre–road friction coefficient is essential for future autonomous vehicles. The environmental conditions, and the tyre–road friction in particular, determine both the braking distance and the maximum cornering velocity and thus set the boundaries for the vehicle. Tyre–road friction is difficult to estimate during normal driving due to low levels of tyre force excitation. This problem can be solved by using active tyre force excitation. A torque is added to one or several wheels in the purpose of estimating the tyre–road friction coefficient. Active tyre force excitation provides the opportunity to design the tyre force excitation freely. This study investigates how the tyre force should be applied to minimise the error of the tyre–road friction estimate. The performance of different excitation strategies was found to be dependent on both tyre model choice and noise level. Furthermore, the advantage with using tyre models with more parameters decreased when noise was added to the force and slip ratio.     

A finite-element-based approach to characterising FTire model for extended range of operation conditions

In order to accurately predict vehicle dynamic responses when traversing high obstacles or large bumps, appropriate tyre models need to be developed and characterised. Tyre models used in vehicle ride and durability are usually characterised by experimental tests on the tyre. However, limitations in rig design and operating conditions restrict the range of test conditions under which the tyre can be tested, hence characterisation of the tyre behaviour during extreme manoeuvres may not be possible using physical tests. In this study, a combination of experimental tests and finite-element (FE) modelling is used in deriving Flexible Ring Tire (FTire) Models appropriate for different levels of tyre/road interaction severity. It is shown that FE modelling can be used to accurately characterise the behaviour of a tyre where limitations in experimental facilities prevent tyre characterisation using the required level of input severity in physical tests. Multi-body simulation is used to demonstrate that the FTire model derived using extended range of obstacles produces more accurate transient dynamic response when traversing low and high road obstacles.     

Comparison of various double-lane change manoeuvre specifications

One of the commonly used performance measures to quantify a vehicle's handling transient dynamics is the maximum forward speed (MFS) while passing a certain specified double-lane change (DLC) manoeuvre without violating the boundary and tyre lift-off. The MFS is directly associated with the minimum curvature radius (MCR) of the vehicle centre of gravity (CG) trajectory controlled by the driver during the manoeuvre. The MCR is further affected by the vehicle dimensions to meet the boundary condition. In this study, a single heavy vehicle CG trajectory is assumed to be a combination of three straight lines and two third-order spline curves. A heavy vehicle multi-body system model established with ADAMS/Car is correlated with test data for step-steer and constant radius cornering events, and then the model is used to demonstrate that the assumptions considered in the formulation applied in this paper are valid for this specific vehicle category. The MCRs of four heavy vehicles are maximised among all the possible choices of the vehicle CG trajectory during each of five specific DLC manoeuvres, including North Atlantic Treaty Organization (Allied Vehicle Testing Publication 03-160W), International Organization for Standardization (ISO) 3888-1, ISO 3888-2, Consumer Union Short Course and Test Operations Procedure 2-2-609. The maximised MCR (MMCR), considered as the best possible choice of vehicle CG trajectories, is further solved as a function of the vehicle width and length. The results will show the sensitivity of the MMCR to the vehicle length and width, thus the impact on the vehicle transient handling dynamics. Finally, the comparison of five DLC specifications may help users to correlate a vehicle's MFS from one specification to others.