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.     

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.     

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.     

轮胎稳态模型的分析综述

轮胎稳态模型描述了轮胎稳态运动过程的纵滑、侧偏特性,可以分为理论模型、经验模型和半经验模型.常用的理论模型包括线性模型、UA模型、Dugoff模型、刷子模型和LuGre模型;常用的经验模型包括多项式模型、Burckhardt模型、K-D模型和LC模型;常用的半经验模型包括魔术公式模型和UniTire模型.通过仿真分析,对各种模型描述轮胎运动状态的能力、复杂性、准确性和适用范围进行了比较和研究,对各种模型存在的局限性进行了分析,可为轮胎稳态特性分析和汽车控制系统设计选择轮胎模型提供依据.     

Emergency Braking Control with an Observer-based Dynamic Tire/Road Friction Model and Wheel Angular Velocity Measurement

Summary A control scheme for emergency braking of vehicles is designed. The tire/road friction is described by a LuGre dynamic friction model. The control system output is the pressure in the master cylinder of the brake system. The controller utilizes estimated states for a feedback control law that achieves a near maximum deceleration. The state observer is designed using linear matrix inequality (LMI) techniques. The analysis shows that using the wheel angular speed information exclusively is not sufficient to rapidly estimate the velocity and relative velocity, due to the fact that the dynamical system is almost unobservable with this measurement as output. Findings are confirmed by simulation results that show that the estimated vehicle velocity and relative velocity converge slowly to their true values, even though the internal friction state and friction parameters converge quickly. The proposed control system has two main advantages when compared with an antilock braking system (ABS): (1) it produces a source of a priori information regarding safe spacing between vehicles that can be used to increase safety levels in the highway; and (2) it achieves a near optimal braking strategy with less chattering.     

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.     

Emergency Braking Control with an Observer-based Dynamic Tire/Road Friction Model and Wheel Angular Velocity Measurement

Summary A control scheme for emergency braking of vehicles is designed. The tire/road friction is described by a LuGre dynamic friction model. The control system output is the pressure in the master cylinder of the brake system. The controller utilizes estimated states for a feedback control law that achieves a near maximum deceleration. The state observer is designed using linear matrix inequality (LMI) techniques. The analysis shows that using the wheel angular speed information exclusively is not sufficient to rapidly estimate the velocity and relative velocity, due to the fact that the dynamical system is almost unobservable with this measurement as output. Findings are confirmed by simulation results that show that the estimated vehicle velocity and relative velocity converge slowly to their true values, even though the internal friction state and friction parameters converge quickly. The proposed control system has two main advantages when compared with an antilock braking system (ABS): (1) it produces a source of a priori information regarding safe spacing between vehicles that can be used to increase safety levels in the highway; and (2) it achieves a near optimal braking strategy with less chattering.     

基于车路系统的事故现场轮胎印迹强度参数化研究

为明确事故现场可视轮胎印迹强度与车辆动力学特性、轮胎橡胶磨损特征及道路表面灰度之间的关联特性,提出基于车路耦合的事故现场轮胎印迹强度参数化研究方法。通过结合动态滑动摩擦因数模型及轮胎非线性模型,建立车辆路面9 DOF非线性系统动力学模型,运用VBOX惯性测量技术验证模型的有效性。运用胎面磨损能量模型,从车路系统角度确定车辆、轮胎和路面特性对轮胎全局摩擦力及胎面磨损特性的影响。结合印迹强度特征模型提出轮胎印迹强度参数研究方法,选取不同制动、转向角工况及3组路面、胎面特性对轮胎路面接地力学特性、胎面橡胶磨损量、可视轮胎印迹特征进行仿真分析。结果表明：印迹强度仅与全局摩擦力大小有关,与轮胎路面滑移方向无关;滑移工况下胎面橡胶磨损量随着全局摩擦力和滑移速度的增大而增大,而印迹强度变化不明显;制动力矩和道路表面灰度对产生可视轮胎印迹起决定作用,转向角主要影响不规则可视轮胎印迹的产生;前轮轮胎最先出现可视印迹,且可视印迹长度和强度均高于后轮轮胎;采取可视印迹起点作为事故车辆速度判定具有一定的误差,应根据具体情况进行具体分析;研究成果能够为基于可视轮胎印迹的交通事故重建提供理论基础。     

轮胎半经验模型中摩擦系数切换问题

介绍了2种简单的摩擦系数模型(库仑摩擦定律、静摩擦力模型)，并将其与轮胎统一刷子理论模型结合起来分析轮胎的摩擦问题。在此基础上引入了满足轮胎刷子理论模型边界条件的轮胎稳态半经验模型，给出了应用轮胎半经验模型实现摩擦状态切换的方法。通过轮胎在2种滚动速度下的侧偏试验，证明了轮胎半经验模型可实现2种速度下的摩擦系数切换。     

Analysis of handling performance based on simplified lateral vehicle dynamics

In this article, the analysis methods for vehicle handling performance are studied. Using simple models, dynamic characteristic parameters such as yaw, natural frequency, and the damping coefficient of a vehicle can be theoretically formulated. Here, the vehicle is simplified by a bicycle (single-track) model, and the tire is modeled by an equivalent cornering stiffness and first order lag. From the experimental road data, the tire model parameters (equivalent cornering stiffness and time lag constant) are extracted. These parameters are then inserted into the theoretically formulated equations of dynamic characteristic parameters. For the purpose of validating the efficiency of the suggested methods, experimental road tests (where the cars have different handling performances) are performed. The results show that vehicle handling performance can be sufficiently represented by the suggested dynamic characteristic parameters. So, it is concluded that the proposed method has practical use for the development of new cars or for the comparison of similar cars since the evaluations of the vehicle handling performance can be efficiently determined by the suggested dynamic characteristic parameters.     

Sliding-mode-based parameter identification with application to tire pressure and tire-road friction

This paper presents a method of simultaneous estimation of tire pressure and tire-road friction. A sliding-mode scheme is designed to identify the system state and the parameter variation of a torsional tire system, which greatly depend on the change in tire pressure. Then, the recursive least-squares method with a forgetting facto is used to estimate the parameter variations of the tire system and the tire-road friction force without a friction model using the information retrieved from the equivalent input for sliding motion. A simulation study is performed to illustrate the effectiveness of the proposed method.     

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.     

用于公路车-桥系统振动分析的精细化轮胎模型

为了解决当前公路车桥耦合振动模型中轮胎模型过于简化、车轮-路面接触力与桥梁响应计算结果不够精确的问题,提出了一种精细化轮胎模型.首先基于车辆橡胶轮胎的几何、力学特征,建立了径向弹簧力学模型并进行了理论推导;然后考虑轮胎与路面接触面的刚度分布特征和高速状况下轮胎的惯性力,提出了轮胎接触面分布刚度的计算方法,保证了轮胎接触...     

Integrated Control of Active Rear Wheel Steering and Direct Yaw Moment Control

An integrated control system of active rear wheel steering (4WS) and direct yaw moment control (DYC) is presented in this paper. Because of the tire nonlinearity that is mainly due to the saturation of cornering forces, vehicle handling performance is improved but limited to a certain extent only by steering control. Direct yaw moment control using braking and/or driving forces is effective not only in linear but also nonlinear ranges of tire friction circle. The proposed control system is a model matching controller which makes the vehicle follow the desired dynamic model by the state feedback of both yaw rate and side slip angle. Various computer simulations are carried out and show that vehicle handling performance is much improved by the integrated control system.     

Integrated Control of Active Rear Wheel Steering and Direct Yaw Moment Control

SUMMARY

An integrated control system of active rear wheel steering (4WS) and direct yaw moment control (DYC) is presented in this paper. Because of the tire nonlinearity that is mainly due to the saturation of cornering forces, vehicle handling performance is improved but limited to a certain extent only by steering control. Direct yaw moment control using braking and/or driving forces is effective not only in linear but also nonlinear ranges of tire friction circle. The proposed control system is a model matching controller which makes the vehicle follow the desired dynamic model by the state feedback of both yaw rate and side slip angle. Various computer simulations are carried out and show that vehicle handling performance is much improved by the integrated control system.     

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.     

Parameter identification for a LuGre model based on steady-state tire conditions

The purpose of this study was to effectively identify parameters for a LuGre friction model based on experimental measures. In earlier work related to this study (Yang et al., 2009), which was based on the characters of polygonal wear (Sueoka and Ryu, 1997), we showed a frictional vibration model for a mass on a moving belt. This model reflected lateral vibrations caused by velocity and toe-in angle. An important aspect of the present study is the improved friction model. A previous friction model, which divided the process into four parts, expressed the sable excited vibration well but failed to reflect the hysteresis loop change when vehicles accelerated or decelerated continuously. A LuGre friction model can solve this problem, but several model parameters must be obtained experimentally. We measured contact width and length of tires as vertical pressure changed; this provided a theoretical basis for apparent stiffness of a unit of tire tread. Based on tire data from Bakker E’s article in a SAE paper from 1987, we identified the Stribeck exponent and Stribeck velocity in LuGre. Then, the results were implemented in a vibration system that verified the rationality of the data.     

Estimation of Tire-Road Friction Using Observer Based Identifiers

This paper presents methods for identifying the tire-road friction coefficient. The proposed methods are: an observer-based least square method and an observer/filtered-regressor-based method. These methods were designed assuming that some of the states are not available since physical parameter identification methods developed assuming that the system states are available are not attractive from a practical point of view. The observer is used to estimate signals which are difficult or expensive to measure. Using the estimated states of the system and the filtered-regressor, the parameter estimates are obtained. The proposed methods are evaluated on an eight state nonlinear vehicle/transmission simulation model with a Bakker-Pacejka's formula tire model. Vehicle tests have been performed on dry and wet roads to verify the performance of the methods. It has been shown through simulations and vehicle tests how the RPM sensors can be used with observer based identification methods to estimate the tire-road friction from measurements of engine rpm, transmission output speed and wheel speed. The proposed methods will be useful in the implementation and adaptation of vehicle collision warning/avoidance algorithm since the tire-road friction can be estimated only using the RPM sensors which are currently being used in production vehicles.