A Study on Construction of Tire Model with Viscoelasticity

<div class="htmlview paragraph">Various vehicle dynamic simulation software programs have been developed for use in reconstructing accidents. Typically these are used to analyze and reconstruct preimpact and postimpact vehicle motion. These simulation programs range from proprietary programs to commercially available packages. While the basic theory behind these simulations is Newton's laws of motion, some component modeling techniques differ from one program to another. This is particularly true of the modeling of tire force mechanics. Since tire forces control the vehicle motion predicted by a simulation, the tire mechanics model is a critical feature in simulation use, performance and accuracy. This is particularly true for accident reconstruction applications where vehicle motions can occur over wide ranging kinematic wheel conditions. Therefore a thorough understanding of the nature of tire forces is a necessary aspect of the proper formulation and use of a vehicle dynamics program.</div> <div class="htmlview paragraph">This paper includes a discussion of tire force terminology, tire force mechanics, the measurement and modeling of tire force components and combined tire force models currently used in simulation software for the reconstruction of accidents. The paper discusses the difference between the idealized tire force ellipse and an actual tire friction ellipse. Equations are presented for five tire force models from three different simulation programs. Each model uses a different method for computing tire forces for combined braking and steering. Some experimentally measured light vehicle tire properties are examined.</div> <div class="htmlview paragraph">Some tire force models begin with a specified level of braking force and use the friction ellipse to determine the corresponding steering force; this produces steering forces and a resultant tire force equal in magnitude to full skidding for combined steering and braking. Comparisons are presented of results from simulation programs using different tire models for vehicle motions involving two types of severe yaw. The comparisons in this paper are not of reconstructions where the user seeks initial conditions to match an existing trajectory. The first comparison is a hypothetical postimpact motion with a given initial velocity and initial angular velocity and the other is a sudden steer maneuver. In some cases, the simulations and their tire models predict the vehicle motion closely. In most cases, however, the results differ significantly between simulation programs.</div> <div class="htmlview paragraph">The example simulations presented in this paper are not intended to reflect the way vehicle dynamic simulation programs are used typically in accident reconstruction.</div>

<p>The objective of this study is to integrate a continuum-based deformable tire and terrain interaction model into a general-use physics-based simulation environment capable of off-road vehicle mobility analysis and high-performance computing potential. Specifically, the physics-based deformable tire and terrain models which were recently proposed and validated by Yamashita, et al. will be implemented into the structure of the multi-physics simulation engine Chrono. In off-road vehicle mobility analysis, empirical and analytical models have been commonly used for vehicle-terrain interaction. While these models utilize experimental data or terramechanics theories to create quick predictive mobility models, they are unable to capture the highly nonlinear behavior of soft soil deformation, which can lead to inaccurate or unreliable results. In order to resolve these limitations, the use of physics-based numerical approaches have been proposed. These methods make use of finite element and discrete element simulations to describe the interaction between the vehicle and deformable terrain. Continuum-based finite element models transfer tire forces to the terrain and model the deformation with elasto-plastic constitutive models. Discrete element soil uses a large number of small rigid body particles to describe the microscale behavior of granular terrain, with the deformation of the soil represented by the motion and contact of the particles. While these physics-based models offer a more accurate vehicle-terrain interaction model, the solution procedure can become complex and computationally expensive since co-simulation techniques are often used.</p><p>To address these issues, the analysis of physics-based full vehicle dynamics simulations utilizing high-fidelity deformable tire and terrain models in a multi-physics engine with high-performance computing capability is desired. To this end, a continuum mechanics based shear deformable laminated composite shell element proposed by Yamashita, et al. was integrated into the flexible body dynamics simulation framework of Chrono. This element was based on the absolute nodal coordinate formulation and is defined by the global position coordinates and the transverse gradient coordinates of its four nodes. Element lockings are eliminated with the incorporation of the enhanced assumed strain (EAS) and assumed natural strain approaches (ANS). The element formulation includes an extension to model laminated composite materials. Additionally, a locking-free 9-node brick element was integrated into the Chrono framework that makes use of the curvature coordinates at the center of the element. This element is formulated with the Hencky strain measure such that multiplicative finite strain plasticity theory can be used to incorporate soil plasticity models, such as the capped Drucker-Prager failed criterion.</p><p>With the shear deformable laminated composite shell element and plastic soil brick element integrated into the Chrono multi-physics simulation engine, an off-road deformable tire and terrain interaction model was developed using the vehicle dynamics simulation module Chrono::Vehicle. An off-road deformable tire model was parameterized based on commercial tire properties and generated as an interchangeable tire model option in the full vehicle dynamics system. Benchmark verification tests were performed to ensure the accuracy of tire deformation and tire force characteristics. Further tests were performed to validate a deformable tire model with a deformable tread pattern constructed from shear deformable shell elements and co-rotational tetrahedral elements. The deformable soil model was also integrated as a terrain option in Chrono::Vehicle and numerical tests were carried out to demonstrate its interaction with rigid and deformable tire models. To make use of the computational performance enhancements available in Chrono, Open Multi-Processing (OpenMP) and Advanced Vector Extensions (AVX) were applied to the evaluation of the elastic force/Jacobian matrix and large matrix operations of flexible bodies, respectively, in order to reduce the computation time by nearly 60%.</p>

<div class="section abstract"><div class="htmlview paragraph">Tread depth, inflation pressure, tire temperature, and road surface condition are among the most notable factors that have a noticeable effect on the tire force and moment characteristics. They can vary significantly during the operation of a tire and can effectively modify tire (and thus vehicle) performance. This study presents details of an adaptive magic formula (MF) tire model capable of coping with changes to the tire operating condition. More specifically, extensions have been made to the magic formula expressions for tire cornering stiffness and peak grip level, to account for variations in the tire inflation pressure, load, tread-depth and temperature. As a next step, the benefits of using an adaptive tire model for vehicle control system applications is demonstrated through simulation studies for enhanced vehicle control systems using an adaptive tire model in comparison to traditional control systems based on a non-adaptive tire model with fixed model parameters. In particular, this paper investigates the improvements an adaptive tire model would bring into the following vehicle control system strategies: Electronic Stability Control (ESC) system, Brake Torque Vectoring (BTV) system, and Rear Wheel Steer (RWS) system.</div></div>

Longitudinal and lateral forces acting on tyres are known to be closely related to tractive ability, braking characteristics, handling stability and manoeuvrability of ground vehicles. However, it is not feasible in the operating vehicles to measure the tyre forces directly because of the high cost of sensors, the limitations in sensor technology, interference with tyre rotation and the harsh environment. Another approach is the indirect monitoring technique that can estimate the tyre forces based on remote sensor outputs and vehicle dynamics models. In this paper, in order to develop tyre force-monitoring systems, a monitoring model is proposed utilizing not only the yawing motion but also the roll motion. Based on the monitoring model, a monitoring system is designed to estimate the lateral tyre force acting on each tyre. The monitoring system is constructed on the basis of a new scaled Kalman filter with model error compensator (SKFMEC) technique that is developed in this study to improve the robustness performance of Kalman filter methods. The SKFMEC technique adopts both the well-conditioned observer and the model error compensator concepts. Tyre force estimation performance of the monitoring system is evaluated in the MATLAB simulations where true tyre force data are generated from a 14-degree-of-freedom vehicle model with the combined-slip ‘magic formula’ tyre model.

Off-road tyre modelling I: the multi-spoke tyre model modified to include the effect of straight lugs

Under extreme conditions, a vehicle's handling performance and yaw stability can significantly deteriorate. This paper proposes a handling and stability decoupled control strategy for four-wheel independent drive (4WID) and four-wheel independent steering (4WIS) vehicles based on tire force vector control (TFVC). To address the conflicting objectives of handling and stability, the turning curvature is employed as the lateral reference, while the sideslip angle serves as the stability indicator, thereby decoupling handling and stability control. Additionally, a driver intention remapping method is introduced to maximise the preview turning curvature, effectively balancing longitudinal and lateral objectives. A yaw controller is developed using non-singular terminal sliding mode control (NTSMC) to maintain vehicle stability. Pentagon constraints are formulated to integrate all tire force constraints. Deviations from the driver's expectations are incorporated into the objective function to ensure the solvability of tire force allocation. The TFVC is implemented by integrating a UniTire-based inverse tire model and a saturated tire model to accurately regulate tire forces under both combined and saturated conditions. Finally, hardware-in-the-loop (HIL) and real vehicle test results confirm that the proposed handling and stability decoupled control strategy not only ensures vehicle stability but also enhances handling performance under extreme conditions.

This paper describes a model for motorcycle tires based on a physical interpretation of experimental data. In this model the real shape of the tire carcass is accurately described and its deformability is taken into account. The actual position of the contact point, that is, the center of the contact patch, is calculated. The concept of instantaneous slip is defined by calculating the longitudinal slip and sideslip angles using the velocity of the actual contact point, which moves with respect to the rim. Tire forces and torques are applied on the actual contact point and calculated according to Pacejka’s magic formula. The coupling of sliding properties with elastic ones and the use of the instantaneous slip concept make it possible to properly describe both steady state and transient behavior using the same relations, thus avoiding the use of any auxiliary equations.

Advanced vehicle dynamics control systems, such as integrated chassis control systems, are generally designed in a two layers structure: a tire force determination layer, and a tire force generation layer. The tire force generation layer works as a servo function that follows the force commands from the force determination layer. Therefore, the tire force generation layer should not require a lot of computing resources. This paper presents an inverse tire model that calculates required slip ratios and slip angles when tire forces are commanded. The inverse tire model is based on a nonlinear tire model that is sufficiently complex and, thus, can capture the nonlinear characteristics of tires. Because mathematical inversion of the nonlinear tire model is challenging, recursive methods can be considered for an inverse model. However, recursive methods, such as Newton Raphson method, do not guarantee the fixed time span required for finishing the computation. This paper presents a method for inverse tire model that does not require recursive computations.

The optimal control for anti-lock system proposed in this paper consists to search the wheel slip corresponding to the maximum of the tyre-road characteristic curve. This allows maximising the braking tyre forces. In this strategy of control the wheel slip is not imposed but used an algorithm to generated desired optimal wheel sleep. This algorithm is based on a nominal Pacejka tyre model which uses two parameters to give an updated of the tyre model with respect to the real tyre force characteristic. The proposed control method is verified through one-wheel simulation model with a "magic formula" tyre model. Simulations results show the effectiveness of this controller scheme.

Vehicle motion and tire force histories are estimated from an incomplete, noise-corrupted measurement set using an extended Kalman filter. A nine degree-of-freedom vehicle model and an analytic tire force model are used to simulate true vehicle motion, and a five degree-of-freedom vehicle model is used in the estimator. The filtered histories of forces and motion can be used to construct tire force models through off-line analysis, and both tire force estimates and state estimates are available for real time control. No prior knowledge of tire force characteristics or external factors that affect vehicle motion is required for the nonlinear estimation procedure. Simulation of a simple slip control braking system using slip and slip angle estimates for feedback demonstrates the effectiveness of the extended Kalman filter in providing adequate state estimates for advanced control of ground vehicles.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Estimation of Nonlinear Tire Forces and Road Friction for Modeling and Control

Compared with passenger cars, heavy vehicles have larger loads, longer bodies, more steering modes, and are more likely to cause improper steering angle relationships among the wheels, resulting in uncoordinated steering. This uncoordinated steering makes the tyres roll and drag, which results in abnormal lateral and longitudinal slips, changing the lateral and longitudinal mechanical characteristics of tyres and leading to low path tracking accuracy and poor steering performance of vehicles. Therefore, a tyre model containing lateral and longitudinal mechanical characteristics under heavy load is the key to solving the problem of uncoordinated steering in heavy vehicles. Firstly, with the physical tyre model, the lateral and longitudinal slip is described as carcass deflections. With the carcass deflections and contact pressure distribution of heavy load, the tyre forces under uncoordinated steering conditions are derived. Then, the tyre imprint parameters are obtained with a test bench, and a two-dimensional contact pressure distribution model is established to reproduce the contact pressure distribution of tyre under heavy loads and dynamic slips. The comparison results with Magic Formula (MF) and Trucksim are generally consistent, proving the effectiveness of the model. Finally, a lateral and longitudinal tyre force distribution control of vehicles is carried out based on the established model. The results of the Trucksim-Simulink co-simulation show that the path tracking accuracy of vehicles is significantly improved, which demonstrates that the tyre model incorporating lateral and longitudinal mechanical characteristics under heavy loads can provide a theoretical basis for solving the problem of uncoordinated steering in heavy vehicles.

This article introduces an efficient and high-precision estimation framework for four-wheel independently actuated (FWIA) autonomous vehicles based on a novel tire model and adaptive square-root cubature Kalman filter (SCKF) estimation strategy. Firstly, a reliable and concise tire model that considers the tire’s nonlinear mechanics characteristics under combined conditions through the piecewise affine (PWA) identification method is established to improve the accuracy of the lateral dynamics model of FWIA autonomous vehicles. On this basis, the longitudinal relaxation length of each tire is integrated into the lateral dynamics modeling of FWIA autonomous vehicle. A novel nonlinear state function, including the PWA tire model, is proposed in this paper. To reduce the impact of the uncertainty of noise statistics on the estimation accuracy, an adaptive SCKF estimation algorithm based on the maximum a posteriori (MAP) criterion is proposed in the estimation framework. Finally, the estimation accuracy and stability of the adaptive SCKF algorithm are verified by the co-simulation of CarSim and Simulink. The simulation results show that when the statistical characteristics of noise are unknown and the target state changes suddenly under critical maneuvers, the estimation framework proposed in this paper still maintains high accuracy and stability.

Modelling of road vehicles for calculation of traffic-induced ground vibration as a random process

&lt;div class="htmlview paragraph"&gt;This paper describes a tire model designed for the full range of operating conditions under both on- and off-road surface conditions. The operating conditions include longitudinal and lateral slip, camber angle and normal load. The model produces tire forces throughout the adhesion range up through peak coefficient of friction, and throughout the saturation region to limit slide coefficient of friction. Beyond the peak coefficient of friction region, the off-road portion of the model simulates plowing of deformable surfaces at large side slip angles which can result in side forces significantly above the normal load (e.g., equivalent coefficients of friction greatly exceeding unity).&lt;/div&gt; &lt;div class="htmlview paragraph"&gt;The model allows changing the saturation function depending the surface currently encountered by a given tire in the vehicle dynamics model. Saturation functions can vary from the sharp peak function associated with radial tires on paved surfaces, to the exponential like saturation associated with various off-road surfaces. Smooth transition functions are provided between the adhesion and saturation regions, and logic is provided to ensure that tire force always opposes slip velocity no matter what operating condition is encountered.&lt;/div&gt; &lt;div class="htmlview paragraph"&gt;This paper describes the model functional characteristics and response to a range of operating conditions. Data sources for the model are discussed. Cases are run with a vehicle dynamics model to illustrate model response to changing surface conditions.&lt;/div&gt;