Vehicle Dynamics Simulation Using Elliptical Combined-Slip Tire Model

<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>

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.

<div class="section abstract"><div class="htmlview paragraph">Accurate parameters of vehicle motion state are very important to the active safety of a vehicle. Currently the extended Kalman filter and unscented Kalman filter are widely used in estimation of the key state parameters, such as speed. In this situation, tire model must be used. The Magic Formula Tire Model is widely used in vehicle dynamics simulation because of its high versatility and accuracy. However, it requires a large number of parameters, which make the key state parameters of a real vehicle difficult to accurately obtain. Therefore, it is limited in real-time control of a vehicle. Firstly, the original Magic Formula Tire Model is simplified in this paper; then Jin Chi's Tire Model is introduced; thirdly, parameters of both the simplified Magic Formula and Jin Chi's Tire Model are identified using PSO (Particle Swarm Optimization) algorithm. Finally, Jin Chi's Tire Model is also used in parameters identification of experimental data.</div></div>

The paper presents an additive unscented Kalman filter (AUKF) based modular approach to estimate lateral vehicle dynamics and tyre forces. In this approach, a simplified single-track vehicular model and a dynamics-oriented tyre model are adopted to represent the vehicular motion. Subsequently, multiple observer modules for each dynamical state are designed and integrated into a unified estimation scheme (UES-AUKF). Two additional non-modular observers using AUKF and extended Kalman filter (EKF) are designed for comparative analysis of estimation accuracy and computational efficiency of the designed scheme. The simplified model and the designed estimators are simulated using double lane change (DLC) and sinusoidal (Sine) manoeuvres for high and low µ surfaces, respectively, and the results are analysed. Thereby the scheme is further validated using real vehicle dataset for estimation accuracy. Simulation results for the simplified vehicle and the tyre model conform with the standard results with acceptable deviations. Also, the proposed scheme exhibits improved accuracy with reduced computational time as compared to the non-modular observers.

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.

<div class="section abstract"><div class="htmlview paragraph">Vehicle dynamics simulation is an essential stage of product development. It is a low cost alternative to the expensive and time-consuming experimental tests performed during the design phase. The ability to accurately predict the dynamics of vehicles can, for instance, provide useful information about their stability characteristics. Yaw instability, in particular, can cause divergent yaw behavior in conventional passenger cars and jackknife in articulated vehicles, like tractor-semitrailer combinations. These instabilities are associated with safety issues since their occurrence can cause accidents with fatal losses, injuries and property damage. Therefore, the development of vehicle models and simulation environments is necessary. This paper presents the open source package <i>Vehicle Dynamics - Lateral</i>. Developed using the object oriented programming paradigm, the package provides a collection of vehicle and tire models, represented by a set of base parameters and characteristic functions covering both simple and articulated vehicle simulations, as well as specific graphics generation. The package is hosted on <i>GitHub</i>, an online platform where software code can be shared and collaboratively developed. The goal of the package is to serve as a free, open and expandable toolbox for vehicle dynamics studies, as an academic alternative to the high cost closed commercial programs. This approach ensures thorough access to the source code and allows researchers and developers to verify all details of the implementation, as well as collaborate to the project. The code is written in <i>MATLAB</i>, a well-spread programming language with a significant number of adepts both in academia and industry.</div></div>

Finite Element (FE) tire models are increasingly being used for tire design, vehicle design studies and dynamic investigations. Such tire models have the inherent advantage of being able to cover a wide range of tire design variables (such as detailed tire geometry and material composition), in addition to an extensive coverage of operational conditions (such as tire load, inflation pressure and driving speed). A variety of road input disturbances can also be considered. This paper presents a three‐dimensional (3D) Finite Element tire model developed using ABAQUS, a commercial finite element code for use in the development of new tire designs and simulation of vehicle dynamics. Of particular interest are rolling tire output responses, such as lateral forces and self‐aligning moment generated due to steering input during vehicle maneuvering and vibration responses to road disturbances. The model will also be applicable in investigating other tire design issues, such as the heat generated in the carcass of the rolling tire. This can present particular problems in heavily loaded tires, for example aircraft tires during take‐off. This model should enable the rapid development of new tire designs to satisfy specific requirements.

<div class="htmlview paragraph">This paper describes the development and validation of an advanced microcomputer based simulation developed to investigate the complex dynamic responses of light vehicles in a variety of maneuvers. These maneuvers include cornering with braking or acceleration as well as vehicle skidding, spinning, and rolling over.</div> <div class="htmlview paragraph">A three-dimensional, non-linear vehicle dynamic model having eight degrees-of-freedom is generated according to user specifications. Multiple masses are connected through suspension elements to properly account for vehicle forward, lateral, vertical, roll, pitch, and yaw directional dynamics. To build an accurate model, any one of 8 different commonly used front suspension systems can be employed in conjunction with any one of 19 different rear independent, semi-independent, and dependent suspension systems. The capability of implementing various suspension systems allows the utilization of important kinematic and dynamic effects of these suspensions on vehicle response. The tire model utilized is based on the non-dimensional slip angle approach and is capable of working over a wide range of normal to extreme operating conditions. These conditions include normal loads of up to 400% of standard load, slip angles from -90 to 90 degrees, slip ratios from -1 to 1, camber angles to 60 degrees, and an unlimited friction regime. The tire forces and moments generated are non-linear functions of actual normal load, longitudinal slip, slip angle, camber angle, and tire-roadway frictional characteristics. The tire model predicts wheel locking, spinning and/or saturation, and automatically simulates the changes in tire behavior caused by these conditions.</div> <div class="htmlview paragraph">The results presented in this paper demonstrate that the Light Vehicle Dynamics Simulation (LVDS) is capable of accurately predicting a wide spectrum of light vehicle maneuvers ranging from low “g” cornering to severe spin out and rollover. The LVDS simulation was successfully validated using experimental results obtained from several vehicles (cars, utility vehicles, vans, and pickup trucks) in a variety of handling and rollover maneuvers and in both unloaded and loaded conditions. It has proven to be a reliable tool for the investigation of the dynamic behavior of light vehicles.</div>

For the first time, this paper introduces the energy deviation method for quantitative analysis of sensitivity between the stable region boundary and tire lateral force. The vehicle model adopts a three degree of freedom vehicle model and unified tire model suitable for large side-slip angle. Through comprehensive analysis from vehicle handling diagram, dissipation of energy, equilibrium points, and energy deviation, the results demonstrate that this method can not only reveal the sensitive areas on the stable region boundary but also quantitatively characterize the sensitivity. It is more intuitive, concise, and convenient compared to traditional methods. Additionally, this research for the first time quantitatively demonstrates the spiral phenomenon occurring at the boundary between stable and unstable region due to fluctuations in tire force from energy perspective. Furthermore, it explains that this method based on conserved quantity, making it applicable to higher-dimensional vehicle models and has a wide range of applications.

<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>

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.

The Maxwell damper model has been adopted in the simulation of railway vehicle dynamics for a long time and became a basic force element in the multi-body dynamics software. The main purpose of this article is to investigate the predictive accuracy of this model for a railway yaw damper in damping characteristics and vehicle stability assessment. To achieve this objective, a simplified physical damper model was developed for comparisons, which is accurate and at the same time presents low computational costs, being able to be adopted in the multi-body dynamics model of a railway vehicle to speed up the numerical simulations. It was suitably validated in the damper rig test, and simulation results for those two damper models were compared with measured data. Results showed that the traditional Maxwell model of this typical yaw damper cannot accurately describe its dynamic behaviors, while the simplified physical model can accurately reproduce the complex behavior of the actual damper system with a high computational efficiency. Comparisons of vehicle dynamics relevant to vehicle stability were then carried out, by integrating the Maxwell model and the simplified physical model presented in this work into a multi-body dynamics model of a high-speed train in China. It was found that the Maxwell model can reproduce the phenomenon of carbody instability accurately if an appropriate value of the dynamic stiffness is set in the Maxwell model. In the simulation of bogie instability, the Maxwell model overestimates the dynamic damping of this typical yaw damper, and as a result it leads to large errors in stability calculations.

As an important component of vehicle design and energy conservation, electric vehicle dynamics simulation is essential, especially under complicated testing conditions. The current commercial vehicle simulation software is mostly used for fuel vehicle dynamics simulation, which lacks accurate electric powertrain parts and open sources. To address this problem, this paper proposes an open-source and flexible vehicle dynamics simulation platform that includes 27 degrees of freedom (DOFs) based on Simulink, which can compatibly support both traditional vehicle dynamics simulations and electric vehicle dynamics simulations. In addition, the platform can support module customization, which is convenient for researchers. Although this platform still needs some iterations to reach industrial and commercial standards, it can already achieve parameter consistency under the stability demands in general scenarios. We believe this work should receive research attention and participation to provide lower thresholds and more references to the dynamic simulation of electric vehicles to reduce vehicle energy consumption.

This paper describes a semi-physical tyre model that enables studies of rolling loss in combination with vehicle dynamic simulations. The proposed model, named extended brush tyre model (EBM), take ...