Nonlinear tire inverse model for integrated chassis control system

Recent research suggests the various applications of Model Predictive Control on vehicle systems. In numerous cases, nonlinear tire models such as the Magic Formula, which are highly complex and are more detailed than necessary, are used. This paper presents a nonlinear tire model that excludes the region of negative slope but expresses the nonlinear properties of tire well enough for tracking the lane of a racing course. The proposed inverse tire model can also be used to calculate the slip angle from the tire force. Thus, the model can be utilized to design the Model Predictive Controller.

Nonlinear inverse dynamic models can be utilized in various parts of advanced model-based control system design: reference trajectory optimization, feedforward control and feedback linearization. In this paper, a new synthesis approach for nonlinear inverse dynamic models of satellites with flexible structures is presented. For satellite configurations with unstable zero dynamics, a stable inverse model approximation is proposed which has been successfully applied to robots with flexible bodies. This inverse modeling approach is part of the newly developed DLR Space Systems Library for objectoriented modeling and simulation of satellites and launchers in a detailed space environment. For satellites with flexible structures, the library provides models for normal simulation mode and the necessary tools to directly generate approximate inverse models. In this paper, trajectory optimization is shown to be an important use case for inverse dynamic models. By inversion based reformulation of the trajectory optimization problem, the optimal reference motion of the control system can be determined in a reliable and efficient way.

The tire forces of vehicles will fall into the non-linear region under extreme handling conditions, which cause poor path tracking performance. In this paper, a model predictive controller based on a nonlinear tire model is designed. The tire forces are characterized with nonlinear composite functions of the magic formula instead of a simple linear relation model. Taylor expansion is used to linearize the controller, the first-order difference quotient method is used for discretization, and the partial derivative of the composite function is used for matrix transformation. Constant velocity and variable velocity conditions are selected to compare the designed controller with the conventional controller in Carsim/Simulink. The results show that when the tire forces fall in the nonlinear region, two controllers have good stability, and the tracking accuracy of the controller designed in this paper is slightly better. However, after the tire forces become nonlinear, the controller with linear tire force becomes worse, the tracking accuracy is far worse than the controller with the nonlinear tire model, and the vehicle stability is also degraded. In addition, an active steering test platform based on LabVIEW-RT is established, and hardware-in-the-loop tests are carried out. The effectiveness of the designed controller is verified.

Many vehicle state parameters will be used in the electronic stability control (ESC) system. ESC system must estimate some important parameters based on a few sensors and help to develop the control strategies. ESC can estimate the work state of vehicle with tire forces based on the estimating of adhesion coefficient and wheel slip ratio, then determine the control strategy on wheels. The tire forces to be estimated include vertical force, lateral force and longitudinal force on tires. In this article, a method of estimating vertical tire force based on vehicle geometric parameters and vehicle working state, a method of estimating tire lateral force based on vehicle yaw rate and the change of tire longitudinal forces caused by the change of vertical forces and lateral forces have researched. Co-simulation shows that the trend of the tire forces changing is correct and small error on high and low adhesive pavement. The ESC system is available on HIL bench and testing vehicle, so the estimates of tire forces are practical methods.

<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>

<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>

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.

In this investigation, a non-linear elastic ring tyre model is developed. The elastic deformation of the tyre belt is modelled using the finite element absolute nodal coordinate formulation that allows for describing large rotational motion and the non-linear inertia effects; the curved structure of the tyre belt; and moving boundary resulting from the tread and road interaction. Using a concept of elastic ring tyre models, the sidewall flexibility of a tyre is modelled using circumferential and radial springs and dampers defined between the belt and rim, while the tangential tyre force is modelled using friction elements defined at contact nodes defined within the curved belt elements. Numerical examples are presented in order to demonstrate the use of the flexible tyre model developed in this investigation. Good agreements are demonstrated in the tyre vibration characteristics obtained using the experiments and the proposed model. It is presented that the proposed tyre model can be used for assessing dynamic characteristics of tyres in high frequency ranges resulting from the interaction to uneven road surfaces.

<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>

This paper examines how changes in the tyre and suspension impact the dynamic spindle force response of an automotive tyre rolling over an obstacle in the roadway. The investigation is conducted with a simple vehicle dynamics simulation coupled to a comprehensive 2D dynamic nonlinear finite tyre model. The tyre model uses an inextensible, circular membrane to approximate the shape, and an empirical, nonlinear term to account for forces that result from the elastic deformation of the sidewall. Both the tyre structure and contact between the tyre and roadway are modelled with the finite element method. This approach allows the tyre model to reproduce the tyre response accurately during large deformations in a computationally efficient manner. This paper presents a brief overview of the models and their limited validation of the tyre model. Included in this discussion is a comparison between measured and simulated tyre contact forces for the case of a slowly rolling tyre. The effect of vehicle speed, suspension stiffness, suspension damping, tyre mass, tyre damping, and tyre inflation pressure on the spindle forces is also presented.

SUMMARYA four wheel steering (4WS) control strategy is derived and evaluated using a nonlinear vehicle model along with a nonlinear tire model. The vehicle model used is a.three degree-of-freedom model. Yaw rate, lateral velocity and longitudinal velocity are considered with the roll effects included in the calculation of the normal loads for each of the four wheels. These normal forces are used in the tire model to calculate the tire forces and moments for each wheel. The results of simulation indicate the potential benefits gained by using 4WS. These benefits include stability and performance at high speeds and maneuverability at low speeds.

Coastal bathymetry is the most essential tool for marine planning, monitoring and management, modelling, nautical navigation and scientific studies of marine environments. The techniques have been developed over the last decade to derive bathymetry using remote sensing technology with efficiently and low costly. Log linear bathymetric inversion model and non-linear bathymetric inversion model provides two empirical approaches for deriving bathymetry from multispectral satellite imagery, which have been refined and widely applied. This paper compares these two approaches by means of a geographical error analysis for the site Kankesanturai using WorldView-2 satellite imagery. In order to calibrate both models; Single Beam Echo Sounding (SBES) data in this study area were used as reference points. Corrections for atmospheric and sun-glint effects are applied prior to the water depth algorithm. The algorithm was tuned and both models were calibrated by performing the necessary algorithm with available single beam echo sounding data in the study area. The coefficients of standard R2 is estimated as 0.846 for log-linear and 0.692 for non-linear model. Log linear model performs better than the non-linear model. The model residuals were mapped and the spatial auto-correlation was calculated based on the bathymetric estimation model. A spatial error model was constructed to generate more reliable estimates of bathymetry by calculating the spatial autocorrelation of model error and integrating this into an improved regression model. Finally, the spatial error model improved the bathymetric estimates of R2 up to 0.854 for log-linear and 0.704 non-linear model respectively. The Root Mean Square Error (RMSE) was calculated for the different depth ranges and also for all reference points. The overall accuracy for the log linear and the non-linear inversion model after the geographical error analysis is estimated as ±1.532 m and ±2.089 m for this study. The spatial error model improved bathymetric estimates than those derived from a conventional log-linear and non-linear technique although these methods perform very similar estimates overall.

<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>

The active safety systems, such as Electronic Stability Program (ESP) and Traction Control System (TCS), have been used widely in the modern production auto-mobile. These safety control systems function well only when the vehicle motion information is known. To achieve this purpose, various sensors are equipped in the production cars in order to detect vehicle motion information like yaw, wheel velocities and accelerations. However some key variables of the vehicle dynamic and road conditions, such as the sideslip angle, the tire road forces and the tire road friction coefficient are difficult to measure directly for technology or economic reasons. A new observer which uses the unscented Kalman filter to estimate simultaneously the sideslip angle, lateral tire road forces and the tire road friction coefficient has been proposed in this paper. The advantage of this observer is that no additional sensors is needed when the ESP has been installed in the auto-mobile. The performance of the proposed observer is demonstrated through the Chicane test.