2024-05-23

Tire Lateral Force Estimation Using Kalman Filter

Introduction

Vehicle stability evaluation has traditionally relied on subjective tests by trained evaluators. Three common parameters that have been widely used by numerous tire-manufacturing companies for indoor tests are three parameters.

• cornering stiffness: $C_\alpha$
• lateral stiffness: $K_L$
• distortion stiffness: $K_D$

Theses three tire parameters comprise the standard indices to determine the lateral motion of a tire by providing specific physical meanings of the tire lateral motion. Then, theses parameters can be implemented in a tire relaxation length model. In the aspects of lateral tire force estimation, use of advanced tire relaxation length model can help improve the accuracy of tire lateral force estimatino especially in transient response.

This paper suggests the improved method of lateral tire force estimation by applying modified relaxation length and modified cornering stiffness in the dynamic tire model.

Planar Full Car Model

3-DoF model of vehicle motion to estimate lateral tire forces

Where $V_x,V_y,\gamma, m, I_z, \delta_1, \delta_2, t, l_f, l_r$ are the longitudinal velocity, lateral velocity, yaw rate, vehicle mass, moment of inertia about yaw axis, left wheel steering angle, right wheel steering angle, half of track width, distance from front axle to the CG and distance from rear axle to the CG

$F_x, F_y$ are longitudinal tire force and lateral tire force followed by subscripts $fl, fr, rl, rr$ representing front left, front right, rear left, rear right, respectively. $C_{av}$ is the lumped parameter to represent the drag resistance, which is expressed as follows:

Where $\rho_{air}, C_d, A$ are density of air, coefficient of air drag and front corss sectional area, respectively.

Front and rear slip angle at each wheel are represented as follows:

Where $\phi, \epsilon_f, \epsilon_r$ are roll angle, front roll steer compliance, and rear roll steer compliance, respectively.

Vertical Load Calculation

Lateral tire force is influenced by load transfer during cornering. The amount of transfer is determined by the vehicle geometry, stiffness of the suspension, and the sprint rate of the tires. The model calculates vertical load changes to improve the accuracy of lateral tire force estimation.

Couplings between the vertical load and lateral force are describes as follows:

Where $l, h_{cg}, a_x, a_y, g, c_{\phi f}, c_{\phi r}, h_r$ are the wheel base length, height from ground to CG, longitudinal acceleration, lateral acceleration, graviational acceleration, front roll stiffness, rear roll stiffness, and distance from CG to roll axis, respectively.

Wheel Dynamics Model

Wheel dynamics can be dereived as:

Where $\omega_i, I_w, T_{di}, T_{bi}, R_e, F_{xi}, R_r$ are the wheel angular velocity, wheel moment of inertia, driving torque, braking torque, effective radius, longitudinal force, and rolling resistance coefficient, respectively

Output Shaft Torque Dynamics Model

The information of driving torque transffered from the engine is essential. To consider dynamic model responses such as torque converter slip and transmission gear shifting event, torque observer can be designed using engine toque and other vehicle data from CAN signal. Here output shaft torque is divided into load torque and inertia torque, and inertia torque is expressed by considering drivetrain as a lumped mass as delineated below:

Where $T_0$ and $J_v$ are the output shaft torque and lumped moment of inertia. Here, Equations (11) and (12) apply to front wheel drive type.

Tire Dynamics for Vehicle Model Integration

By negelecting longitudinal force, the simplified nonlinear lateral force is given by:

where $C_{yi}$ is the cornering stiffness at each tire position. Here, $f(\lambda_i)$ is expressed as follows:

Here, $\mu$ is the road friction coefficient. In this paper, the nominal $\mu$ is assumed as 0.9 to represent a nominal high mu surface.

Weight Shifting Effect in Cornering Stiffness

Cornering stiffness varies with vertical load changes during maneuvers like lane changes. An empirical model is used to reconstruct cornering stiffness, improving lateral force accuracy. The cornering stiffness from the simple empirical lateral tire force model that includes both weight shifting and side slip angle term given by:

Where $C_1, C_2$ are the cornering stiffness relation coefficients and $F_{z,n}, \Delta F_z$ are the static vertical load and the difference between dynamic vertical load that includes vehicle acceleration and static vertical load. To improce the accuracy of the lateral foce, adjustment factors, $k_1$ and $k_2$, are applied to the cornering stiffness equation. Theses values can be changed according to the test tire characteristics. From the equation (16), following modified cornering stiffness $C_y$ is adopted in this study:

Dynamic Tire Model

A first-order dynamic model representing the lagged behavior of lateral tire force is used. The model includes a modified relaxation length for better estimation accuracy.

Modified relaxation length model

Where $C_\alpha$ is exactly same as $C_1F_{z, n}$ and $K_L$is lateral stiffness. However, relaxation length model can be further elaborated through considering three tire characteristics parameters as follows:

Where $K_D$ is distortion stiffness. This study suggets the improved tire force estimation by applying modified relaxation length model. Experiment result of indoor testing machine is used as nominal values of $C_\alpha, K_L, K_D$.

Where $\sigma$ is the relaxation length discussed below

Estimator Design