The Influence of Transient Tire Force Transmission on Sideslip Angle Estimation

Abstract

Knowledge of the sideslip angle is significant in development, testing and validation of advanced driver assistance systems and vehicle control systems, to enhance vehicle performance and stability. This article investigates the influence of having additional information about transient tire behavior during estimation of the sideslip angle. Since transient behavior depends on the dynamic excitation, the effect during different driving maneuvers (excitations) such as steady-state cornering, slalom and general handling as well as on different road surfaces is investigated. The results show that considering transient force transmission even by a pragmatic approach leads to a significant improvement.

1 Introduction

The sideslip angle $\beta$, as a main indicator of dynamic stability, describes the orientation of the horizontal velocity in the center of gravity (COG) of a vehicle and is given as a function of longitudinal $v_x$ and lateral velocity $v_y$ by $\beta =\arctan {\left( v_y/v_x\right) }$. Estimating $\beta$ has garnered significant attention from researchers over the past few decades and observer-based methods such as Kalman filters (KF) or sliding-mode observers (SMO) are often used. Within such observers, different types of uncertainties in the lateral tire forces are typically modeled without distinction. These uncertainties include deviations from the real cornering stiffness, deviations from the real vertical force, transient tire behavior, effects based on nonlinear tire behavior, e.g. assuming linear force transmission, as well as imperfect or missing knowledge of the maximum coefficient of tire-road friction. Being able to distinguish these effects will enable improved modeling of the uncertainties in the observers, especially of effects such as transient tire forces and vertical load, that can be linked to a certain dynamic excitation.

2 Transient Tire Force Transmission

The theory of steady-state (static) tire force transmission $F_{x,y}^\textrm{S}$ is based on the fundamental assumption that when the sliding condition in the contact patch changes, a certain steady-state force value is reached immediately. In general, a certain time delay between a change of the sliding velocity and the build up of a steady-state tire force value occurs. Consequently, transient effects and a hysteresis behavior of horizontal force characteristics $F_{x,y}^\textrm{D}$ occur, see Fig. 1 (right). A simple and widely used approach in vehicle handling dynamics analyses is to approximate the dynamic behavior of tire forces by first-order systems

$$\begin{aligned} \tau _{x,y}\,\dot{F}^\textrm{D}_{x,y}+F^\textrm{D}_{x,y}=F^\textrm{S}_{x,y} & \text {with} & \tau _{x,y}=\frac{r_{x,y}}{v_\textrm{t}}\,. \end{aligned}$$

(1)

The key parameter in such approaches is the relaxation time $\tau _{x,y}$, which is linked to the relaxation length $r_{x,y}$ and the transport velocity of the tire tread elements through the contact patch $v_\textrm{t}$. Relaxation time and length are only constant in a specific operating point. Several literature sources deal with a detailed modeling of these two parameters and their dependencies on e.g. slip $s_{x,y}$, vertical load $F_z$ and visco-elastic effects $d_y$ and $c_y$ of tire compliance $y_\textrm{e}$, see Fig. 1 (left) and e.g. [5, 6]. The determination of relaxation lengths based on experimental investigations with a certain accuracy is still a cumbersome and challenging task as well as a research topic, see e.g. [2].

Fig. 1.

Principles of modeling visco-elastic tire compliance $y_\textrm{e}$ by a Kelvin-Voigt element (left), based on [6], and resulting transient tire force transmission $F_y^\textrm{D}$ (right)

Based on the enhanced first-order tire force dynamics of the semi-physical TMeasy 5 handling tire model, [6], relaxation time and length for pure lateral slip are given by

$$\begin{aligned} \tau _{y}=\frac{v_\textrm{t}\,d_{y}+f_y}{v_\textrm{t}\,c_{y}}=\frac{d_y}{c_y}+\frac{f_y}{v_\textrm{t}\,c_{y}} & \text {and} & r_{y}=v_\textrm{t}\,\tau _y\,, \end{aligned}$$

(2)

where nonlinear steady-state tire force characteristics $F_y^\textrm{S}=F_y^\textrm{S}(s_y,F_z)$ are described by the global derivative $f_y=F_y^\textrm{S}/s_y$ with respect to the lateral slip $s_y=-v_y/v_\textrm{t}=\tan {\alpha }$ or slip angle $\alpha$. For pure lateral slip, $v_\textrm{t}=|v_x|$ holds. Based on the considerations in Eqs. (1)–(2), the transient behavior of the lateral tire force transmission $F_y^\textrm{D}$ is fully determined by nonlinear steady-state contact force characteristics $f_y$ as well as visco-elastic stiffness $c_y$ and damping properties $d_y$ of the tire structure. Thus, tire force dynamics are approximated in an effective and efficient semi-physical manner by a Kelvin-Voigt element.

Fig. 2.

Identified relaxation length $r_y=r_y(\alpha ,F_z)$ and time $\tau _y=\tau _y(\alpha ,F_z)$ (bottom left and right) of a tire of size 245/40 R 20 99 W based on measured lateral tire force responses $F_y^\textrm{meas}$ (top) at different vertical loads $F_z=\mathrm {const.}$ and $v_\textrm{t}=|v_x|=60$ km/h

An extensive tire testing series with five different tires was conducted on an industrial flat track tire test rig, as presented in [4] for two tires. An example of measured dynamic responses of the lateral tire force $F_y^\textrm{meas}$ of a tire of size of 245/40 R 20 99 W due to a frequency based sine sweep maneuver of the slip angle $\alpha$ from $0\rightarrow 5$ Hz at constant tire load $F_z$ and velocity $v_\textrm{t}$ is presented in Fig. 2 (top). Based on such measurements, the Kelvin-Voigt parameters, lateral tire stiffness $c_y$ and damping $d_y$, can be identified by optimization of the computed transient lateral tire force $F_y^\textrm{D}$, see also [4]. Consequently, a reasonable prediction of the behavior of relaxation length $r_y=r_y(\alpha ,F_z)$ and time $\tau _y=\tau _y(\alpha ,F_z)$ according to Eq. (2) is possible, see Fig. 2 (bottom left and right).

3 Sideslip Angle Estimation

The cascaded observer structure from [1] was implemented. First, a discrete-time extended Kalman filter (EKF) is used to estimate front longitudinal $F_{x1}$, front and rear lateral tire forces $F_{yi}$. System and measurement models for the force observer are given by Eqs. (1)–(10) in [1]. Subsequently, these estimates are treated as inputs to the sideslip angle observer.

The sideslip angle observer is based on a combination of a single track vehicle model and a linear adaptive tire force model, where the cornering stiffness $C_{\alpha i}$ is corrected by $\mathrm {\Delta }C_{\alpha i}(t)$. The lateral tire force is $F_{yi}(t) = \left( C_{\alpha i} + \mathrm {\Delta }C_{\alpha i}(t) \right) \, \alpha _i(t)$ with $i=1,2$ distinguishing between front and rear axle, respectively. By adjusting the cornering stiffness, $\mathrm {\Delta }C_{\alpha i}(t)$ accounts for nonlinear effects as well as different uncertainties that can arise. This adjustment is included as a state variable and modeled as the random walk. The system model is given in Table 1 (first set) with the state $\boldsymbol{x}=[ \beta \;\mathrm {\Delta }C_{\alpha 1}\;\mathrm {\Delta }C_{\alpha 2}]^\textrm{T}$ and input vector $\boldsymbol{u}= [ \delta \;\dot{\psi }\;v\;F_{x1} ]^\textrm{T}$, where $\delta$ is the steering angle, $\dot{\psi }$ the yaw rate, and v the vehicle velocity in the COG. The measurement model is given in Table 1 (second set) with measurement vector $\boldsymbol{y}= [ F_{y1}\;F_{y2}\;a_y ]^\textrm{T}$, where $a_y$ is the lateral acceleration. Together with the EKF, this forms the basis of the first observer named $\textrm{O}_1$. The model covariance matrix is set to $\boldsymbol{Q}=\textrm{diag}(10^{-12}, 10^{9}, 10^{9})$, whereas measurement covariance matrix is set to $\boldsymbol{R}=\textrm{diag}(10^{3}, 10^{3}, 10^{-5})$.

Table 1. System model for both sideslip angle observers (first set) and measurement models for observer $\textrm{O}_1$ (second set) and $\textrm{O}_2$ (third set)

The present work investigates the impact of additional information regarding transient tire forces, represented by $\tau _{yi}$, on the sideslip angle estimation. The estimated lateral force $F_{yi}$ is the dynamic force in Eq. (1), i.e. $F_{yi}=F_{yi}^\textrm{D}$. With this information and the knowledge of $\tau _{yi}$, the static force $F_{yi}^\textrm{S}$ can be obtained using the discrete version of Eq. (1). The static force in the linear region can be described by a linear model, $F_{yi, k}^\textrm{S} = C_{\alpha i}\,\alpha _{i, k}$, where k represents the current time step. Consequently, $\alpha _{i,k}=\alpha _{i,k}^\textrm{meas}$ can be calculated in every time step by

$$\begin{aligned} F_{yi,k}^\textrm{S} = \tau _{yi}\,\frac{F_{yi,k}^\textrm{D}-F_{yi,k-1}^\textrm{D}}{\mathrm {\Delta }t}+F_{yi,k}^\textrm{D} & \text {and} & \alpha _{i,k}^\textrm{meas}=\frac{F_{yi,k}^\textrm{S}}{C_{\alpha i}}\,, \qquad i=1,2\,. \end{aligned}$$

(3)

Therefore, if the observer is operating within the linear region of tire force characteristics, sufficiently accurate values of the slip angle $\alpha _{i,k}$ are obtained. Hence, the measurement vector can be augmented by incorporating this values, for both front and rear, $\boldsymbol{y} = [ F_{y1}\;F_{y2}\;a_y\;\alpha _{1}\;\alpha _{2}]^\textrm{T}$. The index k is omitted but implied. Relationships between the slip angles $\alpha _{i}$ and the sideslip angle $\beta$ read $\alpha _1 = \delta - \beta - l_1\,\dot{\psi }/v$ and $\alpha _2 = - \beta + l_2\,\dot{\psi }/v$. These equations are combined with the measurement model from $\textrm{O}_1$ to form the novel measurement model of the second observer named $\textrm{O}_2$, see Table 1 (third set). The system model is the same for both observers. Discrete-time EKF is employed and the algorithm structure with $\textrm{O}_2$ is shown in Fig. 3. Using the Popov-Belevitch-Hautus (PBH) criterion, [3], it is proven that the system is locally observable. However, only in the linear region, sufficiently accurate slip angle values are obtained. Consequently, the values of $\boldsymbol{R}$ corresponding to slip angles are changed depending on the region. In the linear region, the corresponding value is set to $10^{-7}\,\textrm{rad}^2$, while in the nonlinear region, it is $5\cdot 10^{-1}\,\textrm{rad}^2$. This is done for both the front and rear separately. To decide whether the observer is in the linear or nonlinear region, pragmatic threshold values of $F_{yi}=\{5,3.5\}$ kN are implemented.

Fig. 3.

Algorithm structure with the proposed adapted sideslip angle observer $\textrm{O}_2$

4 Results and Discussion

Observers $\textrm{O}_1$ and $\textrm{O}_2$ were tested based on vehicle dynamics measurements obtained with a front wheel driven standard VW Golf 7 GTI TCR with tires of size 235/35 R 19 91 Y. The total vehicle mass is $m=1574$ kg, the distances between the COG and the front and rear axle are $l_i=\{0.992,1.634\}$ m. The cornering stiffnessess of the front and rear axle are $C_{\alpha i}=\{150,142.4\}$ kN/rad.

Different driving maneuvers with a mean vehicle speed of $v\approx 60$ km/h for a wide range of sideslip angle values were performed on road surfaces with different maximum tire-road friction. Although the tires of the test vehicle were not part of the tire testing series, similar tire properties were assumed. Based on the measured vertical tire load $F_{zi}$ and slip angle $\alpha _i$ variations during different sinusoidal steering maneuvers, overall mean values of the relaxation times were determined with $\tau _{yi}=\{38,31\}$ ms by Eq. (2), cf. Figure 2 (bottom right), and were assumed constant for all maneuvers.

The results show that the proposed adapted observer $\textrm{O}_2$ can improve the estimate of $\beta$, see Fig. 4. Significant improvement can be seen in steady-state cornering (top right), where the result of $\textrm{O}_1$ drifts from the real value. Due to the steady-state inputs, the errors accumulate over time. However, $\textrm{O}_2$ is able to correct this error and track the true value of $\beta$. Additionally, the performance of $\textrm{O}_2$ is quite robust against uncertainties of $\tau _{yi}$. This is supported by tests on different road surfaces. Figure 4 (bottom) shows that both $\textrm{O}_1$ and $\textrm{O}_2$ perform well on the ice-like surface around $t=20$ s, although no information on the surface is available to the observers. In both observers, the change in tire-road friction is compensated in the estimates of $\mathrm {\Delta }C_{\alpha i}(t)$. However, during the low excitation on wet asphalt, $\textrm{O}_1$ again drifts and is not able to converge again until around $t\approx 115$ s. In contrast, $\textrm{O}_2$ is again able to correct the error.

To summarize, the overall results show that including even simplified and pragmatic assumptions on the transient tire force transmission can significantly improve the estimation of the sideslip angle.

Fig. 4.

Comparison of measurement and estimation results of observers $\textrm{O}_{1}$ and $\textrm{O}_{2}$ for different maneuvers (excitation) and on different road surfaces (tire-road friction)