An Adaptive Vehicle Stability Enhancement Controller Based on Tire Cornering Stiffness Adaptations

Abstract

This study presents an adaptive integrated chassis control strategy for enhancing vehicle stability under different road conditions, specifically through the real-time estimation of tire cornering stiffness. A hierarchical control architecture is developed, combining active front steering (AFS) and direct yaw moment control (DYC). A recursive regularized weighted least squares algorithm is designed to estimate tire cornering stiffness from measurable vehicle states, eliminating the need for additional tire sensors. Leveraging this estimation, an adaptive sliding mode controller (ASMC) is proposed in the upper layer, where a novel self-tuning mechanism adjusts control parameters based on tire saturation levels and cornering stiffness variation trends. The lower-layer controller employs a weighted least squares allocation method to distribute control efforts while respecting physical and friction constraints. Co-simulations using MATLAB 2018a/Simulink and CarSim validate the effectiveness of the proposed framework under both high- and low-friction scenarios. Compared with conventional ASMC and DYC strategies, the proposed controller exhibits improved robustness, reduced sideslip, and enhanced trajectory tracking performance. The results demonstrate the significance of the real-time integration of tire dynamics into chassis control in improving vehicle handling and stability.

1. Introduction

With the rapid advancement of automotive electrification and intelligence, chassis-by-wire technology has significantly expanded the degrees of freedom in vehicle dynamics control [1]. However, this evolution introduces new challenges for vehicle stability, particularly under varying road conditions [2,3,4]. Traditional stability control strategies, such as phase plane analysis [5] and experimental fitting of control envelopes [6,7], rely on indirect stability indicators (e.g., yaw rate and sideslip angle) and predefined control thresholds. These methods often lack adaptability to real-time changes in tire–road interaction states, leading to compromised robustness in dynamic environments.

As a critical parameter reflecting the lateral force generation capability of tires, tire cornering stiffness directly correlates with the tire–road friction state [8]. Its real-time estimation offers a promising pathway to enhance stability control by bridging the gap between indirect vehicle state measurements and actual road conditions. Despite its potential, existing studies rarely integrate tire cornering stiffness estimation directly into chassis integrated control frameworks. For instance, Lu et al. [9] developed a stability controller based on model predictive control (MPC) and an online cornering stiffness estimation; the estimation was only utilized to deal with the model mismatch problem, instead of direct involvement in stability control. A few research studies [10,11,12,13] have integrated a tire cornering stiffness online estimation into the control frameworks, but none of them made full use of the inherent dynamics nonlinearity and tire–road interactions within the estimation itself. For instance, Qi et al. [11] applied the online estimation result of tire cornering stiffness directly to the coefficient matrix of the vehicle dynamics model, facilitating the construction of the MPC framework. This method only takes tire cornering stiffness as a token of instantaneous dynamics, rather than a representation of the vehicle stability state. This limitation underscores the need for a novel control strategy that combines adaptive parameter adjustments with real-time tire dynamics response quantification.

Vehicle stability control has been extensively studied, with mainstream approaches including sliding mode control (SMC) [14], model predictive control [15,16], and rule-based methods [5,17]. Model predictive control and its variations can deal with complex linear or nonlinear constraints [16]; it optimizes control inputs over a finite horizon, but faces challenges in balancing computational efficiency with real-time performance [18], especially for multi-actuator systems like electronic stability control (ESC) and active front steering (AFS). Rule-based methods are simple in structure and easy to implement, but commonly limited by their adaptability to external uncertainty [18]. SMC-based strategies, such as adaptive sliding mode control (ASMC) [14], are widely adopted for their robustness against model uncertainties. However, conventional ASMC designs often rely on a phase plane analysis to define stability boundaries, which are static and fail to adapt to varying tire–road conditions [19,20]. The control methods listed above predominantly focus on hierarchical control architectures with fixed parameters or gains. Compared with those methods, adaptive controllers with parameters or gains incorporating a self-tuning mechanism are preferable and more robust to external or implicit uncertainties.

This study addresses the aforementioned challenges by proposing an adaptive integrated chassis stability control strategy for AFS and direct yaw motion control (DYC) on the basis of real-time tire cornering stiffness estimation. The primary contributions are twofold. (1) A recursive regularized weighted least squares (RRWLS) algorithm in the form of a batch formulation is developed to estimate tire cornering stiffness using easy-to-obtain vehicle state measurements (e.g., lateral acceleration and yaw rate). This model bypasses the need for complex tire sensors and adapts to different road adhesion conditions. (2) Stability evaluation metrics, including tire force saturation levels and cornering stiffness variation rates, are formulated to dynamically adjust the control parameters, from which an adaptive sliding mode controller (ASMC) is designed for the upper-layer control, while a weighted-least-squares-based allocation algorithm optimizes actuator commands in the lower layer. The proposed strategy is validated through co-simulations in MATLAB/Simulink and CarSim under two typical maneuvers on varied μ roads. The comparative results for the DYC and conventional ASMC demonstrate superior robustness and stability margin improvements.

The remaining sections of this paper are organized as follows: first, a simplified vehicle dynamics model is constructed to facilitate the cornering stiffness online adaption and the controller design; second, a hierarchical controller consisting of an upper level and a lower level is elaborated; third, a self-tuning mechanism for the adaptive parameter in the upper-level controller is proposed; fourth, some co-simulations in MATLAB/Simulink and CarSim are conducted to verify the effectiveness of the proposed control framework; and finally, we present the conclusions of this research.

2. Vehicle Dynamics Modeling for Control

To simplify the construction of the controller, the reference states of a vehicle are generated with a simplified single-track vehicle model, as shown in Figure 1. This model can effectively reduce the computation effort and generate a fairly accurate steady-state response.

The mathematical representation of the single-track vehicle model can be constructed as follows [21]:

$$\left[\begin{array}{c}\dot{\beta }\\ \dot{\gamma }\end{array}\right]=\left[\begin{array}{cc}-{\displaystyle \frac{C_f+C_r}{m{v}_x}}& {\displaystyle \frac{l_r{C}_r-l_f{C}_f}{m{v}_x^2}}-1\\ {\displaystyle \frac{l_r{C}_r-l_f{C}_f}{I_z}}& -{\displaystyle \frac{l_r^2{C}_r+l_f^2{C}_f}{I_z{v}_x}}\end{array}\right]\left[\begin{array}{c}\beta \\ \gamma \end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{C_f}{m{v}_x}}\\ {\displaystyle \frac{l_f{C}_f}{I_z}}\end{array}\right]{\delta }_f+\left[\begin{array}{c}0\\ {\displaystyle \frac{1}{I_z}}\end{array}\right]M_z$$

(1)

$$a_y=v_x(\dot{\beta }+\gamma )$$

(2)

where $m$ is the vehicle mass; $I_z$ is the yaw inertia; $l_f,l_r$ are the distances from the center of mass to the front and rear axle, respectively; $v_x$ is the longitudinal velocity at the center of mass; $a_y$ is the lateral acceleration; $\gamma$ is the yaw rate; ${\delta }_f$ is the front-wheel steering angle exerted by the driver; $C_f,C_r$ are the cornering stiffnesses at the front and rear axle, respectively; $\beta$ is the sideslip angle of the vehicle; and $M_z$ is the additional yaw moment exerted by the controller. In Figure 1, $F_{yf},F_{yr}$ are lateral forces on the front and rear wheels, respectively.

Based on the vehicle dynamics model, the steady-state desired yaw rate can be deduced as follows [22]:

$${\dot{\gamma }}_{des}=-{\displaystyle \frac{l{C}_f{C}_r}{I_z(C_f+C_r)}}\cdot \left[{\displaystyle \frac{l}{v_x}}+({\displaystyle \frac{m{l}_r}{C_{f}l}}-{\displaystyle \frac{m{l}_f}{C_{r}l}})\cdot v_x\right]\cdot {\gamma }_{des}+{\displaystyle \frac{l{C}_f{C}_f}{I_z(C_f+C_r)}}\cdot {\delta }_f$$

(3)

Taking the constraint of the tire–road friction, the ultimate target yaw rate can be obtained as follows:

$${\gamma }_{target}=\min ({\displaystyle \frac{{\gamma }_{des}}{1+{\tau }_{e}s}},0.85\cdot {\displaystyle \frac{\mu g}{v_x}})$$

(4)

where $\mu$ is the road friction coefficient. The target sideslip angle can be simply set as ${\beta }_{\mathrm{target}}=0$.

3. Overall Controller Design

Aiming at enhancing the stability and handling performance of a vehicle, this paper proposes an integrated chassis control method based on the online adaptation of tire cornering stiffness. The overall controller design is illustrated in Figure 2.

3.1. Upper-Level Controller

The upper-level controller consists of a PI velocity regulator and an ASMC, which is responsible for generating the target longitudinal tire force and the additional yaw moment based on the driver input, i.e., the front-wheel steering angle ${\delta }_f$ and the braking/driving pedal displacement $\mathrm{BP}$. First, the target longitudinal tire force $F_x^{*}$ can be obtained by a PI velocity regulator as follows:

$$F_x^{*}=k_{pv}\left(v_x^{*}-v_x\right)+k_{iv}{\displaystyle {\int }_0^t\left(v_x^{*}-v_x\right)d\tau }$$

(5)

where $k_{pv}$ and $k_{iv}$ denote the proportional and integral coefficients, respectively, and $v_x^{*}$ denotes the target longitudinal velocity calculated by the pedal displacement.

Then, for the generation of the additional yaw moment, an adaptive sliding mode control with a self-tuning parameter is proposed in this research. The sliding surface and the corresponding sliding condition are designed as follows:

$$s=\gamma -{\gamma }_{target}-\lambda \left(\beta -{\beta }_{target}\right), \dot{s}=-\xi s$$

(6)

where $\lambda$ is the weight, designed to trade off the relative weights between tracking the target yaw rate and the target sideslip angle. A detailed discussion of the self-tuning mechanism is discussed in Section 3. $\xi >0$ is a constant.

If we take the time derivative of (5), we can get

$$\dot{s}=\dot{\gamma }-{\dot{\gamma }}_{target}-\lambda \left(\dot{\beta }-{\dot{\beta }}_{target}\right)=\dot{\gamma }-{\dot{\gamma }}_{target}-\lambda \dot{\beta }$$

(7)

From (1) and (2), we have

$$\left\{\begin{array}{c}\dot{\beta }={\displaystyle \frac{a_y}{v_x}}-\gamma \hfill \\ \dot{\gamma }=-{\displaystyle \frac{2C_f{C}_r}{C_f+C_r}}\cdot {\displaystyle \frac{{(l_f+l_r)}^2}{I_z{v}_x}}\cdot \gamma +{\displaystyle \frac{m(l_f{C}_f-C_r{l}_r)}{(C_f+C_r)I_z}}{a}_y+{\displaystyle \frac{2C_f{C}_r}{C_f+C_r}}\cdot {\displaystyle \frac{(l_f+l_r)}{I_z}}{\delta }_f+{\displaystyle \frac{1}{I_z}}{M}_z\hfill \end{array}\right.$$

(8)

Then, substituting (8) into (7), we can obtain the additional yaw moment $M_z$ as follows:

$$\begin{array}{l}{M}_z=\left({\displaystyle \frac{2C_f{C}_r}{C_f+C_r}}\cdot {\displaystyle \frac{{(l_f+l_r)}^2}{v_x}}-I_z\left(\lambda +\xi \right)\right)\cdot \gamma -{\displaystyle \frac{2C_f{C}_r(l_f+l_r)}{C_f+C_r}}{\delta }_f\\ -\left({\displaystyle \frac{m(l_f{C}_f-C_r{l}_r)}{C_f+C_r}}-{\displaystyle \frac{I_z\lambda }{v_x}}\right)\cdot a_y+I_z\cdot \left({\dot{\gamma }}_{\mathrm{target}}+\xi {\gamma }_{\mathrm{target}}+\xi \lambda \beta \right)-K_{\gamma }\cdot \mathrm{sat}({\displaystyle \frac{s}{\Phi }})\end{array}$$

(9)

where $K_{\gamma }\ge \xi$ is a constant and $\Phi$ is the thickness of the boundary layer.

3.2. Lower-Level Controller

The primary goal of a lower-level controller is to map the virtual control input to corresponding actuators. In this process, there are some factors that need to be taken into consideration, including (1) the reasonable distribution of all tire forces, (2) matching the resultant yaw moment with the virtual control input from the upper-level controller, and (3) the physical constraint of the chassis system and road–tire friction.

The first factor inherently points to avoiding excessive tire wear and reasonable usage of all tires to make full use of friction. To achieve this goal, we apply the concept of tire dissipation energy [22] to automatically distribute the necessary tire force to non-saturated tires.

First, we define the relative longitudinal and lateral wheel speed as follows:

$$\left\{\begin{array}{c}{\overline{v}}_{xi}={\omega }_{i}r-V_i\cos {\alpha }_i\hfill \\ {\overline{v}}_{yi}=-V_i\sin {\alpha }_i\hfill \end{array},\right.i=fl,fr,rl,rr$$

(10)

where $V_i$ is the speed at the center of the wheel, derived from the speed of the center of mass of the vehicle. Then, based on the tire dissipation power $P_i={\overline{v}}_{xi}{F}_{xi}+{\overline{v}}_{yi}{F}_{yi}$, the cost function can be constructed as follows:

$$J_1={‖W_{u1}U(t)‖}_2^2$$

(11)

where $U(t)$ is the tire force variation and $W_{u1}$ is the corresponding weight; they are expressed as

$$U(t)={\left[\Delta F_{xfl},\Delta F_{yfl},\Delta F_{xfr},\Delta F_{yfr},\Delta F_{xrl},\Delta F_{xrr}\right]}^T, W_{u1}=diag({\overline{v}}_{xi},{\overline{v}}_{yfl},{\overline{v}}_{xfr},{\overline{v}}_{yfr},{\overline{v}}_{xrl},{\overline{v}}_{xrr})$$

(12)

The second factor is to ensure a minimum error in mapping the virtual control from the upper-level controller, i.e., the distribution of tire forces should match the additional yaw moment and the longitudinal force. Then, the second cost function can be constructed as follows:

$$J_2={‖W_{u2}\left(BU(t)-C_U\right)‖}_2^2$$

(13)

where $W_{u2}$ is the diagonal weight matrix; $C_U={\left[F_x^{*},M_z\right]}^T$ denotes the target output; and $B$ is the control effectiveness matrix, represented as

$$B=\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 1\\ -0.5t_w& l_f& 0.5t_w& l_f& -0.5t_w& 0.5t_w\end{array}\right]$$

The third factor represents the constraint of each control input. If the longitudinal tire force is subject to the limitation of tire–road friction and the physical constraint of the braking/driving system itself, then the maximum longitudinal force $F_{xi,\max }$ can be expressed as follows:

$$F_{xi,\max }=\min \left(\sqrt{{\left(\mu F_{zi}\right)}^2-F_{yi}^2},p_{\max }\cdot {\varphi }_{ESC},T_{m,\max }/r_e\right)$$

(14)

where $\mu$ is the road friction coefficient, $F_{zi}$ is the normal load of a certain tire, $p_{\max }$ is the maximum braking pressure of the cylinder, ${\varphi }_{ESC}$ is the gain between the brake cylinder and the actual brake force, $T_{m,\max }$ is the maximum driving force of the tire hub motor, and $r_e$ is the effective radius of the tire. The upper limit of the front-wheel lateral force $F_{yi,\max }$ can be deduced by the linear tire model as follows:

$$F_{yi,\max }=C_f{\alpha }_{f,\max }$$

(15)

where ${\alpha }_{f,\max }$ is the upper limit of the wheel slip angle for different roads, whose value can be obtained by experiments.

Combining (14) and (15), the constraints for the lower-level control input can be expressed as

$$0=\underset{\_}{U}\le U(t)\le \overline{U}(F_{xi,\max },F_{yi,\max })$$

(16)

As a final step, the lower-level control input can be solved with a weighted least squares (WLS) problem, constructed as follows:

$$\begin{array}{l}u(t)=\arg \underset{\underset{\_}{U}\le u\le \overline{U}}{\min }\left[{\xi }_1\cdot J_1+{\xi }_2\cdot J_2\right]\\ =\arg \underset{\underset{\_}{U}\le u\le \overline{U}}{\min }{‖\underset{A}{\underbrace{\left(\begin{array}{c}{\xi }_1^{1/2}\cdot W_{u1}\\ {\xi }_2^{1/2}\cdot W_{u2}\cdot B\end{array}\right)}}\cdot u-\underset{b}{\underbrace{\left(\begin{array}{c}0\\ {\xi }_2^{1/2}\cdot W_{u2}\cdot C_U\end{array}\right)}}‖}_2^2\\ =\arg \underset{\underset{\_}{U}\le u\le \overline{U}}{\min }({‖A\cdot u-b‖}^2)\end{array}$$

(17)

The WLS problem can be solved with the interior-point method to obtain the control input. The incremental longitudinal force can be exerted to the braking/driving actuators, but the lateral force components in the lower-level control input cannot be directly applied to the actuators; there is a gap between the output of the control algorithm and the actual actuation input. A further step is necessary for the lateral forces to be converted to the direct actuation input, i.e., front-wheel steering angle. Here, we apply the inverse tire model to complete this process.

The traditional Dugoff tire model is mostly utilized in a forward manner [23]; it generates corresponding longitudinal and lateral tire forces by feeding such tire states as the slip rate and tire slip angle, so it seems no direct linkage can be found between tire lateral force and the optimal steering angle input. However, the slip angle is closely related to both the road wheel steer angle and the tire lateral force; thus, the tire slip angle can be the bridge from the tire lateral force to the appropriate road wheel steer angle. The first step is to build an inverse Dugoff tire model as follows:

$$\left\{\begin{array}{c}{\alpha }_{i,d}=\arctan \left({\displaystyle \frac{F_{yi}}{C_i}}\left(1+{\displaystyle \frac{F_{xi}}{C_i^s-F_{xi}}}\right)\right),\varsigma \in \left[0,\mu /2\right]\hfill \\ {\alpha }_{i,d}=\arctan \left({\displaystyle \frac{F_{yi}{C}_i^s}{C_i}}\cdot {\displaystyle \frac{{\mu }^2{F}_{zi}^2}{4\mu F_{zi}{C}_i^s\sqrt{F_{xi}^2+F_{yi}^2}-4F_{xi}{C}_i^s\sqrt{F_{xi}^2+F_{yi}^2}-{\mu }^2{F}_{zi}^2}}\right),\varsigma \in (\mu /2,\mu ]\hfill \end{array}\right.$$

(18)

where $i=f,r$; $\varsigma ={\displaystyle \frac{\sqrt{F_{xi}^2+F_{yi}^2}}{F_{zi}}}$; and $C_i^s$ denotes the longitudinal stiffness of the tire.

Then, based on the achieved target tire slip angle ${\alpha }_{i,d}$, we can obtain the actual front-wheel steer angle by simply designing a PID controller as follows:

$${\delta }_{f,d}=k_{p\delta }\left({\alpha }_{i,d}-{\alpha }_i\right)+k_{i\delta }{\displaystyle {\int }_0^t\left({\alpha }_{i,d}-{\alpha }_i\right)d\tau }+k_{d\delta }\left({\dot{\alpha }}_{i,d}-{\dot{\alpha }}_i\right)$$

(19)

4. Parameter Self-Tuning Mechanism

For different conditions, different control endeavors with different parameters or gains will be needed. For vehicle stability control, the state of the yaw rate should be more stressed in the condition of a mild cornering scenario, whereas the emphasis should be put on the sideslip angle in more severe conditions. Thus, the trade-off parameter $\lambda$ in (6) is essential for this condition-adjustment process. The self-tuning mechanism of the parameter is elaborated upon in this section.

4.1. Online Adaptation of Tire Cornering Stiffness

As previously discussed, we adopt tire cornering stiffness as the indicator of conditions. Then, an online adaptation of tire cornering stiffness is necessary for the self-tuning mechanism. First, the vehicle regression model can be built based on a single-track dynamics model as follows:

$$\underset{Y}{\underbrace{\left[\begin{array}{c}{I}_z\dot{\gamma }\\ m{a}_{y,m}\end{array}\right]}}=\underset{\mathrm{P}}{\underbrace{\left[\begin{array}{cc}{\displaystyle \frac{-l_f^2\gamma -l_f{\widehat{v}}_y(k)}{v_x}}+l_f{\delta }_f& {\displaystyle \frac{-l_r^2\gamma +l_r{\widehat{v}}_y(k)}{v_x}}\\ {\displaystyle \frac{-l_f\gamma -{\widehat{v}}_y(k)}{v_x}}+{\delta }_f& {\displaystyle \frac{l_r\gamma -{\widehat{v}}_y(k)}{v_x}}\end{array}\right]}}\underset{\mathrm{T}}{\underbrace{\left[\begin{array}{c}{C}_f\\ C_r\end{array}\right]}}$$

(20)

where $a_{y,m}$ is the lateral acceleration measured by the sensor and ${\widehat{v}}_y(k)$ is the estimated vehicle lateral velocity at instant $k$. The estimation method can be found in detail in [24]. Then, a regularized weighted least squares problem can be constructed as follows:

$$J\left({\mathrm{T}}_k\right)={\displaystyle \sum _{i=1}^k{\varphi }^{\left(k-i\right)}}{‖Y_i-{\mathrm{P}}_i{\mathrm{T}}_i‖}_2^2+\vartheta {‖{\mathrm{T}}_k-{\left(\mathrm{T}\right)}_n‖}_2^2$$

(21)

where ${\mathrm{T}}_k$ is the value of coefficient matrix $\mathrm{T}$ at instant $k$; $0\ll \varphi <1$ denotes the forgetting factor, the setting of which relies more on the latest measurement; ${\left(\mathrm{T}\right)}_n={\left[\begin{array}{cc}{\left(C_f\right)}_n& {\left(C_r\right)}_n\end{array}\right]}^{\mathrm{T}}$ is the nominal value of tire cornering stiffness, which can be calculated in static scenarios; and $\vartheta >0$ is the weight between the output error and the nominal value deviation. Then, the current instant optimal solution ${\mathrm{T}}_k^{*}$ can be obtained by solving the equation $\partial J\left({{\rm N}}_k\right)/\partial {{\rm N}}_k=0$ as follows:

$${\mathrm{T}}_k^{*}={\left({\displaystyle \sum _{i=1}^k{\varphi }^{\left(k-i\right)}}{\mathrm{P}}_i^T{\mathrm{P}}_i+\vartheta {\mathrm{I}}_2\right)}^{-1}\left({\displaystyle \sum _{i=1}^k{\varphi }^{\left(k-i\right)}}{\mathrm{P}}_i^T{Y}_i+\vartheta {\left(\mathrm{T}\right)}_n\right)$$

(22)

If ${\tilde{\mathrm{T}}}_k^{*}={\mathrm{T}}_k^{*}-{\left(\mathrm{T}\right)}_n$ and ${\tilde{Y}}_i=Y_i-{\mathrm{P}}_i{\left(\mathrm{T}\right)}_n$, and they are substituted into (22), we have

$${\tilde{\mathrm{T}}}_k^{*}={\left({\displaystyle \sum _{i=1}^k{\varphi }^{\left(k-i\right)}}{\mathrm{T}}_i^T{\mathrm{T}}_i+\vartheta {\mathrm{I}}_2\right)}^{-1}{\displaystyle \sum _{i=1}^k{\varphi }^{\left(k-i\right)}}{\mathrm{T}}_i^{T}Y$$

(23)

The expressions in (22) and (23) are called batch formulations. These can then be written in a recursive form as follows:

$${\tilde{\mathrm{T}}}_k^{*}={\tilde{\mathrm{T}}}_{k-1}^{*}+{\left({\Gamma }_k+\vartheta {\mathrm{I}}_2\right)}^{-1}\left[\vartheta \left(\varphi -1\right){\tilde{\mathrm{T}}}_{k-1}^{*}+{\mathrm{P}}_i^T{e}_k\right]$$

(24)

where ${\Gamma }_k={\displaystyle \sum _{i=1}^k{\varphi }^{\left(k-i\right)}}{\mathrm{P}}_i^T{\mathrm{P}}_i=\varphi {\Gamma }_{k-1}+{\mathrm{P}}_i^T{\mathrm{P}}_i$ and $e_k={\tilde{Y}}_k-{\mathrm{P}}_k{\tilde{\mathrm{T}}}_{k-1}^{*}$.

There exists a regularization term ${\left({\Gamma }_k+\vartheta {\mathrm{I}}_2\right)}^{-1}$ in (24); this term guarantees that a possible solution exists for this formulation. The matrix ${\Gamma }_k$ serves as the estimation gain; it is formulated in a recursive way to make the adaptation process stable. This completes the cornering stiffness adaptation model.

4.2. Parameter Self-Tuning Based on a Stability Indicator

As tire cornering stiffness can directly reflect the stability states of a vehicle, it is intuitive to formulate a stability indicator based on the real-time estimation of tire cornering stiffness. Following that, the parameter self-tuning mechanism can be constructed.

First, we correlate the tire saturation extent with tire cornering stiffness by designing the weights as follows:

$${\kappa }_i=\mathrm{sat}\left({\displaystyle \frac{1}{2{\xi }_t}}\left(C_i-{\sigma }_i{F}_{zi}+{\xi }_t\right)\right),i=f,r$$

(25)

where ${\xi }_t>0$ can be obtained by dividing the tire slip conditions into certain areas. As shown in Figure 3, assuming a certain normal load on the tire, the tire cornering dynamics can be divided into a linear section, a minor nonlinear section, a major nonlinear section, and a saturated section based on the Magic Formula tire model, where the colors of different sections match each other. We define the thresholds of cornering stiffness for linear and saturated sections as $C_{l\_thr}$ and $C_{s\_thr}$; then, the parameter ${\xi }_t$ is calculated as ${\xi }_t=\left(C_{s\_thr}-C_{l\_thr}\right)/2$. The two thresholds can be iteratively obtained from empirical knowledge or using the trial-and-error method.

In (25), ${\sigma }_i$ is a parameter that varies with the variation in the normal load of a certain tire, the value of which can be obtained by fitting a certain normal load into the following equation:

$$\left\{\begin{array}{c} {\sigma }_i=-0.1086\times {\stackrel{⌢}{F}}_{zi}^2+1.733\times {\stackrel{⌢}{F}}_{zi}+6.956\hfill \\ {\stackrel{⌢}{F}}_{z,i}=0.001\times F_{z,i}, F_{z,i}\in \left[1500,9000\right]\hfill \end{array}\right.$$

(26)

The weights in (25) can then indicate the current saturation state of a certain tire to a large extent. To make use of the weights, we define the following stability indicator as

$$\chi =\left\{\begin{array}{c}4\left(1-{\kappa }_f\right)\left(1-{\kappa }_r\right), {\kappa }_f\in \left[0.5,1\right]\&{\kappa }_r\in \left[0.5,1\right]\hfill \\ 4\left({\kappa }_f-0.5\right)\left(1-{\kappa }_r\right), {\kappa }_f\in \left[0,0.5\right)\&{\kappa }_r\in \left[0.5,1\right]\hfill \\ 4\left({\kappa }_r-0.5\right)\left(1-{\kappa }_f\right), {\kappa }_r\in \left[0,0.5\right)\&{\kappa }_f\in \left[0.5,1\right]\hfill \\ -4\left(0.5-{\kappa }_f\right)\left(0.5-{\kappa }_f\right)-1,{\kappa }_f\in \left[0,0.5\right)\&{\kappa }_r\in \left[0,0.5\right)\hfill \end{array}\right.$$

(27)

As can be inferred from (27), the vehicle is in the stable region when $\chi \in \left[0,1\right]$ is satisfied and the front and rear tires are all in linear or minor nonlinear sections. When one certain axle is in a linear or minor nonlinear section, no matter what region the other axle is in, the chassis is deemed to be capable of falling back into the stable region and the condition $\chi \in \left[-1,0\right]$ holds. For the most unstable condition, both axles slip into the major nonlinear or saturated sections; then, the algorithm set is $\chi \in \left[-2,-1\right]$. This is the complete rationale behind the quantification of the vehicle status.

On the other hand, the variation rate of tire cornering stiffness can be an indicator of the stability trend of the vehicle, and it can further indicate the next-step stability state. To achieve this goal, we define two vectors, $\overrightarrow{C}=\left[{\left(C_f\right)}_n-C_f,{\left(C_r\right)}_n-C_r\right]$ and ${\overrightarrow{C}}_{dot}=\left[-{\dot{C}}_f,-{\dot{C}}_r\right]$, where ${\left(C_f\right)}_n$ and ${\left(C_r\right)}_n$ are the nominal cornering stiffnesses of the front and rear axles, respectively. As shown in Figure 4, the angle between the two vectors can be calculated as

$$\vartheta =\arccos \left({\displaystyle \frac{\overrightarrow{C}\cdot {\overrightarrow{C}}_{dot}}{‖\overrightarrow{C}‖‖{\overrightarrow{C}}_{dot}‖}}\right)$$

(28)

The two vectors are symbols for the current tire state and the trend of the tire state. A vehicle tends to maintain stability when the angle $\vartheta$ is acute, i.e., $\vartheta \in \left[0,\pi /2\right)$, because the operation point $O_p$ will move toward the stable origin $O_s$. When the angle $\vartheta$ is obtuse, i.e., $\vartheta \in \left[\pi /2,\pi \right]$, the tire state shows a trend of moving away from the stable region. A smaller acute $\vartheta$ implies a faster convergence to stability, and a bigger obtuse $\vartheta$ means a faster unstable trend. To exploit this property of $\vartheta$ and to achieve a smooth variation in the stability indicator, we propose the following modified Fal function [25] as

$$\mathrm{Fal}\left(\vartheta \right)=\left\{\begin{array}{c}{\phi }_1\vartheta +{\phi }_2\arctan \vartheta , \vartheta \le {\vartheta }_{thr}\\ {\vartheta }^a\mathrm{sign}(\vartheta ) {\vartheta }_{thr}\le \vartheta <1\\ 1, \theta \ge 1\hfill \end{array}\right.$$

(29)

where ${\phi }_1=a{\vartheta }_{thr}^{a-1}-{\displaystyle \frac{\left(1-a\right){\vartheta }_{thr}^a}{\left(1+{\vartheta }_{thr}^2\right)\arctan {\vartheta }_{thr}-{\vartheta }_{thr}}}$ and ${\phi }_2={\displaystyle \frac{\left(1-a\right)\left(1+{\vartheta }_{thr}^2\right){\vartheta }_{thr}^a}{\left(1+{\vartheta }_{thr}^2\right)\arctan {\vartheta }_{thr}-{\vartheta }_{thr}}}$; $a\in \left(0,1\right)$ is a constant and ${\vartheta }_{thr}$ is a tunable threshold. The comparison between the modified Fal and the traditional Fal functions is illustrated in Figure 5a.

As a final step of the parameter self-tuning mechanism, an updated law is devised by combining the weights in (27) and the Fal function in (29) as follows:

$$\lambda =\left\{\begin{array}{c}0, \chi \cdot \mathrm{Fal}\left(\vartheta \right)\in \left[0,1\right]\hfill \\ {\chi }^2\cdot {\mathrm{Fal}}^2\left(\vartheta \right), \chi \cdot \mathrm{Fal}\left(\vartheta \right)\in \left[-2,0\right)\end{array}\right.$$

(30)

The rationale behind the settings of (30) lies in the judgement of the stable state of the vehicle. The vehicle is considered to be in a stable region when the multiplication $\chi \cdot \mathrm{Fal}\left(\vartheta \right)$ falls into the set $\left[0,1\right]$, as can be implied by combining (30) and (27); then, the algorithm sets $\lambda =0$ to rely solely on the tracking of the target yaw rate. The vehicle is deemed to be outreaching into an unstable region if $\chi \cdot \mathrm{Fal}\left(\vartheta \right)<0$; then, minimizing the sideslip angle is necessary. As shown in Figure 5b, the updated law of $\lambda$ presented in (30) can effectively avoid an abrupt change in control command as it combines the smooth and continuous property of the modified Fal function and the parabolic function; thus, the performance of the controller can be accordingly and smoothly improved.

5. Simulation Results and Discussion

To evaluate the effectiveness of the proposed control framework, two scenarios were simulated on a co-simulation platform built with CarSim and MATLAB/Simulink. Aiming at testing the performance of the controller in normal and emergency obstacle avoidance modes, the two cases were set as follows: (1) the initial longitudinal velocity of the vehicle was 100 km/h; the driver exerted a double lane change (DLC) maneuver at the time of 1.5 s; the brake force exerted on the brake pedal caused a cylinder pressure of 2.5 MPa; and the road friction was set to 0.85, meaning a high-friction road; and (2) the initial longitudinal velocity was 90 km/h, the driver conducted a slalom maneuver at about 1.5 s; and the road friction was set to 0.3, meaning a low-friction road.

To validate the superiority of the proposed controller, two previously published control algorithms are presented for comparison. The first controller adopts an adaptive sliding mode control based on the phase plane method [14], marked as “ASMC” in the following figures. The major difference between the proposed controller and the ASMC lies in the self-tuning capability of the controller parameter $\lambda$: the proposed controller can adjust the value of $\lambda$ accordingly, while the ASMC cannot. The second controller for comparison only relies on the direct yaw motion control to stabilize the vehicle [26], and is marked as “DYC”; this controller focuses on stabilizing the yaw rate error. Thus, the parameter $\lambda$ was set to 0 throughout the simulations. The proposed controller in this paper is marked as “Proposed” in the following figures.

The high-fidelity vehicle dynamics model was built using CarSim. The main parameters of the prototype vehicle utilized in the simulation are listed in

Table 1. Main parameters of the prototype vehicle.

Type	Unit	Value
$m$	kg	2162
$l_f$	mm	1104.3
$l_r$	mm	1595.7
$I_z$	$\mathrm{kg}\cdot {\mathrm{m}}^2$	3234.0
$\left[\begin{array}{cc}{\left(C_f\right)}_n,& {\left(C_r\right)}_n\end{array}\right]$	$\mathrm{N}/\mathrm{rad}$	$\left[\mathrm{62,690} \mathrm{43,200}\right]$
${\tau }_e$	ms	10
$r_e$	mm	375
$t_w$	mm	1555

.

5.1. High-Friction Road DLC Maneuver

This case was set to verify the effectiveness of the controller in a normal driving scenario. To simulate a real-vehicle scenario, a Gaussian-type sensor noise with an amplitude of 5% of the nominal values was added to the accurate states, plus a sensor delay of 3 ms was also exerted. The simulation results are shown in Figure 6.

As the controllers selected for comparison are all capable of handling a vehicle on high-friction roads under normal driving conditions, the driver input steering wheel angles for the three controllers were basically identical, as shown in Figure 6a; the AFS angle, however, differed for the three controllers. As shown in Figure 6b, the AFS angle for the DYC controller is 0 because the auxiliary steering mechanism was not applied to this controller. Compared with the ASMC, the AFS angle for the proposed controller was closer to 0, implying less corrective effort in steering actuation. The longitudinal deceleration performance is shown in Figure 6c. Although the brake effort of the driver remained identical for the three controllers, there existed subtle differences in the longitudinal velocity profiles. This may be because of the braking and steering coupling effect. The vehicle under the proposed controller and the ASMC could brake more effectively than that under the DYC controller, which implies the implicit effect of the steering system on braking performance.

The main performance comparison for vehicle stability and handling ability was conducted in terms of DLC trajectory tracking as well as vehicle sideslip and yaw rate error minimization. Although the yaw rate error shrank faster for the “DYC” controller, the overshoot of its sideslip angle was significantly larger than the other two controllers. This is logical because the “DYC” controller is designed to solely track the target yaw rate rather than the sideslip angle. Comprehensively, the vehicle under the proposed controller could track the desired trajectory and reduce the vehicle sideslip angle/yaw rate error better than vehicles under the other two controllers, implying that the proposed controller could maintain vehicle stability and effectively improve the handling performance of the vehicle, as shown in Figure 6d–f.

This is closely related to the self-tuning mechanism of the controller parameter $\lambda$, as shown in Figure 6h. The parameter value varied with the real-time cornering stiffness adaptation, as shown in Figure 6g. Compared with that, the ASMC relies solely on the variable structure control capability of the sliding mode control algorithm, resulting in a larger fluctuation and greater overshoot in tracking the target states.

5.2. Low-Friction Road Slalom Maneuver

This case represented a more severe scenario simulating when a vehicle needs to avoid consecutive obstacles on a low-friction road. It served to test the limit handling stability of the vehicle. The overall simulation results are illustrated in Figure 7.

In this case, the driver started a slalom maneuver at about 1.5 s, while deceleration started at about 2 s, as shown in Figure 7a,c, respectively. Similarly, the velocity profiles under the three controllers slightly differed from each other, although the driver exerted the same braking effort. The variation in the AFS angles for the ASMC and the proposed controller, however, was not as large as in case (1); this is because the vehicle state was mostly close to the stability envelope in this case and there was not much of a margin left for an active variation in the steer angle to boost the overall performance. As for the yaw rate error, as shown in Figure 7d, the DYC controller managed the smallest error compared with the other two controllers, aligning with the analysis in case (1). However, the proposed controller was superior to the ASMC and the DYC controller regarding overall performance; for instance, the yaw rate error under the proposed controller was only slightly larger than the of DYC, but smaller than that of the ASMC; the sideslip angle under the proposed controller was obviously the smallest of those under the three controllers. The ${\alpha }_f-{\alpha }_r$ phase plane portrait is illustrated in Figure 7f. It can be seen that all three controllers could maintain vehicle stability in this case, but the proposed controller managed the best performance in terms of the tire sideslip motion. Figure 7g,h illustrate the tire cornering stiffness adaptation and the corresponding parameter real-time variation in case (2) for the proposed controller.

5.3. Comparative Analysis

This section presents the overall comparative results of the two simulation cases regarding the algorithm response time and control accuracy, followed by some conclusive discussions. The indicator of control accuracy is designated as the sum of the average absolute error of the yaw rate and the vehicle sideslip angle during a maneuver, and the calculation time of each control step can be measured with the timing function of MATLAB. All the controllers were implemented with an Intel Core i5-8265U processor. The results are listed in

Table 2. Comparative performance of all controllers under the two cases.

Case No.	Aspects	DYC Controller	ASMC	Proposed Controller
Case 1	Accuracy	1.8253	1.2650	0.9487
	Response Time (ms)	1.2592	1.2675	1.2802
	$\mathrm{TDE} (\times {10}^4\mathrm{J}$)	2.1370	2.0605	2.0194
Case 2	Accuracy	2.7375	2.2601	2.1203
	Response Time (ms)	1.2359	1.2378	1.2961
	$\mathrm{TDE} (\times {10}^4\mathrm{J}$)	3.2054	2.6786	2.4233

.

$\mathrm{TDE} (\times {10}^{4}J$$\mathrm{TDE} (\times {10}^{4}J$ According to the definition of accuracy, a larger value indicates the lower accuracy of a specific controller. It can be seen from the table that, taking the DYC controller as the baseline, the ASMC and the proposed controller improved control accuracy up to 30.7% and 48.02%, and 17.44% and 22.55%, for the two cases, respectively. Although the response times showed no evident statistical difference, all the controllers listed could respond swiftly against the 20 ms of the control interval.

From Table 2, the tire dissipation energy (TDE) [22] was chosen as an indicator to reflect the efficiency of the different controllers. This is the multiplication of tire dissipation power (TDP) with each control step. The TDP is calculated as follows:

$$\mathrm{TDP}={\displaystyle \sum _{ii=\left\{fl,fr,rl,rr\right\}}\left(\left|{\overline{v}}_{xii}\cdot F_{xii}\right|+\left|{\overline{v}}_{yii}\cdot F_{yii}\right|\right)}$$

(31)

where ${\overline{v}}_{xii}$ and ${\overline{v}}_{yii}$ are the longitudinal and lateral slip velocity, which can be obtained by multiplication of the tire slip ratio and slip angle with the longitudinal vehicle speed, respectively. $F_{xii}$ and $F_{yii}$ are the longitudinal and lateral tire forces, respectively.

The TDE values for the different controllers under the two cases are shown in Figure 8.

From (31), it is easy to conclude that the quantity of TDE is a direct indicator of tire wear and an indirect indicator of the vehicle stability status. As an energy type of quantity, it links closely to the efficiency of controllers; a larger TDE indicates a lower efficiency. Compared with the baseline DYC controller, the proposed controller generated 5.5% and 24.4% less TDE under the two cases. The proposed controller clearly generated the least TDE and yielded the best efficiency performance.

In summary, the proposed controller could enhance vehicle performance regarding handling and overall stability, which is closely related to the parameter self-tuning mechanism devised in the upper-level controller. Then, combined with the settings of the lower-level controller, the proposed controller helped to exploit the control potential and the control degrees of freedom of the vehicle, and pushed the vehicle performance to the control envelope to a large extent.

6. Conclusions

In this paper, a vehicle stability control framework is proposed based on the concept of a hierarchical controller. The key to the controller lies in a parameter self-tuning mechanism based on a novel tire cornering stiffness adaptation method. By quantitating the vehicle stability status, the vehicle can accordingly adjust the control input in real time. Simulations were conducted to mimic real driving scenarios, validating that the proposed controller can enhance control accuracy by 48.02% and 22.55%, and generate 5.5% and 24.4% less TDE under the two cases, without resulting in a significant response time. This control framework could help to reduce the parameter tuning time and costs for engineers and researchers to a large extent.

At present, the proposed controller is designed for a particular vehicle type; the threshold determination and section separation are not expected to be transplantable. Modeling, including data-driven modeling, methods have risen as hot topics for vehicle dynamics modeling; tire cornering stiffness estimation can benefit from these kinds of methods. Thus, these aspects represent promising directions for future research.