Acceleration Slip Regulation Control Method for Distributed Electric Drive Vehicles under Icy and Snowy Road Conditions

Abstract

To achieve a rapid and stable dynamic response of the drive anti-slip system for distributed electric vehicles on low-friction surfaces, this paper proposes an adaptive acceleration slip regulation control strategy based on wheel slip rate. An attachment coefficient fusion estimation algorithm based on an improved singular value decomposition unscented Kalman filter is designed. This algorithm combines Sage–Husa with the unscented Kalman filter for adaptive improvement, allowing for the quick and accurate determination of the road friction coefficient and, subsequently, the optimal slip rate. Additionally, a slip rate control strategy based on dynamic adaptive compensation sliding mode control is designed, which introduces a dynamic weight integral function into the control rate to adaptively adjust the integral effect based on errors, with its stability proven. To verify the performance of the road estimator and slip rate controller, a model is built with vehicle simulation software, and simulations are conducted. The results show that under icy and snowy road conditions, the designed estimator can reduce estimation errors and respond rapidly to sudden changes. Compared to traditional equivalent controllers, the designed controller can effectively reduce chattering, decrease overshoot, and shorten response time. Especially during road transitions, the designed controller demonstrates better dynamic performance and stability.

1. Introduction

Affected by the climate, roads in the cold regions during winter are often covered with ice and snow, creating various complex low-friction surfaces. This easily leads to vehicle slipping, loss of control, and unstable handling, which severely impacts urban traffic development [1,2]. Therefore, the anti-skid performance of urban snow removal vehicles, especially specialized vehicles for rapid snow clearance, has become an urgent issue to address [3]. In recent years, with the development of fully by-wire vehicle chassis and distributed drive technology, distributed electric drive vehicles have provided the technical basis for the acceleration slip regulation (ASR) system through the rapid and precise torque response of their in-wheel motors [3,4,5,6,7]. The key to achieving this technology lies in the rapid estimation of the road surface friction coefficient to determine the optimal slip ratio. With the swift response of the controller, the slip ratio of each wheel can be maintained near the optimal value under various driving conditions.

In recent years, the theories and algorithms for road friction coefficient estimation have been extensively studied by many researchers, resulting in a substantial number of published papers [8,9,10,11]. The recursive least squares algorithm, a forgetting factor, and a PI estimator were integrated by Gu to estimate the real-time road surface friction coefficient. Then, using fuzzy analysis methods, the current road conditions were classified, and the optimal slip ratio for these conditions was calculated. However, the paper only identified three types of road conditions and did not account for sudden changes in the friction coefficient [12]. A road identification method proposed by Wang is based on the standard road u-S curve, which can accurately obtain the optimal slip ratio. Although the method accounts for variable friction coefficients, it overlooks changes in the longitudinal force of the wheels, resulting in an excessive slip ratio overshoot of nearly 250% [13]. The current road conditions were identified in real time by Guo using slip ratio and tire-road friction coefficient, and the optimal slip ratio for the current conditions was calculated. However, the friction coefficient is significantly influenced by the vehicle’s longitudinal force state, leading to inaccurate calculations [14].

As road estimation algorithms have evolved, the design of estimators now combines both vehicle dynamic responses and external sensor signals [15,16,17,18]. For example, a normalized Dugoff tire model was developed by Wang, and a road friction coefficient observer was designed based on the unscented Kalman filter (UKF) algorithm and a 7-degrees-of-freedom vehicle model [19]. A dynamic joint estimation method for road friction coefficients based on inertial measurement units and built-in sensors was proposed by Li. This method employs machine learning and dual radial basis function neural networks, utilizing the extended Kalman filter to establish the road friction coefficient estimator. However, simulation results indicate significant fluctuations in the estimated road friction coefficient [20]. Overall, current methods for estimating road friction coefficients still face challenges such as low estimation accuracy, significant impact from road surface random noise on the estimation results, and limited research on conditions with sudden changes in the friction coefficient.

The existing ASR control strategies can be categorized into three types based on their control objectives: slip ratio-based control, torque-based control, and a combination of both [21]. The corresponding control algorithms mainly include PID control, model predictive control (MPC), fuzzy control, adaptive control, and sliding mode control [22,23,24,25,26,27]. For example, a variable universe fuzzy PID (VFPID) controller for ASR control was proposed by Chen based on an improved fuzzy PID (FPID) algorithm [28].

However, PID control exhibits poor robustness when dealing with complex and variable road conditions, making it difficult to handle snow-covered surfaces. In contrast, sliding mode controllers and their improved versions are widely used for controlling low friction coefficient roads due to their excellent robustness. A conditional integrator sliding mode control strategy, combining anti-slip torque with feedforward control torque based on driver commands to achieve coordinated control of drive torque was adopted by Zhang [22]. Building on this, a saturation-resistant sliding mode control strategy based on conditional integration was proposed by Kang, and the control variable was redesigned as wheel angular velocity to achieve faster vehicle dynamic response [23]. An ASR control strategy based on saturation-resistant sliding mode control law, better addressing wheel model uncertainties, slip ratio estimation errors, and disturbances, was designed by Xiong [24]. To optimize control parameters, a fuzzy algorithm for adaptive adjustment of the sliding mode controller’s switching function was introduced by Wang [25]. Addressing issues such as multi-actuator coupling, nonlinearity, uncertainty, and actuator faults in ASR systems, a multi-agent system (MAS) based on an adaptive non-singular terminal sliding mode (NTSM) fault-tolerant control method was proposed by Zhang. This method employs an adaptive estimation mechanism to select the gain for the controller’s switching term, ensuring the smoothness of the actual control signal and reducing chattering and energy consumption [29]. Research indicates that sliding mode control has the advantages of strong disturbance rejection and low dependency on system parameters. Nevertheless, it inevitably encounters issues such as chattering and steady-state errors. Therefore, under conditions of low and varying friction coefficients, it remains challenging to minimize errors and chattering, achieve minimal overshoot and rapid response during the startup process, and ensure quick reaction and smooth transitions during switching.

The main contributions of this paper are as follows:

• In terms of road friction coefficient estimation, to reduce the impact of road surface random noise, the Sage–Husa method was combined with the UKF method and adaptively improved. Additionally, the longitudinal force derived from the tire model and the longitudinal force estimated based on wheel dynamics were integrated, which mitigated the impact of estimation errors and was used to derive the road friction coefficient and the optimal slip ratio.
• In terms of slip ratio control algorithms, a dynamic adaptive compensation sliding mode control (DASMC) algorithm based on wheel slip ratio was proposed. This algorithm introduces a dynamically weighted integral function into the control law, adaptively adjusting the integral action based on the error to achieve low overshoot, rapid response, and smooth transitions under varying road friction conditions.
• Joint simulations under two typical conditions were conducted to verify the effectiveness of the algorithm. Discussion of the simulation results indicates that the improved estimation algorithm can mitigate the impact of noise errors, while the DASMC outperforms traditional sliding mode control algorithms in terms of chattering resistance and steady-state response.

The remaining content of this paper is organized as follows: Section 2 introduces the vehicle dynamics model, tire dynamics model, and motor model. In Section 3, an adaptive UKF road estimator is established to estimate the adhesion coefficient of the road surface under working conditions and obtain the optimal slip rate. Section 4 designs a dynamic adaptive compensation control strategy for slip rate to stabilize the wheel slip rate at the target value. The overall structure of the anti-slip system is shown in Figure 1. Section 5 conducts simulation verification and discusses the simulation results based on performance indicators. Finally, the conclusions are given in Section 6.  

2. Vehicle Model

2.1. Vehicle Dynamics Model

In this paper, the four-wheel independent drive engineering vehicle dynamics model describes the vehicle’s dynamic response using longitudinal and yaw motions, neglecting lateral motion, without considering air resistance and gradient resistance, as shown in Figure 2 [30,31]. According to Newton’s second law, the equations of motion are given by:

$$a_x={\displaystyle \frac{\left(F_{x1}+F_{x2}+F_{x3}+F_{x4}\right)}{m}}$$

(1)

$${\dot{\omega }}_z=-{\displaystyle \frac{B_f{F}_{x1}}{2I}}+{\displaystyle \frac{B_f{F}_{x2}}{2I}}-{\displaystyle \frac{B_r{F}_{x3}}{2I}}+{\displaystyle \frac{B_r{F}_{x4}}{2I}}$$

(2)

where $F_{xi}$ is the longitudinal reaction force from the ground, with the subscript $i\in [1,2,3,4]$ representing the left front wheel, right front wheel, left rear wheel, and right rear wheel, respectively; $a_x$ is the vehicle’s longitudinal acceleration; m is the total mass of the vehicle; ${\dot{\omega }}_z$ is the vehicle’s yaw acceleration; $B_f$ and $B_r$ are the front and rear wheel tracks, and I is the vehicle’s moment of inertia. 

To describe the kinematic characteristics of the tires, a single-wheel model is used, assuming that the mechanical properties of all four wheels are identical, as shown in Figure 3. The equations of motion for each wheel are given by:

$$J{\dot{\omega }}_i=T_i-R{F}_{xi}$$

(3)

where J is the rotational inertia of the wheel, $T_i$ is the driving torque on the wheel, and R is the rolling radius of the wheel; $F_{zi}$ is the vertical force on a single wheel, which can be calculated by the following equation:

$$F_{z1}=F_{z2}=m\mathrm{g}{\displaystyle \frac{L_r}{2l}}-m{a}_x{\displaystyle \frac{H_g}{2l}}$$

(4)

$$F_{z3}=F_{z4}=m\mathrm{g}{\displaystyle \frac{L_f}{2l}}+m{a}_x{\displaystyle \frac{H_g}{2l}}$$

(5)

where, in Equations (4) and (5), $L_f$ is the distance from the center of mass to the front axle, $L_r$ is the distance from the center of mass to the rear axle and is the height of the center of mass.

During vehicle acceleration, the wheel slip ratio ${\lambda }_i$ can be defined as [21]:

$${\lambda }_i={\displaystyle \frac{{\omega }_{i}R-v_i}{{\omega }_{i}R}}$$

(6)

where ${\omega }_i$ and $v_i$ are the wheel angular velocity and longitudinal speed, respectively.

2.2. Tire Model

For the study of parameter estimation, considering the complexity, real-time performance, and model accuracy of the algorithm, the Dugoff tire model with fewer parameters is used. The mathematical expression for the longitudinal tire force $F_x$ of a single wheel is:

$$F_x=\mathsf{\mu }{F}_x^0=\mathsf{\mu }{F}_z{C}_x{\displaystyle \frac{\lambda }{1-\lambda }}f\left(L\right)$$

(7)

$$f\left(L\right)=\left\{\begin{array}{c}L(2-L),L<1\\ 1,L\ge 1\end{array}\right.$$

(8)

$$L={\displaystyle \frac{1-\lambda }{2\sqrt{C_x^2{\lambda }^2+C_y^{2}tan{\alpha }^2}}}\left(1-\epsilon v_x\sqrt{{\lambda }^2+tan{\alpha }^2}\right)$$

(9)

where $F_x^0$ is the normalized longitudinal tire force, $\mathsf{\mu }$ is the road adhesion coefficient, and $f\left(L\right)$ is the tire model correction factor; $c_x$ and $c_y$ are the longitudinal and lateral stiffness of the tire, whose values can be fitted based on the tire characteristics data provided by Carsim; $\alpha$ is the slip angle of the tire, $\epsilon$ is the speed influence coefficient used to correct the effect of tire slip speed on the tire force, and $v_x$ is the longitudinal speed of the wheel.

2.3. Motor Model

To relatively accurately reflect the dynamic response characteristics of the wheel motor, this paper uses a simplified first-order equivalent model to describe the motor:

$$G\left(s\right)={\displaystyle \frac{T_{\mathrm{out}}}{T_d}}={\displaystyle \frac{1}{\tau s+1}}$$

(10)

where $T_d$ and $T_{\mathrm{out}}$ are the torque demand from the controller and the actual output torque of the motor, respectively.

3. Adhesion Coefficient Fusion Estimation Based on Improved Singular Value Decomposition UKF

3.1. State–Space Model

The four longitudinal forces $F_{xi}$ on the front and rear wheels of the vehicle are selected as the state variables $x_{k\mid 0}$, and the vehicle’s longitudinal acceleration $a_x$ and yaw acceleration ${\dot{\omega }}_z$ form the observations y.

The estimation system model is:

$$x_{k+1}=f_k{x}_{k\mid 0}+{\omega }_k$$

(11)

$$y_{k+1}=h_k{x}_{k\mid 0}+v_k$$

(12)

where $f_k$ is the state transition matrix, $h_k$ is the observation matrix, ${\dot{\omega }}_k$ is the state noise, and $v_k$ is the observation noise. The state–space and observation–space representations are as follows:

$$\begin{array}{c}{\mathit{x}}_k={\left[F_{x1}{F}_{x2}{F}_{x3}{F}_{x4}\right]}^T\\ {\mathit{y}}_k={\left[a_x{\dot{\omega }}_z\right]}^T\end{array}$$

(13)

Equation (10) can be equivalently expressed as:

$$\left[\begin{array}{c}\hfill F_{x1,k+1}\hfill \\ \hfill F_{x2,k+1}\hfill \\ \hfill F_{x3,k+1}\hfill \\ \hfill F_{x4,k+1}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill F_{x1}\hfill \\ \hfill F_{x2}\hfill \\ \hfill F_{x3}\hfill \\ \hfill F_{x4}\hfill \end{array}\right]+\left[\begin{array}{c}{\omega }_{k+1}^1\\ {\omega }_{k+1}^2\\ {\omega }_{k+1}^3\\ {\omega }_{k+1}^4\end{array}\right]$$

(14)

The observation equation is shown in Equation (15):

$$y_k=\left[\begin{array}{cccc}{\displaystyle \frac{1}{m}}& {\displaystyle \frac{1}{m}}& {\displaystyle \frac{1}{m}}& {\displaystyle \frac{1}{m}}\\ {\displaystyle \frac{-0.5B_f}{Iz}}& {\displaystyle \frac{0.5B_f}{Iz}}& {\displaystyle \frac{-0.5B_r}{Iz}}& {\displaystyle \frac{0.5B_r}{Iz}}\end{array}\right]\left[\begin{array}{c}\hfill F_{x1}\hfill \\ \hfill F_{x2}\hfill \\ \hfill F_{x3}\hfill \\ \hfill F_{x4}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill v_{k+1}^1\hfill \\ \hfill v_{k+1}^2\hfill \end{array}\right]$$

(15)

3.2. Improved Singular Value Decomposition Unscented Kalman Filter

Due to the potential for the unscented Kalman filter (UKF) method to encounter non-positive definite covariance issues and non-positive definite Cholesky factors, which can interrupt the algorithm, this paper employs the singular value decomposition unscented Kalman filter (SVD-UKF) method to estimate the longitudinal tire force of engineering vehicles on bumpy roads. The steps of the SVD-UKF algorithm are introduced below.

First, assign initial values to the state variables and the covariance matrix, then calculate the Sigma points and their corresponding weights. Assuming there are n sampling points, there will be a total of 2n+1 Sigma points, which are:

$$x_{k\mid k}=\left\{\begin{array}{c}{x}^{\left(0\right)}=\widehat{x},i=0\\ x^{\left(i\right)}=\widehat{x}+{\left(\sqrt{(n+\lambda ){\mathit{P}}_k}\right)}_i,i=1\sim n\\ x^{\left(i\right)}=\widehat{x}-{\left(\sqrt{(n+\lambda ){\mathit{P}}_k}\right)}_i,i=n+1\sim 2n\end{array}\right.$$

(16)

where ${\left(\sqrt{(n+\lambda ){\mathit{P}}_k}\right)}_i$ represents the column of the matrix. When calculating the weights, multiple intrinsic parameters are involved, including $\lambda ,\alpha , \mathrm{and} \beta$, and it is generally required to ensure that $(n+\lambda ){\mathit{P}}_k$ is a semi-positive definite matrix. However, in many cases, it is not possible to ensure the positive definiteness of the covariance matrix by controlling the intrinsic parameters, so SVD decomposition is a simple and efficient method. SVD decomposition does not require the decomposed matrix to be a positive definite square matrix. Suppose $\mathit{A}$ is an $m\times n$ matrix, then we define the SVD of the matrix $\mathit{A}$ as:

$$\mathit{A}=\mathit{U}\mathsf{\Sigma }{\mathit{V}}^T$$

(17)

where $\mathit{U}$ is an $m\times m$ matrix, $\mathsf{\Sigma }$ is an $m\times n$ matrix with all elements except those on the main diagonal being zero, and each element on the main diagonal is called a singular value. $\mathit{V}$ is an $n\times n$ matrix. Both $\mathit{U}$ and $\mathit{V}$ are unitary matrices, satisfying ${\mathit{U}}^T\mathit{U}=\mathit{I}$ and ${\mathit{V}}^T\mathit{V}=\mathit{I}$.

Therefore, according to Tan [32]’s research, Equation (16) is updated to:

$$x_{k\mid k}=\left\{\begin{array}{c}{x}^{\left(0\right)}=\widehat{x},i=0\\ x^{\left(i\right)}=\widehat{x}+{\left(\sqrt{(n+\lambda )}{\mathit{U}}_{k-1}\sqrt{{\mathsf{\Sigma }}_{k-1}}\right)}_i,i=1\sim n\\ x^{\left(i\right)}=\widehat{x}-{\left(\sqrt{(n+\lambda )}{\mathit{U}}_{k-1}\sqrt{{\mathsf{\Sigma }}_{k-1}}\right)}_i,i=n+1\sim 2n\end{array}\right.$$

(18)

Calculate the next step prediction and covariance matrix for the state variables:

$${\widehat{x}}_{k+1\mid k}=\sum _{i=0}^{2n}{\omega }_m^{\left(i\right)}{x}_{k+1\mid k}^{\left(i\right)}$$

(19)

$$P_{k+1\mid k}=\sum _{i=0}^{2n}{\omega }_c^{\left(i\right)}\left(x_{k+1\mid k}^i-{\widehat{x}}_{k+1\mid k}\right){\left(x_{k+1\mid k}^i-{\widehat{x}}_{k+1\mid k}\right)}^T+{\mathit{Q}}_k$$

(20)

where ${\mathit{Q}}_k$ is the state noise matrix, and its value affects the confidence level in the state values; a smaller value indicates greater trust in the state values. Based on the time-updated ${\widehat{x}}_{k+1\mid k}$ and ${\mathit{P}}_{k+1\mid k}$, and the 2n+1 Sigma points and their weights obtained through the sampling strategy, the measurement function yields:

$$y_{k+1\mid k}^i={\mathit{H}}_k{x}_{k+1\mid k}^i$$

(21)

Similar to Equations (19) and (20), the measurement means ${\widehat{y}}_{k+1\mid k}^i$, variance ${\mathit{P}}_{y,k+1}$, and covariance ${\mathit{P}}_{xy,k+1}$ can be obtained.

$${\widehat{y}}_{k+1\mid k}=\sum _{i=0}^{2n}{\omega }_m^{\left(i\right)}{y}_{k+1\mid k}^{\left(i\right)}$$

(22)

$${\mathit{P}}_{y,k+1}=\sum _{i=0}^{2n}{\omega }_c^{\left(i\right)}\left(y_{k+1\mid k}^i-{\widehat{y}}_{k+1\mid k}\right){\left(y_{k+1\mid k}^i-{\widehat{y}}_{k+1\mid k}\right)}^T+{\mathit{Q}}_{k+1}$$

(23)

$${\mathit{P}}_{xy,k+1}=\sum _{i=0}^{2n}{\omega }_c^{\left(i\right)}\left(x_{k+1\mid k}^i-{\widehat{x}}_{k+1\mid k}\right){\left(y_{k+1\mid k}^i-{\widehat{y}}_{k+1\mid k}\right)}^T$$

(24)

where ${\mathit{R}}_{k+1}$ is the observation noise matrix, and its value affects the confidence level in the observations; a smaller value indicates greater trust in the observations. The traditional UKF process assumes that the noise covariance matrix is fixed, but this can lead to significant estimation errors. To account for the impact of varying road noise on the estimation results, we introduce the Sage–Husa noise-adaptive filtering method. This method adjusts the observation noise covariance for changing noise conditions. By adaptively adjusting the noise covariance in real-time, this filtering method improves the estimation accuracy and reduces the impact of bumpy road noise on the estimation. The Sage–Husa adaptive filtering algorithm is as follows:

The estimated covariance value of the system observation noise is calculated as:

$$\begin{array}{c}{\mathit{R}}_{k+1}=\left(1-d_k\right){\mathit{R}}_k+d_k\left(e_{k+1}{e}_{k+1}^T-{\mathit{P}}_{y,k+1}{\mathit{P}}_{y,k+1}^T\right)\\ d_k={\displaystyle \frac{1-b}{1-b^2}}\end{array}$$

(25)

where $d_k$ is the forgetting factor, and b is the forgetting factor, usually ranging between 0.95 and 1. $e_k$ represents the difference between the true observation value and the estimated observation value at time k, i.e., the observation residual. It is challenging to ensure that the process noise covariance matrix and the observation noise covariance matrix remain non-negative definite during vehicle operation. Therefore, the improvements to ${\mathit{Q}}_k$ and ${\mathit{R}}_k$ are as follows [33]:

$$\begin{array}{c}\hfill {\mathit{Q}}_k=\sqrt{diag\left(diag\left({\mathit{Q}}_k{\mathit{Q}}_k^T\right)\right)}\hfill \\ \hfill {\mathit{R}}_k=\sqrt{diag\left(diag\left({\mathit{R}}_k{\mathit{R}}_k^T\right)\right)}\hfill \end{array}$$

(26)

The Kalman filter gain matrix is:

$${\mathit{K}}_{k+1}={\mathit{P}}_{xy,k+1}{\left({\mathit{P}}_{y,k+1}\right)}^{-1}$$

(27)

The system state is updated as:

$${\widehat{x}}_{k+1\mid k+1}={\widehat{x}}_{k+1\mid k}+{\mathit{K}}_k\left(y_{k+1}-{\widehat{y}}_{k+1\mid k}\right)$$

(28)

The system covariance is updated as:

$${\mathit{P}}_{k+1\mid k+1}={\mathit{P}}_{k+1\mid k}-{\mathit{K}}_{k+1}{\mathit{P}}_{y,k+1}{\mathit{K}}_{k+1}^{-1}$$

(29)

The above calculation process constitutes the complete AUKF algorithm.

3.3. Longitudinal Force Fusion Estimation of Adhesion Coefficient

The Dugoff tire model requires fewer parameters to be fitted, but it cannot reflect the change in friction force with slip ratio when the tire enters the nonlinear region, leading to significant errors in the calculated longitudinal tire force. Similarly, when the longitudinal force is known, back-calculating the adhesion coefficient also results in substantial errors. Therefore, this paper will fuse the longitudinal force calculated from tire dynamics with the longitudinal force estimated from vehicle dynamics to calibrate the deviation, thereby making the final obtained adhesion coefficient more accurate.

Considering that the vehicle dynamics model has errors, while the tire dynamics formula is highly reliable on high-adhesion roads when the vehicle is not slipping, but its reliability decreases significantly when slipping occurs, the estimated results will be fused with the tire dynamics.

Let the estimated longitudinal force be $F_e$, and the longitudinal force calculated by the tire dynamics model be $F_m$. Then, the fused longitudinal force $F_x$ is:

$$F_x=u{F}_m+(1-u)F_e$$

(30)

where u is the confidence level, ranging from [0,1], with higher values indicating greater trust. On high-adhesion roads (non-slip conditions), u should be set to a higher value, relying more on the tire dynamics calculation results. On low-adhesion, easily slidable roads, u should be set to a lower value, relying more on the vehicle dynamics estimation results. According to the Dugoff tire model, the normalized longitudinal force $F_x^0$ can be calculated, and the adhesion coefficient can be obtained by the following equation:

$${\mathsf{\mu }}_l={\displaystyle \frac{F_x}{F_x^0}}$$

(31)

According to the Burckhardt model, the optimal slip ratio typically varies between 3% and 17%, depending on different road adhesion conditions. To reduce the impact of road tire adhesion estimation accuracy on ASR performance, Ding [21] proposed a fixed slip ratio method based on the Burckhardt model. The fixed slip ratio can be optimized through the following method:

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathsf{\mu }}_l^i\left({\lambda }_i^o\right)}{{\mathsf{\mu }}_{max}^i\left({\lambda }_i^{opt}\right)}}\ge 95\%\\ f\left({\lambda }_i\right)=\sum \left(1-{\displaystyle \frac{{\mathsf{\mu }}_i^i\left({\lambda }_i^o\right)}{{\mathsf{\mu }}_{max}^i\left({\lambda }_i^{opt}\right)}}\right)\\ f\left({\lambda }_i^o\right)=minf\left({\lambda }_i\right),{\lambda }_i\in [0,1]\end{array}\right.$$

(32)

${\lambda }_i^o$ can be obtained by solving the nonlinear programming problem:

$$\begin{array}{c}\hfill minf\left({\lambda }_i^o\right)=\sum \left(1-{\displaystyle \frac{{\mathsf{\mu }}_i^i}{{\lambda }_{max}^i}}\right)\hfill \\ \hfill \mathrm{S}.\mathrm{t}.\left\{\begin{array}{c}{\displaystyle \frac{{\mathsf{\mu }}_i^i\left({\lambda }_i^o\right)}{{\mathsf{\mu }}_{max}^i\left({\lambda }_i^{opt}\right)}}\ge 95\%\\ {\lambda }_i\in [0,1]\end{array}\right.\hfill \end{array}$$

(33)

The pseudocode of the algorithm is shown as Algorithm 1. First, estimate the longitudinal force $F_e$ based on system input, concurrently calculate $F_m$ using the tire dynamics model, and fuse them together considering the confidence level to obtain $F_x$. Subsequently, calculate the adhesion coefficient based on the Dugoff tire model, and finally incorporate it into the Burckhardt model to determine the optimal slip ratio.  

Algorithm 1: Fusion estimation of adhesion coefficient and identification of optimal slip ratio

Input: measurement: ax, ωz, T; confidence level: u, Sampling times: M.

Initialization: F, P, Q, R

for k = 1: M do

Fe is calculated by estimation model;

Fm was calculated by tire dynamics model;

Calculate the fusion longitudinal force by $F_x=u{F}_m+(1-u)F_e$

Calculate the adhesion coefficient by ${\mathsf{\mu }}_l={\displaystyle \frac{F_x}{F_x^0}}$

Calculate the optimal slip rate by $\begin{array}{c}\hfill minf\left({\lambda }_i^o\right)=\sum \left(1-{\displaystyle \frac{{\mathsf{\mu }}_i^i}{{\lambda }_{max}^i}}\right)\hfill \\ \hfill \mathrm{S}.\mathrm{t}.\left\{\begin{array}{c}{\displaystyle \frac{{\mathsf{\mu }}_i^i\left({\lambda }_i^o\right)}{{\mathsf{\mu }}_{max}^i\left({\lambda }_i^{opt}\right)}}\ge 95\%\\ {\lambda }_i\in [0,1]\end{array}\right.\hfill \end{array}$

End

Output: the optimal slip rate.

The algorithm flowchart is shown in Figure 4.

4. Slip Ratio Control Strategy Based on Dynamic Adaptive Compensation Sliding Mode Control

4.1. Design of the Slip Ratio Controller

In the traditional design of wheel slip ratio sliding mode controllers, the control law for the sliding mode control output torque generally consists of two parts: the equivalent control $T_{eq}$ and the switching robust control $T_{sw}$. The equivalent control ensures that the system state remains on the sliding surface, while the switching control ensures that the system state does not leave the sliding surface. The switching function is designed as the error e between the actual slip ratio and the optimal slip ratio:

$$S=e={\lambda }_i-{\lambda }_t$$

(34)

where ${\lambda }_t$ is the optimal slip ratio.

Setting $\dot{S}=0$, the equivalent control $T_{eq}$ is obtained from Equations (3) and (6):

$$T_{eq}=F_{xi}R+{\displaystyle \frac{J{\dot{v}}_i}{\left(1-{\lambda }_i\right)R}}$$

(35)

To ensure that the system state can reach the sliding surface, the switching control is designed as follows:

$$T_{sw}=\epsilon sgn\left(S\right)$$

(36)

where $\epsilon$ is the control gain and $\epsilon >0$, the sliding mode control law is:

$$T=T_{eq}-T_{sw}=F_{xi}R+{\displaystyle \frac{J{\dot{v}}_i}{\left(1-{\lambda }_i\right)R}}-\epsilon sgn\left(S\right)$$

(37)

To alleviate slip ratio oscillations caused by low adhesion and low signal-to-noise ratio at low speeds during vehicle start-up on icy and snowy roads, as well as chattering and steady-state errors in sliding mode control due to wheel speed signal delay and adhesion coefficient estimation errors, it is necessary to correct and compensate for the errors. This aims to improve the transient characteristics of sliding mode control, reduce system chattering, and decrease system response time.

Redesign the integral sliding surface as:

$$S\left(t\right)=c{\int }_0^{t}e\left(\tau \right)d\tau +e\left(t\right)$$

(38)

Taking the derivative of the above equation, we obtain:

$$\dot{S}\left(t\right)=ce\left(t\right)+{\dot{\lambda }}_i$$

(39)

Next, we design a modified reaching law based on the hyperbolic tangent function. This reaching law introduces an integral term with a dynamic weight based on the error state to adaptively compensate for persistent errors and disturbances. The dynamic weight function $\omega \left(t\right)$ will dynamically adjust according to the current state of the system. When the tracking error is large, it enhances the integral effect, helping to reduce the error quickly. When the error is small, it weakens the integral effect, avoiding over-regulation. The modified reaching law is given by the following equation:

$$\left\{\begin{array}{c}\dot{S}=-\epsilon tanh\left({\displaystyle \frac{S}{\sigma }}\right)-kS-{\int }_0^t\omega \left(\tau \right)S\left(\tau \right)d\tau \\ \omega \left(t\right)=k_{\omega }·ex{p}^{-\beta \left|e\right(t\left)\right|}\end{array}\right.$$

(40)

where $\epsilon >0,\sigma >0,0<k_{\omega }<1,k>0,\beta$ is a positive tuning coefficient, and $tanh\left({\displaystyle \frac{S}{\sigma }}\right)$ is the hyperbolic tangent function, which is expressed as:

$$tanh\left({\displaystyle \frac{e^{\frac{s}{\sigma }}-e^{-\frac{s}{\sigma }}}{e^{\frac{s}{\sigma }}+e^{-\frac{s}{\sigma }}}}\right)$$

(41)

Compared to the severe chattering problem in the control signal-switching process with the sign function, the saturation function can perform linear adjustment within the boundary layer but is still a discontinuous function. Therefore, a smooth and continuous hyperbolic tangent function is introduced to replace the original switching function, using the smoothness of this function to alleviate the chattering during control signal switching.

Next, solve for the control law. From the slip ratio calculation formula, we obtain:

$${\dot{\lambda }}_i={\dot{\omega }}_i{\displaystyle \frac{v_i}{{\omega }_i^{2}R}}-{\displaystyle \frac{{\dot{v}}_i}{{\omega }_{i}R}}$$

(42)

Substitute the single-wheel model into:

$${\dot{\lambda }}_i=\left({\displaystyle \frac{1}{J}}T-{\displaystyle \frac{R}{J}}{F}_{xi}\right){\displaystyle \frac{v_i}{{\omega }_i^{2}R}}-{\displaystyle \frac{{\dot{v}}_i}{{\omega }_{i}R}}$$

(43)

Combining the sliding mode control surface with the reaching law, we obtain:

$$\dot{S\left(t\right)}=ce\left(t\right)+{\dot{\lambda }}_i=-\epsilon tanh\left({\displaystyle \frac{S}{\sigma }}\right)-kS-{\int }_0^t\omega \left(\tau \right)S\left(\tau \right)d\tau$$

(44)

Substituting Equation (43), the sliding mode control for the torque is:

$$T={\displaystyle \frac{J{\omega }_i^{2}R}{v_i}}\left(-ce-\epsilon tanh\left(S\right)-kS-{\int }_0^t\omega \left(\tau \right)S\left(\tau \right)d\tau \right)+F_{xi}·R+{\displaystyle \frac{J{\omega }_i{\dot{v}}_i}{v_i}}$$

(45)

The stability and convergence of this control are proven as follows.

Construct the Lyapunov function for the control:

$$V={\displaystyle \frac{1}{2}}{S}^2$$

(46)

Taking the derivative of Equation (46), we obtain:

$$\dot{V}=S\dot{S}=S\left(ce+{\dot{\lambda }}_i\right)$$

(47)

Lemma 1 

([34]). For $\forall S>0$, there exists $\sigma >0$ such that the following inequality always holds: $\left|s\right|-stanh\left({\displaystyle \frac{s}{\sigma }}\right)\le \mathsf{\mu }\sigma ,\mathsf{\mu }=0.2785$.

After transformation, the following is obtained:

$$\epsilon stanh\left({\displaystyle \frac{s}{\sigma }}\right)\le -\epsilon \left|s\right|+\epsilon \mathsf{\mu }\sigma$$

(48)

Substituting the transformed Lemma 1 into Equation (47), we obtain:

$$\begin{array}{l}S\dot{S}=-\epsilon Stanh\left({\displaystyle \frac{s}{a}}\right)-k{S}^2-k_{\omega }{\int }_0^{t}ex{p}^{-\beta \left|e\right(\tau \left)\right|}d\tau ·S^2\\ \le -K{S}^2+\epsilon \mathsf{\mu }\sigma =-2KV+b\end{array}$$

(49)

where $K=k+k_{\omega }{\int }_0^t{exp}^{-\beta \left|e\right(\tau \left)\right|}d\tau$ is a constant determined by $k,k_{\omega }$ and $\beta$, and $b=\epsilon \mathsf{\mu }\sigma$.

The solution to the inequality $\dot{V}\le -2KV+b$ is:

$$V\left(t\right)\le e^{-2K\left(t-t_0\right)}V\left(t_0\right)+{\displaystyle \frac{\epsilon \mathsf{\mu }\sigma }{2K}}\left(1-e^{-2K\left(t-t_0\right)}\right)$$

(50)

We continue solving and obtain:

$$\underset{t\to \infty }{lim}V\left(t\right)\le {\displaystyle \frac{\epsilon \mathsf{\mu }\sigma }{2K}}$$

(51)

Therefore, the system is asymptotically stable and exponentially convergent, with the convergence rate depending on $\epsilon ,\sigma$, and K.

4.2. ASR Triggering

As a practical active safety driving assistance system, ASR needs to have reasonable intervention and exit conditions to ensure driving safety while meeting the driver’s operational needs, which is worth an in-depth study. The focus of this paper is on parameter estimation and control algorithms, thus a relatively simple intervention and exit mechanism is designed. The conditions for ASR participation ($fla{g}_{ASR}=1$) are: (1) when the difference between the speed at the wheel center calculated from the wheel angular velocity and the vehicle longitudinal speed is greater than a set threshol d (${\mathsf{\Delta }}_a$) (2) when the torque demand from the accelerator pedal exceeds the maximum torque that can be provided by the current road adhesion. Otherwise, there is no intervention ($fla{g}_{ASR}=0$).

5. Simulation and Discussion

To verify the effectiveness of the improved AUKF estimation algorithm and the superiority of the designed dynamic adaptive compensation sliding mode control (DASMC) algorithm compared to the conventional equivalent sliding mode control (CESMC) algorithm under low-adhesion icy and snowy road conditions, slip ratio controller models based on DASMC and CESMC, as well as adhesion coefficient estimation models, were established in MATLAB/Simulink and Carsim for joint simulation analysis.

Under icy and snowy conditions, the road surface can take on different forms, with compacted snow and ice being the most common. In this study, compacted snow with low adhesion and slippery asphalt roads are used as typical roads to evaluate the control performance of the two ASR schemes [35]. Additionally, to test whether the controller can respond quickly, two simulation scenarios are selected where the vehicle accelerates from a standstill to the target speed on split roads and joint roads. In the first scenario, the test vehicle model accelerates on a split road, with the left side on low-adhesion compacted snow and the right side on slippery asphalt. In the second scenario, the test vehicle model accelerates from slippery asphalt to compacted snow. To ensure fairness in the simulations, all other parameters are set identically. The vehicle model parameters are shown in

Table 1. Basic vehicle parameters.

Parameters	Symbol	Value (Unit)
Vehicle mass	m	1998 kg
Distance from center of mass to front axis	$Lf$	1.4 m
Distance from center of mass to rear axis	$Lr$	1.85 m
Height of the center of mass	$Hg$	0.795 m
Wheel track	B	1.7 m
Effective radius of tire	R	0.385 m
Moment of inertia about the yaw axis	I	5757 kg · ${\mathrm{m}}^2$
Wheel inertia	J	1.5 kg · ${\mathrm{m}}^2$

.

5.1. Split Road Driving Simulation

In the split road driving scenario, the adhesion coefficient on the left side is set to 0.18 for compacted snow and 0.5 for slippery asphalt on the right side. The reference speed accelerates from 0 km/h to 70 km/h within 10 s, with the optimal slip ratio set to 0.12 on the left side and 0.18 on the right side.

The estimation results and errors of the adhesion coefficients on both sides of the split road are shown in Figure 5. Analysis shows that the adhesion coefficients converge to values close to the true values within 1 s of vehicle start-up. Since the right-side adhesion coefficient is smaller, it converges to the true value faster and remains stable around it. The error graph shows that the adaptive improvement of the UKF significantly enhances the estimation accuracy of the adhesion coefficients on both sides and makes them stable. In contrast, the traditional UKF algorithm cannot mitigate the impact of process noise, resulting in large estimation errors that increase as the vehicle progresses. To quantitatively analyze the errors, we use the mean absolute error (MAE) to measure the estimation performance:

$$MAE={\displaystyle \frac{1}{n}}\sum abs\left({\mathsf{\mu }}_{est}-{\mathsf{\mu }}_{\mathrm{real}}\right)$$

(52)

The statistical results are shown in

Table 2. Average estimation error on split roads.

Estimation Strategy	Condition1 (L)	Condition1 (R)
UKF	0.0468	0.0437
AUKF	0.0173	0.0068

. From the table, it can be seen that the AUKF algorithm significantly outperforms the UKF in estimating adhesion coefficients on split roads, with estimation accuracy improvements of 63% and 84% for the left and right sides, respectively, while maintaining stability.

Figure 6 shows the simulation results of the split road scenario. It can be seen that after the vehicle starts, all wheels quickly slip. The right rear wheel, being on the side with relatively better adhesion and having a slightly larger vertical force than the front wheel, experiences a delay in slip time. When the vehicle detects wheel slip, CESMC and DASMC start to intervene to control the slip of each wheel. From the slip ratio curves shown in Figure 6b,c, it can be seen that as the vehicle starts and accelerates, the slip ratio under CESMC control exhibits relatively severe overshoot and response delay, with a steady-state time of 3 s and an overshoot of 0.32. Under DASMC control, the slip ratio quickly responds and reaches the desired slip ratio value in less than 1 s. The improved response steady-state time is reduced by 66.6%, and the overshoot is decreased by 0.26. The average overshoot for each wheel control is only about 0.06 above the desired value, and both the left and right wheels fully utilize the road adhesion, significantly improving the control effect. This indicates that the DASMC control algorithm has good robustness and response speed. According to Figure 6d,e, which show the output torque of each wheel, it can be seen that after the vehicle starts to slip, the motor output torque directly reaches the maximum limit. Although the torque under traditional control can also reach a stable effective value, there is noticeable severe chattering. In contrast, the output torque of the motor under the improved control is smooth and stable, with almost no chattering, and can quickly respond to the required anti-slip control torque. Figure 6f shows the vehicle speed variation curve. It can be seen that without control, the vehicle speed can slowly track up to about 50 km/h. With CESMC control, the tracking speed increases to 60 km/h but still cannot fully reach the desired speed. Under DASMC control, the speed curve almost coincides with the target tracking speed, indicating that the improved algorithm allows the vehicle to drive stably in this scenario, significantly improving the vehicle’s dynamic performance on icy and snowy roads.

5.2. Docking Road Driving Simulation

The joint road is set with the first half having an adhesion coefficient of 0.5 on a slippery asphalt road, and the second half having an adhesion coefficient of 0.18 on a compacted snow road. The simulation speed accelerates from 0 km/h to 70 km/h within 10 s, with the optimal slip ratio being 0.18 in the first half and 0.12 in the second half.

Figure 7 shows the estimation results and errors of the adhesion coefficient on the joint road. The analysis reveals that, similar to the split road scenario, the adhesion coefficient converges to a value close to the true value within 1 s of vehicle start-up. Due to the sudden change in the road adhesion coefficient at the 5th second, the estimated value also changes abruptly, resulting in a significant estimation error at this point. After the road adhesion coefficient changes, the estimated value stabilizes again. Compared to the UKF, the AUKF still demonstrates excellent performance on the joint road.

Table 3. Average estimation error on joint road.

Estimation Strategy	Condition1 (L)
UKF	0.0660
AUKF	0.0201

shows the average estimation error on the joint road. The analysis shows that the average estimation error of the AUKF is reduced by 69% compared to the UKF, indicating a significant improvement in estimation performance.

Figure 8 shows the simulation results for the joint road scenario. Since the simulation results for the left and right wheels are almost identical, the analysis focuses on the left front wheel and right rear wheel. As shown in Figure 8a, due to the relatively better adhesion coefficient in the first half of the road, the rear wheels start to slip and lose stability after 5 s, but the first 5 s do not fully utilize the road adhesion coefficient. Figure 8b,c indicate that after the intervention of both control algorithms, the slip ratio under CESMC control experiences severe chattering due to the sudden change in road conditions, especially at the transition between the two adhesion coefficients. The overshoot for both front and rear wheels in the first half reaches 0.34, with the steady-state time between 0 and 3 s. At the transition, the overshoot is around 0.1, with the steady-state time between 5 and 7 s. With the DASMC control algorithm, the initial overshoot is controlled within 0.08, with a steady-state time within 1 s. At the road transition, the overshoot is almost reduced to 0, with a response time within 0.1 s. The rear wheel control shows a slight delay due to the vehicle’s length. The results demonstrate that the improved algorithm can quickly respond to sudden changes in conditions, effectively reducing chattering and steady-state errors, promptly controlling when there are significant errors, and avoiding over-control when there are minor errors, indicating good disturbance resistance. Figure 8d,e show that the motor torque under DASMC control can quickly, smoothly, and stably apply the anti-slip control torque to each wheel, significantly improving over traditional equivalent control. The speed curve in Figure 8f shows that without control, the vehicle speed can barely track up to 45 km/h within 0 to 5 s. With CESMC control, the speed tracking improves, and the speed fluctuations at the transition reduces, but the final speed only reaches 59 km/h. Using the improved control algorithm, the speed tracking curve almost coincides with the target speed, as the DASMC control algorithm can appropriately control wheel slip in the first half and suppress excessive wheel slip in the second half, achieving stable speed tracking. The simulation results show that the improved algorithm can adapt to changing conditions and provide stable robust control.

6. Conclusions

In this paper, to achieve anti-slip control for distributed electric drive vehicles on low-adhesion icy and snowy roads, an adhesion coefficient estimator based on improved singular value decomposition unscented Kalman filter and a slip ratio controller based on dynamic adaptive compensation sliding mode control was designed, and a simulation model was built for validation. The conclusions are as follows:

• The proposed improved estimation algorithm and tire longitudinal force fusion estimation method can effectively estimate the adhesion coefficients on split roads and joint roads and can track well even when there are sudden changes in the road adhesion coefficients. Through adaptive improvements and fusion of the algorithm, the average estimation error of the road adhesion coefficients for the left and right wheels on split roads was reduced by 63% and 84%, respectively. On joint roads, the average estimation error decreased by 69%, demonstrating the effectiveness of this estimator in estimating road adhesion coefficients and its quick response to sudden changes.
• The proposed DASMC slip ratio control algorithm can significantly reduce chattering compared to the CESMC control algorithm. The controller’s response time is shortened by 66%, and the overshoot is reduced by more than 0.25. In terms of torque output and speed tracking, the improved control algorithm can quickly and smoothly output torque, with the speed curve closely matching the target curve. This demonstrates that the vehicle has excellent acceleration performance on low-adhesion road surfaces.
• At the transition of road adhesion coefficients, the improved DASMC control algorithm can adaptively adjust according to road conditions, keeping the switching time within 0.1 s and reducing the overshoot to nearly zero, ensuring smooth and stable control at the transition.

However, the proposed method is limited to longitudinal vehicle dynamics without considering lateral stability and lacks verification under complex extreme conditions. In future research, coordinated control of longitudinal and lateral anti-slip and stability under complex conditions will be considered to improve the vehicle’s dynamic safety and handling performance. Additionally, experiments will be conducted on real vehicles under icy and snowy road conditions for validation.