Direct Yaw Moment Control of Distributed-Drive Electric Vehicles via Multi-Agent Full-Order Terminal Sliding Mode

Abstract

To improve the yaw-stability tracking accuracy and torque smoothness of distributed-drive electric vehicles (DDEVs) under high-speed double-lane-change maneuvers and crosswind disturbances, this paper proposes a multi-agent-system (MAS)-based direct yaw moment control (DYC) method using full-order terminal sliding mode (FOTSM) control. First, based on the vehicle yaw dynamics model and the vector superposition principle, the whole-vehicle yaw-rate and sideslip-angle responses are decomposed into the local contributions of four wheel agents. A leader–follower MAS tracking framework is then established, in which the yaw-stability reference model acts as the virtual leader, and the four wheel agents act as followers. Second, the yaw-rate error and sideslip-angle error are combined into an aggregated tracking error, thereby transforming yaw-stability control into a second-order nonlinear MAS tracking problem. A FOTSM DYC law is designed, and Lyapunov analysis proves that the closed-loop error system reaches the sliding surface and converges within finite time. Finally, hardware-in-the-loop experiments are conducted under double-lane-change maneuvers with and without crosswind disturbance. Compared with the uncontrolled case and a conventional MAS-based linear sliding mode controller, the proposed method reduces yaw-rate and sideslip-angle deviations, improves trajectory-tracking performance, maintains yaw stability under crosswind disturbance, and suppresses wheel-driving-torque chattering.

1. Introduction

With the rapid development of vehicle electrification and intelligent-vehicle technologies, distributed-drive electric vehicles (DDEVs) provide additional degrees of freedom for vehicle dynamics control because the driving/braking torque of each wheel can be independently regulated. By generating an additional yaw moment through the longitudinal-force difference between the left and right wheels, DDEVs can actively regulate the yaw rate and sideslip angle. Therefore, direct yaw moment control (DYC) has become an important approach for improving vehicle handling stability and active safety [1,2,3].

Under high-speed steering, double-lane-change (DLC) maneuvers, and crosswind disturbances, strong nonlinear coupling exists among tire forces and vehicle yaw motion. When the vehicle approaches the adhesion limit, tire-force saturation may cause the yaw rate and sideslip angle to deviate rapidly from their desired values, resulting in sideslip or instability. Phase-plane-based lateral-stability analysis and coordinated control methods have been used to improve stability under critical driving conditions [4,5]. However, independent wheel torque control also makes the system multi-input, strongly coupled, and sensitive to parameter uncertainties. Therefore, a yaw-stability controller with high tracking accuracy, strong robustness, and smooth torque output is still required for DDEV chassis control.

Existing DYC methods include proportional-integral-derivative (PID) control, linear quadratic regulator (LQR), model predictive control (MPC), and sliding mode control (SMC). PID and LQR are easy to implement but may perform poorly under nonlinear conditions. MPC can handle multivariable constraints and actuator limits [6,7,8], but it requires high computational effort. SMC is robust to parameter perturbations and external disturbances [9,10], whereas conventional linear SMC usually provides only asymptotic convergence and may induce torque chattering. Terminal SMC improves convergence performance and has been applied to DDEV yaw-moment control under emergency conditions [11,12]. Recent learning- and identification-based tracking control methods have also been investigated for systems with uncertain dynamics [13]. Recent neural-network-based and data-driven control methods have also provided alternative approaches for handling uncertain system dynamics [14,15]. However, the distributed wheel-actuation characteristics of DDEVs have not been fully considered in existing modeling and controller design.

Most existing DYC strategies adopt a centralized architecture, in which the desired additional yaw moment is calculated from a whole-vehicle model and then allocated to the wheels. This structure cannot directly describe the local contribution of each wheel to the global yaw rate and sideslip angle. Since DDEV actuators are naturally distributed at the four wheels, multi-agent system (MAS) theory provides a suitable modeling framework by decomposing a complex system into multiple interacting agents [16,17,18,19]. Although MAS methods, torque-distribution strategies, and robust control techniques have been investigated in vehicle dynamics control [20,21,22], combining MAS modeling with nonlinear robust control that guarantees finite-time convergence for DDEV DYC still requires further study. Distributed optimization has also been applied to multi-vehicle coordination problems, demonstrating the effectiveness of agent-level cooperative decision making [23,24,25].

To address these issues, this paper proposes an MAS-based full-order terminal sliding mode (FOTSM) DYC method. Based on the vehicle yaw dynamics model and the vector superposition principle, the whole-vehicle sideslip angle and yaw rate are decomposed into the local contributions of four wheel agents, and a yaw-stability MAS with one virtual leader and four wheel-following agents is established. The yaw-rate error and sideslip-angle error are combined into an aggregated tracking error, transforming yaw-stability control into an error-tracking problem of a second-order nonlinear MAS. An FOTSM DYC law is then designed, and Lyapunov theory is used to prove finite-time arrival at the sliding surface and stable convergence of the closed-loop error system. Finally, experimental tests under DLC maneuvers with and without crosswind disturbance verify the effectiveness of the proposed method.

The main contributions of this paper are summarized as follows:

(1)

A wheel-level MAS yaw-stability model is established for DDEVs. Based on the vector superposition principle, the whole-vehicle yaw-rate and sideslip-angle responses are decomposed into four wheel-related contributions, and the dynamic relationship between each wheel agent and the whole-vehicle yaw response is constructed. This formulation introduces the four wheel actuators directly into the yaw-stability control model rather than treating them only as subsequent torque-allocation targets.

(2)

A coupled yaw-stability error formulation is developed by defining the weighted tracking error $\Delta {\epsilon }_i=\rho \Delta {\beta }_i+\Delta {\gamma }_i$. Together with the leader–follower interaction errors, this reformulates the simultaneous regulation of yaw rate and sideslip angle as a second-order nonlinear MAS tracking problem and provides a unified basis for coordinated control of the four wheel agents.

(3)

A full-order terminal sliding mode controller is designed for the established second-order MAS. The nonlinear terminal terms provide finite-time convergence of the tracking-error dynamics, while the switching action enters the wheel-torque command through an integral term. This structure retains sliding-mode robustness while reducing the persistent high-frequency torque switching observed with conventional MAS-LSM.

2. Materials and Methods

2.1. Yaw-Stability Dynamics Model of Distributed-Drive Electric Vehicles

According to the definition of DYC, the yaw moment is generated by driving the wheels on one side and braking the wheels on the other side. The body dynamics model for yaw stability can be written as follows:

$$\left\{\begin{array}{cc}\hfill \dot{\beta }& ={\displaystyle \frac{1}{m{v}_x}}{\displaystyle \sum _{i=1}^4}{F}_{yi}-\gamma ,\hfill \\ \hfill \dot{\gamma }& ={\displaystyle \frac{1}{I_z}}\left[l_1\left(F_{y1}+F_{y2}\right)-l_2\left(F_{y3}+F_{y4}\right)+\left(T_1+T_3-T_2-T_4\right)\right].\hfill \end{array}\right.$$

(1)

where $\beta$ is the vehicle sideslip angle at the center of mass; $T_1$, $T_2$, $T_3$, and $T_4$ are the additional yaw-control driving torques of the left-front, right-front, left-rear, and right-rear wheels, respectively.

To reduce the order of the body dynamics model from an MAS perspective, ${\beta }_i$ and ${\gamma }_i$ are defined as the sideslip angle and yaw rate at the vehicle center of mass generated by the lateral force of the ith wheel acting alone, respectively. According to the superposition principle and the chassis architecture of a DDEV, the yaw rate and sideslip angle at the vehicle center of mass are assigned to the four body subsystems represented by the four driving wheels:

$$\left\{\begin{array}{c}\beta ={\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4,\hfill \\ \gamma ={\gamma }_1+{\gamma }_2+{\gamma }_3+{\gamma }_4.\hfill \end{array}\right.$$

(2)

Combining the yaw-stability equations in Equation (1) with the decomposition in Equation (2), the differential equations of the yaw rate and sideslip angle of the body subsystems affected by the driving wheels can be decomposed as follows:

$$\left\{\begin{array}{c}{\dot{\gamma }}_1={\displaystyle \frac{l_1{F}_{y1}+T_1}{I_z}},\hfill \\ {\dot{\gamma }}_2={\displaystyle \frac{l_1{F}_{y2}-T_2}{I_z}},\hfill \\ {\dot{\gamma }}_3={\displaystyle \frac{-l_2{F}_{y3}+T_3}{I_z}},\hfill \\ {\dot{\gamma }}_4={\displaystyle \frac{-l_2{F}_{y4}-T_4}{I_z}}.\hfill \end{array}\right.$$

(3)

$$\left\{\begin{array}{c}{\dot{\beta }}_1={\displaystyle \frac{F_{y1}}{m{v}_x}}-{\gamma }_1,\hfill \\ {\dot{\beta }}_2={\displaystyle \frac{F_{y2}}{m{v}_x}}-{\gamma }_2,\hfill \\ {\dot{\beta }}_3={\displaystyle \frac{F_{y3}}{m{v}_x}}-{\gamma }_3,\hfill \\ {\dot{\beta }}_4={\displaystyle \frac{F_{y4}}{m{v}_x}}-{\gamma }_4.\hfill \end{array}\right.$$

(4)

Define ${\eta }_1={\eta }_2=l_1$, ${\eta }_3={\eta }_4=-l_2$, ${\xi }_1={\xi }_3=1$, and ${\xi }_2={\xi }_4=-1$. Then, the dynamics generated by each wheel at the vehicle center of mass can be expressed as

$$\left\{\begin{array}{c}{\dot{\beta }}_i={\displaystyle \frac{1}{m{v}_x}}{F}_{yi}-{\gamma }_i,\hfill \\ {\dot{\gamma }}_i={\displaystyle \frac{1}{I_z}}{\eta }_i{F}_{yi}+{\displaystyle \frac{1}{I_z}}{\xi }_i{T}_i.\hfill \end{array}\right.i=1,2,3,4.$$

(5)

Here, ${\beta }_i$ and ${\gamma }_i$ are model-based wheel-contribution states rather than independently measured physical states. They are defined and propagated by the corresponding wheel-level dynamics and therefore do not require additional wheel-level sensors.

According to Equation (5), because $d\gamma /2$ is much smaller than $v_x$, the slip angles of the four wheels can be approximated as

$$\left\{\begin{array}{c}{\alpha }_1={\delta }_1-\arctan \left({\displaystyle \frac{v_y+l_1\gamma }{v_x-{\displaystyle \frac{d}{2}}\gamma }}\right)\approx {\delta }_1-{\displaystyle \frac{l_1\gamma }{v_x}}-\beta ,\hfill \\ {\alpha }_2={\delta }_2-\arctan \left({\displaystyle \frac{v_y+l_1\gamma }{v_x+{\displaystyle \frac{d}{2}}\gamma }}\right)\approx {\delta }_2-{\displaystyle \frac{l_1\gamma }{v_x}}-\beta ,\hfill \\ {\alpha }_3={\delta }_3-\arctan \left({\displaystyle \frac{v_y-l_2\gamma }{v_x-{\displaystyle \frac{d}{2}}\gamma }}\right)\approx {\delta }_3+{\displaystyle \frac{l_2\gamma }{v_x}}-\beta ,\hfill \\ {\alpha }_4={\delta }_4-\arctan \left({\displaystyle \frac{v_y-l_2\gamma }{v_x+{\displaystyle \frac{d}{2}}\gamma }}\right)\approx {\delta }_4+{\displaystyle \frac{l_2\gamma }{v_x}}-\beta .\hfill \end{array}\right.$$

(6)

where $v_y=v_x\beta$; ${\delta }_1={\delta }_2={\delta }_f$ are the front-wheel steering angles; ${\delta }_f$ denotes the actual front-wheel steering demand obtained from the steering command through the steering ratio; and ${\delta }_3={\delta }_4=0$. During the closed-loop DLC maneuver, the required steering demand depends on the instantaneous vehicle response and therefore reflects the compensating effect of the additional yaw moment generated by DYC.

When the vehicle lateral acceleration is below $0.4 \mathrm{g}$, and the tire slip angles are small, the tire cornering stiffness remains in the linear range. Therefore, the lateral forces of the driving wheels can be approximated as

$$\left\{\begin{array}{c}{F}_{y1}=k_1{\alpha }_1,\hfill \\ F_{y2}=k_2{\alpha }_2,\hfill \\ F_{y3}=k_3{\alpha }_3,\hfill \\ F_{y4}=k_4{\alpha }_4.\hfill \end{array}\right.$$

(7)

where $k_1$, $k_2$, $k_3$, and $k_4$ are the cornering stiffnesses of the four wheels.

During vehicle operation, tires are directly exposed to the external environment and are affected by disturbances. Considering modeling uncertainties caused by changes in tire cornering stiffness, the modeling error is summarized as a coefficient error and an output bias, as arranged in Equation (8):

$$F_{yi}=(k_i+\Delta k_i){\alpha }_i+\Delta F_{yi}=k_i{\alpha }_i+\Delta k_i{\alpha }_i+\Delta F_{yi},i=1,2,3,4.$$

(8)

Combining Equations (7) and (8), the lateral force of a single wheel can be expressed as

$$\begin{array}{cc}\hfill F_{yi}& =(k_i+\Delta k_i){\alpha }_i+\Delta F_{yi}\hfill \\ \hfill & =k_i{\delta }_i-k_i{\beta }_i-{\displaystyle \frac{k_i{\eta }_i}{v_x}}{\gamma }_i+\Delta F_{yi}+\Delta k_i{\alpha }_i\hfill \\ \hfill & =k_i{\delta }_i-k_i{\beta }_i-{\displaystyle \frac{k_i{\eta }_i}{v_x}}{\gamma }_i+d_i.\hfill \end{array}$$

(9)

where $d_i=\Delta k_i{\alpha }_i+\Delta F_{yi}$ denotes the lumped disturbance obtained by combining the coefficient error and output bias of a single wheel in yaw-stability control.

Substituting Equations (5) and (9) into Equation (4) yields the dynamics of each driving wheel for the DYC function of a DDEV achieved through driving/braking on the two sides:

$$\left\{\begin{array}{c}{\dot{\beta }}_i=-{\displaystyle \frac{k_i}{m{v}_x}}{\beta }_i-\left({\displaystyle \frac{k_i{\eta }_i}{m{v}_x^2}}+1\right){\gamma }_i+{\displaystyle \frac{k_i}{m{v}_x}}{\delta }_i+{\displaystyle \frac{1}{m{v}_x}}{d}_i,\hfill \\ {\dot{\gamma }}_i=-{\displaystyle \frac{{\eta }_i{k}_i}{I_z}}{\beta }_i-{\displaystyle \frac{k_i{\eta }_i^2}{I_z{v}_x}}{\gamma }_i+{\displaystyle \frac{{\eta }_i{k}_i}{I_z}}{\delta }_i+{\displaystyle \frac{{\xi }_i}{I_z}}{T}_i+{\displaystyle \frac{{\eta }_i}{I_z}}{d}_i.\hfill \end{array}\right.$$

(10)

The above derivation describes the effect of a single wheel on yaw stability at the center of mass. The MAS-based DYC strategy is developed below on the basis of Equation (10).

2.2. Dynamic Reference Model of the Distributed-Drive Electric Vehicle

According to the current vehicle operating state and based on Equation (1), the yaw-dynamics reference model is defined as

$$\left\{\begin{array}{c}{\displaystyle {\dot{\beta }}^{\ast }={\displaystyle \frac{1}{m{v}_x}}{\displaystyle \sum _{i=1}^4}{F}_{yi}^{\ast }-{\gamma }^{\ast },}\hfill \\ {\dot{\gamma }}^{\ast }={\displaystyle \frac{1}{I_z}}\left[l_1\left(F_{y1}^{\ast }+F_{y2}^{\ast }\right)-l_2\left(F_{y3}^{\ast }+F_{y4}^{\ast }\right)\right].\hfill \end{array}\right.$$

(11)

According to Equations (6) and (7), the wheel slip-angle approximations are written as follows:

$$\left\{\begin{array}{c}{\alpha }_1^{\ast }\approx {\delta }_1^{\ast }-{\displaystyle \frac{l_1{\gamma }^{\ast }}{v_x}}-{\beta }^{\ast },\hfill \\ {\alpha }_2^{\ast }\approx {\delta }_2^{\ast }-{\displaystyle \frac{l_1{\gamma }^{\ast }}{v_x}}-{\beta }^{\ast },\hfill \\ {\alpha }_3^{\ast }\approx {\delta }_3^{\ast }+{\displaystyle \frac{l_2{\gamma }^{\ast }}{v_x}}-{\beta }^{\ast },\hfill \\ {\alpha }_4^{\ast }\approx {\delta }_4^{\ast }+{\displaystyle \frac{l_2{\gamma }^{\ast }}{v_x}}-{\beta }^{\ast }.\hfill \end{array}\right.$$

(12)

where ${\delta }_1^{\ast }={\delta }_2^{\ast }={\delta }_f$, ${\delta }_f$ denotes the actual front-wheel steering demand used during the closed-loop trajectory-tracking maneuver after conversion through the steering ratio, and ${\delta }_3^{\ast }={\delta }_4^{\ast }=0$.

Therefore, the lateral forces of the wheels can be simplified as

$$\left\{\begin{array}{c}{F}_{y1}^{\ast }=k_1{\alpha }_1^{\ast },\hfill \\ F_{y2}^{\ast }=k_2{\alpha }_2^{\ast },\hfill \\ F_{y3}^{\ast }=k_3{\alpha }_3^{\ast },\hfill \\ F_{y4}^{\ast }=k_4{\alpha }_4^{\ast }.\hfill \end{array}\right.$$

(13)

According to the above reference model, the yaw-stability reference dynamics of each driving wheel of the DDEV can be obtained as follows:

$$\left\{\begin{array}{c}{\dot{\beta }}_i^{\ast }=-{\displaystyle \frac{k_i}{m{v}_x}}{\beta }_i^{\ast }-\left({\displaystyle \frac{k_i{\eta }_i}{m{v}_x^2}}+1\right){\gamma }_i^{\ast }+{\displaystyle \frac{k_i}{m{v}_x}}{\delta }_i^{\ast },\hfill \\ {\dot{\gamma }}_i^{\ast }=-{\displaystyle \frac{{\eta }_i{k}_i}{I_z}}{\beta }_i^{\ast }-{\displaystyle \frac{k_i{\eta }_i^2}{I_z{v}_x}}{\gamma }_i^{\ast }+{\displaystyle \frac{{\eta }_i{k}_i}{I_z}}{\delta }_i^{\ast }.\hfill \end{array}\right.i=1,2,3,4.$$

(14)

where ${\beta }_i^{\ast }$ and ${\gamma }_i^{\ast }$ are the ideal sideslip angle and ideal yaw rate generated at the vehicle center of mass by the ith wheel agent acting alone under the current driving condition, respectively.

2.3. Multi-Agent System for Yaw Stability Control

In this section, an MAS consisting of one virtual leader and four wheel-following agents is established as Figure 1. The black arrows represent the communication links among the four follower agents, while the blue arrows indicate the information transmission from the virtual leader to the follower agents. The virtual leader specifies the ideal yaw rate and sideslip angle generated by a single wheel system at the center of mass. The remaining four wheel agents generate the required torque outputs through the controller.

According to the established topology, the adjacency matrix $\mathbf{A}$, degree matrix $\mathbf{D}$, Laplacian matrix $\mathbf{L}$, and matrix $\mathbf{B}$ describing the interaction between the leader and followers are expressed as follows:

$$\mathbf{A}=\left[\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ 1& 1& 1& 0\end{array}\right],\mathbf{D}=\left[\begin{array}{cccc}3& 0& 0& 0\\ 0& 3& 0& 0\\ 0& 0& 3& 0\\ 0& 0& 0& 3\end{array}\right],$$

$$\mathbf{L}=\mathbf{D}-\mathbf{A}=\left[\begin{array}{cccc}3& -1& -1& -1\\ -1& 3& -1& -1\\ -1& -1& 3& -1\\ -1& -1& -1& 3\end{array}\right],\mathbf{B}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$

During the establishment and control of the MAS, the reference model obtained in Equation (14) is treated as the virtual leader of the vehicle yaw-stability system. The tracking-error equations of the ith wheel agent relative to the target yaw rate and sideslip angle are defined as

$$\left\{\begin{array}{c}\Delta {\beta }_i={\beta }_i-{\beta }_i^{\ast },\hfill \\ \Delta {\gamma }_i={\gamma }_i-{\gamma }_i^{\ast }.\hfill \end{array}\right.$$

(15)

where $\Delta {\beta }_i$ and $\Delta {\gamma }_i$ denote the tracking errors of the sideslip angle and yaw rate, respectively.

Substituting Equations (10) and (14) into Equation (15) yields the stability tracking-error equations for a single wheel agent as

$$\left\{\begin{array}{c}\Delta {\dot{\beta }}_i=-{\displaystyle \frac{k_i}{m{v}_x}}\Delta {\beta }_i-\left({\displaystyle \frac{k_i{\eta }_i}{m{v}_x^2}}+1\right)\Delta {\gamma }_i+{\displaystyle \frac{1}{m{v}_x}}{d}_i,\hfill \\ \Delta {\dot{\gamma }}_i=-{\displaystyle \frac{{\eta }_i{k}_i}{I_z}}\Delta {\beta }_i-{\displaystyle \frac{k_i{\eta }_i^2}{I_z{v}_x}}\Delta {\gamma }_i+{\displaystyle \frac{{\xi }_i}{I_z}}{T}_i+{\displaystyle \frac{{\eta }_i}{I_z}}{d}_i.\hfill \end{array}\right.$$

(16)

Conventional DYC controllers usually select either the yaw rate or the sideslip angle as the control variable. In this work, both variables are considered. To facilitate the construction of the MAS-based whole-vehicle yaw dynamics model, the tracking-error system is processed by defining $\Delta {\epsilon }_i=\rho \Delta {\beta }_i+\Delta {\gamma }_i$ as the aggregated tracking error, reducing the model to

$$\begin{array}{cc}\hfill \Delta {\dot{\epsilon }}_i& =\rho \Delta {\dot{\beta }}_i+\Delta {\dot{\gamma }}_i\hfill \\ \hfill & =-\left(\rho {\displaystyle \frac{k_i}{m{v}_x}}+{\displaystyle \frac{{\eta }_i{k}_i}{I_z}}\right)\Delta {\beta }_i-\left[\rho \left({\displaystyle \frac{k_i{\eta }_i}{m{v}_x^2}}+1\right)+{\displaystyle \frac{k_i{\eta }_i^2}{I_z{v}_x}}\right]\Delta {\gamma }_i+{\displaystyle \frac{{\xi }_i}{I_z}}{T}_i+\left({\displaystyle \frac{1}{m{v}_x}}+{\displaystyle \frac{{\eta }_i}{I_z}}\right)d_i\hfill \\ \hfill & =f_i+z_i{u}_i+d_{zi},\hfill \end{array}$$

(17)

where

$$\begin{array}{cc}\hfill f_i& =-\left(\rho {\displaystyle \frac{k_i}{m{v}_x}}+{\displaystyle \frac{{\eta }_i{k}_i}{I_z}}\right)\Delta {\beta }_i-\left[\rho \left({\displaystyle \frac{k_i{\eta }_i}{m{v}_x^2}}+1\right)+{\displaystyle \frac{k_i{\eta }_i^2}{I_z{v}_x}}\right]\Delta {\gamma }_i,\hfill \\ \hfill u_i& =T_i,z_i={\displaystyle \frac{{\xi }_i}{I_z}},d_{zi}=\left({\displaystyle \frac{1}{m{v}_x}}+{\displaystyle \frac{{\eta }_i}{I_z}}\right)d_i.\hfill \end{array}$$

Here, $\rho$ is the weighting coefficient of the sideslip angle, $0<\rho <1$. The disturbance and its derivative are assumed to be bounded.

Considering the interactions among agents, the coordinated tracking error $e_{1i}$ and its time derivative $e_{2i}$ are defined in Equation (18), and the corresponding system states are given in Equation (19).

$$\left\{\begin{array}{c}{\displaystyle e_{1i}={\displaystyle \sum _{j=1}^N}{l}_{ij}\left(\Delta {\epsilon }_j-\Delta {\epsilon }_i\right)+b_i\Delta {\epsilon }_i,}\hfill \\ {\displaystyle e_{2i}={\dot{e}}_{1i}={\displaystyle \sum _{j=1}^N}{l}_{ij}\left(\Delta {\dot{\epsilon }}_j-\Delta {\dot{\epsilon }}_i\right)+b_i\Delta {\dot{\epsilon }}_i.}\hfill \end{array}\right.$$

(18)

$$\left\{\begin{array}{c}{\displaystyle x_{1i}=\int e_{1i}dt,}\hfill \\ x_{2i}=e_{1i}.\hfill \end{array}\right.$$

(19)

Combining Equations (18) and (19), the yaw-stability dynamics of a single wheel can be written as the following second-order nonlinear MAS:

$$\left\{\begin{array}{c}{\dot{x}}_{1i}=x_{2i},\hfill \\ {\displaystyle {\dot{x}}_{2i}=\left({\displaystyle \sum _{j=1}^N{l}_{ij}+b_i}\right)\left(f_i+z_i{u}_i+d_{zi}\right)-{\displaystyle \sum _{\begin{array}{c}j=1\end{array}}^N}{l}_{ij}\left(f_j+z_j{u}_j+d_{zj}\right).}\hfill \end{array}\right.$$

(20)

Define ${\mathbf{X}}_1={[x_{11},x_{12},x_{13},x_{14}]}^T$, ${\mathbf{X}}_2={[x_{21},x_{22},x_{23},x_{24}]}^T$, $\Delta \epsilon ={[\Delta {\epsilon }_1,\Delta {\epsilon }_2,\Delta {\epsilon }_3,\Delta {\epsilon }_4]}^T$, $\Delta \dot{\epsilon }={[\Delta {\dot{\epsilon }}_1,\Delta {\dot{\epsilon }}_2,\Delta {\dot{\epsilon }}_3,\Delta {\dot{\epsilon }}_4]}^T$. According to Equations (18) and (19), the corresponding compact state vectors can be expressed as

$$\left\{\begin{array}{c}{\dot{\mathbf{X}}}_1=(\mathbf{L}+\mathbf{B})\Delta \epsilon \hfill \\ {\dot{\mathbf{X}}}_2=(\mathbf{L}+\mathbf{B})\Delta \dot{\epsilon }.\hfill \end{array}\right.$$

(21)

According to the lumped-disturbance formulation, the error dynamics can be further summarized as

$$\left\{\begin{array}{c}{\dot{\mathbf{X}}}_1={\mathbf{X}}_2,\hfill \\ {\dot{\mathbf{X}}}_2=(\mathbf{L}+\mathbf{B})\left(\mathbf{F}+{\mathbf{Z}}_d\mathbf{U}+{\mathbf{D}}_z\right).\hfill \end{array}\right.$$

(22)

where $\mathbf{F}={[f_1,f_2,f_3,f_4]}^T$, $\mathbf{U}={[u_1,u_2,u_3,u_4]}^T$, ${\mathbf{D}}_z={[d_{z1},d_{z2},d_{z3},d_{z4}]}^T$, and ${\mathbf{Z}}_d=\mathrm{diag}(z_1,z_2,z_3,z_4)$.

The lumped uncertainty is expressed in matrix form in the MAS. It is assumed to be differentiable, and the disturbance and its derivative are bounded as

$$\left\{\begin{array}{c}∥{\mathbf{D}}_z(t,\mathbf{X})∥\le {\overline{\mathbf{D}}}_z,\hfill \\ ∥{\dot{\mathbf{D}}}_z(t,\mathbf{X})∥\le {\overline{\mathbf{D}}}_z^{\ast }.\hfill \end{array}\right.$$

(23)

Thus, the MAS for yaw-stability control has been established, laying the foundation for the DYC controller design.

3. FOTSM-Based Direct Yaw Moment Controller Design

A FOTSM controller is designed in this section, and its stability is proved to realize the final DYC objective.

3.1. Controller Design

For the ith wheel-drive agent, this paper proposes a DYC strategy based on FOTSM control. The FOTSM surface for a single agent is designed as

$$s_i={\dot{x}}_{2i}+c_2{x}_{2i}^{{\mu }_2}+c_1{x}_{1i}^{{\mu }_1},i=1,2,3,4.$$

(24)

where $c_1$ and $c_2$ are positive integers; ${\mu }_1$ and ${\mu }_2$ satisfy ${\mu }_1={\mu }_2{\mu }_3/(2{\mu }_3-{\mu }_2)$, with ${\mu }_3=1$. The sliding surfaces of all agents are extended into compact matrix form, as shown in Equation (25):

$$\mathbf{S}={\dot{\mathbf{X}}}_2+c_2{\mathbf{X}}_2^{{\mu }_2}+c_1{\mathbf{X}}_1^{{\mu }_1}.$$

(25)

where $\mathbf{S}={[s_1,s_2,s_3,s_4]}^T$, ${\mathbf{X}}_1^{{\mu }_1}={[x_{11}^{{\mu }_1},x_{12}^{{\mu }_1},x_{13}^{{\mu }_1},x_{14}^{{\mu }_1}]}^T$, and ${\mathbf{X}}_2^{{\mu }_2}={[x_{21}^{{\mu }_2},x_{22}^{{\mu }_2},x_{23}^{{\mu }_2},x_{24}^{{\mu }_2}]}^T$.

Based on the sliding-mode stability state equation in Equation (20) and the FOTSM surface in Equation (24), the control law for the ith wheel agent is designed as follows:

$$\begin{array}{cc}\hfill u_i& =u_{i\mathrm{eq}}+u_{i\mathrm{s}},\hfill \\ \hfill u_{i\mathrm{eq}}& ={(l_{ii}+b_i)}^{-1}{z}_i^{-1}\left[-c_2{x}_{2i}^{{\mu }_2}-c_1{x}_{1i}^{{\mu }_1}-\left({\displaystyle \sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^N}{l}_{ij}+b_i\right)f_i+{\displaystyle \sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^N}{l}_{ij}\left(f_j+u_j\right)\right],\hfill \\ \hfill u_{i\mathrm{s}}& =z_i^{-1}\left(-\int K_s\mathrm{sgn}\left(s_i\right)dt\right).\hfill \end{array}$$

(26)

where $K_s={\overline{d}}_z^{\ast }+{\epsilon }_s$, and ${\epsilon }_s>0$.

The designed FOTSM control law can be extended into the following compact matrix form:

$$\begin{array}{cc}\hfill \mathbf{U}& ={\mathbf{U}}_{\mathrm{eq}}+{\mathbf{U}}_{\mathrm{s}},\hfill \\ \hfill {\mathbf{U}}_{\mathrm{eq}}& ={\mathbf{Z}}_d^{-1}{(\mathbf{L}+\mathbf{B})}^{-1}\left[-c_2{\mathbf{X}}_2^{{\mu }_2}-c_1{\mathbf{X}}_1^{{\mu }_1}\right]+{\mathbf{Z}}_d^{-1}(-\mathbf{F}),\hfill \\ \hfill {\mathbf{U}}_{\mathrm{s}}& ={\mathbf{Z}}_d^{-1}\left(-\int \mathbf{h}(\mathbf{s})dt\right).\hfill \end{array}$$

(27)

where $\mathbf{h}\left(\mathbf{s}\right)=K_s{[\mathrm{sgn}\left(s_1\right),\mathrm{sgn}\left(s_2\right),\mathrm{sgn}\left(s_3\right),\mathrm{sgn}\left(s_4\right)]}^T$.

Thus, the MAS-based FOTSM DYC controller is designed.

3.2. Stability Analysis

The stability of the proposed FOTSM controller is analyzed in this subsection. Substituting the control law in Equation (27) into the FOTSM surface in Equation (25) gives

$$\begin{array}{cc}\hfill \mathbf{S}& ={\dot{\mathbf{X}}}_2+c_2{\mathbf{X}}_2^{{\mu }_2}+c_1{\mathbf{X}}_1^{{\mu }_1}\hfill \\ \hfill & =(\mathbf{L}+\mathbf{B})\left(\mathbf{F}+{\mathbf{Z}}_d\mathbf{U}+{\mathbf{D}}_z\right)+c_2{\mathbf{X}}_2^{{\mu }_2}+c_1{\mathbf{X}}_1^{{\mu }_1}\hfill \\ \hfill & =(\mathbf{L}+\mathbf{B})\left({\mathbf{Z}}_d{\mathbf{U}}_{\mathrm{s}}+{\mathbf{D}}_z\right).\hfill \end{array}$$

(28)

Let $\mathbf{M}=\mathbf{L}+\mathbf{B}$. For the fixed undirected topology adopted in this study, $\mathbf{L}$ is symmetric positive semidefinite, and $\mathbf{B}=\mathbf{I}$; so, $\mathbf{M}$ is symmetric positive definite. Choose the Lyapunov function as

$$V={\displaystyle \frac{1}{2}}{\mathbf{S}}^T{\mathbf{M}}^{-1}\mathbf{S}.$$

(29)

From Equation (28) and noting that $\mathbf{M}$ and ${\mathbf{Z}}_d$ are constant, the derivative of the sliding variable satisfies

$$\dot{\mathbf{S}}=\mathbf{M}\left[-\mathbf{h}\left(\mathbf{S}\right)+{\dot{\mathbf{D}}}_z\right].$$

(30)

Therefore,

$$\begin{array}{cc}\hfill \dot{V}& ={\mathbf{S}}^T{\mathbf{M}}^{-1}\dot{\mathbf{S}}\hfill \\ \hfill & ={\mathbf{S}}^T\left[-\mathbf{h}\left(\mathbf{S}\right)+{\dot{\mathbf{D}}}_z\right]\hfill \\ \hfill & =-K_s{\parallel \mathbf{S}\parallel }_2+{\mathbf{S}}^T{\dot{\mathbf{D}}}_z\hfill \\ \hfill & \le -K_s{\parallel \mathbf{S}\parallel }_2+{\overline{d}}_z^{\ast }{\parallel \mathbf{S}\parallel }_2\hfill \\ \hfill & \le -\left(K_s-{\overline{d}}_z^{\ast }\right){\parallel \mathbf{S}\parallel }_2\hfill \\ \hfill & =-{\epsilon }_s{\parallel \mathbf{S}\parallel }_2<0.\hfill \end{array}$$

(31)

Since $\mathbf{M}$ is positive definite,

$$V\le {\displaystyle \frac{1}{2{\lambda }_{\min }\left(\mathbf{M}\right)}}{\parallel \mathbf{S}\parallel }_2^2.$$

(32)

Hence,

$$\dot{V}\le -{\epsilon }_s\sqrt{2{\lambda }_{\min }\left(\mathbf{M}\right)}{V}^{1/2}.$$

(33)

Thus, the sliding variable reaches $\mathbf{S}=0$ in finite time, with the reaching time satisfying

$$T_r\le {\displaystyle \frac{\sqrt{2V\left(0\right)}}{{\epsilon }_s\sqrt{{\lambda }_{\min }\left(\mathbf{M}\right)}}}.$$

(34)

The above result indicates that the state trajectory of the tracking-error system reaches the ideal sliding surface $\mathbf{S}=0$ within finite time. Parameters $c_1$ and $c_2$ are selected such that the polynomial $p^2+c_{2}p+c_1$ satisfies the Hurwitz condition. Therefore, when the ideal sliding-mode system $\mathbf{S}=0$ is established, the state of the aggregated tracking-error system converges from any initial condition to the equilibrium point ${[e_1,e_2]}^T={[0,0]}^T$ along the designed FOTSM surface in finite time. For the drive system of a DDEV, the proposed controller enables each agent’s yaw rate and sideslip angle to track their ideal values. Consequently, the whole-vehicle yaw rate and sideslip angle track their references, and DYC is achieved through braking the wheels on one side while driving the wheels on the other side.

4. Experimental Results and Discussion

To verify the effectiveness and real-time applicability of the proposed DYC strategy, HIL tests are conducted on a real-time vehicle control platform. The controller is generated from MATLAB/Simulink(R2023b) and deployed to the VCU, which exchanges vehicle-state and wheel-torque commands with the real-time computer through the CAN interface. The platform consists of an upper computer for test management, a real-time computer for vehicle-dynamics execution, the VCU, and the driver-operation interface. The VCU receives steering, acceleration, and braking inputs and outputs the corresponding wheel-driving-torque commands through CAN.

The experimental platform is shown in Figure 2. The upper computer uses NI VeriStand(2023 Q3) and LabVIEW(2023 Q3) for test management, while the lower real-time computer executes the vehicle dynamics model. The controller developed in MATLAB/Simulink is compiled and deployed to the VCU, which exchanges vehicle states and wheel-torque commands with the real-time computer through the CAN interface, forming a closed-loop HIL environment. The driver-operation interface provides steering, acceleration, and braking inputs.

During the DLC test, the front-wheel steering demand is generated in the closed-loop trajectory-following process according to the instantaneous vehicle response. Therefore, the steering input is not identical for all control cases. When DYC is activated, the additional yaw moment generated by differential wheel torques changes the corrective steering demand required to follow the same target trajectory. The main vehicle parameters used in the HIL experiments are summarized in

Table 1. Main vehicle parameters used in the HIL experiments.

Parameter	Symbol	Value	Unit
Vehicle mass	m	1520	kg
Yaw moment of inertia	$I_z$	2680	kg·m2
Distance from CG to front axle	$l_1$	1.18	m
Distance from CG to rear axle	$l_2$	1.62	m
Track width	d	1.56	m
Left-front cornering stiffness	$k_1$	48,000	N/rad
Right-front cornering stiffness	$k_2$	48,000	N/rad
Left-rear cornering stiffness	$k_3$	52,000	N/rad
Right-rear cornering stiffness	$k_4$	52,000	N/rad

.

The vehicle dynamics model is executed on the real-time computer with a fixed step size of 1 ms. The proposed MAS-FOTSM controller is deployed on the VCU with a control period of 5 ms (200 Hz). Vehicle-state signals and wheel-torque commands are exchanged through the CAN interface within each control cycle. The topology-related matrices and their inverses used in the controller are fixed and are calculated offline. During online execution, the control law mainly involves algebraic operations, matrix–vector operations, and the evaluation of the sliding-mode term, without iterative online optimization. All wheel-torque commands are therefore calculated within the 5 ms control period in the present HIL implementation.

The DLC maneuver is tested at an equivalent vehicle speed of 120 km/h under two cases: without crosswind disturbance and with crosswind disturbance. To evaluate the chattering suppression ability of the proposed FOTSM controller, a conventional MAS-based linear sliding mode controller is implemented on the same HIL platform as a comparative method. The two methods are denoted as MAS-LSM and MAS-FOTSM, and their controller parameters are listed in

Table 2. Direct yaw-moment controller parameters.

Control Method	Controller Parameters
MAS-LSM	$\rho =0.5$, $K=8000$
MAS-FOTSM	$\rho =0.5$, $c_1=15$, $c_2=9$, ${\mu }_1=9/37$, ${\mu }_2=9/23$, $K_s=3000$

.

4.1. Experimental Results Without Crosswind Disturbance

At an equivalent vehicle speed of 120 km/h and without crosswind disturbance, the experimental results are shown in Figure 3. Figure 3a shows that, under the same maneuver, the required front-wheel steering angle is reduced after applying the proposed DYC strategy, indicating improved vehicle stability. Compared with MAS-LSM, MAS-FOTSM markedly attenuates steering-angle chattering. Figure 3b,c show that, with the established second-order MAS and controller, the sideslip angle and yaw rate are reduced compared with the uncontrolled case and approach their desired values, which also indicates improved driving stability. Figure 3d shows that the lateral displacement of the vehicle tracks the target trajectory more closely, demonstrating improved driving performance. Figure 3e,f show that MAS-FOTSM produces a smoother wheel-torque response with less high-frequency switching than MAS-LSM, while maintaining comparable yaw-stability performance.

4.2. Experimental Results Under Crosswind Disturbance

In the crosswind experiment, the wind disturbance is introduced into the vehicle dynamics as an equivalent aerodynamic lateral force. For the lateral crosswind considered in this study, the aerodynamic force is expressed as

$$F_w={\displaystyle \frac{1}{2}}{\rho }_a{C}_Y{A}_s{v}_w^2,$$

(35)

where ${\rho }_a$ is the air density, $C_Y$ is the aerodynamic lateral-force coefficient, $A_s$ is the effective lateral area of the vehicle, and $v_w$ is the applied crosswind speed. Since the aerodynamic center does not generally coincide with the vehicle center of mass, the crosswind lateral force also generates an additional yaw moment, which is expressed as

$$M_w=l_w{F}_w,$$

(36)

where $l_w$ denotes the longitudinal distance between the aerodynamic center and the vehicle center of mass. Accordingly, the nominal vehicle model in Equation (1) can be written under crosswind disturbance as

$$\left\{\begin{array}{cc}\hfill \dot{\beta }& ={\displaystyle \frac{1}{m{v}_x}}\left({\displaystyle \sum _{i=1}^4}{F}_{yi}+F_w\right)-\gamma ,\hfill \\ \hfill \dot{\gamma }& ={\displaystyle \frac{1}{I_z}}\left[l_1(F_{y1}+F_{y2})-l_2(F_{y3}+F_{y4})+(T_1+T_3-T_2-T_4)+M_w\right].\hfill \end{array}\right.$$

(37)

Thus, the crosswind affects both the lateral and yaw motions of the vehicle. Equation (1) represents the nominal vehicle dynamics used for controller development, while Equations (35)–(37) describe the external crosswind disturbance applied in the HIL test. The resulting aerodynamic force and yaw moment are treated as external disturbances within the lumped-disturbance framework of the proposed controller.

The equivalent vehicle speed is maintained at 120 km/h, and the crosswind speed $v_w$ varies according to the profile shown in Figure 4, with a maximum value of 60 km/h. The corresponding lateral aerodynamic force $F_w$ and yaw moment $M_w$ are calculated according to Equations (35) and (36) and applied to the vehicle dynamics during the HIL test.

The experimental results are shown in Figure 5. Figure 5a shows that, under crosswind disturbance, the uncontrolled vehicle exhibits loss of control in the front-wheel steering angle near the end of the maneuver, whereas the vehicle remains stable after applying the proposed DYC strategy. Figure 5b–d show the sideslip angle, yaw rate, and lateral displacement during driving. After the crosswind disturbance is introduced, the uncontrolled vehicle experiences yaw instability, whereas the proposed DYC strategy enables the vehicle to maintain stable motion. Figure 5e,f show the driving torques under the two control methods. Because of the crosswind disturbance, the driving torques increase for both methods; however, MAS-FOTSM exhibits less high-frequency torque-switching than MAS-LSM while maintaining stable yaw responses under the crosswind disturbance. The applied crosswind profile is shown in Figure 4.

Table 3. Quantitative comparison of the HIL responses under the two test conditions.

Condition	Method	$\mathbf{max}\mathbf{\left|}\mathit{\beta }\mathbf{\right|}$ (deg)	Reduction (%)	$\mathbf{max}\mathbf{\left|}\mathit{\gamma }\mathbf{\right|}$ (deg/s)	Reduction (%)	$\mathbf{max}\mathbf{|}{\mathit{\delta }}_{\mathit{f}}\mathbf{|}$ (deg)	${\mathit{\beta }}_{\mathbf{RMS}}$ (deg)	${\mathit{\gamma }}_{\mathbf{RMS}}$ (deg/s)	${\mathit{e}}_{\mathit{y}\mathbf{,}\mathbf{RMS}}$ (m)	${\mathit{e}}_{\mathit{y}\mathbf{,}\mathbf{pk}}$ (m)	${\mathit{\sigma }}_{\mathit{T}}$ (N·m)	$\mathbf{max}\mathbf{|}{\mathit{T}}_{\mathit{i}}\mathbf{|}$ (N·m)
No crosswind	No Control	2.88	–	11.32	–	6.8	–	–	–	–	–	–
	MAS-LSM	1.75	39.2	6.21	45.1	6.5	0.84	3.60	1.26	0.45	37.2	75.7
	MAS-FOTSM	1.58	45.1	6.15	45.7	6.5	0.86	3.64	1.30	0.51	26.0	73.3
Crosswind	No Control	>8.0	–	≈30.0	–	≈6.3	–	–	–	–	–	–
	MAS-LSM	1.68	>79.0	6.54	78.2	6.4	0.78	3.56	1.19	0.38	40.4	74.2
	MAS-FOTSM	1.71	>78.6	6.28	79.1	6.4	0.83	3.72	1.23	0.44	30.3	85.1

provides a quantitative comparison of the two test conditions. Without crosswind, MAS-FOTSM reduces the peak sideslip angle and peak yaw-rate magnitude by 45.1% and 45.7%, respectively, relative to the uncontrolled vehicle. Under crosswind disturbance, the corresponding reductions exceed 78%. The RMS lateral-displacement tracking errors of MAS-FOTSM and MAS-LSM are 1.30 and 1.26 m without crosswind, and 1.23 and 1.19 m under crosswind, showing comparable trajectory-tracking performance. A clearer difference appears in the torque-variation index ${\sigma }_T$: MAS-FOTSM reduces ${\sigma }_T$ from 37.2 to 26.0 N·m without crosswind and from 40.4 to 30.3 N·m under crosswind, corresponding to reductions of 30.1% and 25.0%, respectively. Here, $e_{y,\mathrm{RMS}}$ and $e_{y,\mathrm{pk}}$ denote the RMS and peak lateral-displacement tracking errors, respectively, while ${\sigma }_T$ denotes the standard deviation of the four wheel-torque commands.

5. Discussion

5.1. Interpretation of the Multi-Agent Yaw-Stability Framework

The proposed MAS introduces the four independently actuated wheels directly into the yaw-stability control model. The variables ${\beta }_i$ and ${\gamma }_i$ represent wheel-related contribution states defined by the model decomposition rather than independently measured physical states, and their summation recovers the whole-vehicle sideslip angle and yaw rate. The distributed feature considered in this work refers to the wheel-level agent formulation and the coordination among the four agents through the prescribed topology. In the HIL implementation, the four agent calculations are executed in the VCU, and the resulting wheel-torque commands are transmitted through the CAN network.

5.2. Comparison Between MAS-FOTSM and MAS-LSM

The experimental results show that MAS-FOTSM and MAS-LSM provide similar vehicle-level yaw-stability responses under the tested conditions, while their wheel-torque responses differ noticeably. MAS-LSM exhibits persistent high-frequency torque switching, whereas the torque commands generated by MAS-FOTSM are smoother. In the FOTSM controller, the nonlinear terminal terms modify the error dynamics, and the switching term enters the control input through integration; so, the discontinuous sign function is not applied directly to the wheel-torque command. Therefore, the advantage of MAS-FOTSM observed in the present tests is mainly the reduction in high-frequency torque activity while maintaining comparable yaw-stability performance, rather than a uniformly smaller peak torque.

5.3. Limitations and Future Work

The modeling and stability analysis in this study are established within a specific range of applicability. The tire lateral force is approximated by a linear model under small-slip-angle conditions, while the stability analysis assumes bounded lumped disturbances and their derivatives. When the vehicle approaches the handling limit, large tire slip angles, tire-force saturation, or disturbances exceeding the assumed bounds may reduce the model accuracy and affect the tracking and stability performance. Therefore, the present conclusions should be interpreted within the tested operating range. The current HIL validation is also limited to a 120 km/h DLC maneuver with one crosswind profile, and MAS-LSM is the only active-control baseline. Future work will consider nonlinear tire characteristics, tire-force saturation, different vehicle speeds, road-friction levels, and more severe disturbance conditions. Comparisons with centralized FOTSM, LQR, and MPC, together with repeated HIL tests and vehicle-level experiments, will also be conducted to further evaluate the robustness and practical applicability of the proposed method.

6. Conclusions

This paper investigated an MAS-based DYC strategy for DDEVs. First, a reduced-order yaw-dynamics model of a DDEV was derived based on the vector superposition principle, and a yaw-stability-oriented MAS was established. Then, an FOTSM controller was proposed for the developed MAS, realizing DYC with finite-time convergence and strong disturbance rejection. Finally, HIL experimental tests were conducted under DLC maneuvers with and without crosswind disturbance to verify the effectiveness and superiority of the proposed controller. The experimental results show that the proposed MAS-FOTSM method improves yaw stability, enhances lateral trajectory tracking, and reduces driving-torque chattering compared with the conventional MAS-LSM method. Under crosswind disturbance, the proposed strategy maintains vehicle stability and avoids loss of control, demonstrating its potential for engineering application in DDEV chassis control.