Research on Yaw Stability Control for Distributed-Drive Pure Electric Pickup Trucks

Abstract

To address the issue of poor yaw stability in distributed-drive electric pickup trucks at medium-to-high speeds, particularly under the influence of continuously varying tire forces and road adhesion coefficients, a novel Kalman filter-based method for estimating the road adhesion coefficient, combined with a Tube-based Model Predictive Control (Tube-MPC) algorithm, is proposed. This integrated approach enables real-time estimation of the dynamically changing road adhesion coefficient while simultaneously ensuring vehicle yaw stability is maintained under rapid response requirements. The developed hierarchical yaw stability control architecture for distributed-drive electric pickup trucks employs a square root cubature Kalman filter (SRCKF) in its upper layer for accurate road adhesion coefficient estimation; this estimated coefficient is subsequently fed into the intermediate layer’s corrective yaw moment solver where Tube-based Model Predictive Control (Tube-MPC) tracks desired sideslip angle and yaw rate trajectories to derive the stability-critical corrective yaw moment, while the lower layer utilizes a quadratic programming (QP) algorithm for precise four-wheel torque distribution. The proposed control strategy was verified through co-simulation using Simulink and Carsim, with results demonstrating that, compared to conventional MPC and PID algorithms, it significantly improves both the driving stability and control responsiveness of distributed-drive electric pickup trucks under medium- to high-speed conditions.

1. Introduction

In recent years, research on yaw stability control for distributed drive electric vehicles (DDEVs) has remained a critical topic in vehicle dynamics. Distributed-Drive Electric Vehicles (DDEVs), utilizing multiple in-wheel/out-board motors, fundamentally differ in architecture from conventional centralized-drive vehicles by enabling precise individual torque control and vectoring. The effectiveness of vehicle yaw stability control directly influences operational integrity during driving. Currently applied algorithms for yaw stability control in distributed drive electric vehicles predominantly include fuzzy control, PID control, Direct Yaw Moment Control (DYC) [1], and sliding mode control. While the aforementioned control methods can significantly enhance vehicle yaw stability under specific operating conditions [2]. They lack predictive capability. Different control strategies exhibit varying requirements for system model accuracy and operational suitability. Notably, the pickup truck studied herein—distinct from conventional light passenger vehicles—features greater vehicle weight [3] and more demanding operating conditions. Designed primarily for cargo transport and emergency rescue operations, this vehicle class necessitates higher-fidelity system modeling. Traditional control methods demonstrate effectiveness at low speeds or with lighter vehicle masses. However, when operating at medium-to-high velocities or with increased curb weight, these methods exhibit degraded response speed and robustness. Within the domain of stability control for distributed drive electric vehicles (DDEVs), tire-road friction coefficient estimation constitutes a critical research focus. Current estimation approaches primarily encompass sensor-based measurement, visual recognition, and Kalman filter-based estimation. Whereas the first two methods suffer from prohibitive costs and computational complexity—limiting their applicability in vehicular stability control—Kalman filtering demonstrates distinct advantages in cost-effectiveness, computational efficiency, and estimation accuracy. Consequently, Kalman filter-based methods have been widely adopted for tire-road friction coefficient estimation in production vehicles. Integrating a tire-road friction coefficient observer into the stability control strategy enables real-time adaptation to varying road conditions. For the specific case of distributed drive electric pickup trucks, a dedicated control strategy is developed to operate effectively under medium-to-high speed scenarios with dynamically changing friction coefficients. This approach significantly enhances controller responsiveness and robustness.

Accurate estimation of the tire-road friction coefficient [4,5,6] is critical for yaw stability control in distributed drive electric pickup trucks. Rapid and precise friction coefficient estimation enables enhanced vehicle response to drastically deteriorated adhesion conditions, thereby significantly accelerating the response time of the overall control strategy. The Kalman filter algorithm [7] represents the most prevalent approach for tire-road friction coefficient (TRFC) estimation. Li et al. [8] enhanced estimation accuracy by introducing a refined adaptive Kalman filter for TRFC estimation. Wang et al. [9] proposed a hybrid learning framework integrating event-triggered cubature Kalman filter (ETCKF) with EKFNet Sensor signals are processed via ETCKF with data validity determined by an event-triggered mechanism. Normalized tire forces are computed through a nonlinear tire model, followed by TRFC estimation using EKFNET—a four-layer neural network embedded with extended Kalman filtering. Ye et al. [10] established correlations between vertical load and tire stiffness by analyzing tire contact patch length variations. Tire stiffness was incorporated into system states to capture its interaction with dynamic responses, while a square root cubature Kalman filter with adaptive measurement noise covariance was designed to accelerate convergence and enhance estimation precision. Liu et al. [11] developed a computationally efficient estimator employing fused auxiliary particle filtering (APF) and iterated extended Kalman filtering (IEKF). This approach achieves accurate estimation of front wheel slip angles under strong nonlinearity and non-Gaussian noise, enabling iterative TRFC estimation based on self-aligning torque sensitivity to slip angles. Current research on tire-road friction coefficient (TRFC) estimation frequently entails high dependence on data-intensive neural network training or elevated system complexity. Such approaches invariably expand state dimensionality, substantially increasing computational burden and degrading real-time performance. Consequently, developing a low-complexity TRFC estimator with minimal computational requirements and enhanced real-time capability is imperative.

For yaw stability control in distributed drive vehicles, fuzzy control, sliding mode control, PID control, and Direct Yaw Moment Control (DYC) have been extensively implemented. Zhao et al. [12] addressed the nonlinear longitudinal-lateral-vertical coupling challenge through a hierarchical integrated control framework. This approach combines nonlinear sliding mode control with tire force optimization, overcoming traditional methods’ local optimization limitations. Wang et al. [13] employed coordinated fuzzy-PID and sliding mode control to enhance driving stability under extreme conditions. A simulated annealing algorithm was utilized to establish coordination control zones, resolving actuator coupling between DDAS (Direct Drive Active Steering) and VSC (Vehicle Stability Control). Guo et al. [14] precisely delineated stability boundaries via phase plane analysis and developed an integral three-step method for direct yaw moment computation, improving both stability and energy efficiency. Shi et al. [15] Utilized recursive Bayesian estimation, weights between actual conditions and sub-models were computed. Nonlinear MPC per sub-model optimized wheel slip via a parallel chaotic optimization algorithm (PCOA), ensuring smooth operation via output fusion. Guo et al. [16] introduced a real-time nonlinear MPC strategy integrating single-track and Magic Formula tire models. Constraints were handled through modified C/GMRES algorithm with exterior penalty functions, featuring variable prediction horizons for accelerated initialization and KKT-condition-based torque distribution. However, in current distributed drive vehicle (DDV) yaw stability control systems, significant limitations persist across prevalent control approaches. PID control exhibits inadequate performance for complex nonlinear scenarios, fuzzy control remains heavily dependent on empirical rules, and LQR control necessitates quadratic-form objective functions—restricting its applicability. Although conventional model predictive control (MPC) [17,18,19,20] demonstrates advantages in constrained optimization, it exhibits constrained disturbance rejection capabilities when subjected to external perturbations. Thus, existing MPC methods cannot meet the stringent yaw stability requirements of distributed-drive electric pickup trucks, especially under high-mass conditions during severe maneuvers. Current research on model predictive control (MPC) predominantly focuses on autonomous vehicle trajectory planning [21,22] and passenger car applications. However, investigations into multipurpose distributed-drive electric pickup trucks—characterized by higher curb weights and demanding operational profiles—remain notably scarce.

To address these challenges, this paper proposes a tire-road friction coefficient estimation method based on square root cubature Kalman filtering (SRCKF), integrated with Tube-based model predictive control (Tube-MPC) [23]. The estimated friction coefficient is fed into the Tube-MPC, thereby enhancing yaw stability for distributed drive electric pickup trucks in medium-to-high speed critical scenarios. This integration significantly improves control accuracy and response speed. The primary contributions are summarized as follows.

A hierarchical control architecture for the entire vehicle is proposed, as shown in Figure 1. The upper layer comprises a square root cubature Kalman filter (SRCKF) module, in which the road adhesion coefficients for all four wheels are estimated in real-time. These coefficients are then fed as inputs into the middle-layer Tube-MPC yaw moment solver, enhancing its predictive capability and responsiveness. The additional yaw moment computed by the Tube-MPC solver is subsequently passed to the lower-layer quadratic programming (QP)-based torque distributor [24]. This distributor solves the wheel torque commands required for vehicle yaw stability, thereby establishing closed-loop control of the vehicle’s yaw dynamics.

To address the real-time requirement of the road adhesion estimator and avoid the computational inefficiency associated with complex multi-model coupling [25,26,27], A Square Root Cubature Kalman Filter (SRCKF)-based estimator is developed for the distributed-drive electric pickup truck, effectively addressing its high nonlinearity and model complexity while maintaining computational efficiency without relying on training data. In contrast to existing studies that often rely on predefined adhesion coefficients or single-layer control architectures, this work introduces a novel integrated framework that combines real-time road adhesion estimation via SRCKF with Tube-MPC for robust yaw stability control. This approach not only adapts dynamically to varying road conditions but also enhances control accuracy and robustness under rapid changes. Under typical and low-adhesion conditions, the method achieves notable reductions in root mean square (RMS) errors compared to both PID and conventional MPCs—particularly a 51.1% decrease in lateral acceleration error at 80 km/h and a 33.7% reduction in yaw rate error at 120 km/h—demonstrating its robustness and practical value.

2. Vehicle Dynamics Model

A 7-DOF vehicle dynamics model capturing longitudinal, lateral, and yaw motions is developed to represent the core dynamics of distributed-drive electric pickup trucks, as shown in Figure 2. This modeling approach is adopted because conventional low-DOF models (e.g., 2- or 3-DOF) cannot capture critical behaviors such as vertical load transfer, roll motion, and individual wheel dynamics. These are essential for distributed-drive electric vehicles, where independent torque allocation at each wheel critically affects stability and performance. The 7-DOF model is therefore necessary for accurate controller design and stability analysis. The relevant vehicle parameters are summarized in

Table 1. Vehicle Parameters.

Parameter	Numerical Value
Vehicle mass/kg	1600
Wheelbase/m	3.0
Distance from front axle to CoG/m	1.4
Distance from rear axle to CoG/m	1.6
Track width/m	1.6
Vehicle yaw moment of inertia/(kg·m2)	2700

.

The 7-degree-of-freedom vehicle kinematic equations are formulated as:

$$\left\{\begin{array}{l}m({\dot{v}}_x-v_y\gamma )=(F_{xfl}+F_{xfr})\cos {\delta }_f-(F_{yfl}+F_{yfr})\sin {\delta }_f+F_{xrl}+F_{xrr}-F_f-F_w-F_i\\ m({\dot{v}}_y-v_x\gamma )=(F_{xfl}+F_{xfr})\sin {\delta }_f+(F_{yfl}+F_{yfr})\cos {\delta }_f+F_{yrl}+F_{yrr}\\ I_z\dot{\gamma }=\left[(F_{xfl}+F_{xfr})\sin {\delta }_f+(F_{yfl}+F_{yfr})\cos {\delta }_f\right]l_f+\\ \left[(F_{xfr}-F_{xfl})\cos {\delta }_f+(F_{yfl}-F_{yfr})\sin {\delta }_f\right]{\displaystyle \frac{t_f}{2}}+(F_{xrT}-F_{xr1}){\displaystyle \frac{t_r}{2}}-(F_{yr1}+F_{yrr})l_r+\\ (F_{xrT}-F_{xr1}){\displaystyle \frac{t_r}{2}}-(F_{yr1}+F_{yrr})l_r+(F_{xrr}-F_{xrl}){\displaystyle \frac{t_r}{2}}-\\ (F_{yrl}+F_{yrr})l_r+M_{zfl}+M_{zfr}+M_{zrl}+M_{zrr}\\ I_w{\dot{w}}_i=T_{di}-T_{bi}-F_{xi}{R}_w-M_{yi}\end{array}\right.$$

(1)

where Parameter $\gamma$ designates the yaw rate; Parameter $m$ designates the vehicle mass; Parameters $l_f$ and $l_r$ designate the distances from the front and rear axles to the center of mass, respectively; Parameters $t_f$ and $t_r$ designate the front and rear track widths, respectively; Parameters $v_x$ and $v_y$ designate the longitudinal and lateral velocities, respectively; Parameter $I_z$ designates the vehicle’s yaw moment of inertia about the z-axis; Parameter ${\delta }_f$ designates the front wheel steering angle; Parameters $F_{xi}$ and $F_{yi}$ designate the longitudinal and lateral forces, respectively; Parameter $I_w$ designates the wheel moment of inertia.

To establish a linear two-degree-of-freedom vehicle dynamics model for deriving the desired yaw rate and sideslip angle. The deviation between the measured values and the sideslip angle and yaw rate derived from the 7-DOF model is utilized as the input to the controller.

Vehicle vertical motion is neglected, and the two-degree-of-freedom differential equations are formulated as follows:

$$\left[\begin{array}{l}\stackrel{·}{\beta }\\ \stackrel{·}{\gamma }\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{k_f+k_r}{m{v}_x}}& {\displaystyle \frac{l_f{k}_f-l_r{k}_r}{m{v}_x^2}}-1\\ {\displaystyle \frac{l_f{k}_f-l_r{k}_r}{l_z}}& {\displaystyle \frac{l_f^2{k}_f+l_f^2{k}_r}{l_z{v}_x}}\end{array}\right]\left[\begin{array}{l}\beta \\ \gamma \end{array}\right]+\left[\begin{array}{l}-{\displaystyle \frac{k_f}{m{v}_x}}\\ -{\displaystyle \frac{l_f{k}_f}{I_z}}\end{array}\right]{\delta }_f$$

(2)

where Parameter $\beta$ designates the Sideslip angle; Parameter $k_f$ and $k_r$ designates the sum of rear axle tire cornering stiffnesses, respectively.

During vehicle steering maneuvers, the desired sideslip angle and yaw rate are given as follows:

$$\left\{\begin{array}{l}{\gamma }_{ref}={\displaystyle \frac{v_x{\delta }_f}{l(1+k{v_x}^2)}}\\ {\beta }_{ref}=({\displaystyle \frac{l_r}{{v_x}^2}}+{\displaystyle \frac{m{l}_f}{k_{r}l}}){\displaystyle \frac{{v_x}^2}{l(1+k{v_x}^2)}}{\delta }_f\end{array}\right.$$

(3)

where Parameter ${\beta }_{ref}$ designates the desired sideslip angle; Parameter ${\gamma }_{ref}$ designates the desired yaw rate; Parameter $l$ designates the vehicle wheelbase; Parameter $K$ designates the stability coefficient.

Simultaneously constrained by road adhesion conditions, the lateral acceleration must satisfy the following constraint when deriving the desired yaw rate and sideslip angle.

$$\left|a_y\right|\le \mu g$$

(4)

where $\mu$ designates the road adhesion coefficient.

To validate the effectiveness of the established 7-degree-of-freedom vehicle model for the distributed-drive electric pickup truck, simulation results under sinusoidal steer conditions were compared against the Carsim vehicle model. As illustrated in Figure 3, the responses of sideslip angle, yaw rate, longitudinal velocity, and yaw velocity in the developed 7-DOF model closely match those of the Carsim benchmark.

Based on the Carsim benchmark, the root mean square error analysis demonstrates that the longitudinal velocity error remains within 1.5%, the lateral velocity error within 1.6%, the yaw rate error within 5%, and the sideslip angle error within 3.2%. These results confirm that the 7-DOF model meets the required accuracy for practical application.

3. Kalman Filter-Based Road Adhesion Estimation and Vehicle Tire Modeling

3.1. Vehicle Tire Modeling

Based on the Magic Formula [28], the tire model is established, with the tire lateral force expressed as:

$$F_y=D\sin (C\arctan (Bx-E(Bx-\arctan (Bx))))+S_v$$

(5)

where Parameter $F_y$ designates the lateral force; Parameter D designates the peak factor; Parameter B designates the stiffness factor; Parameter E designates the curvature factor; Parameter $S_v$ designates the vertical shift in the curve.

3.2. Design of Road Adhesion Coefficient Estimator

A square root cubature Kalman filter [29,30,31] is established to accurately estimate the dynamically varying road adhesion coefficient.

Filter Parameter Initialization.

$$\left\{\begin{array}{l}{\widehat{x}}_0=E(x_0)\\ P_0=E\left[(x_0-{\widehat{x}}_0){(x_0-{\widehat{x}}_0)}^T\right]\end{array}\right.$$

(6)

where Parameter $P_0$ designates the initial covariance matrix; Parameter ${\widehat{x}}_0$ designates the initial state vector.

Computing the cubature points.

$$\left\{\begin{array}{l}{P}_k=S_k{S}_k^T\\ x_{i,k}=S_k{\xi }_i+{\widehat{x}}_{k}i=1,2,\dots ,2n\end{array}\right.$$

(7)

where Parameter $P_k$ designates the covariance matrix at time-step k; Parameter x_i and k designates the predicted cubature points of the state, respectively; Parameter n designates the dimension of the state vector, assigned a value of 4.

Cubature Point Propagation.

The one-step propagated values of the cubature points are computed through the state transition equations.

$$x_{i,k+1}^{-}=x_{i,k}$$

(8)

The prior state estimate $x_k^{-}$ and prior covariance matrix $P_k^{-}$ are computed as follows:

$$\left\{\begin{array}{l}{\widehat{x}}_{k+1}^{-}={\displaystyle \frac{1}{2L}}{\displaystyle \sum _{i=1}^{2L}{x}_{i,k+1}^{-}}\\ P_{k+1}^{-}={\displaystyle \frac{1}{2L}}{\displaystyle \sum _{i=1}^{2L}{x}_{i,k+1}^{-}}{x_{i,k+1}^{-}}^T-{\widehat{x}}_{k+1}^{-}{{\widehat{x}}_{k+1}^{-}}^T+Q\end{array}\right.$$

(9)

The cubature Kalman filter is enhanced to a square root formulation, where the prior covariance matrix is reformulated in square root form as:

$$S_{k+1}^{-}=Tria(\left[{\gamma }_{k+1}^{*},S_Q\right])$$

(10)

The cubature points are recomputed as follows:

$$z_{i,k+1}=S_{k+1}^{-}{\xi }_i+{\widehat{x}}_{k+1}^{-}$$

(11)

where Parameter $z_{i,k}$ designates the predicted measurement cubature points.

Cubature point propagation.

$$z_{i,k+1}^{-}=H(z_{i,k+1},u_k)$$

(12)

The predicted measurement, innovation covariance, and cross-covariance matrix are computed as follows:

$${\widehat{z}}_{k+1}^{-}={\displaystyle \frac{1}{2L}}{\displaystyle \sum _{i=1}^{2L}{z}_{i,k+1}^{-}}$$

(13)

$$\left\{\begin{array}{l}{P}_{Z,k+1}={\displaystyle \frac{1}{2L}}{\displaystyle \sum _{i=1}^{2L}{Z}_{i,k+1}^{-}}{Z_{i,k+1}^{-}}^T-{\widehat{Z}}_{k+1}^{-}{Z_{k+1}^{-}}^T+R\\ P_{xz,k+1}={\displaystyle \frac{1}{2L}}{\displaystyle \sum _{i=1}^{2L}{Z}_{i,k+1}^{-}}{Z_{i,k+1}^{-}}^T-{\widehat{X}}_{k+1}^{-}{{\widehat{Z}}_{k+1}^{-}}^T\end{array}\right.$$

(14)

Following the square root cubature filtering algorithm, the square root forms of the aforementioned matrices are derived as:

$$\left\{\begin{array}{l}{S}_{Z,k+1}=Tria\left(\left[z_{k+1}^{*},S_R\right]\right)\\ S_{xz,k+1}={\gamma }_{k+1}^{*}{z_{k+1}^{*}}^T\end{array}\right.$$

(15)

where Parameter $S_{z,k+1}$ designates the square root of the innovation covariance matrix; Parameter $S_{xz,k+1}$ designates the square root of the cross-covariance matrix.

The Kalman gain matrix is computed as follows:

$$K_{k+1}=(S_{xz,k+1}/S_{z,k+1}^T)/S_{z,k+1}$$

(16)

The system state update and covariance square root update are computed as follows:

$${\widehat{x}}_{k+1}={\widehat{x}}_{k+1}+K_{k+1}(Z_{k+1}-{\widehat{z}}_{k+1}^{-})$$

(17)

$$S_{k+1}=Tria(\left[{\gamma }_{k+1}^{*}-K_{k+1}{z}_{k+1}^{*},K_{k+1}{S}_R\right])$$

(18)

To verify the estimation performance of the road adhesion coefficient estimator, simulation validation was conducted using a split-μ road with friction coefficients of 0.3 and 0.8, and a step-μ road with coefficients of 0.8 and 0.5. The estimation results are shown in Figure 4.

4. Tube-MPC Design

During actual vehicle operation, complex and variable road surfaces introduce significant noise due to model inaccuracies and sensor errors, degrading controller solving accuracy. Conventional Model Predictive Control (MPC) algorithms exhibit high sensitivity to such noise. Furthermore, accumulated errors progressively escalate over time, ultimately causing loss of system stability. Meanwhile, traditional Proportional-Integral-Derivative controllers employ fixed-gain parameters lacking adaptive adjustment capabilities for uncertainties. As vehicle dynamics evolve during driving, PID controllers struggle to promptly adapt to these variations, resulting in inaccurate regulation of vehicle velocity and acceleration, thereby compromising driving safety.

Compared to conventional MPC, Tube-based Model Predictive Control (Tube-MPC) solely incorporates optimized handling of model initial states and tightened constraints. Its computational complexity remains comparable to standard MPC. This implies that Tube-MPC achieves effective disturbance rejection against noise without requiring substantial increases in computational resources or algorithmic complexity during practical implementation. For instance, within distributed vehicle control systems where computational resources per vehicle are constrained, the implementation simplicity of Tube-MPC enables rapid deployment on embedded systems while satisfying real-time control requirements. The implementation process of Tube-MPC is structured as follows.

A discrete time state–space equation is formulated.

$$x_{k+1}=A{x}_k+B{u}_k+w_k$$

(19)

where $x_k$ denotes the state vector at time step $k$ (specifically comprising yaw rate, sideslip angle, and velocity), $u_k$ represents the control input, $A$ corresponds to the system matrix, and B corresponds to the input matrix. And where $w_k$ characterizes bounded disturbances satisfying $‖w_k‖\le \overline{w}$.

Nominal Trajectory Optimization.

At each sampling time step $k$ a finite-horizon optimization problem is solved to determine the nominal control trajectory ${\left\{{\overline{x}}_{i|k},{\overline{u}}_{i|k}\right\}}_{i=k}^{k+N-1}$, where $N$ denotes the prediction horizon. The optimization aims to minimize a cost function that balances trajectory tracking performance and control effort over the prediction horizon. The cost function comprises a quadratic term penalizing state deviations from desired references to enhance tracking accuracy, and a control penalty term mitigating excessive inputs to prevent actuator saturation and reduce energy consumption. The formulation incorporates constraints—including system dynamics, input bounds, and state limits—to ensure operational feasibility and safety. Using a receding-horizon strategy, only the first control input in the optimized sequence is applied; the optimization is then reiterated at each time step using updated measurements. This method enhances adaptability and robustness against uncertainties and disturbances. The optimization formulation is given by

$$\left\{\begin{array}{l}\underset{{\overline{x}}_{i|k},{\overline{u}}_{i|k}}{\min }{\displaystyle \sum _{i=k}^{k+N-1}({\overline{x}}_{i|k}^{T}Q{\overline{x}}_{i|k}+{\overline{u}}_{i|k}^{T}R{\overline{u}}_{i|k})}+{\overline{x}}_{k+N|k}^{T}P{\overline{x}}_{k+N|k}\\ {\overline{x}}_{i+1|k}=A{\overline{x}}_{i|k}+B{\overline{u}}_{i|k},i=k,k+1,\cdots ,k+N-1\\ {\overline{x}}_{k|k}=x_k\\ {\overline{x}}_{i|k}\in X,i=k,k+1,\cdots ,k+N\\ {\overline{u}}_{i|k}\in U,i=k,k+1,\cdots ,k+N-1\end{array}\right.$$

(20)

where $Q\ge 0$, $R>0$ and $P\ge 0$ denotes the state, control input, and terminal state weight matrices, respectively, while $\mathcal{X}$ and $\mathcal{U}$ represents the constraint sets for the state and control input. The simulation step size is 0.001 s, the controller prediction horizon is 10 s, and the controller control horizon is 7 s. The yaw moment is constrained to the range of −1200 N·m to 1200 N·m.

Error Feedback Control Design.

An ancillary feedback controller ${\tilde{u}}_k=K{\tilde{x}}_k$ is synthesized to ensure the error state vector ${\tilde{x}}_k$ remains bounded within a robust invariant tube, thereby guaranteeing stable error dynamics. The gain matrix K is computed offline via pole placement to assign the eigenvalues of the closed-loop matrix (A + BK) strictly inside the unit circle, thereby ensuring Jury stability and desired convergence performance of the error dynamics. The linear feedback gain matrix $K$ is typically designed such that the closed-loop system matrix $A+BK$ is Jury-stable.

Actual control input

At each sampling time step $k$, the actual control input $u_k$ is composed of the nominal control input ${\overline{u}}_{k|k}$ and the error feedback control input ${\tilde{u}}_k$

$$u_k={\overline{u}}_{k|k}+K{\tilde{x}}_k={\overline{u}}_{k|k}+K(x_k-{\overline{x}}_{k|k})$$

(21)

5. Torque Distributor Design Based on Quadratic Programming

Following the computation of the required additional yaw moment by the upper-layer Tube-MPC, a lower-layer torque distributor is designed to optimally allocate the driving torques to the four independent in-wheel motors of the distributed-drive electric pickup truck. This allocation is formulated as a quadratic programming (QP) problem to achieve multiple critical objectives in real time. The use of a QP-based optimization framework is motivated by the need to efficiently resolve the high-degree-of-freedom torque distribution problem while explicitly accounting for physical constraints such as motor torque limits, tire-road friction boundaries, and vehicle stability requirements. By minimizing a weighted cost function that penalizes excessive slip and energy consumption while promoting stability, the QP solver ensures rapid torque distribution that enhances actuation response, improves yaw stability, and maximizes the utilization of available road adhesion. Objective Function Formulation.

$$\min {\displaystyle \sum _{i=1}^4\frac{{T_i}^2}{{(\mu F_{Z,i}r)}^2}}$$

(22)

where T_i denotes the wheel torque, μ denotes the tire–road friction coefficient, F_z,i denotes the vertical load on the i-th wheel, and r denotes effective wheel radius.

Constraint Formulation.

$$\left\{\begin{array}{l}{T}_{L1}+T_{R1}+T_{L2}+T_{R2}=T_x^{req}\\ {\displaystyle \frac{-t_f}{2r}}{T}_{L1}+{\displaystyle \frac{t_f}{2r}}{T}_{R1}+{\displaystyle \frac{-t_r}{2r}}{T}_{L2}+{\displaystyle \frac{t_r}{2r}}{T}_{R2}=M_Z\\ T_i\in \left[\max (-\mu F_{z,i}r,T_{d\min }),\min (\mu F_{z,i}r,T_{d\max })\right]\end{array}\right.$$

(23)

6. Simulation and Results Analysis

To validate the accuracy and real-time performance of the proposed hierarchical yaw stability control strategy, co-simulations were conducted using Simulink and Carsim. The double-lane change maneuver, characterized by significant path curvature variations, was selected to rigorously evaluate controller performance under medium-high speed conditions. Comparative simulations were performed at 80 km/h on a high-adhesion road surface (μ = 0.85), testing three controllers PID, conventional MPC, and Tube-MPC with road adhesion estimation. The results are presented in Figure 5.

The high-speed double-lane change maneuver in Carsim simulates real-world scenarios of emergency lane-changing followed by immediate return to the original lane. This test is widely recognized to rigorously validate vehicle handling stability and road-holding safety.

As evidenced by Figure 5, under PID control alone, the vehicle’s actual yaw rate and sideslip angle exhibit significant deviations from their reference values. This control strategy demonstrates sluggish response characteristics and excessive overshoot. While conventional MPC achieves moderate performance improvements over PID, it still suffers from delayed response and substantial overshoot during transient phases of the maneuver. With the integrated road adhesion estimation and Tube-MPC, the proposed strategy demonstrates significantly enhanced response speed compared to both baseline controllers. The sideslip angle, yaw rate, lateral acceleration, and longitudinal velocity closely track their reference values, yielding substantial improvements in overall control performance. The integrated strategy of road adhesion estimation and Tube-MPC substantially enhances vehicle yaw stability. Quantitatively validated in

Table 2. Error Analysis at 80 km/h.

Control Algorithm	Sideslip Angle Error/Rad		Yaw Rate Error/(rad·s−1)
	RMS	Max	RMS	Max
PID	0.0849	0.1987	0.5779	1.2325
MPC	0.0678	0.1304	0.4733	1.0563
Tube-MPC	0.0528	0.1164	0.4395	0.5731

, this approach reduces yaw rate RMS error by 7.1% relative to conventional MPC and 24.0% relative to PID control, while yaw rate peak error decreases by 45.8% compared to conventional MPC and 53.5% compared to PID control. Similarly, lateral acceleration RMS error declines by 17.1% versus conventional MPC and 51.1% versus PID control, with sideslip angle RMS error diminishing by 22.1% against conventional MPC and 37.8% against PID control, and peak error reducing by 10.7% relative to conventional MPC and 41.5% relative to PID control.

To further validate control strategy performance under low-friction and high-speed conditions, simulations were executed at 120 km/h with a road adhesion coefficient of μ = 0.6, implementing a double-lane change maneuver susceptible to instability. Comparative assessments of PID control, MPC without road adhesion estimation, and Tube-MPC with adhesion estimation are shown in Figure 6.

Under high-speed and low-adhesion conditions (μ = 0.6), vehicles are prone to yaw instability during lane-change maneuvers. The double-lane change test at 120 km/h effectively replicates this real-world driving scenario. As evidenced by Figure 6, compared with PID control and conventional MPC without road adhesion estimation, the integrated approach incorporating the adhesion estimator and Tube-MPC coordination enables closer tracking of reference values for yaw rate, sideslip angle, and lateral acceleration. This strategy demonstrates enhanced response speed, reduced overshoot, and significantly improves vehicle handling stability and driving safety under high-speed, low-adhesion (μ = 0.6) conditions. As quantified in

Table 3. Error Analysis at 120 km/h.

Control Algorithm	Sideslip Angle Error/Rad		Yaw Rate Error/(rad s−1)
	RMS	Max	RMS	Max
PID	0.2364	0.3186	0.8764	1.4125
MPC	0.2118	0.2291	0.6119	1.0247
Tube-MPC	0.1907	0.1872	0.5818	0.7942

, the integrated method combining road adhesion estimation and Tube-MPC reduces yaw rate RMS error by 5.0% compared to conventional MPC and 33.7% compared to PID control, while yaw rate peak error decreases by 22.5% relative to conventional MPC and 43.7% relative to PID control. Similarly, lateral acceleration RMS error declines by 13.0% versus conventional MPC and 31.0% versus PID control. For sideslip angle RMS error diminishes by 10.0% against conventional MPC and 19.4% against PID control, with peak error reduced by 18.3% compared to conventional MPC and 41.3% compared to PID control.

7. Conclusions

This study investigates yaw stability control for distributed-drive pure electric pickup trucks, utilizing cubature Kalman filter-based road adhesion estimation and Tube-MPC algorithms.

To enhance vehicle perception of road conditions and computational efficiency of the adhesion estimator, square root cubature Kalman filtering (SRCKF) is employed for real-time estimation of dynamically varying road adhesion coefficients, These estimates are fed as inputs to the mid-layer additional yaw moment solver, thereby enhancing torque distribution accuracy and response speed.

To enhance computational precision and response speed of the stability control algorithm, a tube-based model predictive control (Tube-MPC) strategy was designed to address model uncertainties and external disturbances, significantly improving vehicle yaw stability under medium- and high-speed conditions.

Simulink/Carsim co-simulations demonstrate that the integrated road adhesion estimation and Tube-MPC strategy significantly enhances yaw stability for massive distributed four-wheel-drive pure electric pickup trucks during medium- and high-speed double-lane change maneuvers. At 80 km/h with a road adhesion coefficient of 0.85, the proposed method reduces root mean square (RMS) errors by 37.8% for sideslip angle, 24.0% for yaw rate, and 51.1% for lateral acceleration compared to PID control. Relative to conventional model predictive control (MPC) with adhesion estimation, it further diminishes RMS errors by 22.1% for sideslip angle, 7.1% for yaw rate, and 17.1% for lateral acceleration. At 120 km/h with a road adhesion coefficient of 0.6 under instability-prone conditions, the proposed method reduces root mean square (RMS) errors by 19.4% for sideslip angle, 33.7% for yaw rate, and 31.0% for lateral acceleration compared to PID control. Relative to conventional model predictive control (MPC) with adhesion estimation, it diminishes RMS errors by 10.0% for sideslip angle, 5.0% for yaw rate, and 13.0% for lateral acceleration.

This study provides a preliminary validation of the proposed algorithm’s effectiveness through simulation. However, several limitations should be acknowledged. In particular, the computational efficiency of the quadratic programming (QP) solver used in the torque distribution layer was not quantitatively compared with alternative solver implementations, which may influence real-time performance in more complex scenarios. In addition, the performance of the control strategy was evaluated primarily under double-lane change maneuvers, while other critical scenarios—such as emergency braking, various road adhesion conditions (split-μ), and combined steering-braking maneuvers—were not extensively investigated. To address these limitations and extend the applicability of the control framework, future work will include quantitative analyses of solver efficiency, along with experimental validation under a wider range of operational conditions. These will specifically incorporate challenging scenarios such as emergency braking, split-μ road surfaces, and integrated steering and braking operations, as well as off-road driving environments. Furthermore, we will focus on integrating vertical dynamics control—incorporating active suspension and load transfer effects—with the existing lateral and longitudinal stability management system. This holistic approach aims to enhance overall vehicle agility and safety in real-world unstructured environments.