Fuzzy Rule-Based Optimal Direct Yaw Moment Allocation for Stability Control of Four-Wheel Steering Mining Trucks

Abstract

To address the poor trajectory tracking of mining trucks in narrow, high-curvature paths, this study explores the impact of four-wheel steering (4WS) and direct yaw moment control (DYC) on vehicle stability. A validated two-degree-of-freedom 4WS vehicle model was developed. A fuzzy logic controller with dual inputs (yaw rate and yaw angular acceleration) and a single output (compensatory yaw moment) was designed, alongside an optimal torque distribution controller based on tire friction circle theory to allocate the resultant yaw moment. A co-simulation platform integrating TruckSim and MATLAB/Simulink was established, and experiments were conducted under steady-state and double-lane-change conditions. Comparative analysis with traditional front-wheel steering and alternative control methods reveals that the 4WS mining truck with fuzzy-controlled optimal torque distribution achieves a reduced turning radius, enhancing maneuverability and stability. Hardware-in-the-loop (HIL) testing further validates the controller’s effectiveness in real-time applications.

1. Introduction

In modern automotive engineering, vehicle safety and handling remain critical technical priorities. The advancement of distributed drive technologies and increasing demands for enhanced stability have established Direct Yaw Moment Control (DYC) as a sophisticated dynamic control system. DYC optimizes vehicle performance by precisely modulating yaw moment distribution, thereby improving dynamic behavior across diverse driving conditions while maintaining precise handling [1,2]. Recent research in vehicle dynamics has extensively explored DYC for stability enhancement. Sun et al. [3] developed a non-singular terminal sliding mode (NTSM) control approach, delivering a robust, accurate, and efficient DYC framework. Studies [4,5,6] have integrated Active Front Steering (AFS) with DYC, achieving precise tracking of target velocities and yaw moments, resulting in measurable improvements in lateral dynamics and stability. Wang et al. [7] proposed a combined control strategy incorporating DYC, AFS, and anti-slip modulation to enhance vehicle stability in both longitudinal and yaw dynamics. However, these methodologies largely overlook rear-wheel steering control, leaving the impact of Four-Wheel Steering (4WS) systems on dynamic maneuvers underexplored.

Mining trucks, which typically operate under low-speed, heavy-load conditions, benefit significantly from 4WS systems due to their dual improvements in dynamic performance and kinematic efficiency. This enhancement arises from nonlinear coordinated control of rear-wheel steering angles, which geometrically reduces the turning radius. In navigating high-curvature slopes in mining environments, 4WS systems minimize theoretical turning radii by dynamically coupling front and rear wheel angles [8]. At the dynamic level, 4WS systems [9] actively stabilize vehicle posture by optimizing lateral stress distribution across tire contact patches. This load redistribution enhances tire–ground pressure uniformity, constrains center-of-mass displacement through roll moment compensation, and improves traction efficiency, steering precision, and stability under complex operational scenarios.

4WS systems significantly enhance vehicle lateral dynamics via multi-degree-of-freedom (MDOF) synergistic control. By coordinating front/rear wheel steering inputs, this methodology attenuates the coupling degree of the yaw angular velocity and the lateral acceleration. Experimental analysis confirms that rotation of wheels in the same direction reduces lateral acceleration phase lag while maintaining the response of the yaw angular velocity unchanged. Reference [10] developed a nonlinear dynamic model demonstrating through frequency- and time-domain simulations under identical parameters and operating conditions that 4WS vehicles exhibit a superior amplitude stability of the yaw angular velocity fluctuation and reduced overshoot compared to Front-Wheel Steering (FWS) systems, despite comparable phase characteristics. While confirming enhanced stability, the study inadequately addresses uncertainty factors like tire load variations, revealing limitations in control robustness. Compared to conventional steering architectures, 4WS systems optimize lateral force distribution matrices to mitigate front-axle lateral loading [11]. Wang et al. [12] identified control redundancy issues in current 4WS vehicle control architectures. To address this, they developed an allocation control layer strategy for rear-wheel steering angles and drive torque distribution, implemented via tire force optimization algorithms. However, the oversimplified modeling framework exhibits limited generalizability, restricting applicability to basic operational scenarios. While 4WS systems enhance stability through coordinated yaw and lateral motion control, excessive reliance on yaw regulation in high-curvature turns may trigger nonlinear instability phenomena. Consequently, Chu [13] proposes a stability enhancement methodology for 4WS vehicles under large-curvature turning conditions. Nevertheless, the model’s precision and robustness remain constrained by unaccounted longitudinal velocity variations and road adhesion fluctuations.

In 4WS mining trucks, current control strategies include Proportional–Integral–Derivative (PID) control [14] for its simplicity and robustness, fuzzy logic control [15,16] to handle system nonlinearities and uncertainties, neural network-based control [17] for its adaptive learning capabilities, sliding mode control [18,19,20] renowned for its disturbance rejection and robustness, and model predictive control (MPC) [21,22] which optimizes system performance based on a dynamic model. Fuzzy control demonstrates distinct advantages for nonlinear systems due to its strong adaptability, operating without precise mathematical models by dynamically adjusting control parameters via membership functions and rule bases. This effectively addresses parameter variations in mining environments, such as abrupt load shifts, fluctuating road friction coefficients, and high-curvature turns. To mitigate lateral instability risks, Reference [23] proposed a fuzzy sliding mode DYC strategy. The upper-layer controller employs a fuzzy sliding mode algorithm to compute required yaw moments, which are then allocated as longitudinal forces across four wheels. Experimental results demonstrate reduced standard deviation of the centroid lateral deviation angle and enhanced disturbance rejection on complex terrains. However, its lower-layer static weight-based torque allocation exhibits limitations in dynamic adaptability and energy efficiency. Reference [24] implemented a hierarchical control framework: the upper layer calculates additional yaw moments using the centroid lateral deviation angle and the yaw angular velocity deviations via fuzzy logic, while the lower layer distributes wheel torque proportionally to vertical loads. Compared to AFS, this 4WS approach improves stability and trajectory tracking accuracy in large-curvature turns, though real-time torque allocation and scenario adaptability require refinement. Reference [25] developed a particle swarm-optimized PID-based DYC algorithm, combining load estimation with motor lookup tables to achieve optimal torque output, effectively enhancing stability on low-adhesion surfaces. Reference [26] also uses a hierarchical framework to solve the lateral trajectory tracking control problem. Reference [27] introduced a dual-steering mode DYC method featuring an upper-layer active disturbance rejection yaw moment controller and a rule-based lower-layer torque allocator, successfully minimizing turning radii and improving maneuverability. Zhang [28] proposed a dual-layer fuzzy DYC system that optimizes yaw moment distribution to boost handling and stability. While existing studies [29,30,31] have advanced torque allocation strategies for 4WS mining trucks, three critical challenges persist in mining-specific applications: dynamic adaptability, real-time performance, and energy efficiency optimization. Future advancements must integrate operational condition awareness and optimal allocation mechanisms into control algorithms, prioritizing synergistic improvements in stability, energy economy, and operational reliability tailored to mining vehicle characteristics.

The implementation of these advanced control strategies necessitates robust and programmable hardware platforms. Modern electronic control units (ECUs), often based on high-performance microprocessors (MPUs), provide the computational power and flexibility required for complex algorithm execution. For validation and testing purposes, Hardware-in-the-Loop (HIL) simulation has become an indispensable tool. HIL systems allow for the rigorous testing of controller hardware and software against a real-time simulated vehicle model, enabling comprehensive evaluation of performance, stability, and safety under a wide range of conditions without the cost and risks associated with full-scale physical prototypes. This approach is critical for developing reliable control systems for Four-Wheel Steering Mining Trucks.

This article aims to solve the problem of poor stability of four-wheel steering mining vehicles in narrow and high-curvature curved conditions. The main content focuses on the low-speed and heavy-duty characteristics of mining vehicles, and proposes a hierarchical control architecture that combines four-wheel steering with fuzzy-based direct yaw moment control and optimal torque distribution to enhance adaptability to nonlinear open-pit mining environments without relying entirely on accurate mathematical models. Experimental validation under steady-state cornering and double-lane-change conditions confirms that fuzzy-controlled torque optimization significantly enhances the stabilization performance of 4WS mining trucks. This article establishes and validates a two degree of freedom four-wheel steering vehicle dynamics model in Section 2, and introduces the design of a hierarchical control strategy in Section 3, including an upper-level yaw moment controller based on fuzzy logic and a lower-level controller based on optimal torque allocation. Section 4 presents the simulation results and hardware in the loop test results under steady-state rotation and double line shifting conditions.

2. Dynamic Model of the Vehicle

Four-Wheel Steering Two-Degree-of-Freedom Vehicle Dynamics Modeling

In the process of mathematical modeling, it is essential to establish appropriate assumptions for vehicle simplification. Considering the characteristics of four-wheel steering mining trucks and the research objective of enhancing handling stability, the following assumptions are formulated:

(1)

The influence of suspension vibration on vehicle handling stability is neglected. The vehicle is assumed to have zero displacement along the Z-axis and only perform planar motion.

(2)

The front and rear wheel steering angles are considered as system inputs, while dynamic lag in the steering system is neglected.

(3)

Uneven load distribution between left and right wheels is disregarded.

(4)

Aerodynamic forces and lateral wind effects are excluded from consideration.

(5)

Nonlinear tire characteristics and aligning moments are not accounted for in the model.

(6)

The road surface is presumed to be flat and even, with exclusion of load variations induced by road irregularities.

This study focuses on analyzing the effects of four-wheel differential drive torque and 4WS on yaw and lateral dynamics of mining trucks. The two-degree-of-freedom (2-DOF) 4WS vehicle dynamics model incorporates structural parameters (e.g., vehicle mass, tire cornering stiffness) and stability evaluation metrics, thereby satisfying the requirements for comprehensive stability analysis. Figure 1 depicts the 2-DOF 4WS vehicle model, with the following key parameters:

$F_{Y1}$, $F_{Y2}$: Lateral reaction force at front/rear wheels;

${\delta }_f$, ${\delta }_r$: Front/rear wheel steering angles;

${\alpha }_f$, ${\alpha }_r$: Front/rear tire slip angles;

${\omega }_{rd}$: Desired yaw angular velocity;

${\beta }_d$: Desired centroid sideslip angle;

$u$, $v$: Lateral/longitudinal velocity components at the center of gravity;

$a$, $b$: Distance from centroid to front/rear axles.

Application of Newton’s Second Law of Motion yields the following formula:

$$\left\{\begin{array}{c}{F}_{Y1}\cos {\delta }_f+F_{Y2}\cos {\delta }_r=m{a}_y\\ a{F}_{Y1}\cos {\delta }_f-b{F}_{Y2}\cos {\delta }_r=I_z{\dot{\omega }}_{rd}\end{array}\right.$$

(1)

where $a_y$ denotes lateral acceleration, $I_z$ represents the moment of inertia about the z-axis, and $a$, $b$ specify the longitudinal distances from the center of gravity to the front/rear axles, respectively. Leveraging tire lateral deflection characteristics yields the following formulations:

$$\left\{\begin{array}{c}{F}_{Y1}=k_1{\alpha }_f\\ F_{Y2}=k_2{\alpha }_r\end{array}\right.$$

(2)

where $k_1$ and $k_2$ denote the front/rear axle tire cornering stiffness coefficients, respectively. Under small steering angles with $\cos {\delta }_f\approx 1$ and $\cos {\delta }_r\approx 1$, and assuming sideslip angle at the center of gravity is also very small, the following formulation applies:

$${\beta }_d\approx \tan {\beta }_d={\displaystyle \frac{u}{v}}$$

(3)

Through geometric and kinematic analytical derivation, the following equations are established:

$$\left\{\begin{array}{c}{\alpha }_f={\beta }_d+{\displaystyle \frac{a{\omega }_{rd}}{u}}-{\delta }_f\\ {\alpha }_r={\beta }_d-{\displaystyle \frac{b{\omega }_{rd}}{u}}-{\delta }_r\end{array}\right.$$

(4)

Substituting Equations (2) and (4) into (1) yields the differential equations of motion for the two-degree-of-freedom four-wheel steering model:

$$\left\{\begin{array}{c} (k_1+k_2){\beta }_d+{\displaystyle \frac{{\omega }_{rd}}{u}}(a{k}_1-b{k}_2)-k_1{\delta }_f-k_2{\delta }_r=m(\dot{v}+u{\omega }_{rd})\\ (a{k}_1-b{k}_2){\beta }_d+{\displaystyle \frac{{\omega }_{rd}}{u}}(a^2{k}_1+b^2{k}_2)-a{k}_1{\delta }_f+b{k}_2{\delta }_r=I_Z{\dot{\omega }}_{rd}\end{array}\right.$$

(5)

This study investigates a mining vehicle with a maximum payload capacity of 150 tons. A distributed-drive four-wheel-steering dump mining vehicle model was constructed in TruckSim software (v.2019.0). The fundamental parameters of the vehicle model are presented in the

Table 1. Vehicle Parameters.

Parameters	Data (m)
Sprung mass m/ $\mathrm{kg}$	91,300
Distance from centroid to front axles $a/\mathrm{m}$	2.8
Distance from centroid to rear axles $b/\mathrm{m}$	2.8
Wheelbase $L/\mathrm{m}$	6.1
Centroid height $h/\mathrm{m}$	2.9
Wheel track $\mathrm{d}/\mathrm{m}$	8.0
Wheel center height $\mathrm{r}/\mathrm{m}$	1.8
Moment of inertia about the vertical-axis $I_z/(\mathrm{kg}\cdot {\mathrm{m}}^2)$	1,050,548

.

TruckSim provides users with a comprehensive modeling and simulation environment that enables efficient vehicle performance evaluation. The software architecture allows direct parameter modification through its graphical interface based on research requirements, utilizing built-in vehicle templates to streamline modeling processes. This integrated solution facilitates simulation, analysis, and testing procedures significantly reducing development costs. The software platform comprises three core functional modules (as illustrated in Figure 2): the Graphical Data Pre-processing Module, Model Solver Module, and Post-processing Module.

In conventional automotive systems, front wheels typically serve as the steering wheels while rear wheels remain non-steering. However, 4WS vehicles enable coordinated steering angle generation at both front and rear axles. The steering process of 4WS systems must comply with the Ackermann Steering Principle, which dictates that during cornering maneuvers, the inner wheel maintains a smaller turning radius than the outer wheel. All four wheels perform circular rotation about a common instantaneous center.

The research subject is a mining vehicle with anti-Ackermann four-wheel steering, where the front and rear axles exhibit the following relationship:

$${\delta }_f=-{\delta }_r$$

(6)

From the structural parameters of the mining vehicle, it is evident that the distance from the center of mass to the front axle equals that to the rear axle. Additionally, the front and rear axles employ identical tire models, resulting in equivalent cornering stiffness values for both axles. Under steady-state conditions, parameters $\dot{\beta }$ and $\dot{\omega }$ are both zero. Incorporating these constraints and substituting Equation (7) into Equation (6) yields:

$$\left\{\begin{array}{c}{\beta }_{z\_\exp }={\displaystyle \frac{m{u}^2{\delta }_f}{2a{k}_1}}\\ {\omega }_{z\_\exp }={\displaystyle \frac{{\delta }_{f}u}{a}}\end{array}\right.$$

(7)

The desired yaw rate must comply with the ground adhesion limit constraint:

$$\left|{\omega }_{z\_\max }\right|=k{\displaystyle \frac{\mu g}{u}}$$

(8)

The safety coefficient $k$ is typically assigned a value of 0.85, while $\mu$ denotes the road adhesion coefficient and $g$ represents gravitational acceleration (9.8 $\mathrm{m}/{\mathrm{s}}^2$). Consequently, the final desired yaw rate is defined as:

$${\omega }_{rd}=\min \left[\left|{\displaystyle \frac{{\delta }_{f}u}{a}}\right|,\left|{\omega }_{z\_\max }\right|\right]·sign({\delta }_f)$$

(9)

To enhance overall lateral stability and minimize vehicle sideslip during operation, thereby ensuring stable trajectory tracking, the ideal centroid sideslip angle is defined as zero.

3. Design of Direct Yaw Moment Control Strategy

The Figure 3 presents the Simulink strategy framework for vehicle yaw moment control, which is used to achieve precise control of vehicle motion. Input the target vehicle speed and steering wheel angle, first process the front wheel angle through the Ackermann steering model, simulate the actual motion of the TruckSim vehicle, and output the real-time status; the ideal vehicle model outputs reference values such as ideal yaw rate and ideal yaw acceleration based on classical theory. The deviation of the two states is input into the direct yaw moment control module based on fuzzy control, which infers the expected yaw moment correction amount. Combined with speed control feedback, the torque distribution control module decomposes it into four wheel torque commands. By optimizing the actuator distribution, it minimizes the tire load ratio and improves the lateral force margin, forming a “target model deviation fuzzy correction torque distribution” closed-loop. Through fuzzy control, it adapts to the nonlinearity of the vehicle, making the actual motion approach the ideal state and improving handling and stability.

3.1. Design of Yaw Moment-Based Upper-Layer Controller

Vehicles constitute complex nonlinear systems whose dynamic behaviors can vary significantly across operating points, potentially leading to suboptimal performance with conventional control methodologies. In contrast, fuzzy control offers distinct advantages, particularly in managing highly nonlinear systems and scenarios demanding rapid response. This approach maps input signals to fuzzy sets using membership functions to quantify their degrees of belonging, applies rule-based inference mechanisms, and converts resultant fuzzy outputs into precise control actions through defuzzification.

This study designs a dual-input single-output (DISO) second-order fuzzy controller, where the yaw rate error $\mathrm{\Delta }{\omega }_r$ and yaw angular acceleration $d\mathrm{\Delta }{\omega }_r$ serve as inputs, and the additional yaw moment $M_z$ constitutes the output. Triangular membership functions are adopted for input/output variables. Both inputs are defined by 7 linguistic variables: {NB, NM, NS, Z, PS, PM, PB} = {Negative Big, Negative Medium, Negative Small, Zero, Positive Small, Positive Medium, Positive Big}. The output is similarly defined with 7 linguistic variables: {NB, NM, NS, ZE, PS, PM, PB} = {Negative Big, Negative Medium, Negative Small, Zero, Positive Small, Positive Medium, Positive Big}.

Angular step and double lane change maneuvers were simulated in TruckSim to analyze simulation results. The universe of discourse for input variables was configured as follows: yaw rate error $\mathrm{\Delta }{\omega }_r$: [−15, 15], yaw angular acceleration $d\mathrm{\Delta }{\omega }_r$: [−15, 15], and output variable $M_z$: [−1, 1]. Membership functions for input and output variables are illustrated in the figure below.

Fuzzy control rules are established to define linguistic relationships between input and output variables. Based on expert knowledge, rules are formulated in the “IF $\mathrm{\Delta }{\omega }_r$ AND $d\mathrm{\Delta }{\omega }_r$ THEN $M_z$” format, as presented in

Table 2. Fuzzy control rule table.

$\mathit{d}\Delta {\mathit{\omega }}_{\mathit{r}}$	$\Delta {\mathit{\omega }}_{\mathit{r}}$
	NB	NM	NS	ZE	PS	PM	PB
NB	PB	PB	PM	PM	PS	NS	NS
NM	PB	PB	PS	PS	PS	NS	NM
NS	PB	PM	PS	PS	Z	NM	NM
Z	PM	PM	PS	Z	NS	NM	NM
PS	PM	PM	Z	NS	NM	NM	NB
PM	PM	PM	NS	NS	NM	NB	NB
PB	PS	PS	NS	NM	NB	NB	NB

. The input–output relational surface characterizing these variables is illustrated in Figure 4.

This study employs the center of gravity method for defuzzification, which determines the crisp output value by calculating the centroid of the area enclosed by the membership function curve of the fuzzy set and the base variable axis. This method is the most prevalent in fuzzy control system design. For continuous output membership functions, the defuzzied output $u^{\ast }$ is mathematically expressed as:

$$u^{\ast }={\displaystyle \frac{{\displaystyle \int u\mu (u)du}}{{\displaystyle \int \mu (u)du}}}$$

(10)

3.2. Design of Optimal Torque Distribution-Based Controller

Based on the tire friction circle theory, the total frictional force of a tire is constrained at any given moment. Different quadrants of the friction circle represent distinct operational states (e.g., acceleration, braking, steering), defined by combinations of positive/negative longitudinal and lateral forces. When longitudinal forces increase, lateral force capacity decreases. To enhance lateral force safety margins, longitudinal forces should be allocated judiciously to meet compensatory yaw moment requirements and velocity targets while minimizing their magnitude, thereby improving mining vehicle stability.

Objective Function Formulation:

$$\min J=\min {\displaystyle \sum \frac{F_{x\_ij}^2+F_{y\_ij}^2}{{({\mu }_{ij}{F}_{z\_ij})}^2}}$$

(11)

where $i$ denotes front/rear axles and $j$ indicates left/right wheels. Given that wheel edge motors exclusively govern tire longitudinal forces, Equation (11) reduces to:

$$\min J=\min {\displaystyle \sum \frac{F_{x\_ij}^2}{{({\mu }_{ij}{F}_{z\_ij})}^2}}$$

(12)

By substituting $F_{x\_ij}={\displaystyle \frac{T_{ij}}{r{\mu }_{ij}}}$ into Equation (12), the final optimized objective function is formulated as:

$$\min J=\min {\displaystyle \sum \frac{T_{ij}^2}{{({\mu }_{ij}{F}_{z\_ij}r)}^2}}$$

(13)

When lateral forces approach saturation during mining vehicle cornering, wheel steering angles should be reduced while increasing tire longitudinal forces. By neglecting steering angles, compensatory yaw moment $\mathrm{\Delta }M$ and driving torque $T$ are derived. The driving torque $T$ is primarily designed to track the desired target speed by processing the error between the actual and desired velocities through a PI controller.

To enhance the adaptability of the PI controller under different driving speeds of vehicles, a segmented PI parameter tuning strategy was designed to address the dynamic characteristic differences between low-speed and high-speed working conditions of mining vehicles. When the vehicle is in a high-speed driving state, appropriately reducing the proportional coefficient and integral coefficient: the sensitivity of the system to the control signal is significantly increased within this speed range. Excessive Kp can easily cause frequent fluctuations in torque output, leading to unstable longitudinal force. However, reducing Ki can avoid excessive accumulation of integral effects, effectively suppress high-frequency vibrations, ensure smooth driving torque output, and form synergy with lateral force control after reducing steering angle.

When the vehicle is in a low-speed driving state, priority should be given to ensuring the response efficiency of speed tracking. The proportional coefficient should be appropriately increased to accelerate the adjustment speed of the driving torque on the speed error and shorten the time for the actual speed to approach the target speed; at the same time, considering that the duration of speed error is relatively long under low-speed conditions, the integration effect is prone to saturation, which can lead to torque output overshoot and cause the phenomenon of “low-speed crawling” of the vehicle. Therefore, an integration limiting mechanism is specially introduced to strictly limit the maximum value of the integration coefficient. While retaining the advantage of eliminating steady-state errors through integration, the negative impact of integration saturation on control accuracy is avoided.

$$T=T_{fl}+T_{fr}+T_{rl}+T_{rr}$$

(14)

Vehicle dynamics principles establish that:

$$\mathrm{\Delta }{M}_z=-{\displaystyle \frac{d}{2r}}{T}_{fl}+{\displaystyle \frac{d}{2r}}{T}_{fr}-{\displaystyle \frac{d}{2r}}{T}_{rl}+{\displaystyle \frac{d}{2r}}{T}_{rr}$$

(15)

Further derivation of Equation (15) yields:

$$\left\{\begin{array}{c}{T}_{fl}={\displaystyle \frac{T}{2}}-{\displaystyle \frac{\mathrm{\Delta }{M}_z}{d}}r-T_{rl}\\ T_{fr}={\displaystyle \frac{T}{2}}-{\displaystyle \frac{\mathrm{\Delta }{M}_z}{d}}r-T_{rr}\end{array}\right.$$

(16)

Substituting Equation (16) into Equation (13) yields:

$$J={\displaystyle \frac{{(\frac{T}{2}-\frac{\mathrm{\Delta }{M}_z}{d}r-T_{rl})}^2}{{({\mu }_{fl}{F}_{z\_fl}r)}^2}}+{\displaystyle \frac{{(\frac{T}{2}+\frac{\mathrm{\Delta }{M}_z}{d}r-T_{rr})}^2}{{({\mu }_{fr}{F}_{z\_fr}r)}^2}}+{\displaystyle \frac{T_{rl}^2}{{({\mu }_{rl}{F}_{z\_rl}r)}^2}}+{\displaystyle \frac{T_{rr}^2}{{({\mu }_{rr}{F}_{z\_rr}r)}^2}}$$

(17)

Partial derivatives of Equation (17) with respect to $T_{rl}$ and $T_{rr}$ yield:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }{T}_{rl}}}=-{\displaystyle \frac{2(\frac{T}{2}-\frac{\mathrm{\Delta }{M}_z}{d}r-T_{rl})}{{({\mu }_{fl}{F}_{z\_fl}r)}^2}}+{\displaystyle \frac{2T_{rl}}{{({\mu }_{rl}{F}_{z\_rl}r)}^2}}\\ {\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }{T}_{rr}}}=-{\displaystyle \frac{2(\frac{T}{2}+\frac{\mathrm{\Delta }{M}_z}{d}r-T_{rr})}{{({\mu }_{fr}{F}_{z\_fr}r)}^2}}+{\displaystyle \frac{2T_{rr}}{{({\mu }_{rr}{F}_{z\_rr}r)}^2}}\end{array}\right.$$

(18)

Setting ${\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }{T}_{rl}}}$ and ${\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }{T}_{rr}}}$ to 0 in Equation (18) solves for the minimum value:

$$\left\{\begin{array}{c}{T}_{rl}={\displaystyle \frac{\frac{{\mu }_{rl}^2{F}_{z\_rl}^{2}T}{2}-\frac{{\mu }_{rl}^2{F}_{z\_rl}^2\mathrm{\Delta }{M}_z}{d}r}{{\mu }_{rl}^2{F}_{z\_rl}^2+{\mu }_{fl}^2{F}_{z\_fl}^2}}\\ T_{rr}={\displaystyle \frac{\frac{{\mu }_{rr}^2{F}_{z\_rr}^{2}T}{2}-\frac{{\mu }_{rr}^2{F}_{z\_rr}^2\mathrm{\Delta }{M}_z}{d}r}{{\mu }_{rr}^2{F}_{z\_rr}^2+{\mu }_{fr}^2{F}_{z\_fr}^2}}\end{array}\right.$$

(19)

Substituting Equation (19) into Equation (14) yields:

$$\left\{\begin{array}{c}{T}_{fl}={\displaystyle \frac{\frac{{\mu }_{fl}^2{F}_{z\_fl}^{2}T}{2}-\frac{{\mu }_{fl}^2{F}_{z\_fl}^2\mathrm{\Delta }{M}_z}{d}r}{{\mu }_{rl}^2{F}_{z\_rl}^2+{\mu }_{fl}^2{F}_{z\_fl}^2}}\\ T_{fr}={\displaystyle \frac{\frac{{\mu }_{fr}^2{F}_{z\_fr}^{2}T}{2}-\frac{{\mu }_{fr}^2{F}_{z\_fr}^2\mathrm{\Delta }{M}_z}{d}r}{{\mu }_{rr}^2{F}_{z\_rr}^2+{\mu }_{fr}^2{F}_{z\_fr}^2}}\end{array}\right.$$

(20)

4. Simulation Experiments and Results Analysis

A high-fidelity vehicle model was constructed in TruckSim, and a control algorithm model was developed in MATLABR2022b/Simulink, as illustrated in Figure 5. Algorithm validation was conducted under two conditions: steady-state steering with a 15-degree front-wheel angle and double lane change maneuvers, with a road adhesion coefficient of 0.85. The analysis evaluated the control efficacy and stability impacts of fuzzy logic-based four-wheel steering stability control, four-wheel-steering mining trucks, and active front-wheel-steering mining trucks.

4.1. Steady-State Steering Simulation Experiment

In TruckSim, the front-wheel steering angle was set to 15 degrees with a vehicle speed of 20 km/h and a simulation duration of 30 s. This configuration simulates the mining vehicle increasing its steering angle during high-curvature turns. The resulting minimum turning radius and handling stability are illustrated in the figure below.

As shown in Figure 6a, compared to the FWS mining vehicle, the 4WS configuration reduces the turning radius, enhancing maneuverability in tight, high-curvature turns. The directional deviations in centroid sideslip angle between FWS and 4WS mining trucks are attributable to vehicle dynamics principles and tire force distribution characteristics. Within the two-degree-of-freedom linear framework, the centroid sideslip angle is governed by the coupled interaction of lateral and yaw motions. For front-wheel-steering (FWS) mining trucks, lateral forces are generated exclusively at the front wheels, aligning with steering inputs. This induces a centroid sideslip angle opposing the yaw rate direction. In contrast, four-wheel-steering (4WS) systems introduce additional lateral forces via active rear-wheel steering. Under a counter-phase steering strategy, these rear-wheel lateral forces partially offset front-wheel contributions, significantly altering the resultant lateral force vector. This compensatory effect enlarges the trajectory curvature radius at the centroid, thereby reversing the sideslip angle direction. The analysis reveals that 4WS mining trucks employing optimal torque distribution achieve a 12% smaller turning radius compared to those using equal torque allocation under a 15-degree steering condition. Furthermore, fuzzy logic-based direct yaw moment control reduces the turning radius by 18.3% relative to uncontrolled 4WS systems. As shown in Figure 6b, compared to the front-wheel-steering (FWS) mining vehicle, the four-wheel-steering (4WS) configuration doubles the yaw rate magnitude, significantly enhancing navigability in high-curvature turns. This improvement allows the vehicle to achieve target trajectories with reduced steering inputs and tighter paths. Optimal torque allocation enhances the yaw rate by 2.2 deg/s compared to equal distribution, while fuzzy logic-based direct yaw moment control further increases it by 4.2 deg/s relative to uncontrolled 4WS systems. These results confirm that fuzzy logic-optimized torque allocation amplifies yaw rate and minimizes turning radii during sharp cornering. As illustrated in Figure 6c, the centroid sideslip angles of the FWS and 4WS mining trucks exhibit opposing directions. Under identical operational conditions, the 4WS configuration achieves faster trajectory tracking. Compared to the FWS vehicle, the 4WS system demonstrates a marginally larger centroid sideslip angle magnitude while maintaining stability, yet significantly elevates the yaw rate, thereby markedly enhancing turning agility.

4.2. Double Lane Change Simulation Experiment

A double-lane-change maneuver was set up in TruckSim to assess obstacle avoidance capabilities under emergency conditions, with a vehicle speed of 30 km/h and a simulation duration of 30 s. The results for centroid trajectory and handling stability are depicted in the accompanying figure.

As shown in Figure 7a, the mining vehicle with optimal torque allocation demonstrates superior trajectory tracking performance and enhanced hazard response capability compared to the vehicle with equal torque distribution control. Figure 7b shows that the peak yaw rate of the four-wheel-steering mining vehicle with fuzzy logic-based direct yaw moment optimal allocation control is 10.8, whereas with equal torque distribution it is 17.8. These results demonstrate the superior target trajectory tracking performance of the fuzzy logic-based control strategy. Figure 7c shows that the peak centroid sideslip angle of the fuzzy logic-based, optimally allocated direct yaw moment, four-wheel steering (4WS) mining vehicle is 1.8, compared to 2.7 for the conventional four-wheel steering (FWS) counterpart. This represents a 33.3% improvement in stability and handling. Under emergency obstacle avoidance conditions, the 4WS configuration optimised by fuzzy logic provides greater safety margins and enhanced stability through coordinated lateral force management.

4.3. HIL Test

To validate the real-time performance of the designed control system, a HIL test involving real actuators is conducted, as illustrated in Figure 8. The HIL testing platform is structured as follows.

The vehicle and road models in TruckSim are executed on the host PC, enabling real-time operation and signal interaction through co-simulation with Simulink, leveraging the Desktop Real-Time and CAN Communication modules from the Simulink library. The motion control algorithm is compiled and deployed to the NI chassis via Ethernet for real-time execution. The NI chassis acquires vehicle status signals from the CAN bus, computes the desired front wheel steering angle and driving/braking torque, and transmits these back to the CAN bus. The calculated driving/braking torque and steering angle are directly sent to TruckSim on the host PC.

To evaluate vehicle stability during sharp turns, a HIL test was conducted under an emergency double-lane-change scenario. The road friction coefficient was set to 0.85, and the vehicle speed was maintained at 20 km/h. The HIL experimental results are presented in Figure 9.

As shown in Figure 9a,c, the yaw rate tracking error remains within 2.17 deg/s. Similarly, Figure 9b,d demonstrate that the sideslip angle at the vehicle’s center of gravity has a tracking error within 0.80 degrees. Given these minimal tracking errors, the proposed motion control strategy is suitable for implementation with real actuators.

5. Conclusions

To address the challenges of trajectory tracking for mining trucks in narrow, high-curvature bends, co-simulation was performed using TruckSim and MATLAB/Simulink under various operating conditions. The simulations evaluated the effectiveness of different control strategies on trajectory tracking, yaw rate, and centroid sideslip angle. Key findings are summarized as follows:

(1)

Compared to conventional front-wheel-steering (FWS) mining trucks, the four-wheel-steering (4WS) configuration with counter-phase Ackermann geometry significantly enhances maneuverability in tight, high-curvature bends. Under extreme conditions, the 4WS system effectively overcomes the poor trajectory tracking and limited stability inherent in FWS vehicles.

(2)

A 4WS mining vehicle equipped with a fuzzy logic-based direct yaw moment control (DYC) system, utilizing dual inputs (yaw rate and yaw angular acceleration) and a single output (compensatory yaw moment), achieves a smaller turning radius, improved maneuverability, and enhanced stability during steady-state steering compared to conventional 4WS vehicles without DYC.

(3)

The 4WS mining truck with optimal torque distribution control accounts for the physical constraints of electric motors and road adhesion. By dynamically allocating wheel-specific torque based on friction circle theory, this approach markedly improves trajectory tracking accuracy and vehicle stability compared to conventional equal torque distribution methods.

Future work will primarily focus on experimental validation through real-vehicle testing. Field tests with a physical 4WS mining truck prototype will be conducted to evaluate trajectory tracking performance, lateral stability, and steering responsiveness under various realistic conditions. Data will be collected for key states such as yaw rate, sideslip angle, and path deviation, allowing for direct comparison with co-simulation results and further refinement of the control strategies.