Time-domain optimization based NMPC path tracking control for underground LHD

Abstract

The path tracking control of underground Load-Haul-Dump (LHD) is essential for reducing operators’ working time in underground environments and enhancing mining efficiency. Existing NMPC-based controllers often exhibit suboptimal performance, such as oscillatory movements and poor tracking accuracy in large-curvature turns, increasing the risk of collision with tunnel walls. To address these limitations, this study proposes an improved NMPC controller specifically designed to enhance path tracking accuracy and stability under these conditions. Comparative simulations between the proposed controller, traditional NMPCs, and NMPCs enhanced with other optimization algorithms demonstrate significant performance improvements. The proposed strategy reduces the maximum absolute lateral deviation by an average of 55.06% across all tested trajectories, with a peak improvement of 85.19% for a specific trajectory. These results demonstrate that the proposed controller achieves high tracking accuracy and effectiveness under the tested reference trajectories. The finding establishes the proposed controller as a promising solution for optimizing the path tracking of underground vehicles.

Introduction

Underground mining heavily relies on advanced equipment, most of which has traditionally been operated manually¹. However, the emergence of intelligent equipment technology has sparked a paradigm shift in industrial automation, positioning intelligent control as the cornerstone of modern machinery. The mining industry, a prime example of this transformation, is transitioning from manual to automated operations, a shift that is crucial for improving both the efficiency and safety of mining activities^2,3. In this context, the study of path tracking control for LHD vehicles, which are indispensable intelligent devices in underground mining, is of particular importance. The LHD is shown in Fig. 1.

Fig. 1.

The load-haul-dump (LHD) (a heavy-duty machine specifically designed for underground mines and similar applications.)

The inherently high-risk nature of mining, coupled with the urgent need to improve operational efficiency and safety, has been a major driver for the adoption of autonomous and automated equipment⁴. The use of intelligent devices, such as LHDs, to replace traditional manual mining and transportation operations has become essential. This shift has not only resulted in significant improvements in the safety of underground mining but has also greatly enhanced operational efficiency. The advancement of unmanned mining technology has given rise to a crucial research area: the improved control of LHD devices to ensure precise tracking of a predetermined reference trajectory, thereby safeguarding mining operations. This study aims to further explore tracking performance.

The underground LHD is a distinctive articulated vehicle, comprising a front and rear body connected by a central articulation point and a swing ring, which together form a rigid structure⁵. Additionally, the LHD is a low-profile vehicle equipped with a front bucket, which consists of a front and rear section connected by a central articulation point, providing high maneuverability in narrow mine shafts⁶. This unique design necessitates specific angle restrictions when turning. As intelligent equipment continues to evolve, the use of LHDs in unmanned mining has become increasingly important. Research is increasingly focused on enhancing the unmanned capabilities of LHDs, particularly in improving their data collection ability for tracking reference paths and employing path-tracking control algorithms for trajectory tracking. These algorithms calculate the optimal cost based on predetermined reference paths. The development of such technologies is crucial for improving both the safety and efficiency of underground mining operations.

Autonomous operation is a crucial prerequisite for unmanned mining, and robust path-tracking control is key to the development of autonomous vehicles like LHDs⁷. The primary objective of path tracking is to ensure precise control of the vehicle along a designated trajectory, thereby maximizing tracking accuracy. Path tracking algorithms typically assume the availability of a pre-recorded path composed of reference points, derived from manual vehicle operation. Each reference point includes specific information, such as global coordinates, heading angle, and the desired vehicle speed associated with that point. The algorithm uses the vehicle’s state, determined by localization, to calculate control inputs that bring the LHD as close as possible to the reference point. However, in underground mining operations, complex tunnels, and harsh environmental conditions can complicate this process, posing a significant challenge to accurately tracking underground lane trajectories. Therefore, path-tracking control methods are critical for achieving accurate control. Although path tracking control techniques—such as pure tracking algorithms⁸, Stanley tracking algorithms⁹, PID control¹⁰, sliding mode control¹¹, linear quadratic regulator (LQR) control¹², and model predictive control (MPC) algorithms ^13–16—have been extensively studied, further research is still needed in the field of underground special engineering vehicle control. Among these, NMPC, an advanced vehicle model-based control algorithm, has emerged as a key research method in the domain of path tracking control¹⁷. However, most existing studies are still focused on ground-based vehicles, with only a limited number of articulated vehicle types (e.g., LHDs) being the subject of research. Nayl, a prominent scholar in this field, initiated research on articulated vehicles early on and proposed a switching model predictive control scheme, among other contributions^5,18,19. Other scholars have also conducted research on articulated vehicles, including agricultural and underground vehicles, though their research focuses on different directions. For instance, Wang proposed a synchronized path planning method for the front and rear axles, which improves path tracking accuracy in articulated vehicles using RL controllers²⁰. Mruillo et al. implemented an optimal control strategy for articulated vehicle speed and path tracking based on MPC, using the relationship between vehicle attitude and position as constraints. They optimized the articulated vehicle with the objectives of minimizing heading angular deviation and lateral tracking deviation²¹. Since then, Nayl et al²² proposed an improved SMC controller based on continuous sliding surfaces, derived from a nonlinear kinematic model of articulated vehicles and Lyapunov stability conditions, to achieve faster tracking while reducing jitter. Based on the hybrid Tunicate Swarm Algorithm (TSA) and Bacterial Foraging Algorithm (BFA), along with an improved kinematic model, Guan designed an adaptive PID tracking controller, named TSABFA-PID, for the LHD. This controller achieved excellent results in both simulation and field tests²³. With ongoing advancements in path tracking technology, NMPC has emerged as an increasingly prevalent and effective technique for controlling articulated vehicles. Bai et al. enhanced the existing MPC controller and proposed a new controller based on NMPC, which significantly improved path tracking accuracy²⁴. Li proposed an innovative path planning and tracking framework that significantly enhances tracking accuracy and steering smoothness in complex curvature scenarios, using batch notification trees and nonlinear model predictive control with pre-steering²⁵. Wisell proposed a nonlinear model-based predictive control approach to optimize braking and power regeneration in electric heavy-duty vehicles by calculating approximations of control invariant sets, while ensuring vehicle stability and safety under various tunnel conditions²⁶.

In summary, while significant progress has been made in developing novel path tracking methods like NMPC for articulated vehicles, further in-depth research on underground LHDs is still needed. These methods have not yet been fully considered in both academic studies and practical applications. Articulated LHDs are well-suited for narrow underground tunnels; however, they face unique control challenges, including complex path tracking due to incomplete constraints and articulated steering, dynamic response issues during large-angle turns, high-precision requirements in confined spaces, and perception and control errors in underground environments. To overcome these challenges, this paper proposes an NMPC control method that utilizes Particle Swarm Optimization (PSO) to adjust time-domain parameters. This method enables flexible tuning of prediction and control time-domain parameters based on varying tunnel geometries, thereby achieving high-precision and smooth path tracking.

The objective of this study is to enhance the NMPC control method by proposing a novel optimization strategy. In NMPC controllers, time-domain parameters, specifically the prediction horizon (Np) and the control horizon (Nc), significantly impact performance. Np defines the future time window over which the system’s state is predicted. A longer Np can improve control accuracy and stability but at the cost of increased computational demands, requiring a balance between accuracy and real-time performance. The control time domain specifies the length of the control input sequence in each optimization step. Typically, the control time domain is shorter than the prediction time domain to reduce computational load and ensure real-time performance. At the same time, its proper setting is crucial for maintaining control effectiveness and system stability. This strategy aims to improve the time-domain parameters of the NMPC by employing the PSO, which we have designated as PT-NMPC. Additionally, the strategy will establish segmented paths, aligned with the actual tunnel trajectories of the LHD—i.e., segmented local paths—to achieve greater adaptability and improved tracking performance.

This paper presents a novel NMPC control strategy (PT-NMPC) developed to enhance path tracking control for underground LHDs. The strategy optimizes both the prediction and control time domains in the NMPC controller using the PSO optimization algorithm. It undergoes multiple verifications of path tracking performance to identify the optimal parameters. These optimal time domain parameters are then incorporated into the controller for final validation. The results demonstrate the effectiveness of the proposed strategy in achieving precise path tracking.

The main contributions of this paper are as follows:

(1) To address the current challenges faced by LHDs using NMPC controllers, such as turning delays that lead to collisions in narrow tunnels, this paper investigates the kinematic model of articulated LHD bodies. Based on the developed kinematic model, mathematical relationships are derived between the vehicle’s reference point velocity, heading angular velocity, and turning angular velocity.

(2) This paper proposes a PT-NMPC strategy that incorporates the PSO algorithm to optimize the time-domain parameters of the NMPC controller, to enhance the tracking efficiency of various reference trajectories. By optimizing the controller’s control time-domain and prediction time-domain parameters, the LHD can achieve improved path tracking control performance, thereby contributing to the advancement of unmanned mining operations.

(3) The reference trajectories for the experiments were extracted from various sections of an actual mine tunnel, ensuring that the simulations realistically reflect operational conditions. This approach validates the practical relevance and effectiveness of the proposed strategy.

The remainder of the paper is structured as follows: "Design of NMPC controller" analyzes the design of the NMPC controller, including the construction of the kinematic model and related components, as well as stability analysis. "Methodology" introduces the PT-NMPC optimization strategy with segmented local paths setting. The next section presents an analysis of the simulation experiment results, comparing the path tracking efficacy of the proposed strategy with that of the traditional NMPC (T-NMPC) and alternative optimization algorithms, namely the Gray Wolf Optimization Algorithm (GWO) and the Whale Optimization Algorithm (WOA). The final section offers a comprehensive summary of the key findings and outlines directions for future research, based on a thorough data analysis and a discussion of the paper’s limitations.

Design of NMPC controller

This section mainly explains the design of the NMPC controller, including steering radius analysis and stability analysis.

Turning radius analysis

The use of articulated vehicles in underground mining is of significant importance. The front and rear bodies of these vehicles can deflect relative to one another, providing enhanced passing capacity, a smaller turning radius, and improved maneuverability²⁷. The steering system of the LHD is controlled by adjusting the angle of articulation between the front and rear axles. This adjustment is proportional to the stroke of the steering cylinder’s piston rod and is limited by its maximum stroke. The movement of the loader during steering consists of two primary components: first, a translational movement, in which the front and rear bodies remain in close proximity; and second, a rotational movement, where the bodies rotate around their articulation points. This combined motion allows the LHD to exhibit flexibility in its steering operations^28,29.

As shown in Fig. 2, O represents the instantaneous steering center of the LHD, while and denote the reference points of the front and rear bodies, respectively. The articulation angle is represented by , and and indicate the distances from the front and rear body reference points to the articulation point, respectively. represents the width of the LHD vehicle. The steering radius of the outer front wheel can be expressed as , as follows:

1

2

3

4

Fig. 2.

LHD turning radius (schematic illustrating the geometric definition of the turning radius.)

The same can be obtained for the rear wheel steering radius of the LHD :

5

6

7

8

To simplify the modeling of the LHD’s steering process, we divide its steering action into two main phases: steady-state steering and in-situ steering. During the steady-state steering phase, the articulation angle remains constant, and the longitudinal velocity of the LHD is non-zero. In contrast, during the in-situ steering phase, the longitudinal speed of the loader is reduced to zero, and only the articulation angle is adjusted. A simplified schematic diagram of the LHD during steady-state steering is shown in Fig. 3, where and represent the heading angles of the front and rear bodies of the LHD, respectively. The specific physical parameters of the front and rear bodies are detailed in Table 1.

Fig. 3.

Steady State Steering Diagram of the LHD. (Key parameters, including vehicle speed and angular velocity, are indicated.)

Table 1.

The physical meaning of parameters.

Parameters	Physical meaning
	The axle center of the front body
	The axle center of the rear body
	The distance between the point of articulation and the center of the front vehicle
	The distance between the point of articulation and the center of the rear vehicle
	The steering angle of the front vehicle
	The steering angle of the rear vehicle
	Articulation angle
	Vehicle speed
	Steering center

In determining the reference point that represents the state of the LHD, we follow the principle that the velocity at this reference point should align with the forward direction of the LHD. This approach ensures ease of calculation and analysis. Based on this principle, the front vehicle reference point, , is chosen as the reference point for the entire vehicle in this study. Therefore, in this paper, the velocity at this reference point, denoted as , is defined as the forward speed of the LHD.

9

The translational velocity of the reference point of the front vehicle is:

10

where and are the horizontal and vertical coordinates of .

To achieve stable steering, the rate of change of the front body heading angle is considered, assuming that the front and rear bodies are not deflected relative to each other.

11

The state of the front vehicle reference point during smooth steering of the LHD can be expressed in terms of , , and :

12

Figure 4 illustrates the in situ steering process of the loader. Since is the reference point, we can consider that the loader is stationary concerning the ground when it performs the in-situ steering; in this case, then we have.

13

Fig. 4.

In-situ steering diagram of LHD. (Visualization of the vehicle’s turning maneuver at a near-stationary state.)

The rate of change of the forward body heading angle is:

14

According to Eqs. (13) and (14), the attitude of the front axle of the LHD during steering can be expressed as :

15

Combining Eqs. (12) and (15) yields the state of the LHD during the steering :

16

where denotes the state of the vehicle, denotes the coordinate position of the vehicle, denotes the heading angle, denotes the articulation angle between the front and rear hulls, is the reference speed of the whole vehicle, is the steering rate between the two hulls, and and are the distances from the reference point of the front and rear hulls to the articulation point, respectively.

NMPC path tracking controller design

NMPC can explicitly handle constraints by incorporating them into future inputs, outputs, or state variables, based on model-based predictions of the system’s future dynamic behavior. These constraints can then be represented in nonlinear programming problems and solved online³⁰. The NMPC closed-loop system is illustrated in Fig. 5. The NMPC controller obtains the vehicle’s state and localization by fusing sensor data. Using a pre-provided reference path, the controller predicts the future state and calculates the tracking error relative to the reference path. The controller then determines an optimal set of control outputs by solving the optimization problem under constraints. The vehicle applies the first set of control outputs, after which a new state is measured, and subsequent control actions are executed.

Fig. 5.

NMPC closed-loop control flowchart (the diagram outlines the process flow for the path tracking control task).

Through the kinematic analysis of the LHD in "Turning radius analysis", the control variables for path tracking should be and . To ensure that the LHD can realize path tracking quickly and smoothly, the errors of the front and rear axles need to be considered comprehensively.

In this paper, an improved nonlinear model predictive control strategy (PT-NMPC) is proposed, which enhances the path tracking performance by optimizing both the prediction time domain and control time domain of the NMPC through the incorporation of an optimization algorithm.

Predictive model

In the field of model predictive control, selecting a model that is both accurate and computationally efficient is crucial. When choosing between a kinematic model and a dynamic model, it is essential to strike a balance between prediction accuracy and computational efficiency. The kinematic model serves as a predictive tool that can compute future states quickly, thereby improving the controller’s computational efficiency. However, its prediction accuracy may decrease at higher vehicle speeds. On the other hand, dynamic models, while offering more precise predictions, are often hindered by the need for greater computational resources and additional input parameters, such as vehicle acceleration and slip angle³¹. These parameters are crucial for accurate path tracking, yet not all of them can be easily obtained directly from sensors, with the slip angle being a prime example. Therefore, when designing the MPC controller, it is essential to strike the best balance between model accuracy and real-time performance, ensuring optimal path tracking under all driving conditions³².

In underground metal mines, unmanned vehicles are typically operated at low speeds, not exceeding 5 m/s, due to factors such as tunnel width, tunnel conditions, and guidelines outlined in the underground LHD operation manual. For the stability of LHD testing in this paper, a fixed speed of 1.5 m/s was selected. Bai’s Ph.D. dissertation emphasizes the critical role of LHDs in specific industries, such as underground mining, and examines the applicability of kinematic models in these contexts³³. At this velocity, the kinematic model of the LHD can predict future states with a reasonable degree of accuracy while maintaining real-time computation. As a result, the nonlinear kinematic model of the loader is used as the predictive model³⁴.

The kinematic model is defined within the global coordinate system, as shown in Fig. 6. In this paper, the front vehicle reference point is chosen as the reference point for the entire vehicle state. The derivation of the LHD’s kinematic model can be found in existing literature^24,35. The nonlinear state space system of LHD can be represented by Eq. (16).

Fig. 6.

Kinematic Model of LHD (the model highlights the geometric and velocity relationships between the front and rear axles.)

Assuming and , the system kinematics can be simply expressed as:

17

To facilitate the implementation of NMPC on a computer, an Eulerian integrator is used in this paper to discretize the continuous nonlinear system.

18

19

where is the time step of the discretization and is the increment of the control input, which is introduced to ensure that the subah constraints are kept within during the solution.

Controller design

The NMPC controller is designed to ensure that the vehicle accurately follows the desired trajectory. To achieve this, the error between the predicted state and the reference path must be minimized. Additionally, smooth control is required, meaning that the variation in control inputs should be minimized. To address the potential effects of nonlinearities on convex optimization problems, the Sequential Quadratic Programming (SQP) algorithm is introduced in our controller design. To solve the nonlinear optimization problem efficiently, this study incorporates the Sequential Quadratic Programming (SQP) algorithm into the controller. SQP is well-suited for this application due to its robustness in handling complex nonlinear constraints. Its core mechanism involves linearizing the objective function and constraints in each iteration, thereby constructing a quadratic programming subproblem. This iterative process ensures stable and accurate convergence to the global optimal solution, which is crucial for the NMPC controller’s performance. This approach allows the algorithm to construct a quadratic programming subproblem at every iteration, gradually converging towards the global optimal solution. The SQP algorithm maintains solution stability and accuracy, even in the presence of complex nonlinear dynamic behavior. For convex optimization problems, the SQP design ensures convergence to the global optimal solution, which is crucial for the performance of the NMPC controller. Thus, our NMPC controller ensures both the efficiency of the algorithm and precise control, optimizing system behavior.

Meanwhile, the objective function of the NMPC controller can be expressed as follows³²:

20

where denotes the predicted state of the vehicle based on the current measured state, is the state information of the reference path, is the prediction time domain, and is the control time domain. is the tracking error weight matrix and is the weight matrix of the control input increment, both of which are positive definite matrices.

To achieve higher tracking accuracy and improve the stability of the closed-loop system, a terminal cost is proposed based on the NMPC controller described above, as follows³⁶:

21

where is the weight matrix of the terminal cost.

In general, the path-following problem can be transformed into an MPC program, such as the following optimization problem³²:

22

s.t.

23

24

25

26

27

28

where and are articulation angle physical limits, , are longitudinal velocity limits, and , are steering rate limits.

After obtaining the optimal solution, a set of control outputs can be obtained as shown in Eq. (29):

29

Applying the first group to vehicles, it can be expressed as Eq. (30):

30

After applying the control inputs, the LHD feeds back its current state to the controller for correction. The NMPC then performs a new round of predictive control based on the updated reference path and the LHD’s new state.

Stability analysis

To rigorously prove the closed-loop system stability of the proposed NMPC controller, we employ the classical Lyapunov stability theory. The core idea is to construct a Lyapunov function and demonstrate that its value monotonically decreases over time, ultimately converging to a stable state, thereby ensuring the system’s convergence and stability.

First, we define the error between the system state and the reference state as . The control objective is to drive the tracking error to gradually stabilize at zero. Therefore, the objective function (22) can be reformulated in terms of the error :

31

The formula can be simplified to:

32

where, is referred to as stage cost. It penalizes the deviation between the vehicle state and the reference path at each prediction step, as well as the change in control input , to ensure smooth control. is referred to as the terminal cost, which specifically penalizes the state deviation at the end of the prediction time domain ).

To ensure stability, NMPC theory typically relies on the following two key assumptions:

Assumption 1 (Terminal Constraint and Terminal Cost): There exists a terminal region containing the origin, and a local feedback controller defined within this region. For all error states within , the following condition is satisfied:

(1) Positive Invariance: If the current state lies within , then under the action of the local controller , the next state will also remain within ; that is . This ensures that once the system enters this “safe zone,” it will not exit.

(2) Local Stability: The terminal cost within this region serves as a Lyapunov function. This implies that, under the influence of the local controller , the terminal cost decreases, and this decrease exceeds the stage cost at that step. Mathematically, this can be expressed as: 。

At each step of the online optimization, a terminal constraint is imposed, requiring the final state in the prediction horizon to lie within the terminal region:

33

This formula, denoted as (33), represents a hard constraint imposed on the optimization problem. It instructs the controller: “Regardless of how future control actions are planned, the predicted state error of the vehicle at the planned endpoint () must lie within the predefined stability region .” This is a crucial step in ensuring system stability.

We select the optimal value function of the optimization problem as a candidate Lyapunov function for the entire closed-loop system, denoted as .

Proof

At time , we solve the optimization problem and obtain the optimal control sequence . At this point, the value of the Lyapunov function is:

34

After the system applies the first control input , the actual state error at time is .

At time , to demonstrate that the Lyapunov function decreases, we construct a feasible but suboptimal control sequence . This sequence consists of the tail portion of the optimal solution at time , followed by the terminal controller. Since this constructed sequence satisfies all constraints, the true optimal value at time must be less than or equal to the cost associated with this suboptimal solution:

35

Expand expression (34) of :

36

Through a series of derivations and using the inequality in Assumption 1, it can be proven that has the following relationship with the part in parentheses in (36):

37

Here, since the stage cost is , and both and are positive definite matrices, as long as the tracking error is , it follows that must be greater than zero. Therefore, Eq. (37) shows that —that is, the Lyapunov function —is strictly monotonically decreasing.

According to Lyapunov stability theory, if there exists a positive definite and monotonically decreasing Lyapunov function, then the system is asymptotically stable at the equilibrium point. Our derivation shows that the optimal value function satisfies these conditions, thereby ensuring that the designed NMPC closed-loop control system is asymptotically stable. This implies that, during path-tracking tasks, the tracking error of the LHD vehicle will eventually converge to zero, thus guaranteeing the stability and reliability of the control system.

Problems with LHDs

The NMPC controller faces several challenges in the actual operation of the LHD, particularly during sharp turns along highly curvilinear paths. In such cases, the controller often fails to adjust the vehicle’s angle in time, leading to significant deviations from the reference trajectory and, consequently, potential collisions with the tunnels. These issues are especially evident when the NMPC controller is used to track the reference path of the LHD. Related studies have highlighted that the prediction time domain and the control time domain are two key factors in NMPC controllers, significantly influencing their performance^37,38. The prediction time domain refers to the future time horizon considered by the controller during the optimization process, while the control time domain represents the time horizon over which the controller applies the control inputs³⁹.

Nevertheless, research on LHD path tracking, particularly regarding the optimization of the NMPC time domain, is still underexplored. Designing an efficient NMPC controller is a challenging task, and optimization algorithms offer an effective and accurate solution for optimizing the time domain⁴⁰. In this paper, we propose improving the efficiency of LHD path tracking in underground tunnels by optimizing the prediction and control time domains of the NMPC controller. The PSO algorithm is employed to determine the optimal values of these parameters, enabling the LHD to achieve its path tracking objectives more effectively.

Methodology

In Eq. (20), the prediction time horizon determines the duration of the rolling optimization process, while the control time horizon influences the tracking performance and control responsiveness of the LHD. Consequently, the values of and significantly affect the LHD’s path tracking performance. In this study, to identify the optimal time-domain parameters, the entire path tracking process is initially optimized based on the system’s sampling frequency. Subsequently, PSO is employed to refine these parameters further. For comparison, the Grey Wolf Optimizer and the Whale Optimization Algorithm are also used in benchmark tests.

The LHD investigated in this study is a complex system with pronounced nonlinearity and multiple constraints. Classical linear controllers such as PID and LQR have inherent limitations in handling such challenges. Therefore, the core contribution of this paper lies in optimizing the NMPC framework itself. A comparison between our proposed approach and the baseline NMPC method would provide a more direct and fair validation of the improvements. For this reason, a benchmark comparison with classical controllers is not included in this paper.

Based on the actual dynamic performance and constraints of the system, the PSO optimization algorithm is chosen to adjust the prediction and control time domains in the NMPC controller, optimizing both control performance and computational efficiency, and thereby enhancing the control of the underground LHD. Additionally, this study integrates real tunnel data to select local features and conducts segmented experiments to validate the overall path tracking performance.

Introduction to optimization algorithms

PSO was first proposed by Kennedy and Eberhart (1995) and has been successfully applied to solve problems with nonlinearity, non-differentiability and multiple optimizations⁴¹. It is found that PSO can generate high-quality solutions with more stable and faster convergence properties and shorter computation times than other stochastic methods⁴². For a standard PSO at time , the update rate and position of the -th particle are represented by the following equations:

38

39

where and are vectors in multidimensional space. , denote the local optimal position and global optimal position, respectively. The particle inertia weights are denoted by . Particle cognitive acceleration and social acceleration are denoted by and , which are defined as positive constants. and are two random parameters in [0, 1].

The Grey Wolf Optimizer, introduced by Mirjalili et al., is one of the fastest-growing swarm intelligence (SI) algorithms, designed to simulate the hunting behavior of grey wolf packs in nature⁴³. It mimics the leadership hierarchy and hunting strategies of grey wolves by categorizing them into four types: alpha, beta, delta, and omega, representing different roles in the hierarchy. During the initial search phase, GWO emphasizes exploration with high intensity, while in the final phase, it shifts focus toward exploitation by gradually adjusting the positions of the three best leaders. This dynamic balance enables GWO to effectively solve a wide variety of optimization problems across diverse research domains⁴⁴. The Whale Optimization Algorithm, proposed by Seyedali Mirjalili and Andrew Lewis in 2016, is a meta-heuristic optimization algorithm inspired by the hunting behavior of humpback whales⁴⁵. It mimics the spiral hunting pattern of humpback whales as they search for prey. WOA has been widely applied in path tracking studies due to its effectiveness and versatility ⁴⁶.

PT-NMPC optimization strategy

Optimizing the prediction and control time domains in NMPC using the PSO algorithm is an effective method for enhancing the performance of the control system⁴⁷. The PSO algorithm can adjust the parameters of the NMPC, thereby improving the LHD’s path tracking performance, particularly in terms of efficient and accurate tracking of the reference path and timely turning. Moreover, the adaptive tuning capability of the PSO algorithm allows the NMPC controller to dynamically adjust its parameters based on real-time system conditions.

First, the PSO algorithm initializes a particle swarm, where each particle represents a candidate set of Np and Nc values. Next, the fitness of each particle is evaluated based on system performance metrics, such as tracking error and control cost, and both individual and global optimal solutions are recorded. In the iterative process, the PSO algorithm uses an inter-particle information-sharing mechanism to dynamically adjust particle positions and velocities to find the local optimum. Each particle undergoes a complete application to the LHD path tracking process, and based on the tracking results at each step, a final comparison is made to determine the locally optimal parameter values. The algorithm then iterates until the global optimal values are found, gradually approaching the optimal prediction and control time domain parameters. This approach enables the NMPC system to adapt to dynamic changes and environmental conditions, thereby enhancing tracking accuracy and system robustness.

The PSO algorithm also addresses nonlinear optimization problems in NMPC, including constraints, by incorporating the SQP algorithm. This improves the efficiency and robustness of solving nonconvex problems and ensures a globally optimal solution for convex problems⁴⁸. At each sampling period, the PSO-optimized NMPC controller solves constrained optimization problems that aim to minimize tracking errors and control input variations, while ensuring control smoothness. This approach enables the controller to update the control inputs in each iteration, responding dynamically to immediate changes in the system. In summary, the integration of the PSO algorithm into NMPC provides a flexible and effective method for optimizing both the prediction and control time domains, thereby significantly enhancing the performance and adaptability of the control system. The PSO application in the NMPC optimization process is illustrated in Fig. 7.

Fig. 7.

Flowchart of the proposed PT-NMPC strategy (the process illustrates the integration of the PSO algorithm for objective function optimization) .

Based on the flowchart, it is evident that this paper applies the PSO algorithm to each NMPC cycle, enabling the derivation of optimal time-domain parameters. In each iteration of the PSO, the current parameters of each particle are applied to the path tracking process of the LHD, and local optimal values are obtained based on the particle iteration loop. This process is repeated for each generation until the global optimal values are achieved. The optimal values of the prediction and control horizons ( and ) are then returned to the NMPC controller for the final path tracking process, resulting in improved optimization performance. The pseudo-code for this control strategy is provided below:

During the implementation of the PSO algorithm, careful selection of key parameters is crucial to ensure the algorithm’s effectiveness and adaptability. Specifically, the self-cognition coefficient () and the social cognition coefficient () play a pivotal role in guiding the particles toward the optimal solution. Based on extensive studies and experiments in the literature⁴¹, these coefficients are commonly set to 2. This choice has been shown to yield strong performance across a wide range of application scenarios and has been verified through numerous experiments, demonstrating its reliability and effectiveness.

Inertia weight () is another key parameter in PSO, playing a crucial role in balancing the exploration of the global solution with the exploitation of local information. The recommended range for inertia weight is between 0.4 and 0.95, which helps the algorithm maintain a balance between exploring the global optimum and utilizing previous search results efficiently⁴⁹. In this study, through a series of experiments, we determined that an inertia weight of 0.9 was optimal, as it demonstrated effectiveness in several test scenarios.

The number of particles and iterations in PSO are crucial factors that directly influence both the computational efficiency and the search capability of the algorithm⁵⁰. Based on prior experience and experimental validation, we determined that 30 particles strike an optimal balance, avoiding excessive computational burden while ensuring sufficient explorability. Consequently, this study adopts a configuration of 30 particles and 50 iterations, a strategy designed to maximize search space exploration while maintaining computational efficiency, ultimately aiming for optimal optimization results.

Additionally, considering the computational cost and other factors, this paper conducts a comparative analysis of various optimization algorithms, including GWO and WOA. Based on relevant literature and the analysis of experimental results, we employ more general parameters for these two algorithms. Specifically, for GWO, we set the number of population agents and iterations to 30 and 80, respectively, while for WOA, we set them to 50 and 80^51,52.

Through careful parameter tuning and extensive experimental analysis, the PSO algorithm is effectively integrated with NMPC, offering a robust solution to the control problem of complex systems. This parameter selection is grounded not only in theoretical analysis but also in practical testing, ensuring the algorithm’s robustness and efficiency in real-world applications.

Design of the fitness function

PSO is an efficient optimization technique that is particularly suitable for solving nonlinear and nonconvex optimization problems⁴⁸. In the PSO algorithm, the fitness function evaluates the fitness level of each particle and is typically defined according to the specific objective of the optimization problem. In this paper, we design the fitness function based primarily on the square of the deviation of the system state to quantify the degree of deviation. By considering the state variables and other relevant parameters during the operation of the LHD, along with the requirements of the objective function, we construct a suitable fitness function, as shown in Eq. (40):

40

where is the number of samples and the loop order.

Segmented local path setting

Segmented local path setting is a path segmentation method designed for complex tunnel environments. It divides the entire path into multiple local segments based on the specific conditions of the tunnel, significantly enhancing the effectiveness of the proposed strategy for LHD validation. This segmentation considers the local characteristics, structural complexity, and functional requirements of the tunnel, ensuring that each local path can be analyzed and optimized independently. For each local path, we establish multiple simulation reference paths, tailored to their unique length, curvature, and other characteristics. This approach enables the customization of parameter optimization for each local path, ultimately achieving the goal of comprehensive path planning.

Based on the map of an actual underground tunnel in a mine (e.g., Fig. 8), the path can be segmented into multiple sections according to local features such as curvature magnitude, allowing for more efficient simulation and verification tests. The tunnel can be divided into straight segments, combinations of straight lines and circular arcs, and continuous curves, which effectively test the tracking accuracy of the proposed strategy on these local paths.

Fig. 8.

Actual tunnel map of a mine (this environment includes diverse features such as curves and straightaways used to generate reference paths) .

Based on the above tunnel in Fig. 8, we divide the simulation path into five characteristic types, as shown in Fig. 9. Each simulation path can be approximated using the mathematical formulas (41-45). For paths with turns, an arc approximation is applied to smooth the transitions and improve the overall path quality.

Fig. 9.

Segmented Local Reference Paths. (Five distinct paths extracted from the tunnel map for controller performance evaluation.)

Segmented local path formulas can be expressed as follows³⁴:

Path 1:

41

Path 2:

42

Path 3:

43

Path 4:

44

Path 5:

45

The effectiveness of the proposed PT-NMPC can be tested by dividing the actual tunnel into the aforementioned multi-segment local paths, which are then used as simulation paths for the path-tracking control of the LHD.

Results analysis

This section presents an in-depth performance analysis of the PT-NMPC controller for LHD path tracking. To thoroughly evaluate the efficacy of PT-NMPC, we compare it with the traditional NMPC (TNMPC) and investigate the impact of integrating other advanced optimization algorithms into the NMPC framework for LHD path tracking, such as GWO and WOA. By analyzing several sets of experimental data, this paper aims to validate the effectiveness and superiority of the proposed PT-NMPC controller.

To more effectively assess the degree of data dispersion in the lateral deviation of the test results, the root mean square error is introduced as one of the metrics. RMSE serves as a valid measure of the difference between the predicted and actual values of the model, as shown in Eq. (46).

46

where is the number of samples and is the lateral deviation.

Tracking results for different strategy paths

In this paper, based on the simulation paths defined in "Segmented local path setting", the proposed strategy is employed to simulate the path tracking of the LHD. The path-tracking results for each method are illustrated in Figs. 10, 11, 12, 13, 14. To better visualize the changes in lateral errors during the tracking process, we have locally enlarged the path-tracking result graphs for each method.

Fig. 10.

Path Tracking Performance on Path 1. (Three strategies were used to test path 1 separately with the original controller, and the path tracking results are shown in the figure.)

Fig. 11.

Path 2 path tracking performance on path 2 (three strategies were used to test path 2 separately with the original controller, and the path tracking results are shown in the figure) .

Fig. 12.

Path tracking performance on path 3 (three strategies were used to test path 3 separately with the original controller, and the path tracking results are shown in the figure) .

Fig. 13.

Path tracking performance on path 4 (three strategies were used to test path 4 separately with the original controller, and the path tracking results are shown in the figure) .

Fig. 14.

Path tracking performance on path 5 (three strategies were used to test path 5 separately with the original controller, and the path tracking results are shown in the figure) .

The optimal time-domain parameter values determined for each path are summarized in Table 2.

Table 2.

Optimal time domain parameters for different paths for each strategy.

Control Strategies	Time Domain Parameters	Path1	Path2	Path3	Path4	Path5
TNMPC	Np	5	5	5	5	5
	Nc	50	50	50	50	50
GWO-NMPC	Np	53	54	44	51	40
	Nc	32	49	35	46	35
WOA-NMPC	Np	35	50	35	50	39
	Nc	35	49	35	44	35
PT-NMPC	Np	51	51	51	36	49
	Nc	47	36	35	36	35

To more comprehensively analyze the tracking data of the improvement strategy across each path, we also introduce the mean absolute deviation (MAD), which reflects the average distance between individual data points in the lateral deviation dataset and the mean of the dataset. Specifically, the mean absolute deviation indicates the average degree to which the values in the lateral deviation dataset differ from the mean, i.e., the extent to which the data points deviate from the mean. Since the mean absolute deviation is less sensitive to extreme values (outliers) — as it calculates absolute values rather than squared values — it provides a more robust characterization of the data. This is shown in Eq. (47):

47

Here, represents the lateral deviation at the -th sample point, and represents the average lateral deviation.

From Fig. 10 and Table 3, it is evident that for the straight-line segmented simulation path, the absence of excessive steering results in minimal changes to the heading angle. Consequently, there is negligible fluctuation in lateral deviation, allowing the LHD to travel more smoothly. Furthermore, the performance differences among the control strategies are relatively insignificant in this scenario.

Table 3.

Path1 path tracking lateral deviation analysis.

Controller	Parameter	Value	Unit
TNMPC	Max lateral deviation	0.0165	m
	Negative max lateral deviation	-0.0146	m
GWO-NMPC	Max lateral deviation	0.0165	m
	Negative max lateral deviation	-0.0158	m
WOA-NMPC	Max lateral deviation	0.0165	m
	Negative max lateral deviation	-0.0157	m
PT-NMPC	Max lateral deviation	0.0152	m
	Negative max lateral deviation	-0.0146	m

For reference path 2, the second half of the simulation path consists of a semicircular segment. This feature effectively highlights the control performance of the proposed strategy. In Table 4, the experimental path tracking results demonstrate that, compared to TNMPC, the proposed strategy achieves a significantly greater reduction in lateral deviation when using other optimization algorithms. Under PT-NMPC control, the maximum absolute lateral deviation is only 0.0464 m. Although WOA-NMPC and GWO-NMPC outperform PT-NMPC in terms of lateral deviation data, the difference is relatively small overall, and the proposed strategy still demonstrates significant improvement compared to TNMPC. Furthermore, when considering the tracking results across several other reference paths, the proposed strategy exhibits clear superiority.

Table 4.

Path2 path tracking lateral deviation analysis.

Controller	Parameter	Value	Unit
TNMPC	Max lateral deviation	0.0398	m
	Negative max lateral deviation	-0.3130	m
GWO-NMPC	Max lateral deviation	0.0479	m
	Negative max lateral deviation	-0.1478	m
WOA-NMPC	Max lateral deviation	0.0434	m
	Negative max lateral deviation	-0.0296	m
PT-NMPC	Max lateral deviation	0.0462	m
	Negative max lateral deviation	-0.0464	m

Path 3 introduces a steering section with a larger radius of curvature than paths 1 and 2, posing a significant control challenge. As illustrated in Fig. 12 and detailed in Table 5, the PT-NMPC strategy successfully limits the maximum absolute lateral deviation to only 0.1456 m. This result stands in stark contrast to the outcomes achieved by TNMPC and other optimization strategies, highlighting the better performance of PT-NMPC even under complex tunnel conditions.

Table 5.

Path3 path tracking lateral deviation analysis.

Controller	Parameter	Value	Unit
TNMPC	Max lateral deviation	0.3664	m
	Negative max lateral deviation	-0.2729	m
GWO-NMPC	Max lateral deviation	0.1630	m
	Negative max lateral deviation	-0.0665	m
WOA-NMPC	Max lateral deviation	0.2406	m
	Negative max lateral deviation	-0.1266	m
PT-NMPC	Max lateral deviation	0.1456	m
	Negative max lateral deviation	-0.0766	m

Simulation path 4 represents a continuous turning tunnel condition, making it an effective scenario for evaluating the timely response capability of each strategy. Under PT-NMPC control, the maximum absolute lateral deviation is 0.1292 m, which is a significant improvement compared to TNMPC tracking in Table 6. Moreover, the tracking results obtained using GWO and WOA optimization algorithms reveal that the lateral deviation under the WOA strategy is closer to that achieved using PSO. This further underscores the outstanding advantages of the PT-NMPC strategy.

Table 6.

Path 4 path tracking lateral deviation analysis.

Controller	Parameter	Value	Unit
TNMPC	Max lateral deviation	0.1863	m
	Negative max lateral deviation	-0.2489	m
GWO-NMPC	Max lateral deviation	0.1863	m
	Negative max lateral deviation	-0.2489	m
WOA-NMPC	Max lateral deviation	0.1356	m
	Negative max lateral deviation	-0.0934	m
PT-NMPC	Max lateral deviation	0.1292	m
	Negative max lateral deviation	-0.0661	m

The key difference between path 5 and path 4 lies in the opposing steering directions during the second turn, providing an effective scenario to evaluate the loader’s tracking performance under various strategies. This helps validate the accuracy and stability of the proposed strategy. Based on the tracking data analysis, it is evident that under the PT-NMPC strategy, the maximum absolute lateral deviation of the LHD is 0.1355 m, representing a significant improvement compared to 0.5186 m with TNMPC and 0.3028 m with WOA-NMPC in Table 7. This demonstrates that the LHD can promptly adjust to upcoming curves using the proposed strategy. Consequently, the optimal time-domain parameters under the PT-NMPC strategy effectively enhance tracking performance.

Table 7.

Path 5 path tracking lateral deviation analysis.

Controller	Parameter	Value	Unit
TNMPC	Max lateral deviation	0.1181	m
	Negative max lateral deviation	-0.5186	m
GWO-NMPC	Max lateral deviation	0.0502	m
	Negative max lateral deviation	-0.3504	m
WOA-NMPC	Max lateral deviation	0.0362	m
	Negative max lateral deviation	-0.3028	m
PT-NMPC	Max lateral deviation	0.0864	m
	Negative max lateral deviation	-0.1355	m

The path tracking effect diagram allows us to visualize the LHD’s tracking performance on the simulation path for each method. However, it may not provide a quantitative description of the specific results. Therefore, we processed the simulation data to analyze the tracking results of these methods under different paths. This analysis includes a comparison of the LHD’s lateral deviation during the tracking process, changes in the heading angle, changes in the articulation angle, and the rate of change of the articulation angle. The results for the five simulation paths are presented in Figs.15, 16, 17, 18, 19.

Fig. 15.

Comparison of Changes in Path 1 Indicators. (Analysis of key indicators, including lateral deviation and heading error, for each strategy.)

Fig. 16.

Comparison of Changes in Path 2 Indicators. (Analysis of key indicators, including lateral deviation and heading error, for each strategy.)

Fig. 17.

Comparison of Changes in Path 3 Indicators. (Analysis of key indicators, including lateral deviation and heading error, for each strategy.)

Fig. 18.

Comparison of Changes in Path 4 Indicators. (Analysis of key indicators, including lateral deviation and heading error, for each strategy.)

Fig. 19.

Comparison of Changes in Path 5 Indicators. (Analysis of key indicators, including lateral deviation and heading error, for each strategy.)

As shown in Fig. 15, under straight tunnel conditions, the LHD’s performance across different strategies exhibits minimal variation. In this scenario, LHD requires little to no steering adjustments, resulting in a stable heading angle and other related parameters.

Path 2 features a larger turn compared to the straight-line of path 1, making it effective for comparing the performance of various strategies. From Fig. 16(a), it is evident that WOA-NMPC demonstrates a clear advantage on this path in terms of lateral deviation changes. However, PT-NMPC also achieves smaller lateral deviation fluctuations compared to TNMPC. Figure 16(b) and (c) further show that the proposed strategy can adjust the heading angle and articulation angle in time before the corner, enabling smoother navigation through the turn. This highlights the effectiveness of the optimal time-domain parameters incorporated in the strategy.

Path 3 has a larger radius of curvature compared to path 2. However, as shown in the data result plots, the PT-NMPC strategy results in relatively smaller changes in lateral deviation compared to several other strategies, with a maximum absolute lateral deviation smaller than that of the other methods. For heading angles, the angles can be effectively adjusted by combining optimal time-domain parameters before steering.

Mining articulated vehicles, such as LHDs, use a hydraulic steering system in which the hydraulic cylinder drives the relative rotation of the vehicle’s front and rear articulated frames to achieve steering. The steering angle and speed are determined by factors such as hydraulic oil flow and cylinder size. When the hydraulic system flow rate is high, the cylinder response becomes more sensitive, and components like the steering control valve can quickly output sufficient fluid to facilitate the rotation of the articulated frame. This enables rapid adjustment of the steering angle and angular speed.

As shown in Fig. 17(d), the controller makes prompt adjustments to the articulated angular velocity and turning angle, enabling a timely steering response to upcoming curvatures and effectively preventing collisions.

Path 4 represents a continuous curve, making it suitable for testing the stability of the proposed optimization strategy. As shown in Fig. 18(a), the PT-NMPC and WOA-NMPC strategies exhibit similar performance, with their maximum absolute lateral deviations being nearly identical. From Fig. 18(b) and other indicators, it can be observed that the proposed strategy effectively adjusts the heading and turning angles in advance of steering. This ensures timely steering by the LHD, thereby avoiding potential collisions caused by steering delays.

The primary difference between path 5 and path 4 lies in the direction of the second turn. This tunnel condition is designed to test whether the LHD can respond to different turns promptly under the PT-NMPC optimization strategy, thereby avoiding collisions caused by steering delays. As shown in Fig. 19(d), the GWO strategy performs poorly under path 5 conditions. From the analysis of Fig. 19(a) and (c), the PT-NMPC strategy demonstrates superior performance, with a smaller maximum lateral deviation and timely corner adjustments. These results indicate that the LHD achieves better reference path tracking under this tunnel condition, showcasing high tracking accuracy and stability.

Across all tested paths, the PT-NMPC strategy consistently yields smaller lateral deviations compared to TNMPC, indicating enhanced stability and tracking accuracy. Notably, the maximum absolute lateral deviation is greatly reduced, enabling the LHD to track each reference path with higher precision. Similarly, the performance of GWO-NMPC and WOA-NMPC also shows improvement compared to TNMPC.

Equations (46) and (47) enable the calculation of individual tracking results, and the summarized results are presented in Table 8.

Table 8.

Summary of path tracking data calculation under each control strategy.

Control Strategies	Value	Path1	Path2	Path3	Path4	Path5
TNMPC	MAD	0.0093	0.1447	0.0928	0.0594	0.1523
	RMSE	0.0102	0.2074	0.1444	0.0864	0.2065
GWO-NMPC	Max absolute lateral deviation improvement	0.06%	84.70%	55.52%	25.14%	90.31%
	MAD	0.0098	0.0290	0.0467	0.0075	0.1021
	MAD Improved Efficiency	-5.26%	79.97%	49.71%	87.43%	32.97%
	RMSE	0.0105	0.0318	0.0600	0.0764	0.1418
	RMSE Improvement Efficiency	-3.03%	84.65%	58.44%	11.57%	31.32%
WOA-NMPC	Max absolute lateral deviation improvement	-0.06%	86.12%	34.34%	45.54%	42.62%
	MAD	0.0109	0.0131	0.0729	0.0358	0.0789
	MAD Improved Efficiency	-17.4%	90.94%	21.51%	39.83%	32.97%
	RMSE	0.0105	0.0242	0.0936	0.0494	0.1104
	RMSE Improvement Efficiency	-2.99%	88.33%	35.23%	42.86%	46.53%
PT-NMPC	Max absolute lateral deviation improvement	7.87%	85.19%	60.26%	48.10%	73.88%
	MAD	0.0090	0.0272	0.0458	0.0394	0.0419
	MAD Improved Efficiency	2.58%	81.21%	50.60%	33.64%	72.50%
	RMSE	0.0059	0.0308	0.0597	0.0491	0.0532
	RMSE Improvement Efficiency	42.01%	85.15%	58.64%	43.20%	74.23%

Based on the analysis of the data results, the improvements in the maximum absolute lateral deviation, average absolute deviation, and the changes in the RMSE for each strategy across different paths can be determined. Taking paths 2, 3, and 5 as examples, the data in Table 8 indicate that under the PT-NMPC strategy, the maximum absolute lateral deviation improves by 85.19%, 60.26%, and 73.88%, respectively, with MAD values of 0.0272, 0.0458, and 0.0419. This suggests that the absolute difference between each lateral deviation data point and its mean value is small, resulting in data points being more tightly clustered around the mean. The RMSE values are 0.0308, 0.0597, and 0.0532, showing improvements of 85.15%, 58.64%, and 74.23%, respectively. These results further indicate that the prediction results for the optimal time-domain parameters under the PT-NMPC strategy are closer to the actual outcomes.

Comparison of path tracking results with different time domain parameters

Based on the analysis of the results, PT-NMPC demonstrates superior performance compared to other strategies, achieving smaller lateral deviations in LHD path tracking across various scenarios. To ensure that the PT-NMPC strategy identifies optimal time-domain parameters for improved path tracking control of LHDs, this section presents an experimental comparison between TNMPC and PT-NMPC. Specifically, the optimal tracking results of PT-NMPC from "Tracking results for different strategy paths" are compared with the path tracking results of TNMPC using different initial time-domain parameters, thereby validating the proposed strategy and confirming the optimality of its time-domain parameters.

For the various initial time-domain parameter settings used by TNMPC, this section adopts an interval of 10, sequentially selecting values from 10 to 50. The specific time-domain parameter settings for each path are presented in Table 9.

Table 9.

TNMPC different initial time domain parameter settings.

Time-domain Parameter	1	2	3	4	5	6	7	8	9
Np	10	20	20	30	30	40	40	50	50
Nc	10	10	20	20	30	30	40	40	50

According to the different time-domain parameters set, the LHD is made to track different paths under the TNMPC policy, and the obtained path tracking results are compared with the PT-NMPC optimal path tracking results in Table 10, 11, 12, 13 and 14.

Table 10.

Comparison of TNMPC and PT-NMPC path 1 tracking results.

Parameter Indicators	Np	Nc	Max Absolute Lateral	MAD	RMSE
TNMPC	10	10	0.1196	0.0279	0.0906
	20	10	0.0349	0.0098	0.0232
	20	20	0.0349	0.0097	0.0231
	30	20	0.0223	0.0102	0.0138
	30	30	0.0222	0.0104	0.0137
	40	30	0.0181	0.0098	0.0114
	40	40	0.0181	0.0098	0.0114
	50	40	0.0169	0.0099	0.0107
	50	50	0.0169	0.0099	0.0107
PT-NMPC	51	47	0.0152	0.0090	0.0059

Table 11.

Comparison of TNMPC and PT-NMPC path 2 tracking results.

Parameter Indicators	Np	Nc	Max Absolute Lateral	MAD	RMSE
TNMPC	10	10	0.8735	0.2885	0.5254
	20	10	0.3613	0.1421	0.2234
	20	20	0.3596	0.1359	0.2194
	30	20	0.1790	0.0671	0.1130
	30	30	0.1780	0.0668	0.1125
	40	30	0.0876	0.0309	0.0531
	40	40	0.0874	0.0307	0.0530
	50	40	0.1472	0.0508	0.0863
	50	50	0.04639	0.02724	0.03080
PT-NMPC	51	36	0.04636	0.02719	0.03079

Table 12.

Comparison of TNMPC and PT-NMPC Path 3 tracking results.

Parameter Indicators	Np	Nc	Max Absolute Lateral	MAD	RMSE
TNMPC	10	10	2.6254	0.8891	1.3310
	20	10	1.3603	0.4067	0.6279
	20	20	1.4177	0.4194	0.6453
	30	20	0.8941	0.2425	0.3725
	30	30	0.9054	0.2459	0.3767
	40	30	0.5957	0.1411	0.2252
	40	40	0.5595	0.1309	0.2076
	50	40	0.4260	0.0962	0.1554
	50	50	0.4776	0.1083	0.1709
PT-NMPC	51	35	0.1456	0.0458	0.0597

Table 13.

Comparison of TNMPC and PT-NMPC Path 4 tracking results.

Parameter Indicators	Np	Nc	Max Absolute Lateral	MAD	RMSE
TNMPC	10	10	2.1742	0.7729	1.0388
	20	10	1.1402	0.3138	0.4493
	20	20	1.1489	0.3158	0.4529
	30	20	0.7124	0.1514	0.2339
	30	30	0.7097	0.1514	0.2339
	40	30	0.4528	0.0807	0.1304
	40	40	0.4638	0.0796	0.1296
	50	40	0.3254	0.0529	0.0905
	50	50	0.3414	0.0545	0.0941
PT-NMPC	36	36	0.1292	0.0394	0.0491

Table 14.

Comparison of TNMPC and PT-NMPC Path 5 tracking results.

Parameter Indicators	Np	Nc	Max Absolute Lateral	MAD	RMSE
TNMPC	10	10	1.8743	0.6350	0.9429
	20	10	0.8900	0.2850	0.4012
	20	20	0.8893	0.2840	0.3991
	30	20	0.4573	0.1345	0.1877
	30	30	0.4717	0.1407	0.1976
	40	30	0.2406	0.0687	0.0927
	40	40	0.2541	0.0723	0.0992
	50	40	0.2652	0.0907	0.1007
	50	50	0.1359	0.0423	0.0536
PT-NMPC	49	35	0.1355	0.0419	0.0532

As can be seen from Table 10, since the reference path 1 is a straight line, the role played by the different time-domain parameters is not obvious, and the comparisons are similar.

For path 2, the comparison results in Table 11 show that when TNMPC employs time-domain parameters of 50 and 50, the path tracking results are nearly identical to the PT-NMPC tracking results. Upon detailed analysis, the minimal differences in the data become indistinguishable when rounded to four decimal places, with the two metrics appearing identical. Therefore, five decimal places were retained for the comparison. The final result demonstrates that the time-domain parameters derived using the proposed strategy can be considered relatively optimal.

Table 12 shows that the maximum absolute lateral deviation under the proposed strategy is 0.1456 m, whereas the maximum absolute lateral deviations for TNMPC are all greater than 0.1456 m. This indicates that the path tracking performance of the proposed strategy is significantly better than that of TNMPC under various time-domain parameter settings.

Unlike path 3, the tracking results for path 4 reveal that, although the differences in performance metrics are not very pronounced, Table 13 indicates that the path tracking results under the PT-NMPC strategy are slightly better than those of TNMPC with various time-domain parameter settings. This suggests that PT-NMPC achieves optimal performance with the best time-domain parameters.

Our analysis of the path tracking data in Table 14 shows that the best absolute lateral deviation achieved by TNMPC under different time-domain parameter settings is 0.1359 m, with an RMSE value of 0.0536. In comparison, the maximum absolute lateral deviation of the proposed PT-NMPC strategy is 0.1355 m, which is slightly better than the TNMPC tracking results with time-domain parameters of 50 and 50. Although the difference is small, PT-NMPC still demonstrates a marginal improvement over TNMPC.

Figure 20 illustrates the lateral deviation resulting from path tracking of each trajectory by TNMPC with different time-domain parameters, compared to the lateral deviation of PT-NMPC under the optimal time-domain parameters. The PT-NMPC strategy yields the best overall performance. By analyzing Fig. 20 in conjunction with Tables 10, 11, 12, 13 and 14, it can be concluded that the time-domain parameters obtained under the PT-NMPC strategy are optimal, as they yield better tracking results for each path.

Fig. 20.

Influence of time-domain parameters on tracking performance (comparison of lateral deviation across all test paths to identify the optimal parameter configuration for the controller).

Based on the above analysis, it is evident that the tracking results of the proposed strategy, optimized with the best time-domain parameters for different reference trajectories, consistently outperform the path tracking results of TNMPC under various time-domain parameter settings. This confirms that the time-domain parameters listed in Table 2 represent the optimal values for different reference trajectories.

Conclusion

To improve the accuracy of the NMPC controller for path tracking of underground LHDs, this study proposes a PT-NMPC strategy that optimizes the time-domain parameters of the NMPC. The PSO is used to calculate the optimal time-domain parameters for the LHD under different tunnel conditions. Based on the path-tracking performance analysis and experimental results, the proposed PT-NMPC strategy shows substantial improvements over traditional NMPC and other tested optimization algorithms in controlling LHD path-tracking, particularly in scenarios involving sharp turns and continuous curves. These findings highlight the strategy’s applicability and effectiveness in enhancing control performance under complex underground conditions. However, the PT-NMPC control strategy proposed in this paper is primarily applicable to underground articulated engineering vehicles, such as LHDs and loaders, and thus has a strong specificity and inherent limitations. It is not suitable for non-articulated structures or high-speed platforms, such as ground passenger vehicles or tracked tanks.

(1) A kinematic model of the underground LHD is established, and multiple strategies are simulated across various paths with local characteristics of the actual tunnel. Compared with TNMPC, the maximum absolute lateral deviation of PT-NMPC is reduced by 0.0013 m, 0.2666 m, 0.2208 m, 0.1197 m and 0.3832 m, corresponding to improvements of 7.87%, 85.19%, 60.26%, 48.10 and 73.88%, respectively. Additionally, for paths 2, 3 and 5, PT-NMPC exhibits smaller RMSE values compared to TNMPC, GWO-NMPC and WOA-NMPC, with improvements of 85.15%, 58.64% and 74.23%, respectively. Moreover, the data plots show that the PT-NMPC strategy results in relatively smaller changes in lateral deviation compared to several other strategies, with a maximum absolute lateral deviation smaller than that of the other methods. This demonstrates that the time-domain optimization parameters of the proposed strategy can effectively enhance LHD travel accuracy. However, for path 2, WOA-NMPC performs slightly better. Overall, PT-NMPC significantly improves the control accuracy and stability of LHD in tracking the reference trajectory.

(2) Simulations conducted in MATLAB R2020a partially verify the effectiveness of the proposed strategy under real-world conditions, using segmented local paths based on the actual tunnel’s local features, such as straight-line paths. Taking path 3 as an example, the results show a 50.60% reduction in the MAD of PT-NMPC compared to TNMPC, while path 2 is reduced by 81.21%. Under the optimal time-domain parameters of the proposed strategy, the lateral deviation data points are more tightly clustered around the mean, highlighting its stability. The results of the multi-strategy, multi-path simulation tests show that the path tracking performance of the proposed PT-NMPC outperforms that of TNMPC.

(3) Several limitations of this study warrant acknowledgement and provide avenues for future research. First, the validation was performed entirely in simulations at a fixed low speed of 1.5 m/s. While this speed is representative of typical LHD operations, real-world performance may be influenced by unmodeled factors such as sensor noise and actuator delays. Second, the comparative analysis focused only on NMPC-based methods and did not include classical controllers such as PID or LQR. Finally, although the PSO algorithm proved effective, it introduces additional computational overhead that must be considered for real-time implementation on hardware with limited resources. These limitations point to clear directions for future research.

Future work

In this study, we propose a strategy (PT-NMPC) that employs the PSO algorithm to optimize the time-domain parameters of NMPC, combined with segmented local path planning to enhance the path tracking performance of underground loaders. Although simulation results validate the feasibility of this approach, significant room for improvement and extension remains. For instance, our experiments focused solely on NMPC-based controllers and did not include comparisons with other benchmarks, such as PID or LQR. Furthermore, challenges such as uncertainty in perception and localization, communication delays, physical characteristics of actuators, computational resource constraints, and real-time limitations persist. These factors may all impact actual control performance. Future research may focus on the following directions:

(1) Development of adaptive tuning mechanisms: Given the dynamic and uncertain nature of underground mining environments, it is crucial to explore adaptive control parameter tuning mechanisms. These would allow the system to automatically adjust control parameters in response to real-time feedback and environmental changes, thereby ensuring optimal operational performance.

(2) Algorithmic optimization for improved efficiency: Although the PT-NMPC controller demonstrated strong performance in simulations, computational efficiency and real-time responsiveness are vital for real-world applications. Future work should aim to optimize the algorithm and simplify the code to reduce computational overhead and improve the system’s real-time responsiveness.

(3) Experimental Comparative Study: While this paper demonstrates the superiority of the PT-NMPC control strategy through comparisons with the baseline NMPC controller, future research should extend these comparisons to other traditional control methods (e.g., PID, LQR) as well as learning-based approaches, such as deep learning and reinforcement learning control strategies. Such comparative studies will help further validate the advantages and limitations of the proposed method, providing valuable insights for selecting the optimal control strategy in practical applications.

(4) Robustness and Sensitivity Analysis of Control Systems: The simulations in this study were performed under relatively ideal conditions. In practical applications, however, non-ideal factors such as sensor noise, actuator delays (particularly response lag in hydraulic systems), and communication delays can significantly affect control performance. To address this, a comprehensive sensitivity analysis can be conducted to quantitatively evaluate the impact of these uncertainties on path-tracking accuracy. The insights gained from such analysis will further inform the design of more robust controllers, thereby ensuring system reliability and stability under complex real-world operating conditions.

Author contributions

Ya Liu: Writing – original draft, Investigation, Resources, Validation. Pingan Peng: Writing—review & editing, Conceptualization. Liguan Wang: Writing—review & editing. Jiaxi Wu: Writing—review & editing, Validation, Conceptualization. Ru Lei: Writing—review & editing, Conceptualization. Mingyu Lei: Writing—review. Chaowei Zhang: Writing—review & editing. Yongqing Zhang: Writing—review & editing.

Funding

This work was supported by the National Natural Science Foundation of China under Grant 52374168, the Science and Technology Innovation Program of Hunan Province under Grant 2023RC3069, and the National Key Research and Development Program of China under grants 2022YFC2904105 and 2023YFC2907403.

Data availability

The data used and/or analyzed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.