Research on Path Tracking of Intelligent Hybrid Articulated Tractor Based on Corrected Model Predictive Control

Abstract

To improve the path tracking performance of intelligent hybrid articulated tractors in all working conditions in unmanned operation, a path-tracking control method based on corrected model predictive control is proposed. The kinematic model of the tractor is established by analyzing the tractor’s kinematics. Taking the lateral and longitudinal errors as the target and the speed and articulation angular acceleration as the constraints, a tracking control algorithm based on model predictive control is proposed. In addition, to improve the transient performance of the tractor in the path tracking process, the proportional-integral-derivative controller and fuzzy controller are used to correct the model-predicted output articulation angular acceleration, forming a corrected model predictive control path tracking control method. To verify the effectiveness of the control method, model predictive control is used as a comparison method, and the effectiveness of the proposed method is verified based on the MATLAB 2024a simulation platform. The results show that compared with the MPC algorithm, the speed standard deviation is reduced by 2%, the longitudinal tracking error is reduced by 8%, and the lateral tracking error is reduced by 50%. The proposed method can effectively improve the path-tracking accuracy of the intelligent hybrid tractor.

1. Introduction

With the aging of the global population and the deepening of sustainable development, it is imperative to ensure the improvement of agricultural production efficiency under the conditions of agricultural labor shortage and energy conservation and environmental protection [1,2,3]. As an important component of agricultural machinery, the intelligence and greenness of tractors can improve agricultural production efficiency and reduce air pollution emissions [4,5]. Path tracking is a key component of tractor intelligence, and its tracking effect will directly affect the tractor’s operating efficiency. Hilly farmland has the characteristics of a small single-plot arable land area and irregular shape. Traditional tractors cannot adapt to hilly farmland due to their large turning radius and poor terrain adaptability. Compared with traditional tractors, intelligent hybrid articulated tractors have a small turning radius and strong terrain adaptability [6,7,8]. Therefore, it is of great significance to carry out research on path-tracking of intelligent hybrid-articulated tractors.

Path-tracking and path-planning are important components in the field of intelligent driving, but the accuracy of path tracking directly affects the operating efficiency of intelligent hybrid articulated tractors [9,10]. Current path-tracking research can be divided into three categories based on differences in algorithm theory: classical control theory, intelligent control theory, and modern control theory. The main methods of classical control theory in path tracking include: proportional-integral-derivative (PID), feedforward control, etc. Xu et al. proposed a real-time path-tracking control method for mobile robots based on Backstepping time-varying feedback and PID control. By introducing a restricted strategy, it achieved good control results, but it cannot solve the problem caused by the change in path curvature during path-tracking [11]. To this end, Ridley et al. used a state feedback control method to track the path of a load haul dump vehicle and verified the effectiveness of the control strategy, but there was a problem of large fluctuations in the speed curve during the path tracking process [12]. In order to deal with the problem of large speed curve fluctuations, Cheng et al. designed a model-free adaptive predictive path-tracking controller based on the model-free adaptive predictive controller PID method. This method has high adaptability under different paths, but requires a large amount of experimental data to determine the optimal control parameters [13]. Zhao Xuan et al. proposed a feedback linearization control path tracking control strategy, which improved the stability of the articulated dump truck by linearizing the nonlinear error equation. This method can achieve path tracking without an accurate model, but it will still produce large errors at turns [14]. Classical control theory has good control effects on linear systems. However, when faced with complex systems with multiple constraints, problems such as instability and overshoot may occur.

The main methods of intelligent control theory in path tracking include fuzzy control, reinforcement learning, and neural networks. Huang et al. proposed a control method based on fuzzy sliding mode control. By fuzzifying the control parameters, the inherent chattering problem of sliding mode control was effectively suppressed, and the straight-line path tracking stability of agricultural traction vehicles was improved. However, it cannot solve the problems caused by uneven road surfaces and side winds during path-tracking [15]. For complex disturbance scenarios, Hu Jie et al. integrated sliding mode control with the linear quadratic regulator (LQR) algorithm and coordinated the two algorithms through fuzzy control, thereby improving the accuracy of path tracking. However, they did not solve the problem of the impact of slip rate on tracking accuracy [16]. To this end, Zhao et al. proposed a fuzzy control method considering skid and slip, which improved the tracking accuracy. However, this control method lacks a compensation mechanism for path deviation under lateral interference [17]. An event-triggered adaptive path following the control strategy proposed by Yang Yuhang et al. reduces the tracking deviation caused by tire sideslip, but it cannot effectively guarantee the tracking accuracy when the load changes drastically [18]. Based on this, Wang et al. proposed a path-tracking control strategy based on a nonlinear disturbance observer, which significantly improved the tracking accuracy of four-wheel independent driving and four-wheel independent steering in ridge farming mode, However, there is still room for improvement in adaptability to changing road curvatures [19]. For scenes such as orchards with narrow roads and large curvature changes, Liu et al. proposed a path-tracking control algorithm based on the Virtual Radar Model. This algorithm uses a virtual radar map to describe the positional relationship between the vehicle and the path, generates control instructions through a deep neural network, and controls the crawler tractor. This enhances the navigation accuracy of the crawler tractor, but there is still a significant error in the variable curvature path [20]. The Pure Pursuit path-tracking control method based on B-spline optimization proposed by Zhang Wenyu improves the tracking performance of agricultural machinery on variable curvature paths but is susceptible to steady-state errors caused by system disturbances [21]. Wu et al. designed an adaptive slip compensation controller based on fault-tolerant control, achieving accurate compensation for the velocity error caused by wheel slippage, and it exhibits stronger robustness and higher tracking accuracy compared to conventional controllers [22]. Intelligent control theory has good robustness to uncertainties in the system, but in a dynamic environment, it has high requirements for training and computing resources, which will lead to insufficient real-time performance.

The main methods of modern control theory in path tracking include LQR, model predictive control (MPC), robust control, etc. Zhou et al. proposed a lateral control strategy of path tracking based on a linear quadratic regulator. The feed forward control was added to improve the control accuracy of path-tracking, which improved the lateral control accuracy during path tracking. However, the error problem under high-speed conditions has not been solved [23]. To solve this problem, Meng Yu et al. proposed an LQR strategy based on predictive information, which effectively improved the tracking accuracy of articulated vehicles [24,25]. Tang et al. introduced a lateral-longitudinal coordination path tracking control method based on MPC, improving vehicle stability and path-tracking accuracy across various roads. However, the potential understeering issue in autonomous vehicles was not addressed [26]. To address the need for steering compensation, Hu et al. proposed a hierarchical control method based on Robust Model Predictive Control. This method, which considers external disturbances and understeering in trucks, established a two-layer control structure that effectively improved steering control accuracy. However, it suffers from poor real-time performance [27]. To solve this problem, Bai et al. introduced feedforward steering angle information based on an inverse kinematics model and proposed a Feedforward MPC method, significantly improving robot control accuracy and reducing computational time. However, vehicle dynamic response characteristics were not fully considered [28]. To improve accuracy, Zhang et al. built a Multilayer MPC framework for speed control path tracking, combining the advantages of Feedforward MPC and Nonlinear MPC, achieving high path tracking accuracy even at higher average longitudinal speeds. However, further optimization of path-tracking accuracy is still needed [29]. To overcome the limitations of model dependency, Han et al. combined Gaussian process regression to establish an error-fitting model. They used this error correction model as the prediction model, designed a path-tracking cost function, and formulated a quadratic programming optimization problem, proposing a learning-based model predictive path-tracking control method. This reduced the average path-tracking error, but dynamic parameter adaptation for different vehicle models is still required [30]. Qin et al., based on the principle of Extended Kalman Filtering, performed learning and updating on the parameters of unmodeled dynamics, improving the tracking accuracy of intelligent vehicles in the presence of external disturbances. However, control lag remains an issue in high-speed, large-curvature scenarios [31]. To address this issue, He et al. proposed an adaptive prediction horizon control strategy based on optimized dynamic models, effectively improving prediction accuracy for vehicles in large-curvature conditions [32]. In terms of robust control, Chang Sheng et al. proposed a robust control method based on H∞ Ioop shaping. By processing the vehicle dynamics model, the vehicle’s trajectory tracking capability on a variable curvature path is enhanced, but the adaptability to roll conditions is not mentioned [33]. Dong et al. proposed a robust multi-dimensional switched H∞ path tracking control method. This method can ensure the tracking accuracy and roll stability of the vehicle at high speed and large turns [34]. Modern control theory has more advantages than classical control theory in dealing with complex systems with multiple constraints. When performing precise control on known models, it has higher real-time performance than intelligent control methods.

In summary, in order to help understand the characteristics of these control theories more intuitively, Table 1 lists the advantages and disadvantages of several common control methods mentioned above. Intelligent control methods can effectively deal with complex systems with multiple constraints, but are affected by data training time, model update frequency, etc. Modern control methods can not only deal with complex systems with multiple constraints but also have good real-time performance. The intelligent hybrid articulated tractor is a multi-constrained system. In order to improve the path-tracking performance of the intelligent hybrid articulated tractor, this paper adopts MPC combined with fuzzy control and PID control and proposes a path tracking control method based on corrected model predictive control (CMPC). This control method, Compared with MPC, integrates the advantages of PID’s rapid response to small-range errors and the strong robustness of fuzzy control, and can quickly correct the tractor’s articulation angular acceleration, thereby improving tracking accuracy. The specific contributions are as follows:

(1)

Kinematic analysis is conducted to obtain motion state parameters during path tracking, and a kinematic model that reflects articulated steering characteristics is established.

(2)

To improve the path tracking accuracy of intelligent hybrid articulated tractors under all working conditions, this paper uses a PID controller and fuzzy controller to correct the model prediction output articulation angular acceleration based on MPC and proposes a CMPC method to solve the problem of low tracking accuracy of linear MPC-processed tractors in the process of U-shaped variable curvature trajectory tracking.

Table 1. Comparison of control methods.

Control Methods	Advantage	Shortcoming
PID	Responds quickly to small errors, enabling efficient control in simple systems.	Limited performance in large error ranges and complex dynamic systems, struggling with nonlinear characteristics.
Feedforward control	Can proactively respond to known external disturbances or system model variations	Unable to handle unknown disturbances and requires high precision in system modeling.
Fuzzy control	Exhibits strong robustness in uncertain and noisy environments	Control rule design relies on expert knowledge, making parameter tuning and optimization challenging.
Reinforcement Learning	Adapts to complex dynamic environments, optimizing control strategies	Requires extensive training time and large amounts of interaction data.
Neural networks.	Suitable for systems where precise mathematical models are difficult to establish, enabling effective control through training.	Difficult to interpret system behavior, making debugging and optimization complex.
LQR	Provides good stability and optimized performance for known linear system models.	Struggles with nonlinear systems and large disturbances, requiring high model accuracy.
MPC	Effectively handles complex constraints, making it suitable for multi-variable optimization control.	High computational complexity, requiring real-time optimization of control inputs.
Robust control	Ensures system stability and reliability in uncertain environments, making it ideal for complex systems.	Complex design and analysis process, requiring in-depth modeling to handle uncertainties and system variations.

Table 1. Comparison of control methods.

Control Methods	Advantage	Shortcoming
PID	Responds quickly to small errors, enabling efficient control in simple systems.	Limited performance in large error ranges and complex dynamic systems, struggling with nonlinear characteristics.
Feedforward control	Can proactively respond to known external disturbances or system model variations	Unable to handle unknown disturbances and requires high precision in system modeling.
Fuzzy control	Exhibits strong robustness in uncertain and noisy environments	Control rule design relies on expert knowledge, making parameter tuning and optimization challenging.
Reinforcement Learning	Adapts to complex dynamic environments, optimizing control strategies	Requires extensive training time and large amounts of interaction data.
Neural networks.	Suitable for systems where precise mathematical models are difficult to establish, enabling effective control through training.	Difficult to interpret system behavior, making debugging and optimization complex.
LQR	Provides good stability and optimized performance for known linear system models.	Struggles with nonlinear systems and large disturbances, requiring high model accuracy.
MPC	Effectively handles complex constraints, making it suitable for multi-variable optimization control.	High computational complexity, requiring real-time optimization of control inputs.
Robust control	Ensures system stability and reliability in uncertain environments, making it ideal for complex systems.	Complex design and analysis process, requiring in-depth modeling to handle uncertainties and system variations.

The remainder of the paper is organized as follows: Section 2 introduces the kinematic model of the tractor, Section 3 presents the improved MPC path tracking control method, Section 4 validates the proposed method through simulations, and Section 5 summarizes the paper.

2. Intelligent Hybrid Articulated Tractor Kinematic Model

The intelligent hybrid articulated tractor model is divided into a kinematic model and a dynamic model. The vehicle kinematic model is usually used to analyze the relationship between variables such as speed, position, acceleration, etc., during the vehicle’s motion. The vehicle dynamics model is usually used in high-speed driving situations. Since the tractor travels at a slow speed, the kinematic model is used to design the path tracking controller to ensure the reliability of the system.

As shown in Figure 1, during straight-line motion, there is no relative displacement between the front and rear sections of the intelligent hybrid articulated tractor. In the steering process, the hydraulic cylinders on both sides of point J extend and retract, which causes the front and rear sections of the vehicle to form a certain bending angle. This action, in turn, completes the steering process under the influence of the tractor’s driving force. The midpoint of the front axle, P_f (X_f, Y_f), is selected as the reference point for the tractor’s state, and the kinematic equations for the unmanned articulated tractor are derived through kinematic analysis [35].

$$\left\{\begin{array}{l}{\stackrel{·}{x}}_f={\nu }_f\cos {\theta }_f\hfill \\ {\stackrel{·}{y}}_f={\nu }_f\sin {\theta }_f\hfill \\ {\stackrel{·}{\theta }}_f=-{\displaystyle \frac{{\nu }_f\sin \gamma +l_r\stackrel{·}{\gamma }}{l_f\cos \gamma +l_r}}\hfill \\ \stackrel{·}{\gamma }={\omega }_{\gamma }\hfill \end{array}\right.$$

(1)

In Equation (1), θ_f is the heading angle of the front axle center relative to the X-axis, γ is the angle between the front and rear vehicle body centerlines, F(x_f, y_f) is the front axle center, L_f is the distance between the hinge point J and the front roller center, and v_f is the front axle speed. For the rear body, θ_r is the heading angle of the rear body relative to the X-axis, G(x_r,y_r) is the rear axle center, L_r is the distance from the hinge point J to the rear axle center, and v_r is the speed of the rear body.

By converting Equation (1) into matrix form, the kinematic model of the unmanned articulated tractor can be obtained:

$$\left[\begin{array}{c}{\stackrel{·}{x}}_f\\ {\stackrel{·}{y}}_f\\ {\stackrel{·}{\theta }}_f\\ \stackrel{·}{\gamma }\end{array}\right]=\left[\begin{array}{c}\cos {\theta }_f\\ \sin {\theta }_f\\ -{\displaystyle \frac{\sin \gamma }{l_f\cos \gamma +l_r}}\\ 0\end{array}\right]v+\left[\begin{array}{c}0\\ 0\\ {\displaystyle \frac{l_r}{l_f\cos \gamma +l_r}}\\ 1\end{array}\right]\stackrel{·}{\gamma }$$

(2)

3. Intelligent Hybrid Articulated Tractor Control Strategy Design

3.1. Path Tracking Control System Based on Model Predictive Control

MPC is a commonly used optimization control algorithm that effectively handles multi-constrained problems and possesses good robustness. It is widely applied in path tracking. Compared to other algorithms, the MPC based on a linear time-varying model offers high computational efficiency, good real-time performance, and strong stability. Therefore, this paper adopts the linear time-varying model predictive control algorithm.

As shown in Equation (2), the state variables of the control system include the coordinates of the front axle center, the tractor’s heading angle, and the articulation angle of the tractor. The control variables include the front axle speed and the front wheel steering angle, leading to the general form of the control system [36].

$$\stackrel{·}{\chi }=f\left(\chi ,u\right)$$

(3)

Among them,

$$\chi ={\left[x,y,\theta ,\gamma \right]}^T u={\left[v,\omega \right]}^T$$

For a given reference trajectory, the general form of the reference trajectory can be obtained:

$${\stackrel{·}{\chi }}_f=f\left({\chi }_f,u_f\right)$$

(4)

Among them,

In the equation, x_f is the tractor’s front axle center reference x-coordinate, y_f is the tractor’s front axle center reference y-coordinate, θ_f is the tractor’s front axle reference heading angle, v_f is the tractor’s front axle center reference speed, ω_f is the tractor’s reference articulation angle angular velocity, γ_f is the tractor’s reference articulation angle.

Equation (5) is expanded using a Taylor series based on the reference point position, neglecting higher-order terms, and is then discretized [37]:

$$\tilde{\chi }\left(k+1\right)=A_{k,t}\tilde{\chi }\left(k\right)+B_{k,t}\tilde{u}\left(k\right)$$

(5)

Among them,

In the equation, Ts is the Sampling time.

By converting the control variables into control increments and incorporating a relaxation factor, the new discretized state-space representation can be derived:

$$\left\{\begin{array}{l}\xi \left(k+1 | t\right)={\tilde{A}}_{k,t}\xi \left(k | t\right)+{\tilde{B}}_{k,t}\Delta U\left(k | t\right)\\ \eta \left(k | t\right)={\tilde{C}}_{k,t}\xi \left(k | t\right)\end{array}\right.$$

(6)

Among them,

In the equation, N_x is the State variable dimension, N_u is the Control variable dimension.

To simplify the calculations, the following assumptions are made:

$$\left\{\begin{array}{l}{A}_{k,t}=A_{t,t},k=1,2,\cdots ,t+N-1\\ B_{k,t}=B_{t,t},k=1,2,\cdots ,t+N-1\end{array}\right.$$

(7)

Based on Equations (6) and (7), the predicted output expression of the system can be obtained:

$$Y\left(t\right)={\psi }_t\xi \left(t | t\right)+{\Theta }_t\Delta U\left(t\right)$$

(8)

Among them,

$$\begin{array}{c}Y\left(t\right)=\left[\begin{array}{c}\eta \left(t+1 | t\right)\\ \eta \left(t+2 | t\right)\\ \cdots \\ \eta \left(t+N_c | t\right)\\ \cdots \\ \eta \left(t+N_p | t\right)\end{array}\right] {\psi }_t=\left[\begin{array}{c}{\tilde{C}}_{t,t}{\tilde{A}}_{t,t}\\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^2\\ \cdots \\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^{N_c}\\ \cdots \\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^{N_p}\end{array}\right]\\ {\Theta }_t=\left[\begin{array}{cccc}{\tilde{C}}_{t,t}{\tilde{B}}_{t,t}& 0& 0& 0\\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}{\tilde{B}}_{t,t}& {\tilde{C}}_{t,t}{\tilde{B}}_{t,t}& 0& 0\\ \cdots & \cdots & \ddots & \cdots \\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^{N_c-1}{\tilde{B}}_{t,t}& {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^{N_c-2}{\tilde{B}}_{t,t}& \cdots & {\tilde{C}}_{t,t}{\tilde{B}}_{t,t}\\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^{N_c}{\tilde{B}}_{t,t}& {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^{N_c-1}{\tilde{B}}_{t,t}& \cdots & {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}{\tilde{B}}_{t,t}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^{N_p-1}{\tilde{B}}_{t,t}& {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^{N_p-2}{\tilde{B}}_{t,t}& \cdots & {\tilde{C}}_{t,t}{\tilde{A}}_{t,t}^{N_p-N_c-1}{\tilde{B}}_{t,t}\end{array}\right]\end{array}$$

$$\Delta U\left(t\right)=\left[\begin{array}{c}\Delta u\left(t | t\right)\\ \Delta u\left(t+1 | t\right)\\ \cdots \\ \Delta u\left(t+N_c | t\right)\end{array}\right]$$

(9)

In the equation: N_p is the Prediction horizon; N_c is the Control horizon.

For the optimization problem involving the system’s state and control variables, the following objective function is derived:

$$J\left(k\right)={\displaystyle \sum _{i=1}^{N_p}\left|\right|\eta \left(k+i | t\right)-{\eta }_{ref}\left(k+i | t\right)|{|}_Q^2+}{\displaystyle \sum _{i=1}^{N_c}\left|\right|\Delta U\left(k+i | t\right)|{|}_R^2+\rho {\epsilon }^2}$$

(10)

In the equation: Q is the weight matrix for the state variables, R is the weight matrix for the control variables, ρ is the weight coefficient, ε is the relaxation factor.

3.2. Corrected Model Predictive Control Method

The tractor is in a steady-state condition during the path tracking process of straight driving, while the system is in a transient condition during the path tracking process of turning. In order to improve the path tracking accuracy of the tractor in all working conditions during the path tracking process, a correction control method is proposed. The correction control method includes PID and Fuzzy, where PID is used to output the articulation angular acceleration δ₁; Fuzzy is used to provide the weighting coefficient of PID. The error between the predicted position and the actual position of the tractor is transmitted to the PID controller and the fuzzy controller. The fuzzy controller corrects the articulation angular acceleration of the prediction model and the articulation angular acceleration output by PID, thereby improving the accuracy of path tracking.

PID control is a common feedback control system that adjusts the parameters of the three parts, namely the proportional, integral, and differential, to make the system output as close to the set value as possible. As shown in Figure 2, the proportional part can adjust the output according to the current error. The integral part can adjust the output according to the accumulation of historical errors to eliminate steady-state errors. The differential part predicts possible future errors based on the error change rate so as to make adjustments in advance. Through the combination of these three parts, the system output can be adjusted in real time to ensure that the system has good response speed and stability in a dynamic environment. The specific calculation formula of PID is as follows:

$$u(t)=K_{p}e(t)+K_i{\int }_0^{t}e(\tau )d\tau +K_d{\displaystyle \frac{de(t)}{dt}}$$

(11)

In the process of path tracking of the intelligent hybrid articulated tractor, the error between the actual state variable and the expected state variable is calculated and input into the PID controller. The PID controller outputs an articulation angular acceleration δ₁ according to the size of the error. When the articulation angular acceleration output by PID control is simply coupled with the articulation angular acceleration output by MPC; a larger error may be generated to affect the tracking accuracy. Therefore, fuzzy control is introduced to dynamically adjust the combined relationship between PID and MPC for articulation angular acceleration.

As shown in Figure 3 and Figure 4, fuzzy control does not rely on an accurate model but rather achieves control by fuzzifying the input variables and then using fuzzy rules and reasoning. In the process of path tracking, the corresponding fuzzy control rules are formulated by calculating the error between the horizontal coordinate of the expected front axle center position and the horizontal coordinate of the actual front axle position, the error between the vertical coordinate of the expected front axle center position and the vertical coordinate of the actual front axle position, the error between the expected front wheel heading angle and the actual front wheel heading angle, and the error between the expected articulation angle and the actual articulation angle. When the error is input into the fuzzy control system, the fuzzy control system outputs a PID articulation angular velocity weighting coefficient a according to the fuzzy rule, which is used to adjust the articulation angular acceleration δ₁ output by the PID control system. Correspondingly, the system also outputs an MPC articulation angular velocity weighting coefficient 1−a, which is used to adjust the articulation angular acceleration δ₂ output by the MPC system to achieve the weight distribution of different control strategies for the articulation angular acceleration. In the process of path tracking, by formulating reasonable fuzzy rules, the intelligent hybrid articulated tractor can maintain a high accuracy during the tracking process, thereby improving the operating efficiency.

As shown in Figure 5, the errors between the actual and desired state variables of the intelligent hybrid articulated tractor are calculated and fed into both the PID and fuzzy controllers. The PID controller calculates and outputs a steering angle acceleration δ₁ in response to the error, while the fuzzy controller calculates weighting coefficients a and 1 − a for the PID and MPC, respectively. During the tractor’s straight-line driving, the path tracking accuracy is high, and the error between the actual and desired state variables is small, so the fuzzy controller outputs a smaller PID weighting coefficient a. However, during turns, when larger errors may occur, the fuzzy control system increases the PID weighting coefficient a to ensure that the steering angle acceleration is rapidly adjusted within a reasonable range, thereby maintaining the path tracking accuracy of the intelligent hybrid articulated tractor.

The key variables in Figure 5 are described as follows. X_f(t) is the actual tractor front axle center reference horizontal coordinate, Y_f(t) is the actual tractor front axle center reference vertical coordinate, θ_f(t) is the actual tractor front axle reference heading angle, V_f(t) is the actual tractor front axle center reference speed, γ_f(t) is the actual tractor reference articulation angle, ω_f(t) is the actual tractor reference articulation angle angular velocity. X_f(t+1) is the desired tractor front axle center reference horizontal coordinate, Y_f(t+1) is the desired tractor front axle center reference vertical coordinate, θ_f(t+1) is the desired actual tractor front axle reference heading angle, V_f(t+1) is the desired tractor front axle center reference speed, γ_f(t+1) is the desired tractor reference articulation angle, ω_f(t+1) is the desired tractor reference articulation angle angular velocity, δ₂(t+1) is the joint angular acceleration output by MPC, δ₁(t+1) is the joint angular acceleration output by PID, δ(t+1) is used to input the actual model.

Compared with MPC, after the PID controller and fuzzy controller in CMPC receive the error between the actual state variable and the expected state variable, the PID controller outputs the articulation acceleration according to the error, and the fuzzy controller outputs the weighting coefficient according to the error. These outputs are used to correct the articulation angular acceleration output by MPC, which can effectively improve the transient performance of the tractor during path tracking.

4. Simulation Results Analysis

4.1. Experiment Platform Configuration

To evaluate the effectiveness of the CMPC method, this paper investigates control strategies using the intelligent hybrid articulated tractor as the subject of control. A simulation model is built using MATLAB-Simulink, with MPC implemented as the comparison method, and an S-shaped path is used for the simulation. The key parameters of the tractor components during path tracking are listed in

Table 2. Parameters of main components of tractors.

Name	Parameters
Front wheel to connection point length (m)	1.4
Rear wheel to connection point length (m)	1.4
Total weight (Kg)	2500
Front body weight (Kg)	900
Rear body weight (Kg)	1600
Minimum turning radius (m)	2.5
Axle distance (mm)	1000
Maximum turning angle (°)	36

.

The test route is mainly divided into two parts: the straight area and the U-turn area. According to the change in the reference path curvature, it is divided into three working conditions: steady-state working condition, instantaneous working condition, and total working condition. The tractor driving the entire path is the total working condition, the straight driving section is the steady-state working condition, and the U-turn area is the instantaneous working condition. The specific reference trajectory is shown in Figure 6. The intelligent hybrid articulated tractor starts from the origin (0, 0), drives 80 m in the positive direction of the y-axis, turns around, drives 80 m in the negative direction of the y-axis, turns around again, and then continues to drive 80 m in the positive direction of the y-axis to end the movement.

4.2. Tractor Path Tracking Performance and Analysis

In order to facilitate calculation and analysis, the first straight driving path is selected as the steady-state condition, the first U-turn area is selected as the transient condition, and the simulation results of the first 120 s are selected for analysis. The reference speed is set to 7.2 km/h, and the actual speed curves and path tracking errors of the CMPC algorithm and the MPC algorithm are shown in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

As shown in Figure 7, the speed curves of the tractor under the MPC and CMPC methods. Under steady-state conditions, both MPC and CMPC showed good control effects: the speed of CMPC and MPC is stable at 7.23 KM/h. Under transient conditions, the speed fluctuation of CMPC is slightly greater than that of MPC: the speed range of CMPC is 1.05, and the speed standard deviation is 6.2; the speed range of MPC is 1.22, and the speed standard deviation is 6.03. The speed standard deviation of CMPC is 2.7% higher than that of MPC. Under total conditions: the speed standard deviation of CMPC is 0.37, and the speed standard deviation of MPC is 0.36. The speed standard deviation of CMPC is 2% lower than that of MPC.

As shown in Figure 8, the lateral error diagram of the tractor under the MPC and CMPC methods. Under steady-state conditions, the lateral errors produced by MPC and CMPC for path tracking are both small: the lateral errors of CMPC and MPC are stable within 0.01 m. Under transient conditions, the control effect of CMPC is better than that of MPC: the lateral mean absolute error of CMPC is 0.009, and the lateral error standard deviation is 0.031; the lateral mean absolute error of MPC is 0.011, and the lateral error standard deviation is 0.12. The lateral tracking error of CMPC is reduced by 18.2% compared with that of MPC. Under total conditions: the lateral error range of CMPC is within 0.07 m, and the lateral error range of MPC is within 0.14 m. Among them, the standard deviation of the lateral error of CMPC is 0.024, and the standard deviation of the lateral error of the MPC algorithm is 0.071. The lateral mean absolute error of CMPC is 0.0058, and the lateral mean absolute error of the MPC algorithm is 0.0063. The lateral tracking error of CMPC is reduced by 50% compared with that of MPC.

As shown in Figure 9, the longitudinal error diagram of the tractor under the MPC and CMPC methods. Under steady-state conditions, the longitudinal errors of MPC and CMPC in path tracking are both small: the longitudinal errors of CMPC and MPC are stable within 0.01 m. Under transient conditions, the longitudinal mean absolute error of CMPC is 0.026, and the longitudinal error standard deviation is 0.01; the longitudinal mean absolute error of MPC is 0.033, and the longitudinal error standard deviation is 0.01. The longitudinal tracking error of CMPC is reduced by 21.2% compared with that of MPC. Under total conditions, the longitudinal error range of CMPC is within 0.07 m, and the longitudinal error range of MPC is within 0.07 m. Among them, the standard deviation of the longitudinal error of CMPC is 0.008, and the standard deviation of the longitudinal error of the MPC algorithm is 0.009. The longitudinal mean absolute error of CMPC is 0.014, and the longitudinal mean absolute error of the MPC algorithm is 0.051. The longitudinal tracking error of CMPC is reduced by 8% compared with that of MPC.

As shown in Figure 10, the heading angle error diagram of the tractor under the MPC and CMPC methods. Under steady-state conditions, the heading angle errors generated by MPC and CMPC for path tracking are both small: the heading angle errors of CMPC and MPC are both 0. Under transient conditions, CMPC is better than MPC: the heading angle range of CMPC is 0.08, and the heading angle error standard deviation is 0.013; the heading angle range of MPC is 0.1, and the heading angle error standard deviation is 0.016. The heading angle standard deviation of CMPC is reduced by 18.7% compared with that of MPC. Under total conditions: the maximum heading angle error of CMPC is 0.039 rad, and the maximum heading angle error of MPC is 0.049 rad. The standard deviation of the heading angle error of CMPC is 0.007, and the standard deviation of the heading angle error of the MPC algorithm is 0.008. The heading angle standard deviation of CMPC is reduced by 20% compared with that of MPC.

As shown in Figure 11, the articulation angle error diagram of the tractor under the MPC and CMPC methods Under steady-state conditions, both MPC and CMPC have a better control effect on articulation angle when performing path tracking: the articulation angle errors of CMPC and MPC are both 0. Under transient conditions, the control effect of CMPC is slightly better than that of MPC: the articulation angle range of CMPC is 0.69, and the standard deviation of the articulation angle error is 0.008; the articulation angle range of MPC is 0.69, and the standard deviation of the articulation angle error is 0.009. The articulation angle standard deviation of CMPC is reduced by 11.1% compared with that of MPC. Under total conditions, the maximum articulation angle error of CMPC is 0.35 rad, and the maximum articulation angle error of MPC is 0.35 rad. The standard deviation of the articulation angle error of CMPC is 0.039, and the standard deviation of the articulation angle error of the MPC algorithm is 0.041. The standard deviation of the articulation angle of CMPC is reduced by 4.9% compared with that of MPC.

As shown in Figure 12, from the simulation results of CMPC and MPC, it can be seen that both algorithms can complete path tracking, but CMPC is more stable than MPC in the turning area. Both CMPC and MPC can complete the path tracking task well under steady-state conditions. Compared with MPC under transient conditions, CMPC has the speed standard deviation increased by 2.7%, the lateral tracking error decreased by 18.2%, the longitudinal tracking error decreased by 21.2%, the heading angle standard deviation decreased by 18.7%, and the articulation angle standard deviation decreased by 11.1%.

Under the total working condition, compared with MPC, CMPC has a speed standard deviation of 2%, a longitudinal tracking error of 8%, a lateral tracking error of 50%, a heading angle standard deviation of 20%, and an articulation angle standard deviation of 4.9%, which verifies the effectiveness of the CMPC strategy.

5. Conclusions

To improve the path-tracking performance of the intelligent hybrid articulated tractor across all operating conditions during unmanned operations, a CMPC path-tracking control method is proposed. The method demonstrates good path-tracking performance across all conditions.

The results show that in the process of path tracking under instantaneous working conditions, the CMPC increases the speed standard deviation by 2.7%, reduces the lateral tracking error by 18.2%, reduces the longitudinal tracking error by 21.2%, reduces the heading angle standard deviation by 18.7%, and reduces the articulation angle standard deviation by 4.9% compared with MPC. In the process of path tracking under full working conditions, CMPC can stabilize the tractor speed at about 7.2 km/h; compared with MPC, the speed standard deviation is reduced by 2%, the longitudinal tracking error is reduced by 8%, the lateral tracking error is reduced by 50%, and the maximum heading angle error is reduced by 20%, which proves the effectiveness of the corrected MPC strategy in path tracking under full working conditions.

The method proposed in this paper can provide a new reference for the path tracking of intelligent hybrid articulated tractors. In the future, considering the interaction between soil and tire, the proposed method still has a lot of room for improvement in the path-tracking effect of intelligent hybrid articulated tractors. By studying the soil-tire interaction mechanism, the path-tracking accuracy can be further improved.