Tractor Path Tracking Control Method Based on Prescribed Performance and Sliding Mode Control

Abstract

In addressing the challenges of low path tracking accuracy and poor robustness during tractor autonomous operation, this paper proposes a path tracking control method for tractors that integrates prescribed performance with sliding mode control (SMC). A key feature of this control method is its inherent immunity to system parameter perturbations and external disturbances, while ensuring path tracking errors are constrained within a predefined range. First, the tractor is simplified into a two-wheeled vehicle model, and a path tracking error model is established based on the reference operation trajectory. By defining a prescribed performance function, the constrained tracking control problem is transformed into an unconstrained stability control problem, guaranteeing the boundedness of tracking errors. Then, by incorporating SMC theory, a prescribed performance sliding mode path tracking controller is designed to achieve robust path tracking and error constraint for the tractor. Finally, both simulation and field experiments are conducted to validate the method. The results demonstrate that compared with the traditional SMC method, the proposed method effectively mitigates the impact of complex farmland conditions, reducing path tracking errors while enforcing strict error constraints. Field experiment data shows the proposed method achieves an average absolute error of 0.02435 m and a standard deviation of 0.02795 m, confirming its effectiveness and superiority. This research lays a foundation for the intelligent development of agricultural machinery.

1. Introduction

With the advancement of automatic navigation and control technologies, the automation and intelligence of agricultural machinery have been continuously enhanced, emerging as a critical development trend in modern agricultural production. The rise of precision agriculture has imposed stricter requirements on the operational accuracy and efficiency of agricultural machinery, making automatic path tracking control—a key enabler for intelligent operation of agricultural machinery—an area of extensive attention. As the primary power machinery in agricultural production [1], tractors require tracking control accuracy and robustness as core technical indicators to determine the efficiency and quality of their autonomous operations. However, the complex field working environment—such as dynamic terrain slope variations, uncertainties in tire–ground contact mechanics, and actuator clearances—poses significant challenges to tractor path tracking control. Additionally, in scenarios like tractor headland turning and re-alignment, path errors must be strictly constrained, while straight-line path errors must meet operational accuracy requirements to ensure work quality. Therefore, research on tractor path tracking control technology is of great significance for enhancing agricultural mechanization levels and advancing agricultural modernization.

Currently, research on path tracking control for agricultural machinery is predominantly based on geometric principles, with typical approaches including the Pure Pursuit method, Stanley algorithm, and Line-of-Sight (LOS) method. Zhou et al. optimized the Pure Pursuit model by integrating fuzzy control with dynamic look-ahead distance and variable speed strategies [2]. Wu et al. proposed an improved Pure Pursuit model using a simulated annealing-based particle swarm optimization algorithm to enhance path tracking accuracy and robustness [3]. Liu et al. optimized an ultrasonic ridge-tracking method that contains a fuzzy Stanley model [4]. Qin et al. improved the Stanley guidance law and integrated a reduced-order extended state observer to achieve state constraints for underactuated ships [5]. Cui et al. designed a tractor automatic navigation system based on dynamic path search and a fuzzy Stanley model [6]. Zhang optimized a novel LOS guidance law based on fixed-time theory, achieving faster path convergence for USVs [7], while Li attempted to integrate a nonlinear tracking differentiator with an anti-saturation controller to optimize LOS method accuracy [8]. Due to their simple parameters and ease of implementation, these methods have seen preliminary applications in structured farmland. However, these algorithms based on geometric principles suffer from limitations such as large tracking errors on curved paths and poor adaptability, which hinder the improvement of tractor path tracking control performance.

In response to the limitations of geometric principle-based path tracking control methods, scholars have increasingly turned to control research based on the kinematic or dynamic models of agricultural machinery. Among them, sliding mode control (SMC) has emerged as a research focus due to its strong robustness against parameter perturbations and external disturbances [9]. Li et al. proposed a path tracking method for an autonomous rice transplanter based on an adaptive sliding mode variable structure control [10]. Sun et al. proposed a path-tracking control scheme using fixed-time nonsingular terminal sliding mode and adaptive disturbance observer technique [11]. Chen et al. developed a control algorithm based on an improved double-power hyperbolic function integral sliding mode reaching law, improving system convergence speed and ensuring finite-time reaching of the sliding mode surface [12]. Liu et al. investigated high-order sliding mode controller, constructing a new quasi-state saturated second-order sliding mode algorithm using saturation technology and a quasi-backstepping method [13]. Li et al. proposed a novel control strategy combining interval type-2 fuzzy logic system with second-order SMC, enabling rapid state adjustment and chattering reduction [14]. Sun et al. presented an adaptive fractional order sliding mode tracking control method for a piezoelectric platform [15]. Beyond the anti-interference control methods above, some scholars have proposed using disturbance observers to identify and compensate for disturbances, mitigating their impact on the system. To address the challenge of precise path tracking for agricultural vehicles, Taghia et al. developed an SMC with a nonlinear disturbance observer [16]. Ding Chen proposed a composite second-order sliding mode path tracking control method, employing a finite-time disturbance observer to observe and compensate for lumped disturbances in the path tracking system [17]. Dai proposed an SMC with a neural network observer, applied to nonlinear hysteretic systems to eliminate unknown hysteresis and uncertainties [18]. Liu et al. presented a terminal SMC method based on a predefined-time adaptive disturbance observer to avoid chattering [19]. Han et al. proposed an adaptive hyper-screw fast terminal SMC method with a disturbance observer, solving the attitude tracking problem of quadrotor UAVs with mismatched disturbances [20]. Wang et al. proposed a sliding mode synchronization control method based on a disturbance observer [21].

However, the above-mentioned researches on sliding mode path tracking control mainly focus on suppressing chattering and compensating for disturbances, while ignoring the problem of error constraint. In the field of control engineering, the prescribed performance method can achieve explicit constraints on the convergence rate and maximum deviation of tracking errors by prescribed the error evolution trajectory [22]. Introducing this method into path tracking control can solve the problem of unbounded errors when tractors change rows at field heads or realign and improve path tracking accuracy. Therefore, this paper proposes a prescribed performance sliding mode path tracking control method for tractors by integrating prescribed performance with SMC. The main contributions of this paper are as follows:

(1). Based on the rigid body kinematics hypothesis, the kinematic model of the tractor is established. Then, by analyzing the position and attitude deviations between the tractor’s actual state and the reference path, the lateral position error and heading angle error are defined. Through geometric relationship analysis, a tracking error model of the tractor relative to the reference path is constructed, which lays the foundation for the design of the path tracking controller.

(2). To address the error constraint problem, a smooth, continuous, and monotonically decreasing prescribed performance function is constructed, and an error conversion function is introduced to map the original error into a new variable for constraint. Through this transformation, the constrained tracking control problem is converted into an unconstrained stability control problem, ensuring bounded constraint of the path tracking error.

(3). Based on the lateral error and heading error, a sliding mode surface is constructed. By integrating an exponential sliding mode reaching law, a tractor sliding mode path tracker is designed. Furthermore, to address the high-frequency oscillation in classical SMC inputs, this paper introduces a saturation function to replace the traditional sign function, effectively suppressing high-frequency chattering while maintaining disturbance robustness.

(4). Finally, simulation and field experiments are conducted to verify that the proposed method ensures the tractor’s tracking error remains within the prescribed performance range under field operation conditions, significantly improving the path tracking accuracy and reliability of the tractor.

The follow-up structure of the full text is as follows: Section 2 establishes the tractor kinematics model to reflect the dynamic characteristics of the tractor and introduces the definition of path tracking problem; Section 3 describes in detail the design process of the sliding mode controller integrated with prescribed performance theory and analyzes the closed-loop system based on Lyapunov stability theory; Section 4 verifies the effectiveness of the proposed method through multi-condition simulation and field experiments and compares and analyzes the tracking accuracy and robustness of traditional and improved control strategies; and finally, the research results are summarized and the future research directions are prospected.

2. Materials and Methods

2.1. Kinematic Model

The design foundation of the path tracking controller hinges on establishing a kinematic model that accurately illustrates the tractor’s dynamic behaviors while ensuring engineering practicality. This paper formulates the tractor kinematic model based on the following rational assumptions: (1) Treating the tractor body as a rigid system while neglecting the elastic deformation of the suspension system and the effects of component flexibility. (2) Disregarding the impact of load transfer between the front and rear axles on tire mechanical properties during acceleration and deceleration. (3) The tractor motion is constrained within a two-dimensional horizontal plane, excluding vertical (Z-axis) bounce and roll motions. (4) A front-wheel steering control structure is employed, with the centroid sideslip angle assumed to be zero during steering, and the effect of lateral slip on the heading angle is neglected.

The tractor generally maneuvers according to the Ackermann geometric steering principle [23], where the steering centers of the four wheels intersect at a single point, as illustrated in Figure 1. Consequently, the following steering relationship can be derived:

(1)$\cot \alpha +\cot \beta =2\cot \theta$

Since the direction of the rear wheel is always consistent with the direction of the body, the movement velocity direction of the rear axle center is uniquely determined by the tractor heading angle φ, and the corresponding unit direction vector can be expressed as [cos φ, sin φ]^T. Further decomposition shows that the velocity vector v at the center of the rear axle is the sum of the velocity vectors along the abscissa axis and the ordinate axis [23]:

(2)$v=\dot{x}\cos \phi +\dot{y}\sin \phi$

When the tractor is turning, the following kinematic constraints need to be met:

(3)$\left\{\begin{array}{l}\dot{x}\sin \phi -\dot{y}\cos \phi =0\hfill \\ {\dot{x}}_f\sin (\phi +\theta )={\dot{y}}_f\cos (\phi +\theta )\hfill \end{array}\right.$

where (x_f, y_f) represents the center of the front axle. Simultaneous Equations (2) and (3) give the following [24]:

(4)$\left\{\begin{array}{l}\dot{x}=v\cos \phi \hfill \\ \dot{y}=v\sin \phi \hfill \end{array}\right.$

The following kinematic equation is established according to the simplified geometric relationship of the tractor:

(5)$\left\{\begin{array}{l}{\dot{x}}_f=\dot{x}+L\cos \phi \hfill \\ {\dot{y}}_f=\dot{y}+L\sin \phi \hfill \end{array}\right.$

Simultaneous Equations (4) and (5) are substituted into Equation (3) and sorted out to obtain the following:

(6)$\dot{\phi }={\displaystyle \frac{v}{L}}\tan \theta$

From the geometric relationship in Figure 1, it can be deduced that the relationship between tractor turning radius R and front wheel angle θ is as follows:

(7)$\left\{\begin{array}{l}R={\displaystyle \frac{L}{\tan \theta }}\hfill \\ \theta =\arctan {\displaystyle \frac{L}{R}}\hfill \end{array}\right.$

Then, according to Equations (4) and (6), the tractor kinematics model is converted into a state space equation [24]:

(8)$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\phi }\end{array}\right]=\left[\begin{array}{c}\cos \phi \\ \sin \phi \\ \tan \theta /L\end{array}\right]v$

Define system state variables and system control variables as X = [x, y, φ]^T, u = [v, θ]^T.

2.2. Prescribed Performance Control

Let the state vector of the reference path be $q_r={[x_{\mathrm{r}},y_r,{\phi }_r]}^T$, then the tracking error model of tractor can be expressed as $q_e=q-q_r={[q_{e1},q_{e2},q_{e3}]}^T={[x_e,y_e,{\phi }_e]}^T$. Through coordinate transformation, the attitude error between the driving path and the reference path of the tractor can be converted from the geodetic coordinate system (OXY) to the tractor body coordinate system [25]:

(9)$\left[\begin{array}{c}{x}_e\\ y_e\\ {\phi }_e\end{array}\right]=\left[\begin{array}{ccc}\cos \phi & \sin \phi & 0\\ -\sin \phi & \cos \phi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x-x_{\mathrm{r}}\\ y-y_r\\ \phi -{\phi }_r\end{array}\right]$

In order to improve the transient performance of the control system and the final tracking error, the prescribed performance function is introduced to set the performance envelope of the controlled system, so that the tracking error is always within the specified boundary range. To do this, the element of each tracking error vector $q_e(t)\in {\Re }^n$ design is a strictly positive, bounded, smooth, and monotonically decreasing performance function ${\rho }_j(t):{\Re }^{+}\to {\Re }^{+}$, $q_e(t)\in {\Re }^n$, i.e., q_ej (t), j = 1, …, 4, to meet the following conditions [22]:

(10)$-{\underset{\_}{\lambda }}_j{\rho }_j(t)<q_{ej}(t)<{\overline{\lambda }}_j{\rho }_j(t),\forall t>0,j=1,\dots ,4$

where ${\underset{\_}{\lambda }}_j$ and ${\overline{\lambda }}_j$ are positive constants, representing the overshoot suppression parameters. Define the following smooth, continuous, and monotonically decreasing prescribed performance function [22]:

(11)${\rho }_j(t)=({\rho }_{j0}-{\rho }_{j\infty })e^{-l_{j}t}+{\rho }_{j\infty }$

where the prescribed performance function ρ_j (t) is required to satisfy the following conditions: (1) $\underset{t\to 0}{\lim }{\rho }_j(t)={\rho }_{j0}$; (2) ${\rho }_j(0)=({\rho }_{j0}-{\rho }_{j\infty })e^{-l_{j}t}+{\rho }_{j\infty }={\rho }_{j0}$. Among them, ${\rho }_{j\infty }$, ${\rho }_{j0}$, and $lj$ are positive numbers, ${\rho }_{j0}$ represents the prescribed initial value, ${\rho }_{j\infty }$ represents the prescribed maximum allowable steady-state error, and $lj$ signifies the convergence rate of the tracking error.

The inequality constraint in Equation (10) is converted into an equality constraint, and the following error transformation is introduced:

(12)$q_{ej}(t)={\rho }_j(t)S({\zeta }_j)$

where ${\zeta }_j$ represents the new transformed error, and the error transformation function S (${\zeta }_j$) satisfies the following conditions: (1) S (${\zeta }_j$) is a smooth and strictly monotonic increasing function; (2) $-{\underset{\_}{\lambda }}_j<S({\zeta }_j)<{\overline{\lambda }}_j$; and (3) $\underset{{\zeta }_j\to -\infty }{\lim }S({\zeta }_j)=-{\underset{\_}{\lambda }}_j$, $\underset{{\zeta }_j\to \infty }{\lim }S({\zeta }_j)={\overline{\lambda }}_j$.

The selected transformation function is as follows:

(13)$S({\zeta }_j)={\displaystyle \frac{{\overline{\lambda }}_j{e}^{{\zeta }_j}-{\underset{\_}{\lambda }}_j{e}^{-{\zeta }_j}}{e^{{\zeta }_j}+e^{-{\zeta }_j}}}$

The following can be obtained from Equation (13):

(14)${\zeta }_j=S^{-1}({\gamma }_j)={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{{\underset{\_}{\lambda }}_j+{\gamma }_j}{{\overline{\lambda }}_j-{\gamma }_j}}\right)$

where ${\gamma }_j=q_{ej}(t)/{\rho }_j(t)$. Therefore, the new transformed error can be obtained via the error equivalent transformation ${\zeta }_j$. Meanwhile, the tracking error of the tractor can be minimized by designing the path-tracking controller $q_e(t)\in {\Re }^n$, that is, $q_{ej}$(t), j = 1, …, 4 is controlled within the prescribed performance boundary (as shown in Equation (10)). This indicates that the prescribed performance tracking control problem for the tractor (as shown in Equation (9)) can be transformed into a stability problem of the equivalent error system.

2.3. Design of SMC Controller

Sliding mode surface is the key to the design of sliding mode path tracking controller, and its requirements are as follows: (1) The system state can reach the sliding mode surface (s = 0) from any initial position in the state space within a finite time interval, and (2) it can converge to the origin (equilibrium point) on the sliding mode surface [13].

Based on the aforementioned requirements, this paper designs the sliding mode surface by selecting the lateral error and heading error of the tractor, as expressed below [9,11]:

(15)$s=\tau de+\phi e=0$

where τ represents the linear term coefficient, τ > 0. According to the aforementioned kinematic model (Equation (8)), the following can be derived:

(16)$\left\{\begin{array}{l}{\dot{d}}_e=v\sin {\phi }_e\hfill \\ {\dot{\phi }}_e={\displaystyle \frac{v\tan \theta }{L}}\hfill \end{array}\right.$

The first derivative of the sliding mode surface defined in Equation (15) is expressed as follows [9,16]:

(17)$\dot{s}=\tau \dot{d}e+\dot{\phi }e$

Select the following exponential reaching law:

(18)$\dot{s}=-k\mathrm{sgn}(s)-\lambda s$

where k, λ are positive coefficients, and sgn (s) is the signum function characterizing the discontinuous switching behavior in SMC, which is defined as follows [9]:

(19)$\mathrm{sgn}(s)=\left\{\begin{array}{l}-1,s<0\hfill \\ 0,s=0\hfill \\ 1,s>0\hfill \end{array}\right.$

However, the discontinuous nature of the signum function inherently induces chattering when directly employed in sliding mode surface construction [12]. To mitigate this issue, the saturation function sat (s) is adopted in this paper, which is defined as follows:

(20)$\mathrm{sat}(s)=\left\{\begin{array}{l}-1,s<-\Delta \hfill \\ {\displaystyle \frac{s}{\Delta }},\left|s\right|\le \Delta \hfill \\ 1,s>\Delta \hfill \end{array}\right.$

where $\Delta$ > 0.

After replacement, the following expression is obtained:

(21)$\dot{s}=-k\mathrm{sat}(s)-\lambda s$

By integrating Equations (15), (16), and (21), the control law for the tractor’s front wheel angle is derived as follows:

(22)$\theta =\arctan \left[{\displaystyle \frac{L}{v}}\left(-k\mathrm{sat}(s)-\lambda s-\alpha \times (v\sin {\phi }_e)\right)\right]$

In this paper, the Lyapunov second rule is selected to determine the stability of the system, and the Lyapunov function is selected as follows [26]:

(23)$V={\displaystyle \frac{1}{2}}{s}^2$

According to Equations (16), (18) and (22), the following expression can be obtained:

(24)$\dot{V}=s\dot{s}=s\left(-k\mathrm{sat}(s)-\lambda s\right)$

The following expression can be obtained by further derivation:

(25)$\dot{V}=\left\{\begin{array}{l}s\left(k-\lambda s\right),s<-\Delta \hfill \\ -\left(k{\displaystyle \frac{s^2}{\Delta }}+\lambda s^2\right),-\Delta \le s\le 0\hfill \\ 0,s=0\hfill \\ -\left(k{\displaystyle \frac{s^2}{\Delta }}+\lambda s^2\right),0\le s\le \Delta \hfill \\ -\left(ks+\lambda s^2\right),s>\Delta \hfill \end{array}\right.$

Therefore, $\dot{V}$ ≤ 0, the path tracking error converges, and the tractor control system is stable.

In summary, this paper has completed the design of a prescribed performance sliding mode path tracking controller for tractors. The design is based on the established tractor kinematic model, integrated with prescribed performance theory and SMC theory. The control system principle is illustrated in Figure 2.

3. Results

3.1. Simulation and Experimental Platform

(1). Simulation Model

The simulation of tractor path tracking control is carried out on the MATLAB/Simulink R2023b. The lateral deviation and heading deviation are derived by computing the difference between the tractor’s current position and the nearest point on the reference path. These deviations are fed into a prescribed performance controller, which uses a specified prescribed performance function to constrain the deviation from the reference path within a specified range. Then the converted error is input into the sliding mode path tracking controller, and the control value of the front wheel angle obtained by the path tracking controller is input into the kinematic model of the tractor. According to the pose and front wheel angle values output from the tractor kinematic model, the lateral deviation and heading angle deviation at the next moment are calculated to realize the closed-loop tracking control of the reference path. The sampling time interval is set to 0.1 s. To simulate the measurement results of the actual differential Beidou positioning and navigation system, Gaussian white noise with a mean value of 0 is superimposed on the theoretical values, with the position standard deviation set to 0.01 m and the heading angle standard deviation set to 1.0°. Similarly, to simulate the disturbances experienced by the tractor during actual steering, Gaussian white noise with a mean value of 0 and a standard deviation of 1° is superimposed on the tractor’s front wheel angle.

(2). Experiment Platform

In this paper, the DX1204 tractor produced by Jiangsu World Agricultural Machinery Co., Ltd. (Zhenjiang, China). is selected as the experiment platform, and its main performance parameters are shown in Table 1. In addition, the MS-6111 high-precision integrated navigation and positioning system produced by BDStar Navigation Co. (Beijing, China) is used to obtain the position and heading information of the tractor during movement. The specific parameters are shown in Table 2.

The field experiment was carried out in Shiyezhou, Zhenjiang City, China. The “U”-shaped path widely used in tractor field operations was selected to verify the effectiveness of the proposed prescribed performance sliding mode path tracking control method, which was compared with the traditional sliding mode path tracking control method [9]. The traditional sliding mode path tracking controller is denoted as the SMC method, and the prescribed performance sliding mode path tracking controller proposed in this paper is denoted as the proposed PPC-SMC method. Before each experiment, the tractor started from a position with a lateral deviation of approximately 0.4 m from the planned operation path and moved at a speed of about 1 m/s. The control cycle in both the simulation and field experiment are 0.15 s, and the controller parameters are shown in Table 3.

3.2. Simulation Results

The reference path originates at (10, 20), while the tractor starts at (5, 19). Tractor path tracking simulation results are depicted in Figure 3, comparing the performance of the traditional SMC method [9] and the proposed PPC-SMC method. Figure 3 includes enlargements of the straight-line tracking segments following the initial path engagement and the final turn. Results show that the proposed PPC-SMC method effectively tracks the specified reference path. Compared with the traditional SMC method [9], the introduction of prescribed performance constraints significantly reduces overshoot.

Figure 4 illustrates the lateral error during straight-line tracking from the reference path origin to the first turn. Statistical results show that the traditional SMC method [9] exhibits a lateral error overshoot of approximately 0.45 m, requiring 11 s for stable convergence. In contrast, the proposed PPC-SMC method reduces the overshoot to 0.26 m with a convergence time of 8 s, demonstrating a 42.2% reduction in overshoot compared to the conventional SMC approach. This confirms that the prescribed performance function not only minimizes the lateral error magnitude but also suppresses overshoot and significantly shortens the error convergence duration for path tracking.

Figure 5 illustrates the probability distribution of lateral errors for the simulated path, with statistical characteristics tabulated in Table 4. Quantitative results show that the proposed PPC-SMC method yields a mean absolute error of 0.0144 m and standard deviation of 0.0644 m, representing reductions of 6.49% and 6.94% compared to the SMC method’s values of 0.0154 m and 0.0692 m, respectively.

The cumulative path tracking error curve of the tractor is shown in Figure 6. It can be seen that the path tracking error of the traditional SMC method [9] has accumulated to 0.82 m over time. Compared with the proposed PPC-SMC method, the final accumulated error is only 0.69 m, an optimization increase of 15.85%.

3.3. Experimental Results

The common “U”-shaped operation path of a tractor is used for the field experiment. The path tracking control results are shown in Figure 7, and the control amount of tractor front wheel angle is shown in Figure 8. It can be seen that the trajectory of the proposed PPC-SMC method is more suitable for the planned path in straight lines and curves, while the traditional SMC method [9] has poor local tracking effect, and the actual front wheel angle of the proposed PPC-SMC method is consistent with the expected angle, indicating that this method improves the path tracking accuracy and steering control performance.

In order to analyze the accuracy and stability of tractor path tracking, the straight-line paths (path 1 and path 2) of the tractor in Figure 7 are intercepted for comparative analysis, as shown in Figure 9. It can be seen that the proposed PPC-SMC method is more suitable for the planning path with small deviation, and the traditional SMC method [9] has poor local tracking effect. In addition, the proposed PPC-SMC method can maintain the lateral tracking control error of the tractor within the prescribed performance boundary.

The lateral error probability distribution of the tractor based on the traditional SMC method [9] is shown in Figure 10. It can be seen that the lateral error distribution of the traditional SMC method [9] is wide, and the error value is basically more than ±0.10 m. The distribution on path 1 and path 2 shows a relatively dispersed state, indicating that the lateral error fluctuates greatly and the stability of path tracking is poor. The probability distribution of lateral error of the tractor based on the PPC-SMC method proposed in this paper is shown in Figure 11. It can be seen from the figure that the lateral error distribution of the proposed PPC-SMC method is relatively concentrated at 0, and the overall shape is approximately normal distribution, mainly concentrated in the range of ±0.06 m. Compared with Figure 10 and Figure 11, the proposed PPC-SMC method reduces the horizontal error range of path 1 by 60% and path 2 by 30% compared with the traditional SMC method through prescribed performance constraints, and the error dispersion is greatly reduced.

The statistical characteristics of tractor lateral error are presented in Table 5. For the traditional SMC method [9], the overall maximum lateral error is 0.1862 m, the minimum is −0.0970 m, the mean absolute error is 0.04015 m, and the standard deviation is 0.0486 m. In contrast, the proposed PPC-SMC method yields an overall maximum lateral error of 0.0745 m, a minimum of −0.0615 m, a mean absolute error of 0.02435 m, and a standard deviation of 0.02795 m. Compared with the traditional SMC method [9], the proposed PPC-SMC method reduces the mean absolute error and standard deviation by 39.3% and 42.5%, respectively.

4. Conclusions

Aiming at the issues of low accuracy and poor robustness in tractor path tracking within complex farmland environments, this paper proposes a novel path tracking control method integrating prescribed performance with SMC. By designing a smooth monotonic decreasing prescribed performance function and combining it with an error transformation function, the original error is mapped to a bounded space, realizing explicit constraints on the tracking error. Subsequently, a sliding mode surface incorporating lateral error and heading error is constructed, and a tractor prescribed performance sliding mode path tracking controller is designed using an exponential reaching law. Finally, path tracking control simulations and field experiments of the tractor are conducted.

The simulation results indicate that the mean absolute error of the proposed PPC-SMC method is 0.0144 m, which is 6.49% lower than the 0.0154 m of the traditional SMC method. Field experiment results further demonstrate that the proposed PPC-SMC method achieves a mean absolute error of 0.02435 m, representing a 6.49% reduction compared to the traditional SMC method. In terms of error dispersion, the error distribution of the proposed method is more concentrated around zero, with the error range narrowed by approximately 30–40%, effectively verifying its adaptability under complex farmland conditions. These results collectively show that the proposed PPC-SMC method exhibits excellent dynamic accuracy and robustness in tractor path tracking control. For future work, the proposed PPC-SMC algorithm will be further optimized by integrating machine learning and big data analysis technologies. This will involve modeling and analyzing tractor motion characteristics, farmland environmental features, and operational task requirements to enable adaptive adjustment and optimization of the algorithm. Such advancements aim to better address complex farmland operation scenarios and the evolving demands of operational tasks.

Author Contributions

Conceptualization, L.Z. and Q.Z.; methodology, L.Z. and W.S.; software, J.X. and G.S.; investigation, J.X.; writing—original draft preparation, L.Z.; writing—review and editing, E.L. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Data Availability Statement

Data available on request from the authors.

Conflicts of Interest

Author Guohui Sha was employed by the company Weichai Lovol Intelligent Agricultural Technology Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

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Figures and Tables

Figure 1 Structure diagram of tractor path tracking. (α is the angle of the front outer wheel. β is the angle of the front inner wheel. θ is the equivalent angle of the front axle center. φ is the heading angle of the tractor. φ_r is the heading angle of the reference path. φ_e is the heading angle deviation of the tractor, and d_e is the lateral deviation of the tractor. v is the tractor velocity. L is the wheelbase between the front and rear wheels. R is the turning radius of the tractor. O_t is the steering center).

Figure 2 Schematic diagram of the proposed tractor path tracking control method (PPC-SMC).

Figure 3 Simulation results of tractor path tracking control.

Figure 4 Lateral errors in the simulation of tractor straight-line path tracking control.

Figure 5 Lateral error probability distributions in the simulation of tractor path tracking control: (a) traditional SMC method; (b) proposed PPC-SMC method.

Figure 6 Cumulative lateral error curves in the simulation of tractor straight-line path tracking control.

Figure 7 Experiment results of tractor path tracking control.

Figure 8 Front wheel angles in the experiment of tractor path tracking control.

Figure 9 Lateral errors in the experiment of tractor straight-line path tracking control: (a) straight-line path 1; (b) straight-line path 2.

Figure 10 Lateral error probability distributions in the experiment of tractor path tracking control based on traditional SMC method: (a) straight-line path 1; (b) straight-line path 2.

Figure 11 Lateral error probability distributions in the experiment of tractor path tracking control based on proposed PPC-SMC method: (a) straight-line path 1; (b) straight-line path 2.

Table 1

Main performance parameters of DX1204.

Type	Parameter
Size	4650 × 2070 × 2890 mm
Wheelbase	2330 mm
Power output	≥75 kW
Maximum traction	≥31.6 kN
Maximum lifting force	≥21.2 kN
Track width	2020 mm

Table 2

Main performance parameters of MS-6111.

Parameter	Metric (Typical)
Horizontal positioning accuracy	1 cm + 1 ppm (RTK)
Heading accuracy	0.1°
Attitude accuracy	0.1°
GNSS update rate	10 Hz
IMU update rate	100 Hz
Working temperature	−40 °C ~ +85 °C

Table 3

Main parameters of proposed PPC-SMC.

Parameter	Description	Value
L	Tractor wheelbase	2.33 m
ρj 0	Prescribed initial value	1.6
ρj ∞	Prescribed allowable maximum steady-state error	0.025
lj	Tracking error convergence rate	0.3
α	Linear term coefficient	2.1
k	Linear term coefficient	1
λ	Linear term coefficient	1
∆	Thickness of boundary layer	0.05

Table 4

Lateral error statistics in the simulation of tractor straight-line path tracking control.

Controller	Maximum	Minimum	Mean Absolute Error	Standard Deviation
Traditional SMC method [9]	0.4667 m	−0.9977 m	0.0154 m	0.0692 m
Proposed PPC-SMC method	0.2709 m	−0.9977 m	0.0144 m	0.0644 m

Table 5

Lateral error statistics in the experiment of tractor straight-line path tracking control.

Controller	Operation Path	Maximum	Minimum	Mean Absolute Error	Standard Deviation
Traditional SMC method [9]	Straight-line path 1	0.1862 m	−0.0970 m	0.0538 m	0.0639 m
	Straight-line path 2	0.0727 m	−0.1181 m	0.0265 m	0.0333 m
Proposed PPC-SMC method	Straight-line path 1	0.0597 m	−0.0615 m	0.0243 m	0.0270 m
	Straight-line path 2	0.0745 m	−0.0602 m	0.0244 m	0.0289 m

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