Trajectory Tracking Control of an Orchard Robot Based on Improved Integral Sliding Mode Algorithm

Abstract

To address the problems of insufficient trajectory tracking accuracy, pronounced jitter over undulating terrain, and limited disturbance rejection in orchard mobile robots, this paper proposes a trajectory tracking control strategy based on a double-loop adaptive sliding mode. Firstly, a kinematic model of the orchard robot is constructed and a time-varying integral terminal sliding surface is designed to achieve global fast finite-time convergence. Secondly, a sinusoidal saturation switching function with a variable boundary is employed to suppress the high-frequency chattering inherent in sliding mode control. Thirdly, an improved double-power reaching law (Improved DPRL) is introduced to enhance disturbance rejection in the inner loop while ensuring continuity of the outer-loop output. Finally, Lyapunov stability theory is used to prove the asymptotic stability of the double-loop system. The experimental results show that attitude angle error settles within 0.01 rad after 0.144 s, while the position errors in both the x-axis and y-axis directions settle within 0.01 m after 0.966 s and 0.753 s, respectively. Regarding position error convergence, the Integral of Absolute Error (IAE)/Integral of Squared Error (ISE)/Integral of Time-Weighted Absolute Error (ITAE) are 0.7629 m, 0.7698 m, and 0.2754 m, respectively; for the attitude angle error, the IAE/ISE/ITAE are 0.0484 rad, 0.0229 rad, and 0.1545 rad, respectively. These results indicate faster convergence of both position and attitude errors, smoother control inputs, and markedly reduced chattering. Overall, the findings satisfy the real-time and accuracy requirements of fast trajectory tracking for orchard mobile robots.

1. Introduction

With the deep integration of the new generation of electronic information technology and agricultural systems engineering, traditional agricultural production is rapidly evolving toward modernization and intelligence. Against this backdrop, intelligent agricultural production equipment and its control methods have increasingly become research hotspots [1,2,3]. Among them, orchard mobile robots, as representative equipment, have been widely applied in key orchard operations such as weeding, pruning, bagging, and harvesting, significantly reducing labor intensity and improving the efficiency and quality of agricultural operations [4,5]. However, achieving safe and reliable inter-row navigation in unstructured environments such as orchards remains one of the core challenges for ensuring stable operations. A reasonable, reliable, and safe trajectory planning method is the key technology to ensure operational safety [6].

To achieve the objectives of precision agricultural operations, it is essential to design advanced path tracking control schemes for mobile robots. The key lies in enhancing the robot’s capability for precise target recognition, rapid positioning, and efficient trajectory planning [7,8]. Accordingly, researchers worldwide have extensively studied various path tracking control strategies for mobile robots, including Proportional–Integral–Derivative (PID) control, pure pursuit control, Model Predictive Control (MPC), fuzzy control, neural network control, and sliding mode control (SMC) [9]. Given the complex and dynamic orchard environments, the performance and robustness of control systems often degrade significantly under uncertain operating conditions [10]. Therefore, high-performance path tracking control schemes for orchard mobile robots must effectively address the combined effects of system modeling uncertainties and unknown external disturbances. Sliding mode control (SMC), which exhibits inherent robustness against unmodeled dynamics, parameter perturbations, and external disturbances, has been widely applied in the kinematic control of orchard mobile robots [11]. To address the decline in path tracking accuracy caused by uncertainties such as lateral sliding in rice fields, Jinyang Li and colleagues proposed an SMC method with Radial Basis Function (RBF) neural network-based adaptive parameter tuning for unmanned rice transplanters. This approach significantly reduced lateral and heading deviations, effectively suppressed chattering, and improved operational stability [12]. For trajectory tracking challenges in automated tractor–trailer systems under model uncertainties and disturbances, a dual-loop MPC-SMC composite controller was developed. By integrating a Nonlinear Disturbance Observer (NDO) and Prescribed Performance Control (PPC), high robustness and superior transient and steady-state tracking accuracy were achieved [13]. To address the trajectory tracking accuracy challenges faced by agricultural uncrewed helicopters under complex wind disturbances and payload variations, a fuzzy extended state observer-based sliding mode control (FESO-SMC) method is proposed. By expanding the state vector, a unified disturbance model is established; a sliding mode control law that integrates the extended state observer is designed to achieve disturbance estimation and compensation; and fuzzy logic is employed to dynamically optimize the ESO parameters to enhance estimation accuracy, thereby improving the system’s disturbance rejection and robustness [14].

However, the inherent discontinuous switching characteristic of traditional sliding mode control induces significant chattering. Chattering not only causes energy loss in the system but also severely affects control performance, making its effective suppression a critical research objective [15]. Existing suppression methods mainly include High-Order Sliding Mode (HOSM), Non-Singular Terminal Sliding Mode Control, boundary layer methods, and approaches based on reaching laws [16,17,18]. Although HOSM can effectively eliminate chattering, it faces challenges in obtaining higher-order derivatives of sliding mode variables [16]. Compared to gain adjustment, strategies for improving switching functions are more diverse, with the core idea being to replace the sign function with continuous functions (e.g., linear or nonlinear functions). The boundary layer method proposed in [18] replaces the sign function with a saturation function, which maintains the system near the sliding mode surface but at the expense of control accuracy. In the work of Kim et al. [19], a nonlinear hyperbolic function was applied, while Gong et al. [20] adopted a nonlinear S-shaped function to suppress chattering.

Furthermore, Lu et al. [21] conducted a comparative analysis of four switching functions and ultimately selected a sinusoidal saturation function with variable boundaries. Non-Singular Terminal Sliding Mode Control and reaching law methods share structural similarities, with the former focusing on optimizing the dynamic performance on the sliding mode surface and the latter emphasizing the motion characteristics during the reaching phase [17]. The reaching law approach accelerates system convergence and attenuates chattering by designing dedicated reaching laws—such as the constant rate, power rate, exponential rate, and generalized reaching laws—thereby markedly improving the dynamic performance of the reaching phase and laying a solid foundation for subsequent studies. Among them, the exponential reaching law introduces an exponential term based on the constant-speed reaching law to accelerate the reaching process, but chattering still exists when the sliding mode surface is reached. The power reaching law leverages the characteristics of power functions to maintain a smaller control gain when approaching the sliding mode surface, thereby reducing chattering intensity. A single reaching law cannot simultaneously satisfy the dual requirements of fast reaching and complete chattering elimination. In light of this, Zhang et al. [22] introduced a new power term into the power reaching law, allowing parameter configuration to adjust the reaching speed. Kang et al. [23] proposed an improved double-power adaptive reaching law, which utilizes an adaptive mechanism to dynamically adjust the gain dynamically, further enhancing system response speed and control quality. Based on the above analysis, this paper adopts an improved double-power reaching law based on a sinusoidal saturation switching function with variable boundaries.

Orchard mobile robots are a class of complex systems characterized by significant nonlinearity, underactuation, and strong coupling, operating in environments with high levels of disturbance. To further improve the trajectory tracking accuracy and disturbance rejection capability of such systems, this paper proposes a trajectory tracking control strategy for orchard mobile robots based on double closed-loop adaptive sliding mode control. Firstly, a kinematic model of the mobile robot is established, and a time-varying integral terminal sliding mode surface is designed to enhance the controllability of the convergence process and improve the system’s disturbance rejection capability. Secondly, a sinusoidal saturation function with variable boundaries is adopted as the switching function, employing a mechanism of dynamic adjustment via a boundary layer to effectively suppress the chattering in sliding mode control and ensure control accuracy. Furthermore, a novel reaching law is introduced to accelerate the system’s approach to the sliding mode surface while mitigating chattering induced by uncertain disturbances, thereby enhancing the overall robustness of the control system. The stability of the inner- and outer-loop control systems is rigorously proven based on Lyapunov stability theory. Finally, the effectiveness of the proposed method is validated through simulations.

2. Materials and Methods

2.1. Kinematic Model Construction

The orchard mobile robot is a complex, nonlinear, and strongly coupled time-varying system. It relies on two side-driven wheels to move forward or backward at different speeds, using the speed difference between the wheels and their friction with the ground to achieve steering. This allows for operations such as zero-turn steering without dedicated steering components. In the motion analysis, the robot’s motion model is constructed based on the four-wheel-drive chassis, as shown in Figure 1. The kinematic equations are given in Equation (1).

$$\left[\begin{array}{c}v\\ w\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{1}{L}}& -{\displaystyle \frac{1}{L}}\end{array}\right]\left[\begin{array}{c}{v}_r\\ v_l\end{array}\right]$$

(1)

When the linear velocities and movement directions of the left and right wheels of an orchard mobile robot are not completely identical, its forward trajectory will change. At this time, the instantaneous turning radius can theoretically be obtained as follows:

$$r={\displaystyle \frac{v}{w}}={\displaystyle \frac{L}{2}}{\displaystyle \frac{v_l+v_r}{v_l-v_r}}$$

(2)

The turning radius of an orchard mobile robot is determined by the difference in linear velocities between the left and right wheels and the robot’s wheelbase. When the robot moves in a straight line, the linear velocities of the left and right wheels are identical ($v_l=v_r$), resulting in a turning radius of $r=\infty$. When $v_l\ne v_r$, the orchard mobile robot moves along a curved trajectory with a turning radius (r). By adjusting the velocity distribution between the left and right wheels, the robot’s turning radius and angular velocity can be modified, thereby achieving motion posture adjustments.

The core of chassis control is the realization of closed-loop path tracking, which is achieved through the coordinated operation of perception, planning, and control.” At each moment, the pose error information serves as the controller’s input, enabling the robot to reach its target point. The positioning system provides the real-time pose $(x,y,\theta )$ of the orchard mobile robot, offering a current state reference for trajectory tracking. This allows the chassis to determine its absolute position and motion orientation within the orchard plane. The planned path is discretized into a set of desired poses in the target coordinate system, translating the abstract path into quantifiable control objectives executable by the chassis. As illustrated in Figure 2, the principle of trajectory tracking fundamentally involves pose deviation control coordinated across multiple coordinate systems. The global coordinate system $XOY$ anchors the robot’s initial and target positions within the orchard plane, while the vehicle coordinate system $X_C{O}_C{Y}_C$ perceives the orchard mobile robot’s motion state. By comparing the deviations between the real-time pose $(x,y,\theta )$ and the desired pose $(x_d,y_d,{\theta }_d)$, the control algorithm dynamically adjusts the chassis’ velocity distribution and steering commands. This drives the actual trajectory to iteratively converge toward the target trajectory, ultimately accomplishing path tracking [24].

As shown in Figure 2, the transformation relationship between the pose of the orchard mobile robot’s $O_C$ in the global coordinate system and its coordinates in the vehicle coordinate system can be expressed as follows:

$$\left[\begin{array}{c}x\\ y\\ \theta \end{array}\right]=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_c\\ y_c\\ {\theta }_c\end{array}\right]$$

(3)

Based on the above kinematic derivation, the no-slip kinematic constraint of the nonholonomic system is ${\dot{y}}_c=0$. Taking $O_C$ as the reference point, we analyze the relationship between the velocity and pose; in the vehicle coordinate frame $X_C{O}_C{Y}_C$, the relationship between the orchard mobile robot’s pose and its velocity vector is as follows:

$$\left[\begin{array}{c}{\dot{x}}_c\\ {\dot{y}}_c\\ {\dot{\theta }}_c\end{array}\right]=\left[\begin{array}{c}v\\ 0\\ w\end{array}\right]$$

(4)

Its equivalent expression in the global coordinate frame is as follows:

$$\dot{x}\cos \theta -\dot{y}\sin \theta =0$$

(5)

Substituting into Equation (3), the kinematic model of the orchard mobile robot in the global coordinate system $XOY$ can be simplified as follows:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & 0\\ \sin \theta & 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}v\\ w\end{array}\right]$$

(6)

The vehicle moves by controlling the forward velocity $v$ and the rotational angular velocity $w$. When $v\ne 0$ and $w=0$, the vehicle performs linear motion; when $v\ne 0$ and $w\ne 0$, the vehicle moves in a circular arc with a radius of $r = v/w$; and when $v=0$ and $w\ne 0$, the vehicle performs in-place rotational motion.

Based on the aforementioned kinematic control principles, a hierarchical architecture design concept and modular development approach were adopted to construct the overall vehicle control process, aiming to enhance system scalability and maintainability. The system architecture is illustrated in Figure 3. High-level algorithms, such as path tracking, are deployed on the host computer platform, making the path tracking function an independent, modular component that can be flexibly added or removed, thereby facilitating algorithm iteration and functional expansion [25]. Subsequently, data exchange is conducted with the lower-level chassis controller (STM32) via a serial interface, with the forward velocity $v$ and rotational angular velocity $w$ calculated by the path tracking algorithm transmitted as motion commands to complete the closed-loop motion control.

2.2. Dual-Loop Trajectory Tracking Control

According to the kinematic equations of the orchard mobile robot, the system possesses two control degrees of freedom, while the model output includes three state variables. Consequently, the model is categorized as an underactuated system, where only two input variables can be actively controlled to enable certain state variables $(x,y)$ to track the desired trajectory, while the remaining state variable $\theta$ indirectly stabilizes through the coupling relationship of the system [26]. Essentially, this control problem can be reduced to a trajectory tracking constraint problem, wherein the control law $q = \left[v, w\right]$ is designed to drive the robot pose $[x,y,\theta ]$ to approximate the desired trajectory $[x_d,y_d,{\theta }_d]$. Based on the underactuated characteristics, the error model can be decomposed into a position subsystem error and an angular velocity subsystem error, which are addressed hierarchically through dual-loop control.

Firstly, a position control law is designed to ensure that $x$ can follow $x_d$, while $y$ can follow $y_d$. Consequently, the position $[x,y]$ moves along the ideal trajectory $[x_d,y_d]$, which is a time-dependent trajectory $[x_d,y_d]$ satisfying motion constraints. The reference trajectory is defined as follows:

$$x_d(t)=f(t), y_d(t)=g(t)$$

(7)

Let $f(t)$ and $g(t)$ be smooth functions.

Define the virtual velocity components $\left(u_1, u_2\right)$.

$$\left\{\begin{array}{l}{u}_1=v\cos \theta \hfill \\ u_2=v\sin \theta \hfill \end{array}\right.$$

(8)

The actual linear velocity of the orchard mobile robot can then be calculated from the virtual velocities, such that $v=\sqrt{u_1^2+u_2^2}$. The desired orientation angle of the orchard mobile robot is defined as ${\theta }_d$.

$${\theta }_d=\arctan ({\displaystyle \frac{u_2}{u_1}})$$

(9)

where $u_1\ne 0$. Ultimately, the trajectory tracking error $q_e$ can be expressed as follows:

$$q_e=\left[\begin{array}{c}{x}_e\\ y_e\\ {\theta }_e\end{array}\right]=\left[\begin{array}{c}x-x_d\\ y-y_d\\ \theta -{\theta }_d\end{array}\right]$$

(10)

Thus, the trajectory tracking system can be expressed as follows:

$$\left\{\begin{array}{l}{\dot{x}}_e=\dot{x}-{\dot{x}}_d=u_1-{\dot{x}}_d\hfill \\ {\dot{y}}_e=\dot{y}-{\dot{y}}_d=u_2-{\dot{y}}_d\hfill \end{array}\right.$$

(11)

$${\dot{\theta }}_e=\dot{\theta }-{\dot{\theta }}_d=w-{\dot{\theta }}_d+d$$

(12)

where $d$ represents external disturbance signals, $\left|d\right|\le M$, and $M\in R^{+}$. Equation (12) corresponds to the outer-loop position tracking system, while Equation (12) represents the inner-loop attitude tracking system. In Equation (12), it is necessary to compute the derivative of ${\theta }_d$. Given the complexity of the differentiation process, this paper employs a linear second-order differentiator to obtain ${\theta }_d$:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=-2R^2(x_1-n(t))-R{x}_2\hfill \\ \psi =x_2\hfill \end{array}\right.$$

(13)

In the Equation, $\psi$ represents the output of the differentiator, $n(t)$ is the input signal to be differentiated, $x_1$ is the tracking signal, $x_2$ is the estimate of the first-order derivative of the signal, and $R$ is the parameter of the differentiator. The initial values of the differentiator are set as $x_1\left(0\right) = 0$ and $x_2\left(0\right) = 0$.

Figure 4 illustrates the control architecture of the orchard mobile robot under the proposed control method. It primarily consists of three components: the desired tracking trajectory, the kinematics-based controller, and the orchard mobile robot. The closed-loop control system designed within this architecture adopts a dual-loop structure to achieve hierarchical regulation of position tracking and attitude tracking for the orchard mobile robot. Specifically, the inner loop corresponds to the attitude subsystem, while the outer loop corresponds to the position subsystem. The outer loop generates an intermediate command signal ${\theta }_d$ based on the position error $\left(x_e, y_e\right)$ and transmits this signal to the inner loop. The inner loop employs a sliding mode controller to ensure the rapid convergence of the attitude error ${\theta }_e$, thereby providing a foundational support for executing the outer-loop position commands. Meanwhile, the outer loop continuously adjusts the intermediate command to guide the attitude adjustments of the inner loop, ultimately driving the position error $\left(x_e, y_e\right)$ and the attitude error ${\theta }_e$ to asymptotically converge, ensuring precise tracking of the desired trajectory by the robot [13].

The trajectory tracking of an orchard mobile robot based on the dual-loop control strategy requires consideration of the distinct control constraints in the inner and outer loops:

(1)

The inner-loop attitude tracking is sensitive to external disturbances, necessitating attention to the anti-disturbance performance during the tracking process.

(2)

The outer-loop position error has no upper bound, requiring consideration of the stability of the convergence rate when the orchard mobile robot is far from the target trajectory. Once it approaches the target trajectory, the position error should converge rapidly within a finite time.

(3)

To avoid control oscillations and response delays, the convergence rate of the inner-loop system must be significantly faster than that of the outer loop, and the output of the outer loop must ensure continuity.

To satisfy the above constraints, this paper considers applying the fast terminal sliding mode (FTSM) method with further improvements.

The form of the fast terminal sliding mode is as follows:

$$s=\dot{e}+k|e{|}^p\mathrm{sgn}(e)$$

(14)

where $e$ denotes the system error; $k$ and $p$ are sliding surface parameters, and $0 < p < 1$.

Lemma 1 ([27]).

Based on the phase-deficit concept, the parameter choices in (14) lead to different dynamic behaviors on the sliding surface, specifically the following: (1) When $0 < p < 1$, the convergence rate is relatively smooth in regions far from the origin, whereas near the origin, the system converges rapidly; this property ensures finite-time convergence. (2) When$p >1$, the system converges faster far from the origin but slows down in the vicinity of the origin, exhibiting asymptotic convergence. (3) When$p =1$, the terminal sliding mode coincides with the linear sliding mode, and the convergence likewise becomes asymptotic.

In summary, the convergence rate of the system is related to the terminal sliding surface parameter $p$. Considering that the dual-loop control of the orchard mobile robot imposes different convergence constraints on the inner and outer loops, an adaptive regulation strategy for $p$ must be designed to dynamically coordinate the convergence characteristics of the inner and outer loops, satisfying the hierarchical control requirements.

2.3. Design of the Trajectory Tracking Controller

The time-varying integral terminal sliding surface $s$ is constructed as follows:

$$s=e-e_0+{\displaystyle {\int }_0^{t}k}\xi {\left|{\displaystyle \frac{e(\tau )}{\xi }}\right|}^{p(e)}\tanh \left({\displaystyle \frac{e(\tau )}{\phi }}\right)\mathrm{d}\tau$$

(15)

$$p(e)=\left\{\begin{array}{ll}{\rho }_{\infty }[1-\exp (-\epsilon {(|e|+\xi )}^2)]+\exp (-\epsilon {(|e|+\xi )}^2),\hfill & \left|e\right|>\xi \hfill \\ 1-\lambda \exp \left(\epsilon \left(1-{\displaystyle \frac{{\xi }^2}{{\xi }^2-e^2}}\right)\right),\hfill & \left|e\right|<\xi \hfill \\ 1,\hfill & \left|e\right|=\xi \hfill \end{array}\right.$$

(16)

where $e_0$ is the initial system error; $k$, $\epsilon$, $\lambda$, and ${\rho }_{\infty }$ are undetermined parameters, all of which are positive real numbers; $\xi$ is the critical value of the system error at which the convergence characteristics change; $p(e)$ is a continuous time-varying parameter. $\phi >0$, and the value of $\phi$ determines the rate of change in the inflection point in the hyperbolic tangent smoothing function.

$$\tanh \left({\displaystyle \frac{e(\tau )}{\phi }}\right)={\displaystyle \frac{e^{\frac{e(\tau )}{\phi }}-e^{\frac{e(\tau )}{\phi }}}{e^{\frac{e(\tau )}{\phi }}+e^{\frac{e(\tau )}{\phi }}}}$$

(17)

$$\underset{e\to 0}{\lim }p(e)=1-\lambda$$

(18)

$$\underset{e\to \infty }{\lim }p(e)={\rho }_{\infty }$$

(19)

Let ${\rho }_0=1-\lambda$. By selecting appropriate ${\rho }_0$ and ${\rho }_{\infty }$, the system’s convergence characteristics can be designed to correspond to the regions $\left|e\right|<\xi$ and $\left|e\right|>\xi$, respectively. Therefore, based on the control requirements for inner-loop attitude tracking and outer-loop position tracking, the sliding mode controllers for the inner and outer loops are designed as follows.

(1)

The sliding mode variable for outer-loop position tracking control is as follows:

$$\left\{\begin{array}{l}{s}_1=x_e-x_0+{\displaystyle {\int }_0^t{k}_1}{\xi }_1{\left|{\displaystyle \frac{x_e(\tau )}{{\xi }_1}}\right|}^{p_1(x_e)}\tanh \left({\displaystyle \frac{x_e(\tau )}{\phi }}\right)\mathrm{d}\tau \hfill \\ s_2=y_e-y_0+{\displaystyle {\int }_0^t{k}_2}{\xi }_2{\left|{\displaystyle \frac{y_e(\tau )}{{\xi }_2}}\right|}^{p_2(y_e)}\tanh \left({\displaystyle \frac{y_e(\tau )}{\phi }}\right)\mathrm{d}\tau \hfill \end{array}\right.$$

(20)

The sliding surfaces are defined as $s_1 = 0$ and $s_2 = 0$. Taking position tracking in the $x$ direction as an example, ${\rho }_0<{\rho }_{\infty }<1$ is selected, where $0<{\rho }_0<1$. When $x_e>\xi$, ${\rho }_{\infty }<p_1(x_e)<1$, and when $0<x_e<\xi$, ${\rho }_0<p_1(x_e)<1$.

The functional characteristics of $p_1(x_e)$ and its derivative $d{p}_1(x_e)/d{x}_e$ are depicted in Figure 5a. $d{p}_1(x_e)/d{x}_e$ is a continuous and bounded function. The outer-loop sliding surface is shown in Figure 5b. When $\left|x_e\right|>\xi$, $p_1(x_e)$ approaches ${\rho }_{\infty }$ as the absolute value of the error increases, ensuring the stability of the convergence rate when the orchard mobile robot is far from the target trajectory. When $\left|x_e\right|<\xi$, $p_1(x_e)$ varies between ${\rho }_0$ and 1. As the robot approaches the target trajectory, this ensures rapid finite-time convergence of the position error, with $p_1(x_e)$=1 at $\left|x_e\right|>\xi$.

As shown in Figure 5b, a comparison is made between $p=p_1(x_e)$ and the linear sliding mode $p=1$. When $\left|x_e\right|>\xi$, and the system is on the sliding surface, far from the target trajectory, the system exhibits smooth convergence, with a convergence rate strictly lower than exponential convergence (linear sliding mode $p=1$). When $\left|x_e\right|<\xi$, the system’s convergence rate accelerates and exceeds exponential convergence, ensuring finite-time convergence of the system.

(2)

The sliding mode variable for inner-loop attitude tracking control is as follows:

$$s_3={\theta }_e-{\theta }_0+{\displaystyle {\int }_0^t{k}_3}{\xi }_3{\left|{\displaystyle \frac{{\theta }_e(\tau )}{{\xi }_3}}\right|}^{p_3({\theta }_e)}\tanh \left({\displaystyle \frac{{\theta }_e(\tau )}{\phi }}\right)\mathrm{d}\tau$$

(21)

The sliding surface is defined as $s_3 = 0$, with parameters selected as $0<{\rho }_0<1$ and ${\rho }_{\infty }>1$. When ${\theta }_e>\xi$, $1<{\rho }_3({\theta }_e)<{\rho }_{\infty }$, and when $0<{\theta }_c<\xi$, ${\rho }_0<{\rho }_3({\theta }_e)<1$.

As shown in Figure 6a, the function plots of $p_3({\theta }_e)$ and its derivative $d{p}_3({\theta }_e)/d{\theta }_e$ are presented. The first-order derivative function exhibits continuity and boundedness. When $\left|{\theta }_e\right|>\xi$, $p_3({\theta }_e)$ approaches ${\rho }_{\infty }$, the absolute value of the error increases, thereby accelerating the convergence rate when the orchard mobile robot is far from the target trajectory. When $\left|{\theta }_e\right|<\xi$, $p_3({\theta }_e)$ approaches ${\rho }_0$ as the absolute value of the error increases. At $\left|{\theta }_e\right|=\xi$, the value of $p_3({\theta }_e)$ is equal to 1, ensuring parameter continuity and enabling smooth control transitions as the system error crosses the critical value $\xi$. This ensures that the convergence rate of the inner-loop system remains significantly faster than that of the outer loop.

As shown in Figure 6b, when the system operates on the designed sliding surface, global fast convergence is achieved. Compared to the constant value $p=1$, the phase trajectory corresponding to $p_3({\theta }_e)$ demonstrates a convergence rate strictly exceeding exponential convergence. The convergence rate of the inner-loop system is accelerated, ensuring finite-time convergence of the system.

By designing the subsequent reaching law, the system state is ensured to remain within the sliding surface, thereby controlling the system’s convergence characteristics through adjustments to the sliding surface and enhancing the controllability of the trajectory tracking process. Furthermore, by regulating the convergence speeds of the inner and outer loops and dynamically adjusting the critical system error value $\xi$ for both loops, the convergence speed of the inner loop can be guaranteed to be significantly faster than that of the outer loop.

2.4. Design of the Reaching Law

During system operation, the high-frequency switching of the sign function can still cause slight chattering issues. To suppress chattering, this paper proposes a boundary layer-variable sinusoidal saturation piecewise function [21], where $s$ represents the sliding surface.

$$f(s)=\left\{\begin{array}{ll}1\hfill & s\ge B\hfill \\ \sin (\eta s)\hfill & -B<s<B\hfill \\ -1\hfill & s\le -B\hfill \end{array}\right.$$

(22)

where $\eta =\pi /(2B)$, $B$ represents the boundary layer.

As shown in Figure 7, compared to other switching functions, this switching function remains continuous within the boundary layer and exhibits a faster convergence rate. In sliding mode control (SMC), by adjusting the width of the boundary layer $B$ and the slope of the function, the system’s chattering, response, accuracy, and stability can be directly influenced. Therefore, an analysis of $B$ is conducted. The effect of different $B$ values on $f(s)$ is illustrated in Figure 8, where $B$ represents an adjustable boundary layer. By tuning the value of $B$, the system can achieve smoother and more stable transitions during switching. A larger $B$ value makes $f(s)$ smoother, enhancing the system’s anti-chattering performance and ensuring stable operation. Conversely, a smaller $B$ value makes $f(s)$ steeper, resulting in faster system responses but increasing sensitivity to variations in $s$.

In the design of sliding mode controllers, the reaching law plays a crucial role in improving the dynamic performance of sliding mode control. A commonly used reaching law is as follows:

The exponential reaching law (ERL) is designed as follows:

$$\dot{s}=-k_{11}f(s)-k_{12}s$$

(23)

where $k_{11}$ and $k_{12}$ are control parameters, with $k_{11}>0$ and $k_{12}>0$.

The power reaching law (PRL) is designed as follows:

$$\dot{s}=-k_{21}|s{|}^{{\alpha }_{21}}f(s)$$

(24)

where $k_{21}$ and ${\alpha }_{21}$ are control parameters, with $k_{21}>0$ and $0<{\alpha }_{21}<1$.

The dual-power reaching law (DPRL) is expressed as follows:

$$\dot{s}=-k_{31}|s{|}^{{\alpha }_{31}}f(s)-k_{32}|s{|}^{{\beta }_{31}}f(s)$$

(25)

where $k_{31}$, $k_{32}$, ${\alpha }_{31}$, and ${\beta }_{31}$ are control parameters, with $k_{31}>0$, $k_{32}>0$, $0<{\alpha }_{31}<1$, and ${\beta }_{31}>0$.

Compared to the PRL algorithm, the traditional DPRL algorithm has advantages in terms of speed but still suffers from certain chattering issues. The exponential reaching law algorithm ensures good dynamic performance when the system reaches the sliding surface, thereby mitigating the chattering effect to some extent. On this basis, the improved dual-power reaching law (Improved DPRL) is proposed. By combining Equations (24) and (25), the reaching law is designed as follows:

$$\dot{s}=-k_{41}f(s)-k_{42}s-k_{43}|s{|}^{{\alpha }_{41}}f(s)-k_{44}|s{|}^{{\beta }_{41}}f(s)$$

(26)

where $k_{41}$, $k_{42}$, $k_{43}$, $k_{44}$, ${\alpha }_{41}$, and ${\beta }_{41}$ are control parameters, with $k_{41}>0$, $k_{42}>0$, $0<{\alpha }_{41}<1$, ${\beta }_{41}>0$, $k_{43}>0$, and $k_{44}>0$.

(1)

Existence condition validation

Theorem 1.

In the region where the sliding surface s = 0, for the system state to reach the sliding surface within finite time, the condition for the existence of the sliding mode is as follows:

$$\left\{\begin{array}{l}\underset{s\to 0^{+}}{\lim }\dot{s}<0\hfill \\ \underset{s\to 0^{-}}{\lim }\dot{s}>0\hfill \end{array}\right.$$

(27)

To verify whether Equation (26) satisfies Equation (27), we observe the following:

$$\underset{s\to 0^{+}}{\lim }\dot{s}=\underset{s\to 0^{+}}{\lim }\left[k_{41}f(s)-k_{42}s-k_{43}|s{|}^{{\alpha }_{41}}f(s)-k_{44}|s{|}^{{\beta }_{41}}f(s)\right]$$

(28)

$$\underset{s\to 0^{-}}{\lim }\dot{s}=\underset{s\to 0^{-}}{\lim }\left[k_{41}f(s)-k_{42}s-k_{43}|s{|}^{{\alpha }_{41}}f(s)-k_{44}|s{|}^{{\beta }_{41}}f(s)\right]$$

(29)

When $s\to 0^{+}$, $\dot{s}<0$, and when $s\to 0^{-}$, $\dot{s}>0$. Therefore, the Improved DPRL proposed in this paper satisfies the existence condition of the sliding mode.

(2)

Reaching condition validation

Theorem 2.

The proposed reaching law sliding mode s can reach the equilibrium point under its influence.

To verify whether the system satisfies the dynamic reaching conditions of the sliding mode, the Lyapunov function is chosen as follows:

$$V={\displaystyle \frac{1}{2}}s{s}^T$$

(30)

By differentiating Equation (30) and combining it with Equation (26), we obtain the following:

$$\dot{V}=s\dot{s}=-k_{41}sf(s)-k_{42}{s}^2-k_{43}|s{|}^{{\alpha }_{41}+1}f(s)-k_{44}|s{|}^{{\beta }_{41}+1}f(s)<0$$

(31)

From Equation (31), it can be concluded that the system is asymptotically stable and satisfies the reaching condition. Through the validation of the existence and reaching conditions for the sliding mode of the Improved DPRL, this reaching law meets the design requirements.

In the dual closed-loop control design for orchard mobile robots, the inner loop must account for external disturbances to maintain precise attitude tracking, while the outer loop must ensure the continuity of output signals. The reaching law for the outer-loop position controller is designed as follows:

$$u_{x1}=-k_{41}f(s_1)-k_{42}{s}_1-k_{43}|s_1{|}^{{\alpha }_{41}}f(s_1)-k_{44}|s_1{|}^{{\beta }_{41}}f(s_1)$$

(32)

$$u_{y2}=-k_{41}f(s_2)-k_{42}{s}_2-k_{43}|s_2{|}^{{\alpha }_{41}}f(s_2)-k_{44}|s_2{|}^{{\beta }_{41}}f(s_2)$$

(33)

The reaching law for the inner-loop attitude controller is designed as follows:

$$u_{\theta 3}=-k_{41}f(s_3)-k_{42}{s}_3-k_{43}|s_3{|}^{{\alpha }_{41}}f(s_3)-k_{44}|s_3{|}^{{\beta }_{41}}f(s_3)$$

(34)

Based on the control law design of the proposed Improved DPRL-TVITSMC algorithm, the final control laws for the inner and outer loops are obtained as follows:

$$\left\{\begin{array}{l}{u}_1={\dot{x}}_d-k_1{\xi }_1{\left|{\displaystyle \frac{x_e}{{\xi }_1}}\right|}^{p_1(x_e)}\tanh \left({\displaystyle \frac{x_e}{\phi }}\right)+u_{x1}\hfill \\ u_2={\dot{y}}_d-k_2{\xi }_2{\left|{\displaystyle \frac{y_e}{{\xi }_2}}\right|}^{p_2(y_e)}\tanh \left({\displaystyle \frac{y_e}{\phi }}\right)+u_{y2}\hfill \end{array}\right.$$

(35)

$$u_3={\dot{\theta }}_d-k_3{\xi }_3{\left|{\displaystyle \frac{{\theta }_e}{{\zeta }_3}}\right|}^{p_3\left({\theta }_e\right)}\tanh \left({\displaystyle \frac{{\theta }_e}{\phi }}\right)+u_{\theta 3}$$

(36)

2.5. Stability Analysis

The stability analysis of the orchard mobile robot system (6) under the recommended control methods (35) and (36) is presented in two steps:

Step 1: Prove the convergence and stability of the outer-loop system.

When the system state is on the sliding surface,$s_1 = s_2 = 0$. Taking the time derivative of the sliding mode variables $s_1$ and $s_2$ in the outer-loop attitude tracking system as given in Equation (20), we have the following:

$$\left\{\begin{array}{l}{\dot{x}}_e+k_1{\xi }_1{\left|{\displaystyle \frac{x_e}{{\xi }_1}}\right|}^{p_1(x_e)}\tanh \left({\displaystyle \frac{x_e}{\phi }}\right)=0\hfill \\ {\dot{y}}_e+k_2{\xi }_2{\left|{\displaystyle \frac{y_e}{{\xi }_2}}\right|}^{p_2(y_e)}\tanh \left({\displaystyle \frac{y_e}{\phi }}\right)=0\hfill \end{array}\right.$$

(37)

Choose the Lyapunov function $V_1=1/2(s_1^2+s_2^2)$, which clearly satisfies $V_1\ge 0$, and when $s_1 = s_2 = 0$, $V_1=0$. Taking the derivative of $V_1$:

$${\dot{V}}_1=s_1{\dot{s}}_1+s_2{\dot{s}}_2$$

(38)

Substitute the reaching laws $u_{x1}$ and $u_{y2}$ from Equations (32) and (33):

$${\dot{V}}_1=-k_{41}{\displaystyle \sum _{i=1}^2{s}_i}f(s_i)-k_{42}{\displaystyle \sum _{i=1}^2{s}_i^2}-k_{43}{\displaystyle \sum _{i=1}^2|}{s}_i{|}^{{\alpha }_{41}+1}|f(s_i)|-k_{44}{\displaystyle \sum _{i=1}^2|}{s}_i{|}^{{\beta }_{41}+1}|f(s_i)|$$

(39)

When $\left|s_i\right|\ge B$, $f(s_i)=\mathrm{sgn}(s_i)$. When $\left|s_i\right|\le B$, $f(s_i)=\sin (\eta s_i)$ and has the same sign as $s_i$. The components in Equation (39) are defined as follows:

$$k_{41}{\displaystyle \sum _{i=1}^2{s}_i}f(s_i)⩾0$$

(40)

$$k_{42}{\displaystyle \sum _{i=1}^2{s}_i^2}⩾0$$

(41)

$$k_{43}{\displaystyle \sum _{i=1}^2|}{s}_i{|}^{{\alpha }_{41}+1}|f(s_i)|⩾0$$

(42)

$$k_{44}{\displaystyle \sum _{i=1}^2|}{s}_i{|}^{{\beta }_{41}+1}|f(s_i)|⩾0$$

(43)

Through derivation, it can be concluded that ${\dot{V}}_1\le 0$, and ${\dot{V}}_1=0$ if and only if $s_{1 }= s_2 = 0$. According to the Lyapunov stability theorem, since $V_1$ is positive definite and ${\dot{V}}_1$ is negative definite, the outer-loop attitude control subsystem of the orchard mobile robot’s dual closed-loop trajectory tracking control system is asymptotically stable. The position errors $(x_e,y_e)$ asymptotically converge to zero, thereby ensuring the asymptotic stability of the system.

Step 2: Prove the convergence and stability of the inner-loop system, and analyze the steady-state error.

When the system is subjected to uncertain bounded disturbances, this reaching law can drive the sliding mode state to converge only to a neighborhood of the equilibrium at the origin—termed the steady-state error bound. Considering the attitude tracking error dynamics, differentiating the sliding mode variable $s_3$ of the inner-loop attitude tracking system in (21) yields the following:

$${\dot{s}}_3={\dot{\theta }}_e+k_3{\zeta }_3{\left|{\displaystyle \frac{{\theta }_e}{{\zeta }_3}}\right|}^{p_3({\theta }_e)}\tanh \left({\displaystyle \frac{{\theta }_e}{\phi }}\right)=0$$

(44)

Choose the Lyapunov function $V_2=1/2s_3^2$. This function clearly satisfies $V_2\ge 0$, and when $s_3=0$, $V_2=0$. Taking the derivative of $V_2$ and substituting the reaching law $u_{\theta 3}$ from Equation (34):

$$\begin{array}{l}{\dot{V}}_2=s_3{\dot{s}}_3\\ =\left(-k_{41}f(s_3)-k_{42}{s}_3-k_{43}\left|s_3{|}^{{\alpha }_{41}}\right|f(s_3)|-k_{44}|s_3{|}^{{\beta }_{41}}|f(s_3)|+d\right)s_3=\\ \left(-k_{41}{s}_{3}f(s_3)-k_{42}{s}_3^2-k_{43}\left|s_3{|}^{{\alpha }_{41}+1}\right|f(s_3)|-k_{44}|s_3{|}^{{\beta }_{41}+1}|f(s_3)|+M\left|s_3\right|\right)\le \\ -k_{42}{s}_3^2-k_{43}|s_3{|}^{{\alpha }_{41}+1}-k_{44}|s_3{|}^{{\beta }_{41}+1}+M\left|s_3\right|=\\ -\left(k_{42}{s}_3+k_{43}|s_3{|}^{{\alpha }_{41}}+k_{44}|s_3{|}^{{\beta }_{41}}-M\right)\left|s_3\right|\end{array}$$

(45)

To ensure system stability, the following conditions must be satisfied:

$$k_{42}{s}_3+k_{43}|s_3{|}^{{\alpha }_{41}}+k_{44}|s_3{|}^{{\beta }_{41}}-M⩾0$$

(46)

Thus

$$|s_3|⩾{\displaystyle \frac{M}{k_{42}+k_{43}|s_3{|}^{{\alpha }_{41}-1}+k_{44}|s_3{|}^{{\beta }_{41}-1}}}$$

(47)

When the sliding variable satisfies (47), we have ${\dot{V}}_2\le 0$; hence, the sliding variable converges to the bound $\left|s_3\right|=M/(k_{42}+k_{43}|s_3{|}^{{\alpha }_{41}-1}+k_{44}|s_3{|}^{{\beta }_{41}-1})$, i.e., in the presence of an external disturbance $d$, the steady-state error bound of the sliding variable is $\left|s_3\right|\le M/(k_{42}+k_{43}|s_3{|}^{{\alpha }_{41}-1}+k_{44}|s_3{|}^{{\beta }_{41}-1})$. In the absence of the external disturbance $d$, the stability analysis of the inner loop is identical to that of the outer loop; the proof follows the same steps and is therefore omitted. In this case, all terms satisfy ${\dot{V}}_2⩽0$, and ${\dot{V}}_2=0$ holds if and only if $s_3 = 0$. From the positive definiteness of $V_2$ and the negative definiteness of ${\dot{V}}_2$, it follows that the inner-loop attitude control subsystem of the double-loop trajectory tracking controller for the orchard mobile robot is asymptotically stable, and the attitude error ${\theta }_e$ asymptotically converges to zero.

3. Results and Discussion

3.1. Simulation Results of the Control System

MATLAB/Simulink was employed to conduct the simulation experiments, with the kinematic model of the orchard mobile robot serving as the control object. The initial reference attitude of the orchard mobile robot is set to $q$ = [0 m, 0 m, 0 rad], and the parameters are chosen as $k_1 = k_2 = 3$, $k_3 =3$, ${\epsilon }_1={\epsilon }_2=0.1$, ${\epsilon }_3=2$, ${\lambda }_1={\lambda }_2={\lambda }_3=0.4$, ${\rho }_{\infty }^1={\rho }_{\infty }^2=0.6$, ${\rho }_{\infty }^3=1.5$, ${\xi }_1={\xi }_2=1.6 \mathrm{m}$, ${\xi }_3=\pi /36 \mathrm{rad}$, $\phi =0.1$, $B = 0.1$, and $R = 100$ (The selection of the parameter $R$ is given in Appendix A). The reference trajectory is given by Equation (48).

$$\left\{\begin{array}{l}{x}_d=t+1\hfill \\ y_d=\sin (0.5x_d)+0.5x_d+1\hfill \end{array}\right.$$

(48)

To verify the superiority of the proposed Improved DPRL-TVITSMC in the trajectory tracking control of orchard mobile robots, simulation comparisons were conducted with trajectory tracking controllers designed using ERL-TVITSMC, PRL-TVITSMC, and DPRL-TVITSMC. The simulation parameters for the reaching laws are listed in

Table 1. Simulation parameter settings for ERL-TVITSMC.

Parameters	Sliding Surface
	${\mathit{s}}_1$	${\mathit{s}}_2$	${\mathit{s}}_3$
k11	0.2	0.2	1
k12	0.2	0.2	5

,

Table 2. Simulation parameter settings for PRL-TVITSMC.

Parameters	Sliding Surface
	${\mathit{s}}_1$	${\mathit{s}}_2$	${\mathit{s}}_3$
k21	0.3	0.2	1.5
α21	0.5	0.5	0.5

,

Table 3. Simulation parameter settings for DPRL-TVITSMC.

Parameters	Sliding Surface
	${\mathit{s}}_1$	${\mathit{s}}_2$	${\mathit{s}}_3$
k31	0.2	0.2	2
k32	0.2	0.2	2
α31	0.1	0.1	0.2
β31	1.1	1.1	1.2

and

Table 4. Simulation parameter settings for Improved DPRL-TVITSMC.

Parameters	Sliding Surface
	${\mathit{s}}_1$	${\mathit{s}}_2$	${\mathit{s}}_3$
k41	0.2	0.2	1
k42	0.2	0.2	5
k43	0.2	0.2	2
α41	0.1	0.1	0.2
k44	0.2	0.2	2
β41	1.1	1.1	1.2

.

As illustrated in Figure 8a, a comparative simulation of four reaching laws was conducted. The trajectory tracking results indicate that, compared to the other three reaching laws, the Improved DPRL-TVITSMC achieves closer adherence to the desired trajectory with significantly reduced trajectory deviation. This demonstrates the method’s superior accuracy in global trajectory tracking, effectively minimizing tracking errors.

Figure 8b–d present the convergence characteristics of the $x$, $y$, and $\theta$ errors. For the Improved DPRL-TVITSMC, the ${\theta }_e$ stabilizes within 0.01 rad after 0.144 s, $x_e$ stabilizes within 0.01 m after 0.966 s, and $y_e$ stabilizes within 0.01 m after 0.753 s. The convergence times of the Improved DPRL-TVITSMC are faster than those of the other methods, with steady-state errors approaching zero and fluctuation amplitudes lower than those of the other curves. This method enhances the overall tracking performance of the orchard mobile robot in planar motion.

Figure 8e–g analyze the trajectory tracking control output characteristics of the orchard mobile robot. For the Improved DPRL-TVITSMC, the linear velocity $v$ and angular velocity $w$ outputs exhibit significantly lower fluctuation amplitudes compared to the other methods, resulting in smoother outputs. This effectively suppresses control input chattering, ensuring motion stability. Regarding the differentiator output, the reaching law quickly captures the change rates of the attitude angles, stabilizing the output signals in a short period while effectively suppressing high-frequency noise and fluctuations.

Figure 8h–j analyze the variations in the sliding mode variables $s_1$, $s_2$, and $s_3$ under the four reaching laws. For the Improved DPRL-TVITSMC, $s_1$ stabilizes within 0.01 after 0.975 s, $s_2$ stabilizes within 0.01 after 0.966 s, and $s_3$ stabilizes within 0.01 after 0.144 s. The convergence times are faster compared to the other methods, with steady-state errors approaching zero. After reaching the sliding surface, the method adaptively adjusts the chattering generated according to the system’s state variables, ultimately converging to the equilibrium point. In the dual closed-loop structure, the Improved DPRL-TVITSMC demonstrates the superior convergence characteristics of the sliding mode variables for both the outer-loop position tracking and the inner-loop control. It quickly guides the sliding mode variables to the sliding surface and maintains stability, enhancing the system’s robustness and trajectory tracking accuracy.

The performance of the four reaching law control strategies in tracking tasks for the position $(x,y)$ and angle $\theta$ was quantitatively evaluated based on three performance evaluation criteria: the Integral of Absolute Error (IAE), Integral of Squared Error (ISE), and Integral of Time-weighted Absolute Error (ITAE). The results are listed in

Table 5. Convergence Evaluation of Position-Tracking Error for Reaching-Law Comparison.

Control Method	IAE/m	ISE/m	ITAE/m
ERL-TVITSMC	0.8274	0.8245	0.3833
PRL-TVITSMC	1.009	0.8948	2.808
DPRL-TVITSMC	0.8648	0.8452	0.6234
Improved DPRL-TVITSMC	0.7629	0.7698	0.2754

and

Table 6. Convergence Evaluation of Angle-$\theta$ Tracking Error for Reaching-Law Comparison.

Control Method	IAE/rad	ISE/rad	ITAE/rad
ERL-TVITSMC	0.0675	0.0297	0.2321
PRL-TVITSMC	0.1302	0.0429	1.034
DPRL-TVITSMC	0.0763	0.0333	0.2686
Improved DPRL-TVITSMC	0.0484	0.0229	0.1545

.

The IAE reflects the cumulative error, with smaller values indicating better cumulative error control. The ISE is more sensitive to large errors, with smaller values signifying stronger suppression of large errors. The ITAE takes into account both the duration of the error and the regulation efficiency, with smaller values representing higher overall regulation efficiency. From the results across the three performance evaluation criteria, it is evident that the Improved DPRL-TVITSMC exhibits the best performance in position $(x,y)$ and angle $\theta$ tracking tasks.

3.2. Comparative Analysis of Results with Different Control Methods

To further demonstrate the superiority of the proposed improved dual closed-loop SMC method, the improved reaching law was applied and compared with the dual closed-loop SMC method, dual closed-loop integral SMC (ISMC) method, and dual closed-loop integral terminal SMC (ITSMC) method. The equations used in the simulation are shown in

Table 7. Controller settings.

Sliding Mode Controller	Equation
Improved DPRL-SMC	$s=ke$
Improved DPRL-ISMC	$s=e+{\displaystyle {\int }_0^{t}k}\xi \left|{\displaystyle \frac{e(\tau )}{\xi }}\right|\mathrm{sgn}(e(\tau ))dt$
Improved DPRL-ITSMC	$s=e+{\displaystyle {\int }_0^{t}k}\xi {\left|{\displaystyle \frac{e(\tau )}{\xi }}\right|}^p\mathrm{sgn}(e(\tau ))dt$

Where $p=5/7$.

, and the parameters are as previously defined.

The simulation comparison results are presented in Figure 9, which includes the trajectory tracking results, position and angle tracking errors, linear and angular velocities, and the convergence rates of the inner and outer sliding mode surfaces under the SMC, ISMC, ITSMC, and TVITSMC.

The trajectory of the orchard mobile robot under the desired trajectory is shown in Figure 9a. The improved dual closed-loop SMC method achieves the best trajectory tracking performance, with the curve closely following the desired trajectory. This indicates that the proposed method converges more quickly and stably to the desired trajectory in complex trajectory tracking tasks, significantly outperforming the traditional dual closed-loop SMC, dual closed-loop ISMC, and dual closed-loop ITSMC methods.

Figure 9b–d display the comparisons of the tracking errors $x_e$, $y_e$, and ${\theta }_e$. These results show that, compared to other dual closed-loop control methods, the proposed method regulates the error convergence process more efficiently, allowing the system to return to the target trajectory more effectively. The proposed algorithm stabilizes ${\theta }_e$ within 0.01 rad after 0.144 s, $x_e$ within 0.01 m after 0.966 s, and $y_e$ within 0.01 m after 0.753 s. The simulation errors converge continuously over time, eventually approaching zero. Furthermore, the method achieves high accuracy with a small range of variation, further validating the superiority of the proposed trajectory tracking control scheme.

Figure 9e–f illustrate the control signals $v$, $w$, and the differentiator dθ_d of the orchard mobile robot. The control algorithm proposed in this paper exhibits smooth outputs without significant chattering, which is more conducive to the normal operation of actuators. In contrast, other controllers experience noticeable chattering, with the chattering of the Improved DPRL-ITSMC being the most pronounced.

As shown in Figure 9h–j, all four control systems reach the sliding mode surface within a short time without chattering upon reaching it. Compared to the other methods, the proposed method achieves the fastest convergence to the sliding mode surface.

In summary, the orchard mobile robot controlled by the proposed method exhibits a faster convergence speed and smaller tracking errors, demonstrating that the proposed control method offers superior tracking performance and stronger disturbance rejection capability. Using three performance indices—the integral of absolute error (IAE), the integral of squared error (ISE), and the integral of time-weighted absolute error (ITAE)—the performances of four sliding-mode control strategies for the position $\left(x, y\right)$ and angle $\theta$ tracking tasks were quantitatively evaluated; the results are listed in

Table 8. Convergence Evaluation of Position-Tracking Error for a Comparison of Sliding-Mode Controllers.

Control Method	IAE/m	ISE/m	ITAE/m
Improved DPRL-SMC	3.362	4.205	2.904
Improved DPRL-ISMC	1.265	0.8502	1.293
Improved DPRL-TSMC	0.8948	0.7714	0.5837
Improved DPRL-TVITSMC	0.7629	0.7698	0.2754

and

Table 9. Convergence Evaluation of Angle-$\theta$ Tracking Error for a Comparison of Sliding-Mode Controllers.

Control Method	IAE/rad	ISE/rad	ITAE/rad
Improved DPRL-SMC	0.1927	0.1042	0.2089
Improved DPRL-ISMC	0.0926	0.0445	0.5954
Improved DPRL-TSMC	0.1576	0.0442	2.873
Improved DPRL-TVITSMC	0.0484	0.0229	0.1545

. Synthesizing the results across the three indices indicates that Improved DPRL-TVITSMC achieves the best performance in both the position $\left(x, y\right)$ and angle $\theta$ tracking tasks.

To simulate a realistic orchard path, this study selected an 18 m × 25 m rectangular lawn as the working area for the orchard mobile robot, where a bypass path was planned. Considering the presence of uncertain disturbances in the system, an external disturbance $d=0.5\sin (2\pi t)$ was added to the attitude angle $\theta$ between 35 s and 45 s, where $\left|d\right| < 0.5$. The simulation environment and controller parameters were consistent with those used in previous studies. The initial pose of the orchard mobile robot was set to (0 m, 0 m, 0 rad), the starting position of the path was defined as (2 m, 2 m), the straight path length was set to 20 m, and the circular arc radius was set to 1 m. During the bypass path tracking, when the system reached steady state, it was observed that $s<1$. Therefore, the steady-state error boundary was $\left|s\right|⩽M/(k_{42})$. Substituting the simulation parameters, the boundary was determined to be $\left|s\right|⩽0.1$. The simulation results are presented in Figure 10.

As shown in Figure 10a, the proposed controller enables the orchard mobile robot to converge smoothly to the desired trajectory as expected, achieving satisfactory tracking performance. Figure 10b demonstrates that the simulation errors of the tracking trajectory continuously converge over time, with ${\theta }_e$ stabilizing within 0.01 rad after 0.17 s, $x_e$ stabilizing within 0.01 m after 1.23 s, and $y_e$ stabilizing within 0.01 m after 1.04 s. Due to external disturbances present between 35 and 45 s, and with $\left|{\theta }_e\right|<0.015$, the system was unable to converge to the equilibrium point, but ${\theta }_e$ rapidly recovered after 44.91 s. As shown in Figure 10c, the sliding mode variables $s_1$, $s_2$, and $s_3$ exhibit a stable convergence trend after 1.5 s, while $s_3$ is subjected to external disturbances between 35 and 45 s but remains within $\left|s\right|<0.015$. This demonstrates the effective disturbance rejection capability of the inner-loop control. The system’s steady-state error is smaller than the steady-state error boundary derived in this study, indicating that the proposed method can track the target trajectory quickly and stably, with the inner-loop attitude tracking system converging significantly faster than the outer loop.

In conclusion, the Improved DPRL-TVITSMC effectively suppresses the influence of disturbances during path tracking, ensuring trajectory tracking accuracy and system stability. This provides strong support for the reliable operation of orchard mobile robots in complex environments. The impact of system uncertainties and external disturbances may hinder the system from converging to the sliding mode surface, thereby affecting control performance. Introducing a disturbance observer could be a potential solution, and related methods require further investigation.

4. Conclusions

This study focuses on an orchard mobile robot, for which a kinematic model is constructed and a time-varying integral terminal sliding mode controller (TVITSMC) is designed to enhance the tunability of convergence and the disturbance rejection capability; by assigning differentiated parameter settings to the inner and outer sliding mode loops, the system-level requirement is satisfied that the outer loop exhibits continuous, stable convergence while the inner loop converges rapidly; a sinusoidal saturation switching function with a variable boundary is employed, whereby tuning the boundary layer effectively suppresses the high-frequency chattering inherent to sliding mode control; an Improved DPRL is introduced to accelerate the approach to the sliding surface and suppress disturbance-induced chattering, thereby strengthening the overall robustness of the control system. The simulation results are as follows:

(1)

Comparative simulations across reaching laws and sliding mode controller variants indicate that the Improved DPRL-TVITSMC achieves faster convergence and higher accuracy in complex trajectory tracking tasks, reducing both tracking error and chattering.

(2)

For an external disturbance $\left|d\right| < 0.5$ acting in the inner loop, the simulations yield $\left|s\right|<0.015$, and the upper bound on the tracking error derived from the stability analysis satisfies $\left|s\right|⩽0.1$.

(3)

A preset detour trajectory demonstrates that the proposed tracking controller enables the orchard mobile robot to follow the detour effectively; ${\theta }_e$ settles within 0.01 rad after 0.17 s, $x_e$ settles within 0.01 m within 1.23 s, and $y_e$ settles within 0.01 m after 1.04 s. These results highlight favorable rapidity, accuracy, and robustness.

Although the proposed trajectory tracking algorithm exhibits stable performance in simulation, the following directions merit further investigation:

(1)

The present study considers only idealized simulation scenarios and representative disturbances, and lacks hardware experiments on a real orchard robot. As future work, we will build a physical platform and conduct comparative experiments to verify the controller’s robustness and portability under uneven terrain and external disturbances.

(2)

To assess the robustness margins and reproducibility of the control strategy, subsequent experiments will report key implementation details—including the chassis and actuation scheme, sensor types and mounting poses, computing hardware and real-time system, terrain conditions, and path settings—and will evaluate the algorithm’s adaptability in non-ideal scenarios.