Application of Improved Sliding Mode and Artificial Neural Networks in Robot Control

Abstract

Mobile robots are autonomous devices capable of self-motion, and are utilized in applications ranging from surveillance and logistics to healthcare services and planetary exploration. Precise trajectory tracking is a crucial component in robotic applications. This study introduces the use of improved sliding surfaces and artificial neural networks in controlling mobile robots. An enhanced sliding surface, combined with exponential and hyperbolic tangent approach laws, is employed to mitigate chattering phenomena in sliding mode control. Nonlinear components of the sliding control law are estimated using artificial neural networks. The weights of the neural networks are updated online using a gradient descent algorithm. The stability of the system is demonstrated using Lyapunov theory. Simulation results in MATLAB/Simulink R2024a validate the effectiveness of the proposed method, with rise times of 0.071 s, an overshoot of 0.004%, and steady-state errors approaching zero meters. Settling times were 0.0978 s for the x-axis and 0.0902 s for the y-axis, and chattering exhibited low amplitude and frequency.

1. Introduction

Mobile robots, as semi-autonomous or autonomous systems, have the capability to navigate unstructured environments without direct human intervention [1,2,3]. These systems are equipped with sensors, processors, and locomotive mechanisms, enabling them to execute a wide range of tasks from simple to complex, such as cargo transport, space exploration, or providing assistance in medical and industrial applications. Mobile robots can autonomously locate and navigate [4,5,6] using algorithms and technologies such as artificial intelligence (AI), machine learning, and sensor-based mapping, efficiently interacting with their surroundings. They can also be remotely controlled via wireless connections. The field of mobile robotics continues to evolve with ongoing research and development aimed at enhancing automation capabilities and optimizing task performance across various industries.

In recent years, the robotics domain has made significant strides, propelled by the amalgamation of sophisticated control strategies and computational intelligence. Notably, Sliding Mode Control (SMC) [7,8] and Artificial Neural Networks (ANNs) [9] have emerged as key technologies to enhance the precision and adaptability of robotic systems. SMC is known for its robustness and effectiveness in dealing with non-linearities and system uncertainties, making it an ideal choice for dynamic and complex environments. Conversely, ANNs play a crucial role by introducing learning capabilities to robots, thereby facilitating real-time decision-making and adaptation based on sensory input.

Accurate trajectory tracking is a crucial component in robotic applications, underscoring the importance of advanced control methodologies. Various control strategies have been developed and implemented to enhance the precision and responsiveness of robots in following designated paths. Among these, linear trajectory control based on the kinematic model of the robot has been explored in [10], where the principles of linearization help in simplifying the control of complex robotic motions. Adaptive sliding control, detailed in [11], adjusts to changes in robot dynamics and environmental conditions to maintain path accuracy. Fuzzy control, discussed in [12], achieved target precision and maintained it within 2.23 s with a maximum error of less than 1 mm, showcasing its effectiveness in dealing with uncertainties and non-linearities. Furthermore, Proportional-Integral (PI) control presented in [13], and the Proportional-Derivative Plus (PD+) method developed in [14], where the Integral of Squared Error (ISI) was notably high at 8,705,235.5, with Root Mean Squared Error (RMSE) indices of 46.41 and 6.56, respectively, highlight the different approaches to minimizing steady-state error and improving settling time. Dynamic PI sliding control proposed in [15] integrates the robustness of sliding mode control with the simplicity of PI control, enhancing stability and reducing overshoot. Nonlinear adaptive control, designed and simulated in [16], successfully demonstrated a circular robot trajectory within 0.5 s without overshoot, maintaining negligible steady-state error. Additionally, Model Predictive Control (MPC) with friction compensation developed in [17] tested on an eight-shaped trajectory, showed Integral Absolute Error (IAE_xy) and Integral Angular Error (IAE_θ) of 1.517 m and 1.2907 rad, respectively, with corresponding Mean Absolute Errors (MAE_xy and MAE_θ) of 0.0997 m and 0.0655 rad, underscoring MPC’s capability to predict and correct future trajectory errors efficiently. Despite these advancements, the issue of chattering in sliding control laws, particularly noted in studies [11,15], has not been sufficiently addressed, indicating an area for further research and improvement in control strategies for mobile robots. This introduction sets the stage for discussing innovative control solutions that enhance trajectory tracking while minimizing the drawbacks of existing methods.

To address these challenges, this study proposes the use of an improved sliding surface based on Artificial Neural Networks (specifically, Radial Basis Function networks) for robot trajectory tracking control. By integrating these methods, the proposed hybrid control strategy aims to overcome the limitations of each approach while enhancing the overall performance of robotic systems. Specifically, this paper discusses the implementation of this hybrid control mechanism in various robotic applications, detailing its potential to improve stability, reduce chattering, and enhance adaptability to dynamic environmental changes. The improved sliding surface is designed to replace the classical sliding surface combined with exponential and hyperbolic tangent (tanh) [18,19,20] approach functions to minimize chattering phenomena. The RBF network [21,22] is employed to estimate the nonlinear components in the sliding control law, calculated based on Lyapunov stability theory. This paper is organized into four sections: Section 2 presents the application of the improved sliding surface and artificial neural networks to robots, results are presented in Section 3, and Section 4 includes a discussion and concludes the paper.

This study introduces an innovative approach to mobile robot control by integrating improved sliding surfaces with artificial neural networks. The proposed method aims to fill the knowledge gaps left by previous research by addressing the following unresolved issues:

• Reduction in Chattering: Traditional SMC techniques suffer from chattering, which reduces control accuracy and efficiency. This paper proposes the use of exponential and hyperbolic tangent approach laws combined with an improved sliding surface to significantly mitigate chattering.
• Real-Time Adaptation: While ANNs have been used in control systems, their application in real-time adaptation for mobile robots remains underexplored. This study employs Radial Basis Function (RBF) networks to estimate nonlinear components of the control law, updating weights online using a gradient descent algorithm. This real-time learning capability enhances the robot’s adaptability to dynamic changes in the environment.
• Comprehensive Validation: Many existing studies lack extensive validation of their proposed methods. This paper provides thorough simulation results using MATLAB/Simulink to demonstrate the effectiveness of the proposed control strategies. Key performance metrics such as rise time, settling time, steady-state error, and overshoot are evaluated to highlight the improvements achieved.

By addressing these critical gaps, the proposed hybrid control strategy not only enhances the performance of mobile robots, but also opens avenues for research in robotic control systems. The introduction of this innovative control mechanism promises to refine the operational capabilities of robots, making them more adaptable, efficient, and capable of operating autonomously in a wide range of environments.

The introduction of this innovative control strategy not only promises to refine the operational capabilities of robots, but also opens avenues for research in robotic control systems. This paper provides a comprehensive analysis of the theoretical foundations, supplemented by simulation results and practical implementations, to demonstrate the effectiveness and versatility of the combined approach.

2. Application of Improved Sliding Surfaces and Artificial Neural Networks in Robotics

2.1. Mathematical Equations for Robotics

This segment details the dynamic modeling of a tri-wheeled omnidirectional robot (Figure 1), which is based on the reference provided in [23,24,25,26,27,28]. Formulating the robot’s mathematical model is pivotal for the development of its control architecture.

In this study, we examine a three-wheeled omnidirectional mobile robot positioned on a solid plane. The robot, as depicted in the image, has its own coordinate system anchored at its center of gravity, along with a corresponding real-world coordinate system. The arrangement of its three-wheel assemblies endows the robot with flexible mobility, with the driving forces of each wheel denoted as $D_1, D_2,$ and $D_3$. These forces collaborate to enable the robot to move in any direction across the plane. The angle, $\Psi$, indicates the robot’s orientation relative to the real-world coordinate system, while the rotational angles associated with each wheel indicate their specific orientations in relation to the robot’s coordinate framework. The orientation and the forces combined provide a foundation for the robot to move accurately and respond to control directives, which requires precise coordination among the wheels to achieve the desired path.

$$\begin{array}{ll}\left[\begin{array}{c}{\ddot{x}}_W\\ {\ddot{y}}_W\\ \ddot{\varphi }\end{array}\right]& =\left[\begin{array}{ccc}{a}_1& -a_2\dot{\varphi }& 0\\ a_2\dot{\varphi }& a_1& 0\\ 0& 0& a_3\end{array}\right]\left[\begin{array}{c}{\dot{x}}_W\\ {\dot{y}}_W\\ \dot{\varphi }\end{array}\right]+\left[\begin{array}{ccc}{b}_1{\gamma }_1& b_1{\gamma }_2& 2b_1\mathrm{c}\mathrm{o}\mathrm{s}\varphi \\ b_1{\gamma }_3& b_1{\gamma }_4& 2b_1\mathrm{s}\mathrm{i}\mathrm{n}\varphi \\ b_2& b_2& b_2\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]+\left[\begin{array}{c}{D}_{f_x}\\ D_{f_y}\\ D_{f_{\varphi }}\end{array}\right]\\ & ={\mathit{A}}_W\mathit{\beta }+{\mathit{B}}_W\mathit{U}+{\mathit{D}}_f\end{array}$$

(1)

With ${\mathit{D}}_f=\left[\begin{array}{ccc}{D}_{f_x}& D_{f_y}& D_{f_{\varphi }}\end{array}\right]$ is the unknown system noise.

• where:

  $${\mathit{A}}_W=\left[\begin{array}{ccc}{a}_1& -a_2\dot{\varphi }& 0\\ a_2\dot{\varphi }& a_1& 0\\ 0& 0& a_3\end{array}\right]$$

  $${\mathit{B}}_W=\left[\begin{array}{ccc}{b}_1{\gamma }_1& b_1{\gamma }_2& 2b_1\cos \varphi \\ b_1{\gamma }_3& b_1{\gamma }_4& 2b_1\sin \varphi \\ b_2& b_2& b_2\end{array}\right]$$

  $$\mathit{U}={\left[\begin{array}{lll}{u}_1& u_2& u_3\end{array}\right]}^T$$

  $$a_1=\frac{-3c}{\left(3I_W+2M{r}^2\right)},a_2^{\prime }=\frac{2M{r}^2}{\left(3I_W+2M{r}^2\right)},a_3=\frac{-3c{L}^2}{\left(3I_W{L}^2+I_v{r}^2\right)}$$

  $$b_1=\frac{kr}{\left(3I_W+2M{r}^2\right)}{,b}_2=\frac{krL}{\left(3I_W+I_v{r}^2\right)}$$

  $$a_2=1-a_2^{\prime }=\frac{3I_W}{\left(3I_W+2M{r}^2\right)}$$

  $${\gamma }_1=-\sqrt{3}\sin \varphi -\cos \varphi , {\gamma }_2=\sqrt{3}\mathrm{s}\mathrm{i}\mathrm{n}\varphi -\mathrm{c}\mathrm{o}\mathrm{s}\varphi ;{\gamma }_3=\sqrt{3}\mathrm{c}\mathrm{o}\mathrm{s}\varphi -\mathrm{s}\mathrm{i}\mathrm{n}\varphi ;{\gamma }_4=-\sqrt{3}\mathrm{c}\mathrm{o}\mathrm{s}\varphi -\mathrm{s}\mathrm{i}\mathrm{n}\varphi$$

  where $L$ denotes the distance from any wheel to the robot’s center of mass; $k$ is the drive gain factor; $D_i$represents the driving force of each robot wheel; $r$ is the radius of each wheel; $c$ stands for the viscous resistance factor of the wheel; ${\omega }_i$ indicates the angular velocity of the robot; $I_R$ is the moment of inertia of each wheel around its driving shaft; and $u_i$ is the input torque for driving.

2.2. Controlling Robots Using an Improved Slider Interface

Schematic structure of the sliding controller based on an improved sliding mode control (ISMC), as shown in Figure 2.

Figure 2 depicts a schematic structure for a sliding control system applied to a mobile robot, employing an enhanced sliding method. At the heart of the diagram is the ISMC controller, an acronym for “Integral Sliding Mode Control”. This represents a robust control strategy aimed at mitigating the impact of uncertainties and ensuring system stability, despite changes in the robot’s dynamics or configuration. The reference signal (${\mathsf{\Gamma }}_d$) contains the desired parameters that the robot aims to achieve, such as position or orientation. This signal is input into a comparator, where it is subtracted from the actual feedback signal (not directly shown in the diagram) to generate the error signal ($\mathit{e}$). The error signal represents the difference between the current state and the desired state of the robot, providing critical information for the control process. The ISMC then processes the error signal to produce a control signal ($\mathit{U}$), with the primary goal of appropriately adjusting the robot’s actions to reduce this error. The mobile robot, with input parameters including position ($x_w, y_w$) and orientation ($\varphi$), will act based on the control signal, $\mathit{U}$, received. Feedback from the robot post-action ($\mathsf{\Gamma }$) becomes the input for the next control cycle, creating a closed-loop that allows for automatic adjustment and continuous improvement. This ensures that the robot not only reacts to the current controls, but also self-adjusts based on feedback, enhancing its ability to achieve goals accurately and efficiently. The schematic also suggests an improvement to the traditional sliding control mechanism, potentially related to enhanced noise handling or dynamic changes of the robot.

The discrepancy between the actual output, $\mathsf{\Gamma }$, and the reference input, ${\mathsf{\Gamma }}_d$, is as Equation (2):

$$\mathit{e}=\mathsf{\Gamma }-{\mathsf{\Gamma }}_d$$

(2)

where $\mathsf{\Gamma }={\left[\begin{array}{ccc}{x}_w& y_w& \varphi \end{array}\right]}^T$ represents the actual output, and ${\mathsf{\Gamma }}_d={\left[\begin{array}{ccc}{x}_d& y_d& {\varphi }_d\end{array}\right]}^T$ is the reference input of the robot.

The first and second derivatives of Equation (2) yield Equations (3) and (4):

$$\dot{\mathit{e}}=\dot{\mathsf{\Gamma }}-{\dot{\mathsf{\Gamma }}}_d$$

(3)

$$\ddot{\mathit{e}}=\ddot{\mathsf{\Gamma }}-{\ddot{\mathsf{\Gamma }}}_{\mathit{d}}=\left[\begin{array}{c}{\ddot{x}}_w\\ {\ddot{y}}_w\\ \ddot{\varphi }\end{array}\right]-\left[\begin{array}{c}{\ddot{x}}_d\\ {\ddot{y}}_d\\ {\ddot{\varphi }}_d\end{array}\right]=\dot{\mathit{X}}-{\ddot{\mathsf{\Gamma }}}_{\mathit{d}}={\mathit{A}}_w\mathit{X}+{\mathit{B}}_w\mathit{U}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_{\mathit{d}}$$

(4)

The developed sliding surface is as Equation (5) [29]:

$$\mathit{s}=2\dot{\mathit{e}}+\left(\mathit{\lambda }+2\mathit{\alpha }\right)\mathit{e}+{\mathit{\alpha }}^2{\int }_0^t\mathit{e}\left(\tau \right)d\tau$$

(5)

where $\mathit{\lambda }=diag({\lambda }_1,{\lambda }_2,{\lambda }_3)$, with each ${\lambda }_i>0$, and $\mathit{\alpha }=diag({\alpha }_1,{\alpha }_2,{\alpha }_3)$, with each ${\alpha }_j>0$.

Substituting Equation (4) into the derivative of Equation (5), we obtain Equation (6):

$$\dot{\mathit{s}}=2\left({\mathit{A}}_w\mathit{X}+{\mathit{B}}_w\mathit{U}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_{\mathit{d}}\right)+\left(\mathit{\lambda }+2\mathit{\alpha }\right)\dot{\mathit{e}}+{\mathit{\alpha }}^2\mathit{e}$$

(6)

The exponential approach law with the Hyperbolic tangent function (tanh) is as Equation (7) [30]:

$$\dot{\mathit{s}}=-\mathit{\eta }\tanh \left(\frac{\mathit{s}}{\mathit{\epsilon }}\right)-\mathsf{\kappa }\mathbf{s}$$

(7)

where $\mathit{\eta }=diag\left(\left[{\eta }_x,{\eta }_y,{\eta }_{\varphi }\right]\right),\mathit{\kappa }=diag\left(\left[{\kappa }_x,{\kappa }_y,{\kappa }_{\varphi }\right]\right),\mathit{\epsilon }=diag\left(\left[{\epsilon }_x,{\epsilon }_y,{\epsilon }_{\varphi }\right]\right)$, and for all ${\eta }_i,{\kappa }_i,{\epsilon }_i>0$.

The proposed control law for the robot is given by Equation (8):

$$\mathit{U}={\mathit{U}}_{ISMC-\mathrm{Tanh}}=-\frac{1}{2}{\mathit{B}}_W^{-1}\left[2\left({\mathit{A}}_W\mathit{X}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_d\right)+(\mathit{\lambda }+2\mathit{\alpha })\dot{\mathit{e}}+{\mathit{\alpha }}^2\mathit{e}+\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{\epsilon }}\right)+\mathit{\kappa }\mathit{s}\right]$$

(8)

The Lyapunov function is as Equation (9):

$$V=\frac{1}{2}{\mathit{s}}^2$$

(9)

At this point, the derivative of V with the tanh function is represented by Equation (10):

$$\begin{array}{ll}\dot{V}& =\mathit{s}\dot{\mathit{s}}=\mathit{s}\left[2\left({\mathit{A}}_W\mathit{X}+{\mathit{B}}_W\mathit{U}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_d\right)+(\mathit{\lambda }+2\mathit{\alpha })\dot{\mathit{e}}+{\mathit{\alpha }}^2\mathit{e}\right]\\ & =\mathit{s}\left[-\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{\epsilon }}\right)-\mathit{\kappa }\mathit{s}\right]=-\mathit{\eta }\mathit{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{\epsilon }}\right)-\mathit{\kappa }{\mathit{s}}^2\end{array}$$

(10)

According to [30], we have:

$$|\mathit{s}|-\mathit{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{\epsilon }}\right)\le \mathit{\mu }\mathit{\epsilon }$$

(11)

Then, we have Equation (12) and consequently Equation (13):

$$\mathit{\eta }|\mathit{s}|-\mathit{\eta }\mathit{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{\epsilon }}\right)\le \mathit{\mu }\mathit{\epsilon }$$

(12)

$$-\mathit{\eta }\mathit{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{\epsilon }}\right)\le -\mathit{\eta }|\mathit{s}|+\mathit{\eta }\mathit{\mu }\mathit{\epsilon }$$

(13)

Therefore, we can establish Equation (14) for the derivative of V:

$$\dot{V}=\mathit{s}\dot{\mathit{s}}\le -\mathit{\kappa }{\mathit{s}}^2-\mathit{\eta }|\mathit{s}|+\mathit{\eta }\mathit{\mu }\mathit{\epsilon }\le -\mathit{\kappa }{\mathit{s}}^2+\mathit{\eta }\mathit{\mu }\mathit{\epsilon }=-2\mathit{\kappa }V+\mathit{\beta }$$

(14)

Using Lemma 1.3 from [12], we obtain Equation (15):

$$\begin{array}{ll}& \dot{V}\le {\mathit{e}}^{-2\mathit{\kappa }\left(t-t_0\right)}V\left(t_0\right)+\mathit{\beta }{\mathit{e}}^{-2\mathit{\kappa }t}{\int }_{t_0}^t{\mathit{e}}^{2\mathit{\kappa }\tau }d\tau ={\mathit{e}}^{-2\mathit{\kappa }\left(t-t_0\right)}V\left(t_0\right)+\frac{\mathit{\beta }{\mathit{e}}^{-2\mathit{\kappa }t}}{2\mathit{\kappa }}\left({\mathit{e}}^{-2\mathit{\kappa }t}-{\mathit{e}}^{-2\mathit{\kappa }{t}_0}\right)\\ & ={\mathit{e}}^{-2\mathit{\kappa }\left(t-t_0\right)}V\left(t_0\right)+\frac{\mathit{\beta }}{2\mathit{\kappa }}\left(1-{\mathit{e}}^{-2\mathit{\kappa }\left(t-t_0\right)}\right)={\mathit{e}}^{-2\mathit{\kappa }\left(t-t_0\right)}V\left(t_0\right)+\frac{\mathit{\eta }\mathit{\mu }\mathit{\epsilon }}{2\mathit{\kappa }}\left(1-{\mathit{e}}^{-2\mathit{\kappa }\left(t-t_0\right)}\right)\end{array}$$

(15)

Thus, we conclude with Equation (16):

$$\underset{t\to \infty }{\lim }V\left(t\right)\le \frac{\mathit{\eta }\mathit{\mu }\mathit{\epsilon }}{2\mathit{\kappa }}$$

(16)

From inequality (16), it can be deduced that the tracking error converges asymptotically, and the convergence accuracy depends on the values of $\mathit{\epsilon }$, $\mathit{\eta }$, and $\mathit{\kappa }$. Additionally, the error e(t) will approach zero, as s(t) does when t approaches infinity. Therefore, $\mathit{e}\left(t\right)$ and $\dot{\mathit{e}}\left(t\right)$ will both approach zero as $t\to \infty$.

2.3. Estimating Nonlinear Components Using Artificial Neural Networks

This study employs an RBF neural network to estimate the nonlinear components in Equation (8). The RBF neural network has numerous applications, including function approximation, classification, and system control. Its strengths lie in its simple design structure, ease of training, fast convergence, and the ability to effectively fit any nonlinear function without falling into local optimum solutions [31]. The [5-7-1] structure of the RBF neural network is utilized to estimate the components $a_{\left.i\right|i=\mathrm{1,2},3}$ in the matrix $A_W$ of Equation (8) as shown in Figure 3.

The matrix, ${\mathit{A}}_W$, in the control law (8) contains characteristic parameters of the robot such as: the radius of each wheel ($r$); the robot’s moment of inertia ($I_v$); and the mass of the robot ($M$). The RBF neural network uses the Gradient Descent algorithm for online weight value updates. Each neural network contains seven Gaussian functions, and is described as in Equation (17):

$${\left.h_{ij}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{‖{\mathit{x}}_i-{\mathit{c}}_{ij}‖}^2}{2{\mathit{b}}_{ij}^2}\right)\right|}_{i=\overline{\mathrm{1,3}};j=\overline{\mathrm{1,7}}}$$

(17)

where:

$${\mathit{x}}_i=\left[\begin{array}{c}{\mathit{x}}_1\\ {\mathit{x}}_2\\ {\mathit{x}}_3\end{array}\right]=\left[\begin{array}{ccccc}e\left(1\right)& \dot{e}\left(1\right)& {\mathsf{\Gamma }}_d\left(1\right)& {\dot{\mathsf{\Gamma }}}_d\left(1\right)& {\ddot{\mathsf{\Gamma }}}_d\left(1\right)\\ e\left(2\right)& \dot{e}\left(2\right)& {\mathsf{\Gamma }}_d\left(2\right)& {\dot{\mathsf{\Gamma }}}_d\left(2\right)& {\ddot{\mathsf{\Gamma }}}_d\left(2\right)\\ e\left(3\right)& \dot{e}\left(3\right)& {\mathsf{\Gamma }}_d\left(3\right)& {\dot{\mathsf{\Gamma }}}_d\left(3\right)& {\ddot{\mathsf{\Gamma }}}_d\left(3\right)\end{array}\right]$$

$$h_{\left.ij\right|i=\mathrm{1,2},3}=\left[\begin{array}{ccccccc}{h}_{i1}& h_{i2}& h_{i3}& h_{i4}& h_{i5}& h_{i6}& h_{i7}\end{array}\right]$$

$$w_{\left.ij\right|i=\mathrm{1,2},3}=\left[\begin{array}{ccccccc}{w}_{i1}& w_{i2}& w_{i3}& w_{i4}& w_{i5}& w_{i6}& w_{i7}\end{array}\right]$$

The output of the neural network is given by Equation (18):

$${\widehat{a}}_i={\mathit{w}}_{ij}^T{\mathit{h}}_{ij}$$

(18)

The objective function of the neural network is as Equation (19):

$$E_i\left(t\right)=\frac{1}{2}{\left(a_i\left(t\right)-{\widehat{a}}_i\left(t\right)\right)}^2;i=\mathrm{1,2},3$$

(19)

Based on the Gradient Descent method, the weight values of the network are updated as in Equations (20) and (21) [32]:

$$∆w_j\left(t\right)=-\mathsf{\mu }\frac{\partial E}{\partial w_j}=\mathsf{\mu }\left(a_i\left(t\right)-{\widehat{a}}_i\left(t\right)\right)h_j$$

(20)

$$w_j\left(t\right)=w_j\left(t-1\right)+∆w_j\left(t\right)+\delta \left(w_j\left(t-1\right)-w_j\left(t-2\right)\right)$$

(21)

where $\mu \in \left(\mathrm{0,1}\right)$ is the learning rate and $\delta \in \left(\mathrm{0,1}\right)$ is the momentum factor.

At this point, the approximated matrix of ${\mathit{A}}_W$ is as Equation (22):

$${\widehat{\mathit{A}}}_W=\left[\begin{array}{ccc}{\mathit{w}}_{1j}^T{\mathit{h}}_{1j}& -{\mathit{w}}_{2j}^T{\mathit{h}}_{2j}\dot{\varphi }& 0\\ {\mathit{w}}_{2j}^T{\mathit{h}}_{2j}\dot{\varphi }& {\mathit{w}}_{1j}^T{\mathit{h}}_{1j}& 0\\ 0& 0& {\mathit{w}}_{3j}^T{\mathit{h}}_{3j}\end{array}\right]$$

(22)

The sliding control law based on the improved sliding surface and the RBF neural network for the robot is as Equation (23):

$${\mathit{U}}_{ISMC-\mathrm{Tan}\mathrm{h}-\mathrm{R}\mathrm{B}\mathrm{F}}=-\frac{1}{2}{\mathit{B}}_W^{-1}\left[2\left({\widehat{\mathit{A}}}_W\mathit{X}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_d\right)+(\mathit{\lambda }+2\mathit{\alpha })\dot{\mathit{e}}+{\mathit{\alpha }}^2\mathit{e}+\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{\epsilon }}\right)+\mathit{\kappa }\mathit{s}\right]$$

(23)

When the actual trajectory of the robot deviates from the reference trajectory due to conditions such as road friction, changing inertia, etc., the error, $\mathit{e}=\mathsf{\Gamma }-{\mathsf{\Gamma }}_d$, will change. The RBF neural network will then automatically update, leading to changes in ${\mathit{A}}_W$ so that the errors can be minimized. By using the RBF neural network in law (23), the proposed controller can adapt to the conditions of the robot.

To prove the stability of the ISMC-Tanh-RBF control law using the Lyapunov method, we need to show that the Lyapunov function $V=\frac{1}{2}{s}^2$ decreases over time, ensuring $\dot{V}\le 0$. We choose the following Lyapunov function:

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

(24)

where $\mathit{s}=\dot{\mathit{e}}+\mathit{\lambda }\mathit{e}$.

Taking the time derivative of V:

$$\dot{V}={\mathit{s}}^T\dot{\mathit{s}}$$

(25)

The sliding surface is defined as:

$$\mathit{s}=\dot{\mathit{e}}+\mathit{\lambda }\mathit{e}$$

(26)

Taking the derivative:

$$\dot{\mathit{s}}=\ddot{\mathit{e}}+\mathit{\lambda }\dot{\mathit{e}}$$

(27)

From the system dynamics, we have:

$$\ddot{\mathit{e}}={\stackrel{\mathit{ˆ}}{\mathit{A}}}_W\mathit{X}+{\mathit{B}}_W\mathit{U}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_d$$

(28)

Substitute $\ddot{\mathit{e}}$:

$$\dot{\mathit{s}}={\stackrel{\mathit{ˆ}}{\mathit{A}}}_W\mathit{X}+{\mathit{B}}_W\mathit{U}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_d+\mathit{\lambda }\dot{\mathit{e}}$$

(29)

Substitute the control law $U_{\mathrm{ISMC}-\mathrm{Tan}\mathrm{h}-\mathrm{RBF}: }$:

$$\mathit{U}=-\frac{1}{2}{\mathit{B}}_W^{-1}\left[2\left({\widehat{\mathit{A}}}_W\mathit{X}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_d\right)+(\mathit{\lambda }+2\mathit{\alpha })\dot{\mathit{e}}+{\mathit{\alpha }}^2\mathit{e}+\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{\epsilon }}\right)+\mathit{\kappa }\mathit{s}\right]$$

(30)

Substitute $\mathit{U}$ into $\dot{\mathit{s}}$:

$$\begin{array}{ll}& \dot{\mathit{s}}={\stackrel{\mathit{ˆ}}{\mathit{A}}}_W\mathit{X}+\\ & {\mathit{B}}_W\left(-\frac{1}{2}{\mathit{B}}_W^{-1}\left[2\left({\stackrel{\mathit{ˆ}}{\mathit{A}}}_W\mathit{X}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_d\right)+(\mathit{\lambda }+2\mathit{\alpha })\dot{\mathit{e}}+{\mathit{\alpha }}^2\mathit{e}+\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{ϵ}}\right)+\mathit{\kappa }\mathit{s}\right]\right)+{\mathit{D}}_f-\\ & {\ddot{\mathsf{\Gamma }}}_d+\mathit{\lambda }\dot{\mathit{e}}\end{array}$$

(31)

Simplify the expression:

$$\dot{\mathit{s}}={\stackrel{\mathit{ˆ}}{\mathit{A}}}_W\mathit{X}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_d+\mathit{\lambda }\dot{\mathit{e}}-\left({\stackrel{\mathit{ˆ}}{\mathit{A}}}_W\mathit{X}+{\mathit{D}}_f-{\ddot{\mathsf{\Gamma }}}_d+\frac{1}{2}(\mathit{\lambda }+2\mathit{\alpha })\dot{\mathit{e}}+\frac{1}{2}{\mathit{\alpha }}^2\mathit{e}+\frac{1}{2}\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{ϵ}}\right)+\frac{1}{2}\mathit{\kappa }\mathit{s}\right)$$

(32)

$$\dot{\mathit{s}}=\mathit{\lambda }\dot{\mathit{e}}-\frac{1}{2}(\mathit{\lambda }+2\mathit{\alpha })\dot{\mathit{e}}-\frac{1}{2}{\mathit{\alpha }}^2\mathit{e}-\frac{1}{2}\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{ϵ}}\right)-\frac{1}{2}\mathit{\kappa }\mathit{s}$$

(33)

$$\dot{\mathit{s}}=-\mathit{\alpha }\dot{\mathit{e}}-\frac{1}{2}{\mathit{\alpha }}^2\mathit{e}-\frac{1}{2}\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{ϵ}}\right)-\frac{1}{2}\mathit{\kappa }\mathit{s}$$

(34)

Substitute $\dot{\mathit{s}}$ into $\dot{V}$:

$$\dot{V}={\mathit{s}}^T\left(-\mathit{\alpha }\dot{\mathit{e}}-\frac{1}{2}{\mathit{\alpha }}^2\mathit{e}-\frac{1}{2}\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{ϵ}}\right)-\frac{1}{2}\mathit{\kappa }\mathit{s}\right)$$

(35)

Since $\mathit{s}=\dot{\mathit{e}}+\mathit{\lambda }\mathit{e}$:

$$\dot{V}={\mathit{s}}^T\left(-\mathit{\alpha }(\dot{\mathit{e}}+\mathit{\lambda }\mathit{e}-\mathit{\lambda }\mathit{e})-\frac{1}{2}{\mathit{\alpha }}^2\mathit{e}-\frac{1}{2}\mathit{\eta }\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{ϵ}}\right)-\frac{1}{2}\mathit{\kappa }\mathit{s}\right)$$

(36)

Simplify $\dot{V}$:

$$\dot{V}=-\mathit{\alpha }{\mathit{s}}^T(\dot{\mathit{e}}+\mathit{\lambda }\mathit{e})-\frac{1}{2}{\mathit{\alpha }}^2{\mathit{s}}^T\mathit{e}-\frac{1}{2}\mathit{\eta }{\mathit{s}}^T\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{ϵ}}\right)-\frac{1}{2}\mathit{\kappa }{\mathit{s}}^T\mathit{s}$$

(37)

$$\dot{V}=-\mathit{\alpha }{\mathit{s}}^T\mathit{s}-\frac{1}{2}{\mathit{\alpha }}^2{\mathit{s}}^T\mathit{e}-\frac{1}{2}\mathit{\eta }{\mathit{s}}^T\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{ϵ}}\right)-\frac{1}{2}\mathit{\kappa }{\mathit{s}}^T\mathit{s}$$

(38)

Since $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{\mathit{s}}{\mathit{ϵ}}\right)$ is bounded by −1 and 1, and assuming $\mathit{\alpha },\mathit{\eta },\mathit{\kappa }>0$:

$$\dot{V}\le -\frac{1}{2}(\mathit{\alpha }+\mathit{\kappa }){\mathit{s}}^T\mathit{s}$$

(39)

With proper choices of $\mathit{\alpha },\mathit{\eta },\mathit{\kappa }$, the derivative of the Lyapunov function $\dot{V}$ is negative semi-definite, ensuring that $V$ does not increase over time. This guarantees the stability of the system, and that the error, $e$, and its derivative, $\dot{e}$, will converge to zero.

3. Results

The MATLAB/Simulink diagram simulating the proposed method for the Mobile robot is shown in Figure 4.

The block diagram under analysis in Figure 4 in this study likely outlines an advanced control system framework for a mobile robot, integrating Intelligent Sliding Mode Control (ISMC) with a Radial Basis Function (RBF) neural network. This synthesis suggests a sophisticated mechanism that aims to refine the robot’s trajectory tracking in the presence of uncertainties and non-linearities typical of robotic systems.

At the outset, the Input Block defines the desired trajectory or setpoint for the robot, providing a benchmark for its motion. The Summing Junction then computes an error signal by contrasting this reference with the feedback from the robot, signifying the deviation of the robot’s actual path from the intended course.

Subsequently, an Integrator Block accumulates the error, suggesting a possible inclusion of an integral component in the control strategy, which could be indicative of a PID controller’s influence or another integrative control methodology.

The Mux Block consolidates multiple signals into one, implying a synthesis of error information, possibly amalgamating the instantaneous and cumulative error signals for input into the control system.

Central to the diagram is the ISMC-RBF Controller Block, which embodies the core intelligence of the system. It leverages a fusion of ISMC for its robustness against uncertainties and the adaptive nature of RBF neural networks to fine-tune control laws dynamically. This adaptive ability ensures the controller’s responsiveness to the complex dynamics of the mobile robot. The code used in this block is mentioned in Appendix A.

The Robot Block symbolizes the actual mobile robot that acts on the control signals. It reflects the robot’s physical response to the inputs, with its dynamics and movement patterns forming the output of this block.

The Demux Block, situated downstream, disperses the combined input into separate streams, which likely represent distinct state variables of the robot’s movement for subsequent processing or feedback.

A critical component of the system is the Feedback Loop, which circulates the robot’s state back to the Summing Junction. This closed-loop feedback is instrumental in real-time control adjustments, allowing for a self-correcting mechanism that aligns actual performance with targeted outcomes.

Lastly, the bidirectional communication between the robot and the ISMC-RBF Controller indicates a two-way exchange of information: the robot provides its state data to the controller, and the controller, in turn, relays control instructions back to the robot. The diagram encapsulates a dynamic and intelligent control strategy that caters to the demanding requirements of mobile robot navigation, promising enhanced precision, swift response, and steadfastness, even when subjected to external disturbances and intrinsic dynamic complexities.

The parameters of the robot used in the simulation are presented in

Table 1. Parameters of a Three-Wheeled Omnidirectional Mobile Robot model.

Parameters	Description	Value
$I_v$	Robot Moment of Inertia	$11.25 {\mathrm{k}\mathrm{g}\mathrm{m}}^2$
$M$	Robot Mass	$9.4 \mathrm{k}\mathrm{g}$
$L$	Distance Between	$0.178 \mathrm{m}$
$k$	Driving Gain Factor	$0.448$
$c$	Viscous Friction Factor	$0.1889 {\mathrm{k}\mathrm{g}\mathrm{m}}^2{\mathrm{s}}^{-1}$
$I_W$	Moment of Inertia of Wheel	$0.02108 {\mathrm{k}\mathrm{g}\mathrm{m}}^2$
$r$	Radius of Wheel	$0.0245 \mathrm{m}$

.

The parameters of the proposed controller are presented in

Table 2. Parameters of the proposed controller.

Parameters	Value
$\mathit{\kappa }$	$diag\left(\left[15, 15, 8\right]\right)$
$\mathit{\eta }$	$diag\left(\left[200, 200, 200\right]\right)$
$\mathit{\epsilon }$	$diag\left(\left[0, 15, 0, 15, 0, 15\right]\right)$
$\mathsf{\lambda }$	$diag\left(\left[15, 15, 15\right]\right)$
$\mathit{\alpha }$	$diag\left(\left[12, 12, 12\right]\right)$

.

The response of the ISMC-RBF method between $x_d$and $x_W$, and $y_d$ and $y_W$ with the Tricuspoid curve is presented in Figure 5. Observing Figure 5, it can be seen that $x_W$ converges towards $x_d$, with a rise time ($t_{r-x}$) of 0.0710 s, a settling time ($t_{ss-x}$) of 0.0978 s, a steady-state error ($e_{ss-x}$) of 0 m, and a percent overshoot (${POT}_x$) of 0.0040%; and $y_W$ converges towards $y_d$, with a rise time ($t_{r-y}$) of 0.0646 s, a settling time ($t_{ss-y}$) of 0.0902 s, a steady-state error ($e_{ss-y}$) of 0 m, and a percent overshoot (${POT}_y$) of 0.0042%.

Figure 6 displays the control signals of the ISMC-RBF controller. These control signals exhibit small amplitudes and low oscillation frequencies compared to the constant speed approach law and exponential function, with signum in [33] using the same parameters with this robot. This indicates that the proposed control method has successfully mitigated the chattering phenomenon commonly associated with sliding control.

Figure 7 presents the sliding surface $\mathit{s}=[s_1 s_2 s_3]ᵀ$ of the proposed controller. Initially, the surface starts according to the sliding coefficient values. Then, s quickly reaches a convergence point and continues to slide around $\mathit{s}=0$. The Tricuspoind trajectory response of the ISMC-RBF controller is depicted in Figure 8.

Figure 9 and Figure 10 showcase the trajectory responses of the ISMC-RBF controller to the Tricuspoid and Lissajous curves under the influence of noise at the robot’s output (assuming sensor noise with a power of 0.0005 w and a sampling time of 0.0001 s). The robot’s actual trajectory still converges to the reference trajectory within a finite time, with the error converging to zero.

The simulation results presented in Figure 8, Figure 9 and Figure 10 provide compelling evidence of the effectiveness of the proposed ISMC-RBF controller in tracking desired trajectories with high precision and reliability.

Figure 8 illustrates the actual trajectory of the mobile robot compared to the desired Tricuspoid trajectory. The close alignment between the actual trajectory (red) and the desired trajectory (blue dashed lines) demonstrates the controller’s high precision in trajectory tracking. The minimal deviation observed indicates that the controller effectively minimizes tracking errors, ensuring that the robot follows the intended path accurately. The smoothness of the trajectory reflects the controller’s capability to manage the robot’s movements without significant oscillations, further emphasizing its precision.

Figure 9 shows the robot’s trajectory response under the influence of sensor noise while following the same Tricuspoid curve. Despite the presence of noise, the actual trajectory remains close to the desired trajectory, highlighting the robustness of the ISMC-RBF controller against disturbances. Although some oscillations and deviations are introduced due to noise, the controller successfully compensates for these disturbances, maintaining the overall shape and direction of the desired trajectory. This robustness to noise is crucial for practical applications where environmental disturbances are common.

Figure 10 presents the trajectory response for a more complex Lissajous curve under sensor noise. The Lissajous curve represents a more challenging trajectory compared to the Tricuspoid curve. Despite the increased complexity, the actual trajectory closely follows the desired trajectory, showcasing the controller’s ability to handle complex paths accurately. The controller’s performance under these conditions demonstrates its adaptability and robustness, maintaining trajectory accuracy even in the presence of noise and complex navigation requirements.

The detailed analysis of these figures underscores several important aspects of the proposed ISMC-RBF controller:

• Precision and Accuracy: The controller consistently achieves high precision in trajectory tracking, as evidenced by the close alignment of actual and desired trajectories across different path shapes.
• Robustness to Noise: The ability to maintain trajectory accuracy under noisy conditions, as shown in Figure 9 and Figure 10, highlights the controller’s robustness to environmental disturbances.
• Adaptability to Complexity: The accurate tracking of the complex Lissajous curve under noise conditions demonstrates the controller’s versatility and adaptability to various trajectory complexities.
• Reduction in Chattering: The smooth and precise trajectories indicate that the improved sliding surface, along with the exponential and hyperbolic tangent approach laws, successfully reduces the chattering phenomenon commonly associated with a traditional sliding mode control.

The practical implications of these findings are significant. The enhanced performance and robustness of the ISMC-RBF controller make it well-suited for real-world applications, such as industrial automation, autonomous vehicles, and service robots, where precise and reliable control is essential. Future research could focus on real-world implementation and further optimization to enhance the controller’s performance in diverse operational environments.

The simulation results provide strong validation of the proposed ISMC-RBF controller’s effectiveness in achieving precise and robust trajectory tracking, highlighting its potential for advanced mobile robot control applications.

4. Discussion and Conclusions

This study has successfully demonstrated the effectiveness of combining improved sliding surfaces with artificial neural networks in mobile robot control, particularly in precise trajectory tracking. The introduction of an enhanced sliding surface, along with exponential and hyperbolic tangent approach laws, has significantly reduced the chattering phenomenon, commonly encountered in sliding mode control. This improvement is crucial, as chattering can decrease the performance of the control system and lead to increased wear and tear on mechanical components. The use of Radial Basis Function (RBF) neural networks to estimate the nonlinear components of the sliding control law marks a substantial improvement over traditional methods. This approach allows for real-time updates and adaptability to dynamic changes in the environment or operational parameters of the robot, representing a significant advancement in the development of autonomous systems. The real-time learning and adjustment capabilities of the neural network enhance the robustness and flexibility of the control system, making it suitable for complex and unpredictable environments. Simulation results obtained from MATLAB/Simulink confirm the theoretical predictions and demonstrate the practical feasibility of the proposed control strategy. The reduction in the rise and settling times, along with minimal overshoot and steady-state errors, emphasize the precision that this hybrid control strategy can achieve. Moreover, the observed low amplitude and frequency of chattering during the simulations further affirm the effectiveness of the improved sliding surface in minimizing this undesirable phenomenon.

In the research on mobile robot control based on neural networks (NN), a prominent issue is the large computational resource requirement due to the complexity and the continuous data processing needed for adaptation and learning. This study applied a combination of neural networks with improved sliding surfaces to reduce chattering and increase accuracy in trajectory tracking. However, real-time computation still demands a substantial amount of computational resources, posing a significant challenge in practical applications. To address this issue, several low-computation control methods have been developed and tested. Notably, adaptive finite-time fuzzy attitude control based on observers for quadrotor UAVs. Both methods use observers to reduce computational load, thereby maintaining high performance without requiring excessive resources. Additionally, finite-time neural control for nonlinear systems with actuator saturation compensation has proven to be effective in handling complex nonlinear systems. Combining NN with nonlinear control laws and saturation compensation allows the system to quickly adapt to environmental changes while reducing computational resource requirements. The results from this study by the authors show that integrating improved sliding surfaces and neural networks can significantly reduce chattering and increase trajectory tracking accuracy, although the neural network model and real-time learning computation still require considerable resources. Low-computation control methods, such as adaptive fuzzy control and the use of observers, have demonstrated their effectiveness in reducing resource requirements while ensuring system performance. Based on these results, this study suggests optimizing neural network architectures to minimize necessary computational resources, testing the system in various real-world conditions to determine its limitations and potential improvements, and integrating low-computation control methods with NN-based control systems to enhance adaptability and reduce required resources. These recommendations will help expand the application of intelligent control systems in practice, ensuring high performance and autonomous operation in various environments.

However, implementing such advanced control systems is not without challenges. The complexity of the neural network model and the computational demands of real-time learning are significant considerations. Future research could explore optimizing the neural network architecture to reduce computational burdens while maintaining or enhancing performance metrics. Furthermore, while the study focuses on laboratory conditions with controlled disturbances, real-world applications may introduce more complex and unpredictable challenges. Testing the system in various real-world scenarios will be essential to understanding its limitations and potential for improvement in control strategies for mobile robots. The integration of improved sliding surfaces and artificial neural networks represents a promising direction in robot control. This approach not only addresses the shortcomings of existing methods, but also opens avenues for research in making robotic systems more adaptable, efficient, and capable of operating autonomously in a wide range of environments. Future studies should continue to refine this technology, exploring its application across different types of robots and in more diverse operational contexts.