Adaptive Backstepping Hierarchical Sliding Mode Control for 3-Wheeled Mobile Robots Based on RBF Neural Networks

Abstract

This paper proposes a new adaptive controller for three-wheeled mobile robots (3WMRs) called the ABHSMC controller. This ABHSMC controller is developed through a cooperative approach, combining a backstepping controller and a Radial Basis Function (RBF) neural network-based Hierarchical Sliding Mode Controller (HSMC). Notably, the RBF neural network exhibits the remarkable capability to estimate both the uncertainty components of the model and systematically adapt its parameters, leading to enhanced output trajectory responses. A novel navigational model, constructed by the connection to the adaptive BHSMC controller, Timed Elastic Band (TEB) Local Planner, and A-star (A*) Global Planner, is called ABHSMC navigation stack, and it is applied to effectively solve the tracking issue and obstacle avoidance for the 3-Wheeled Mobile Robot (3WMR). The simulation results implemented in the Matlab/Simulink platform demonstrate that the 3WMRs can precisely follow the desired trajectory, even in the presence of disturbances and changes in model parameters. Furthermore, the controller’s reliability is endorsed on our constructed self-driving car model. The achieved experimental results indicate that the proposed navigational structure can effectively control the actual vehicle model to track the desired trajectory with a small enough error and avoid a sudden obstacle simultaneously.

1. Introduction

1.1. Motivations

Today, Wheeled Mobile Robots (WMRs) are becoming increasingly popular due to their mobility and impressive features. Scientists and engineers worldwide are dedicating significant research and attention to the monitoring and autonomous control of these robots. WMRs are used more frequently in various industries, such as the military, space exploration, and industry. They can operate in hazardous environments and complete tasks autonomously without human intervention [1,2].

In industrial production lines, autonomous vehicles are mobile robots with wheels used to move quickly and accurately in multiple directions [3]. WMR offers the advantage of cost savings while ensuring flexibility and speed in design. These vehicles are equipped with two rear wheels, each driven by an independent motor, and a freewheel in front, allowing them to move in all directions. Due to its remarkable structure, it is suitable for various uses, such as transporting goods and materials in factories.

The three-wheeled mobile robot (3WMR) is a nonlinear mechanical model that is difficult to accurately model kinematically and dynamically due to the many unknown parameters. This type of vehicle is also classified as an under-actuated object class, meaning it has fewer actuators than the Degrees of Freedom (DOF) to be controlled. Furthermore, 3WMRs have non-holonomic constraints, meaning their movement is not entirely free, and horizontal movement of the body is impossible. As a result, designing control algorithms for 3WMRs is a challenging task, especially when deploying controllers to real devices, as they are susceptible to mechanical uncertainties and environmental disturbances during operation.

1.2. Related Papers

Over the years, there have been various publications that have focused on studying the 3WMR system, such as robust Proportional-Integral-Derivative (robust PID) in [4], fractional-order PID in [5], nonlinear control methods in [6,7], fuzzy-based adaptive control techniques [8,9,10], disturbance observer-based controllers [11,12,13,14,15] and the deep reinforcement learning-based nonlinear control method [16]. All mentioned methods had been validated, and their effectiveness was demonstrated such that they can make the 3WMR system follow the desired trajectory in a short time interval.

Let us consider some existing methods, for instance, the robust PID controller proposed by Normey Rico et al. in [4] demonstrated its performance when it could make the control quality of the closed-loop scheme more stable and accurate compared to the classical PID controller. Notwithstanding the experimental results in the actual autonomous robot, the authors of paper [4] had not analyzed the stability of the closed system, the impact of the environmental disturbances, as well as the uncertain parameters on the control quality. A double-loop control structure with Fractional Order PID (FOPID) was introduced in paper [5] to improve the closed-loop’s robustness and steer the movement of the robot system in the desired trajectory more accurately. The inner-loop policy was the linear velocity loop, and its mission was to handle all the feedback signals from the encoder to control the robot wheels’ speeds. By utilizing an accelerometer and a six-degree-of-freedom gyroscope, the outer-loop policy could be ensured that the vehicle’s rotation was always within the acceptable angle restriction. Although the tracking performance of the FOPID controller was demonstrated to be better than the PID method, the lack of the FOPID’s parameter procedure, as well as the comparison with the existing methods caused the proposed method in this paper to not be convincing enough. In addition to these disadvantages, the influence of environmental disturbances in the control process, as well as the time-varying of the system’s parameters had not been considered. This can lead to subjectivity in evaluating the effectiveness of the FOPID controller when in actual application. Both the classical PID method and the FOPID method have some restrictions when determining the optimal values for the controller’s parameters without the system’s mathematical model. Manual interference can be time-consuming when it depends entirely on experience. With the idea to adjust the parameters of the PID controller more flexibly, paper [8] developed the PID controller in which its parameters were determined by the fuzzy rules to enhance the control quality, and the convergence of the algorithm was also better than the PID simultaneously. However, the fuzzy-based PID controller in this paper had not been validated in the actual robot, and this method is needed to compare with another existing method for evaluating its performance. Integrating the technique of artificial intelligence such as fuzzy logic and neural networks to improve the performance of the classic controller has been the research subject of many papers, [9] for instance. In this paper, Chiraz Ben Jabeur et al. introduced two intelligent techniques for a two-wheeled non-holonomic mobile robot that was a smart PID-optimized neural network-based controller (SNNPIDC) and a PD fuzzy logic controller (PDFLC). A comparison of the two proposed methods brings an overview of the advantages and restrictions of each method. The PID controller with optimal parameters achieved by the neural network helped the closed-loop energy-consumption to be lower, whereas the PDFLC controller constructed by Gaussian memberships reduced the overshoot value and the time response compared with the SNNPIDC. Nonetheless, all of this paper’s results had not been deployed in the actual system. In the trend of applying the fuzzy inference to the control system, this was a different approach that was to integrate the Deep Reinforcement Learning (DRL) principle into the fuzzy controller proposed by Chin-Tan Lee and Wen-Tsai Sung in [16]. Thanks to the DRL principle, the gain matrices of the Fuzzy-baed PID controller were automatically updated until the system was optimized instead of via manual adjustment. Thereby, this method could enhance the adaption of the closed-loop 3WMR system and save time in the design process. In spite of the advantages it brings to the control system, achieving good enough control quality depended deeply on the reliability of the data. The design procedure of the DRL-fuzzy-based controller in [16] required a complete and accurate input-output data set, which this article had not been able to achieve. All the results of [16] have just been guaranteed in some specific circumstances.

In addition to the PID controller, some nonlinear control methods have also been studied and applied to orbital tracking control for 3WMR, such as backstepping (BSP) control [6], the recursive integral backstepping control method [17], sliding mode control (SMC) [7] and the optimal tolerance method proposed by Peng Huang et al. [18]. However, there are some hindrances when handling most nonlinear control design procedures such as dependence on the system’s mathematical model, unpredictable components, or undesired chattering phenomena. There is a recent study integrating the fuzzy-logic interference into an SMC controller (FASMC), see [10], for trying to eliminate all the uncertainties of the 3WMR system. Different from the robust sliding mode control studied in [11,19], FASMC ensured the finite-time convergence of the state trajectories and reduced the chattering phenomena in inputs as well as the influence of the random disturbances. In spite of its mentioned advantages, this proposed method did not embed into an actual robot and it relied on the model of the system to calculate the control signal. Moreover, paper [10] had not considered the case of changing parameters over time. In many real cases, trying to determine the time-varying parameters, uncertainty nonlinearities, and unpredictable environmental disturbances will confront various rather rigorous technical restrictions and be a waste of money. Thus, a class of adaptive control algorithms arises naturally in control engineering to optimize the control performance, and the developed vehicles for enhancing the adaption of the closed-loop system need to be studied further.

Instead of designing a robust controller to enhance the endurance of the closed-loop system with lumped noises, there is a different and more effective proposed solution. That is to develop equipment that can approximate the external disturbances with a small enough error. This equipment not only can estimate external disturbances but only so that the stability of the closed-loop system is still ensured. The equipment with mentioned properties is called a disturbance observer and is studied in papers [12,13,14,15]. In [12] for instance, an adaptive control method, which was combined with a disturbance observer, had been proposed for WMR in controlling the vehicle following the reference trajectory. This method considered both skidding and sliding factors. The disturbance observer was designed to estimate total disturbances and combined well with adaptive controllers. The convergence of the proposed observer, as well as the stability of the closed-loop system, was proven rigorously by the Lyapunov theorem. Simulation results demonstrated that this was an effective control strategy, but it had yet to be endorsed in practice. Additionally, the design and implementation of an adaptive controller based on a disturbance observer require highly specialized skills. Therefore, installing and implementing the method may not be easy for researchers or engineers.

Based on the nonlinear disturbance observer, Li Li et al. proposed a control scheme for trajectory tracking of a 3WMR with an extended Kalman Filter-based observer to estimate the random noises and reduce non-random disturbances [13]. However, according to remark 5 in this paper, despite the high accuracy and fast convergence of the observer, this proposed method has still some limitations such as the large overshoot value or a high computational cost. Another disturbance observer-based adaptive control for non-holonomic 3-wheeled mobile robots was introduced in [14]. The authors of this paper restructured the original system’s model to establish a unique vector of lumped disturbance which was a time-varying vector including external noises, nonlinear components, and uncertain factors. In the next step, a disturbance observer was designed based on the available restructured model to create the estimated value of one. In this way, the vector of lumped disturbances was compensated in the input, and with the scheme of combining to adaptive sliding mode controller, the robustness and effectiveness of this proposed controller were verified by numerical simulations. In addition, with the idea of developing a disturbance estimator to compensate for all uncertain components at the input signals, paper [15] designed a double-loop scheme for solving the tracking issue of the non-holonomic 3WMR. The inner loop was a scheme for controlling the 3WMR’s velocities by utilizing the sliding mode controller with a reconstructed reaching law to ensure the 3WMR’s velocities were input-to-state stable at the equilibrium values. The outer loop was a scheme for steering the vehicle’s positions. Designing a double-loop scheme brings an advantage that can adjust both the vehicle’s positions and velocities and integrating a disturbance compensation-based observer improves the durability of the closed-loop system with lumped disturbances. Although the stability of the closed-loop system when applying most above-mentioned methods was proven rigorously by the Lyapunov theorem, it had not been verified by an actual mobile robot, and they did not give a solution for the obstacle avoidance problem.

In [20], the authors developed a trajectory-tracking controller for a 3WMR based on BSP and Hierarchical SMC (HSMC) techniques. The BSP controller was used to regulate the vehicle’s direction angle, while the HSMC controller was used to ensure the vehicle tracked its position. The combination of HSMC and BSP techniques has fully exploited the advantages of both. In particular, the BSP technique is known for its ability to control nonlinear objects systematically. It is a recursive design method to build the feedback control law and the Lyapunov function. However, the BSP technique has disadvantages, including low responsiveness to changes in environmental factors or operand explosion in complex systems [21]. The HSMC method is suitable for nonlinear underactuated systems. This technique is developed based on the SMC technique, which provides better noise cancellation because there are many sliding surfaces to minimize the effect of disturbances. However, the HSMC technique depends on the system model, so the controller’s accuracy will be affected if the system model is incorrect [22]. In addition, the vehicle is also affected by external disturbances from the external environment during operation.

From the above analysis, we develop a novel adaptive control algorithm for 3WMR combining backstepping, HSMC techniques, and RBF neural networks. The adaptive controller designed in this paper is also a backstepping HSMC controller which is constructed by the backstepping method for controlling the vehicle’s rotation and RBF network-based HSMC controller for steering the vehicle’s positions. The connection of two nonlinear control methods takes advantage of the advantages of each method. The RBF neural network [21,22] is used here because it can excellently approximate the unknown nonlinear components of the model. In short, compared with the paper [20], we have added an RBF network into the control structure to deal with the system model’s uncertainty and the presence of unknown external disturbances. Moreover, this method allows the RBF neural network to update its parameters for improving the flexibility of the adaptive controller. The stability of the closed system is proven rigorously based on the Lyapunov stability theory and through simulations in Matlab/Simulink software. In addition, we compare the performance of the proposed adaptive controller with other control methods such as the conventional backstepping HSMC (BHSMC) controller and the second-order sliding mode control (RBFN-SOSMC) in [23]. The effectiveness of our proposed method is demonstrated through the numerical results in case model parameters change and external disturbances occur. The simulation results show that the proposed control algorithm brings a better quality than the existing method.

Furthermore, to solve the obstacle avoidance problem for this robotic system, we introduce a novel structure called ABHSMC navigation stack. This structure is constructed by the connection to the adaptive BHSMC controller, Timed Elastic Band (TEB) local planner, and A-star (A*) global planner. According to the paper [24], TEB local planner, which has a low computational cost, is suitable for solving several problems involved in the localization in the environment, vehicle’s ego-motion, and environmental understanding. Meanwhile, A* global planner studied in [25] can bring an optimal solution for the shortest path problem. With the robustness and adaption of the proposed ABHSMC controller, the ABHSMC navigation stack is a novel structure that has various superior features and helps the 3WMR avoid dramatic obstacles in its moving process. Moreover, the navigational capacity is simulated in a new software developed by our research team and embedded in an actual mobile robot depicted in Figure 1 to endorse its control performance. The robot consists of a mechanical frame with motors and wheels, designed to bear the load and hold necessary equipment. Additionally, the robot is equipped with an A1 Lidar sensor, a Human-Machine Interface (HMI) screen, and a control board. The power supply is provided by a battery. The experimental results show that this is an effective control method, so it can be used in practice to help a vehicle work effectively in environments with a lot of noise and fluctuations.

1.3. Contributions, the Structure of Paper and Notations

All novel contributions in this paper are listed below:

• A new Adaptive Backstepping Hierarchical Sliding Mode Control (ABHSMC) scheme for 3WRM is introduced based on an RBF neural network. By aggregating all uncertain components in specific vectors and estimating by RBF neural network, the effect of uncertainties will be minimized, thereby improving the tracking quality of 3WRM, even when the model parameters change and in the presence of unknown external noises.
• For neural network design, a novel Lyapunov candidate function is proposed, thereby provide a systematic update law for parameter of activate function while ensuring the stability of the closed-loop system. This contribution improves the flexibility of the adaptive controller.
• This is the first time that ABHSMC navigation stack is introduced. The cooperation of the ABHSMC controller, TEB local planner, and A* global planner enhances navigation tasks and gives 3WMR a better capacity to avoid unpredictable obstacles. All the experimental results are deployed on the actual mobile robot developed by our research team.

The rest of the paper is structured as follows: Section 2 outlines the mathematical model of a 3WMR, followed by Section 3, which provides an overview of the backstepping HSMC (BHSMC) controller. Section 4 introduces an adaptive neural network BHSMC (ABHSMC) controller for 3WMR, while Section 5 covers 3WMR navigation and obstacle avoidance. The algorithm’s simulation and experimental results are evaluated in Section 6. Finally, Section 7 summarizes the conclusions and highlights future directions for development.

In this paper, ${\mathbb{R}}^{m\times n}$ is a space of matrices having m rows and n columns. The notation ${\mathbb{R}}^n$ is a short presentation of ${\mathbb{R}}^{n\times 1}$. $\mathbf{q}\in {\mathbb{R}}^n$ is a column vector of n real numbers $q_1,q_2,...,q_n$. ${\mathbf{q}}^T\in {\mathbb{R}}^{1\times n}$ is a row vector of n real numbers $q_1,q_2,...,q_n$. $\left|x\right|$ is the absolute value of a real number x. $∥\mathbf{q}∥$ is symbolised for the 2-norm of vector $\mathbf{q}$, and it is calculated by $∥\mathbf{q}∥=\sqrt{{\mathbf{q}}^T\mathbf{q}}$. The Frobenious norm of a vector $\mathbf{q}$ is notated $q_F$ instead of ${∥\mathbf{q}∥}_F$. In addition, $sgn\left(x\right)$ is the signum function of a real number x and $tr\left(A\right)$ is the trace operator of matrix A.

2. Mathematical Model

The 3WMR are autonomous robots designed to move around on three wheels, including two driving wheels and one caster wheel, as shown in Figure 2. Let m and J represent the mass and moment of inertia of the 3WMR. In addition, $T_1$ and $T_2$ are the torque exerted by the right and left motors associated with the driving wheels, r is the radius, L denotes half the distance between the 2 rear wheels, and $\lambda$ is the Lagrange coefficient. The position and direction of the 3WMR in the two-dimensional Descartes coordinate system OXY are determined by the vector $\mathbf{q}={[x,y,\theta ]}^T$, where x, y, and $\theta$ represent the positional coordinates of the robot in terms of the x-axis, y-axis, and navigation angle, respectively, during its operation.

The following Euler-Lagrange equation is employed for describing the kinetics and dynamics of the above wheeled mobile robot [20,26].

$$M\left(\mathbf{q}\right)\ddot{\mathbf{q}}+C(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}=E\left(\mathbf{q}\right)\mathbf{T}-A^T\left(\mathbf{q}\right)\lambda ,$$

(1)

where:

$$\begin{array}{c}\hfill M\left(\mathbf{q}\right)=\left[\begin{array}{ccc}m& 0& 0\\ 0& m& 0\\ 0& 0& J\end{array}\right],C(\mathbf{q},\dot{\mathbf{q}})=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],\mathbf{T}=\left[\begin{array}{c}{T}_1\\ T_2\end{array}\right],\\ \hfill E\left(\mathbf{q}\right)=\frac{1}{r}\left[\begin{array}{cc}cos\theta & cos\theta \\ sin\theta & sin\theta \\ L& -L\end{array}\right],A^T\left(\mathbf{q}\right)=\left[\begin{array}{c}-sin\theta \\ cos\theta \\ 0\end{array}\right].\end{array}$$

Considering non-holonomic constraints and non-slip rolling conditions, we have:

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

The Lagrange binding force factor is determined as follows:

$$\lambda =-m\dot{\theta }(\dot{x}cos\theta +\dot{y}sin\theta ),$$

The dynamic model of the robot is rewritten as follows:

$$\left\{\begin{array}{c}\ddot{\theta }=b_1{u}_1\hfill \\ \ddot{x}=\frac{\lambda }{m}sin\theta +b_2{u}_{2}cos\theta \hfill \\ \ddot{y}=-\frac{\lambda }{m}cos\theta +b_2{u}_{2}sin\theta \hfill \end{array}\right.$$

(2)

Here: $b_1=L/\left(rJ\right)$ and $b_2=1/\left(rm\right)$ are constant, $u_1=T_1-T_2,u_2=T_1+T_2$ are control inputs, where $T_1=0.5(u_1+u_2),T_2=0.5(u_2-u_1)$.

3. Backstepping Hierarchical Sliding Mode Control (BHSMC)

In this section, we present how to design a controller for a 3WMR system. As can be seen in (2), the system model is divided into two subsystems. The first subsystem is solely dependent on the control signal $u_1$, which is used to regulate the 3WMR’s movement angle. The second subsystem is solely dependent on the control signal $u_2$, which is used to control the robot’s position.

The proposed control system structure is depicted in Figure 3, where BSP and HSMC are controllers designed for the first and second subsystems, respectively. The detailed design steps for these controllers are described in the following subsections.

3.1. Backstepping Controller for Angular Tracking

From Equation (2), we have the first subsystem [20]:

$$\ddot{\theta }=b_1{u}_1$$

(3)

The objective is to generate a control signal $u_1$ that will enable the angular output signal $\theta$ to follow the reference angle ${\theta }_r$. To achieve this, we propose a controller design method based on the backstepping technique.

The definition of the error between the output signal $\theta$ and reference signal ${\theta }_r$ is as follows:

$$e_{\theta }=\theta -{\theta }_r$$

(4)

Let us define new state variables:

$$\left\{\begin{array}{c}{z}_1=e_{\theta }\hfill \\ z_2=\dot{\theta }-{\alpha }_1\hfill \end{array}\right.$$

(5)

In order to the error $e_{\theta }\to 0$, we select the function Lyapunov as follows:

$$V_{11}=\frac{1}{2}{z_1}^2$$

(6)

Differentiating $V_{11}$ with respect to time yields:

$${\dot{V}}_{11}=z_1\dot{z_1}=z_1(z_2+{\alpha }_1-{\dot{\theta }}_r)$$

(7)

Select the virtual control signal ${\alpha }_1=-a_1{z}_1+{\dot{\theta }}_r$ with $a_1>0$, then substitute it in (7), and we obtain:

$${\dot{V}}_{11}=-a_1{z}_1^2+z_1{z}_2$$

(8)

where $-a_1{z}_1^2$ makes the system stable, while $z_1{z}_2$ will be removed in the next step.

Consider a Lyapunov function as follows:

$$V_1=V_{11}+\frac{1}{2}{z}_2^2$$

(9)

Differentiating $V_1$ with respect to time yields:

$${\dot{V}}_1={\dot{V}}_{11}+z_2{\dot{z}}_2=-a_1{z}_1^2+z_2(z_1+b_1{u}_1-{\dot{\alpha }}_1)$$

(10)

We choose the control signal as follows:

$$u_1=-\frac{1}{b_1}(z_1+a_2{z}_2-{\dot{\alpha }}_1)$$

(11)

where the constants: $a_1>0,a_2>0$, substituting (11) into (10) we have:

$${\dot{V}}_1=-a_1{z}_1^2-a_2{z}_2^2<0,\forall z_1,z_2\ne 0$$

(12)

According to Lyapunov’s criterion, system (3) is stable asymptotic, meaning that the output angle $\theta$ follows the desired angle ${\theta }_r$.

3.2. Hierarchical Sliding Mode Control for Position Tracking

Considering the second subsystem from Equation (2) [20]:

$$\left\{\begin{array}{c}\ddot{x}=\frac{\lambda }{m}sin\theta +b_2{u}_{2}cos\theta \hfill \\ \ddot{y}=-\frac{\lambda }{m}cos\theta +b_2{u}_{2}sin\theta \hfill \end{array}\right.$$

(13)

Let us define new state variables: $\mathbf{X}={[x_1,x_2,x_3,x_4]}^T={[x,\dot{x},y,\dot{y}]}^T$

As a result, (13) can be written as a system of equations of state as follows:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=f_1\left(\mathbf{X}\right)+g_1\left(\mathbf{X}\right)u_2\hfill \\ {\dot{x}}_3=x_4\hfill \\ {\dot{x}}_4=f_2\left(\mathbf{X}\right)+g_2\left(\mathbf{X}\right)u_2\hfill \end{array}\right.$$

(14)

where:

$$\begin{array}{cc}\hfill & f_1\left(\mathbf{X}\right)=\frac{\lambda }{m}sin\theta ,g_1\left(\mathbf{X}\right)=b_{2}cos\theta ,\hfill \\ \hfill & f_2\left(\mathbf{X}\right)=-\frac{\lambda }{m}cos\theta and{g}_2\left(\mathbf{X}\right)=b_{2}sin\theta .\hfill \end{array}$$

Let us define the error vector between the output value and the preset value:

$$\mathbf{e}\left(t\right)={\left[e_x{e}_y\right]}^T={\left[e_1{e}_3\right]}^T={[x-x_{r}y-y_r]}^T$$

(15)

Therefore, (14) can be rewritten as:

$$\left\{\begin{array}{c}{\dot{e}}_1=e_2\hfill \\ {\dot{e}}_2=f_1\left(\mathbf{X}\right)+g_1\left(\mathbf{X}\right)u_2-{\ddot{x}}_r\hfill \\ {\dot{e}}_3=e_4\hfill \\ {\dot{e}}_4=f_2\left(\mathbf{X}\right)+g_2\left(\mathbf{X}\right)u_2-{\ddot{y}}_r,\hfill \end{array}\right.$$

(16)

Consider the first subsystem:

$$\left\{\begin{array}{c}{\dot{e}}_1=e_2\hfill \\ {\dot{e}}_2=f_1\left(\mathbf{X}\right)+g_1\left(\mathbf{X}\right)u_2-{\ddot{x}}_r\hfill \end{array}\right.$$

(17)

We define the first-level sliding surface as follows:

$$S_1=c_1{e}_1+e_2$$

(18)

where $c_1$ is a positive constant.

Differentiating both sides of (18) with respect to time yields:

$${\dot{S}}_1=c_1{\dot{e}}_1+{\dot{e}}_2=c_1{e}_2+f_1\left(\mathbf{X}\right)+g_1\left(\mathbf{X}\right)u_2-{\ddot{x}}_r=-k_1{S}_1-{\eta }_{1}sgn{S}_1$$

(19)

From (19) with $k_1>0,{\eta }_1>0$, we get:

$$u_{2\left(1\right)}=-\frac{c_1{\dot{e}}_1+f_1\left(\mathbf{X}\right)}{g_1\left(\mathbf{X}\right)}-\frac{k_1{S}_1+{\eta }_{1}sgn{S}_1-{\ddot{x}}_r}{g_1\left(\mathbf{X}\right)}$$

(20)

We define the second-level sliding surface for the system (16):

$$S_2={\lambda }_1{S}_1+{\beta }_1{s}_2$$

(21)

where $s_2=c_2{e}_3+e_4$ and $c_2>0$.

Differentiating both sides of Equation (21) with respect to time, we obtain:

$$\begin{array}{cc}\hfill {\dot{S}}_2& ={\lambda }_1{\dot{S}}_1+{\beta }_1{\dot{S}}_2\hfill \\ \hfill & ={\lambda }_1\left(c_1{e}_2+f_1\left(\mathbf{X}\right)+g_1\left(\mathbf{X}\right)u_2-{\ddot{x}}_r\right)+{\beta }_1\left(c_2{e}_4+f_2\left(\mathbf{X}\right)+g_2\left(\mathbf{X}\right)u_2-{\ddot{y}}_r\right)\hfill \\ \hfill & =-k_2{S}_2-{\eta }_{2}sgn{S}_2\hfill \end{array}$$

(22)

where $k_2>0$ and ${\eta }_2>0$ are positive constants.

Hence, we deduce the control signal from (22):

$$u_2=-\frac{{\lambda }_1{f}_1\left(\mathbf{X}\right)+{\beta }_1{f}_2\left(\mathbf{X}\right)+{\lambda }_1{c}_1{e}_2+{\beta }_1{c}_2{e}_4}{{\lambda }_1{g}_1\left(\mathbf{X}\right)+{\beta }_1{g}_2\left(\mathbf{X}\right)}-\frac{{\eta }_{2}sgn\left(S_2\right)+k_2{S}_2-{\lambda }_1{\ddot{x}}_r-{\beta }_1{\ddot{y}}_r}{{\lambda }_1{g}_1\left(\mathbf{X}\right)+{\beta }_1{g}_2\left(\mathbf{X}\right)}$$

(23)

To reduce chattering phenomenal at high frequencies, sign(.) function is replaced by the function sat(.):

$$sat\left(S_2\right)=\left\{\begin{array}{cc}sgn\left(S_2\right),\hfill & \left|S_2\right|>1\hfill \\ S_2\hfill & \left|S_2\right|\le 1\hfill \end{array}\right.$$

(24)

Therefore, the control signal for the second subsystem can be obtained as follows:

$$u_2=-\frac{{\lambda }_1{f}_1\left(\mathbf{X}\right)+{\beta }_1{f}_2\left(\mathbf{X}\right)+{\lambda }_1{c}_1{e}_2+{\beta }_1{c}_2{e}_4}{{\lambda }_1{g}_1\left(\mathbf{X}\right)+{\beta }_1{g}_2\left(\mathbf{X}\right)}-\frac{{\eta }_{2}sat\left(S_2\right)+k_2{S}_2-{\lambda }_1{\ddot{x}}_r-{\beta }_1{\ddot{y}}_r}{{\lambda }_1{g}_1\left(\mathbf{X}\right)+{\beta }_1{g}_2\left(\mathbf{X}\right)}$$

(25)

3.3. System Stability Analysis

To analyze the stability of the system, the control signal for the 3WMR system is synthesized into:

$$\mathbf{T}={\left[\begin{array}{cc}{T}_1\hfill & T_2\hfill \end{array}\right]}^T={\left[\begin{array}{cc}\frac{u_1+u_2}{2}\hfill & \frac{u_2-u_1}{2}\hfill \end{array}\right]}^T$$

(26)

Theory 1.

If the control laws for the 3WMR system,$u_1$ and $u_2$, as described in (11) and (25) are applied to the system whose mathematical model is described by (2), then the closed system will be asymptotically stable.

Proof. 

Choosing the Lyapunov function for the closed system as follows:

$$V=\frac{1}{2}{z}_1^2+\frac{1}{2}{z}_2^2+\frac{1}{2}{S}_2^2=V_1+V_2$$

(27)

where: $V_1=\frac{1}{2}{z}_1^2+\frac{1}{2}{z}_2^2$ and $V_2=\frac{1}{2}{S}_2^2$

Differentiating V with respect to time, we have:

$$\dot{V}={\dot{V}}_1+{\dot{V}}_2$$

(28)

Differentiating V with respect to time, and then using (22), we can conclude that:

$${\dot{V}}_2=S_2{\dot{S}}_2=-k_2{S}_2^2-{\eta }_2{S}_{2}sgn\left(S_2\right)$$

(29)

From (12) and (29), we can deduce that

$$\dot{V}=\dot{V_1}+\dot{V_2}<0$$

(30)

This proves that the closed system is asymptotically stable according to the Lyapunov method. □

4. Adaptive Backstepping Hierarchical Sliding Mode Control (ABHSMC)

The BHSMC controller performed well under ideal simulation conditions when both the model and adjustment parameters remained constant. However, accurately determining the model parameters can be challenging, requiring precise measurements to avoid errors. Additionally, external factors, such as disturbances, impacting on the vehicle during its operation can negatively affect the system’s control quality. This paper proposes an effective control algorithm utilizing an RBF neural network to address issues in case of parameter component changes and model uncertainties. Figure 4 displays the structure diagram of the proposed control algorithm.

If we assume that the mass of the 3WMR system is satisfied, the bounded condition is as follows: $m\le \left|m_m\right|$, where $\left|m_m\right|$ is the maximum mass of the vehicle. Therefore, the components $f_1\left(\mathbf{X}\right)=\frac{\lambda }{m}sin\theta$ and $f_2\left(\mathbf{X}\right)=-\frac{\lambda }{m}cos\theta$ in (25) are also bounded. In case $f_1\left(\mathbf{X}\right),f_2\left(\mathbf{X}\right)$ cannot be determined exactly, this paper proposes using the neural RBF network, whose structure is shown in Figure 5, to estimate these uncertainty components $f_1\left(\mathbf{X}\right),f_2\left(\mathbf{X}\right)$.

This neural network consists of three distinct components: an input layer, a hidden layer, and an output layer. In which, the inputs of the network are the state vector $\mathbf{q}={\left[\begin{array}{ccc}x\hfill & y\hfill & \theta \hfill \end{array}\right]}^T$ and its derivation $\dot{\mathbf{q}}={\left[\begin{array}{ccc}\dot{x}\hfill & \dot{y}\hfill & \dot{\theta }\hfill \end{array}\right]}^T$, the outputs are the two components to be estimated of the model ${\hat{f}}_1\left(\mathbf{X}\right)$ and ${\hat{f}}_2\left(\mathbf{X}\right)$.

Let us define ${\hat{\mathbf{W}}}_1\in {\mathbb{R}}^{1\times n},{\hat{\mathbf{W}}}_2\in {\mathbb{R}}^{1\times n}$ to be the weight matrices of the network, and $\mathbf{h}=\left[h_1,h_2,\dots ,h_n\right]\in {\mathbb{R}}^{1\times n}$ is the output vector of the hidden layer, whose element is calculated as the formula:

$$h_i=\frac{exp\left(-\frac{{∥\mathbf{q}-{\mathbf{c}}_i∥}^2+{∥\dot{\mathbf{q}}-{\mathbf{c}}_i∥}^2}{b_i^2}\right)}{{\sum }_{j=1}^{n}exp\left(-\frac{{∥\mathbf{q}-{\mathbf{c}}_j∥}^2+{∥\dot{\mathbf{q}}-{\mathbf{c}}_j∥}^2}{b_j^2}\right)}$$

(31)

where $\mathit{n}$ is the number of neurons of the network in the hidden layer; $b_i$ is the i-th standard deviations; ${\mathbf{c}}_i$ is the i-th vector of centroids of the activation function in the RBF network.

We determine two functions ${\hat{f}}_1\left(\mathbf{X}\right),{\hat{f}}_2\left(\mathbf{X}\right)$ to estimate of the components $f_1\left(\mathbf{X}\right),f_2\left(\mathbf{X}\right)$, respectively, as follows:

$$\left\{\begin{array}{c}{\hat{f}}_1\left(\mathbf{X}\right)={\hat{\mathrm{W}}}_1{}^T\mathbf{h}\hfill \\ {\hat{f}}_2\left(\mathbf{X}\right)={\hat{\mathrm{W}}}_2{}^T\mathbf{h}\hfill \end{array}\right.$$

(32)

Designing the HSMC controller by using the values of the estimators ${\hat{f}}_1\left(\mathbf{X}\right)$ and ${\hat{f}}_2\left(\mathbf{X}\right)$, the control signal $u_2$ in (25) is determined by the RBF network as follows:

$$u_2^{\ast }=-\frac{{\lambda }_1{\hat{f}}_1\left(\mathbf{X}\right)+{\beta }_1{\hat{f}}_2\left(\mathbf{X}\right)+{\lambda }_1{c}_1{e}_2+{\beta }_1{c}_2{e}_4+{\eta }_{2}sat\left(S_2\right)+k_2{S}_2-{\lambda }_1{\ddot{x}}_r-{\beta }_1{\ddot{y}}_r}{{\lambda }_1{g}_1\left(\mathbf{X}\right)+{\beta }_1{g}_2\left(\mathbf{X}\right)}$$

(33)

To determine the working torque of the right motor, ${\tau }_R$, and the left motor, ${\tau }_L$, we use the following equations: ${\tau }_R=0.5\left(u_1+u_2^{\ast }\right)$ and ${\tau }_L=0.5\left(u_2^{\ast }-u_1\right)$.

From Equation (33) and Equation (22), the derivation of the second-level sliding manifold $S_2$ is rewritten as the following:

$${\dot{S}}_2={\lambda }_1\left[f_1\left(\mathbf{X}\right)-{\hat{f}}_1\left(\mathbf{X}\right)\right]+{\beta }_1\left[f_2\left(\mathbf{X}\right)-{\hat{f}}_2\left(\mathbf{X}\right)\right]-k_2{S}_2-{\eta }_{2}sgn{S}_2$$

(34)

In the next step, the notation ${\mathbf{W}}_1^{\ast }\in R^{1\times n},{\mathbf{W}}_2^{\ast }\in R^{1\times n}$ are symbolised for the optimal weight matrices of the neural network. Then, the optimal estimators $f_1^{\ast }\left(\mathbf{X}\right),f_2^{\ast }\left(\mathbf{X}\right)$ are determined through the following expression:

$$\left\{\begin{array}{c}{f}_1^{\ast }\left(\mathbf{X}\right)={\mathbf{W}}_1^{\ast T}\mathbf{h}\hfill \\ f_2^{\ast }\left(\mathbf{X}\right)={\mathbf{W}}_2^{\ast T}\mathbf{h}\hfill \end{array}\right.$$

(35)

From Equation (35), we have:

$$\left\{\begin{array}{c}{f}_1\left(\mathbf{X}\right)=f_1^{\ast }\left(\mathbf{X}\right)+{\epsilon }_1={\mathbf{W}}_1^{\ast T}\mathbf{h}+{\epsilon }_1\hfill \\ f_2\left(\mathbf{X}\right)=f_2^{\ast }\left(\mathbf{X}\right)+{\epsilon }_2={\mathbf{W}}_2^{\ast T}\mathbf{h}+{\epsilon }_2\hfill \end{array}\right.$$

(36)

where ${\epsilon }_1$ and ${\epsilon }_2$ are the ideal approximation error of the network that satisfies ${\epsilon }_1\le \left|{\epsilon }_{1m}\right|$ and ${\epsilon }_2\le \left|{\epsilon }_{2m}\right|$. Let us define the errors of the evaluation weight from the ideal weight: ${\tilde{\mathbf{W}}}_1\in R^{1\times n},{\tilde{\mathbf{W}}}_2\in R^{1\times n}$:

$$\left\{\begin{array}{c}{\tilde{\mathbf{W}}}_1={\mathbf{W}}_1^{\ast }-{\hat{\mathbf{W}}}_1\hfill \\ {\tilde{\mathbf{W}}}_2={\mathbf{W}}_2^{\ast }-{\hat{\mathbf{W}}}_2\hfill \end{array}\right.$$

(37)

Combining Equations (34), (36) and (37), we obtain the following result:

$${\dot{S}}_2={\lambda }_1{\tilde{\mathbf{W}}}_1^T\mathbf{h}+{\beta }_1{\tilde{\mathbf{W}}}_2{}^T\mathbf{h}+{\lambda }_1{\epsilon }_1+{\beta }_1{\epsilon }_2-k_2{S}_2-{\eta }_{2}sgn\left(S_2\right)$$

(38)

Let us define $\mathbf{b}={[b_1,b_2,...,b_n]}^T$ as the vector of standard deviations and $\mathbf{c}=col\left({\mathbf{c}}_j\right)$, $j=1,...,n$ is the vector of ${\mathbf{c}}_j$. $\hat{\mathbf{b}}$ and $\hat{\mathbf{c}}$ are symbolised for the two estimates for the standard deviation $\mathbf{b}$ and the centroid of the activation function $\mathbf{c}$ in the RBF network (31). ${\mathbf{b}}^{\ast }$ and ${\mathbf{c}}^{\ast }$ are two ideal estimators for $\mathbf{b}$ and $\mathbf{c}$, respectively. Then, we can express the relationship between these estimators as follows:

$$\left\{\begin{array}{c}\tilde{\mathbf{b}}={\mathbf{b}}^{\ast }-\hat{\mathbf{b}}\hfill \\ \tilde{\mathbf{c}}={\mathbf{c}}^{\ast }-\hat{\mathbf{c}}\hfill \end{array}\right.$$

(39)

where $\tilde{\mathbf{b}}$ and $\tilde{\mathbf{c}}$ s the difference between the ideal estimates ${\mathbf{b}}^{\ast }$ and ${\mathbf{c}}^{\ast }$ with the estimates $\hat{\mathbf{b}}$ and $\hat{\mathbf{c}}$, respectively.

Considering the candidate Lyapunov function as the following:

$$V=\frac{1}{2}{S}_2^2+\frac{1}{2{\gamma }_1}{\tilde{\mathbf{W}}}_1^T{\tilde{\mathbf{W}}}_1+\frac{1}{2{\gamma }_2}{\tilde{\mathbf{W}}}_2^T{\tilde{\mathbf{W}}}_2+\frac{1}{2{\gamma }_c}tr\left({\tilde{\mathbf{c}}}^T\tilde{\mathbf{c}}\right)+\frac{1}{2{\gamma }_d}tr\left({\tilde{\mathbf{b}}}^T\tilde{\mathbf{b}}\right)$$

(40)

where the constants ${\gamma }_1>0,{\gamma }_2>0$ and ${\gamma }_c>0,{\gamma }_d>0$ are the gain of the adaptive update rules, tr(.) is the trace operator. Differentiating from both sides of Equation (40) with respect to time yields:

$$\dot{V}=S_2{\dot{S}}_2+\frac{1}{{\gamma }_1}{\tilde{\mathbf{W}}}_1^T{\dot{\tilde{\mathbf{W}}}}_1+\frac{1}{{\gamma }_2}{\tilde{\mathbf{W}}}_2^T{\dot{\tilde{\mathbf{W}}}}_2+\frac{1}{{\gamma }_c}tr\left({\tilde{\mathbf{c}}}^T\dot{\tilde{\mathbf{c}}}\right)+\frac{1}{{\gamma }_d}tr\left({\tilde{\mathbf{b}}}^T\dot{\tilde{\mathbf{b}}}\right)$$

(41)

Substituting (38)–(40) into (41) gives:

$$\begin{array}{cc}\hfill & \dot{V}=S_2{\lambda }_1{\tilde{\mathbf{W}}}_1^T\mathbf{h}+S_2{\beta }_1{\tilde{\mathbf{W}}}_2^T\mathbf{h}-\frac{1}{{\gamma }_1}{\tilde{\mathbf{W}}}_1^T{\dot{\hat{\mathbf{W}}}}_1-\frac{1}{{\gamma }_2}{\tilde{\mathbf{W}}}_2^T{\dot{\hat{\mathbf{W}}}}_2-\frac{1}{{\gamma }_c}tr\left({\tilde{\mathbf{c}}}^T\widehat{\hat{\mathbf{c}}}\right)-\frac{1}{{\gamma }_d}tr\left({\tilde{\mathbf{b}}}^T\dot{\hat{\mathbf{b}}}\right)\hfill \\ \hfill & -k_2{S}_2^2-{\eta }_2{S}_{2}sgn\left(S_2\right)+S_2\left({\lambda }_1{\epsilon }_1+{\beta }_1{\epsilon }_2\right)\hfill \end{array}$$

(42)

The direct RBF adaptive update rules are chosen as follows:

$$\begin{array}{cc}\hfill & {\dot{\hat{\mathbf{W}}}}_1={\gamma }_1{\lambda }_1{S}_2\mathbf{h},{\dot{\hat{\mathbf{W}}}}_2={\gamma }_2{\beta }_1{S}_2\mathbf{h}\hfill \\ \hfill & \dot{\hat{\mathbf{c}}}=-{\gamma }_c\hat{\mathbf{c}}{S}_2,\dot{\hat{\mathbf{b}}}=-{\gamma }_d\hat{\mathbf{b}}{S}_2\hfill \end{array}$$

(43)

Substituting (43) into (42), we have:

$$\dot{V}=-k_2{S}_2^2-{\eta }_2{S}_{2}sgn\left(S_2\right)+S_2\left({\lambda }_1{\epsilon }_1+{\beta }_1{\epsilon }_2\right)+S_{2}tr\left({\tilde{\mathbf{c}}}^T\hat{\mathbf{c}}\right)+S_{2}tr\left({\hat{\mathbf{b}}}^T\hat{\mathbf{b}}\right)$$

(44)

Combining (39) and (44) yields:

$$\dot{V}=-k_2{S}_2^2-{\eta }_2{S}_{2}sgn\left(S_2\right)+S_2\left({\lambda }_1{\epsilon }_1+{\beta }_1{\epsilon }_2\right)+S_{2}tr\left({\tilde{\mathbf{c}}}^T\left({\mathbf{c}}^{\ast }-\tilde{\mathbf{c}}\right)\right)+S_{2}tr\left({\tilde{\mathbf{b}}}^T\left({\mathbf{b}}^{\ast }-\tilde{\mathbf{b}}\right)\right)$$

(45)

From $tr\left[{\tilde{\mathbf{x}}}^T\left({\mathbf{x}}^{\ast }-\tilde{\mathbf{x}}\right)\right]\le {\tilde{x}}_F{x}_F^{\ast }-{\tilde{x}}_F^2$ and $x_F^{\ast }\le x_{max}$, we have:

$$\left\{\begin{array}{c}tr({\tilde{\mathbf{c}}}^T({\mathbf{c}}^{\ast }-\tilde{\mathbf{c}}))\le {\tilde{c}}_F{c}_F^{\ast }-{\tilde{c}}_F^2\le {\tilde{c}}_F(c_{max}-{\tilde{c}}_F)\hfill \\ tr({\tilde{\mathbf{b}}}^T({\mathbf{b}}^{\ast }-\tilde{\mathbf{b}}))\le {\tilde{b}}_F{b}_F^{\ast }-{\tilde{b}}_F^2\le {\tilde{b}}_F(b_{max}-{\tilde{b}}_F)\hfill \end{array}\right.$$

(46)

By utilizing two inequalities in (46), the derivation of the Lyapunov function calculated in (45) becomes:

$$\begin{array}{cc}\hfill \dot{V}& \le -k_2{S}_2^2-{\eta }_2{S}_{2}sgn\left(S_2\right)+S_2\left({\lambda }_1{\epsilon }_1+{\beta }_1{\epsilon }_2\right)+S_2{\tilde{c}}_F\left(c_{max}-{\tilde{c}}_F\right)+S_2{\tilde{b}}_F\left(b_{max}-{\tilde{b}}_F\right)\hfill \\ \hfill & \le -k_2{S}_2^2-{\eta }_2{S}_{2}sgn\left(S_2\right)+S_2\left({\lambda }_1{\epsilon }_1+{\beta }_1{\epsilon }_2\right)+S_2\left[{\tilde{c}}_F\left(c_{max}-{\tilde{c}}_F\right)+{\tilde{b}}_F\left(b_{max}-{\tilde{b}}_F\right)\right]\hfill \\ \hfill & \le -k_2{S}_2^2-{\eta }_2{S}_{2}sgn\left(S_2\right)+S_2\left({\lambda }_1{\epsilon }_1+{\beta }_1{\epsilon }_2\right)-S_2\left[{\tilde{c}}_F\left({\tilde{c}}_F-c_{max}\right)+{\tilde{b}}_F\left({\tilde{b}}_F-b_{max}\right)\right]\hfill \\ \hfill & \le -k_2{S}_2^2-{\eta }_2{S}_{2}sgn\left(S_2\right)+S_2\left({\lambda }_1{\epsilon }_1+{\beta }_1{\epsilon }_2\right)\hfill \\ \hfill & -S_2\left[{\left({\tilde{c}}_F-\frac{c_{max}}{2}\right)}^2+{\left({\tilde{b}}_F-\frac{b_{max}}{2}\right)}^2-\frac{c_{max}^2}{4}-\frac{b_{max}^2}{4}\right]\hfill \\ \hfill & \le -k_2{S}_2^2-S_2\left[{\eta }_{2}sgn\left(S_2\right)-{\lambda }_1{\epsilon }_1-{\beta }_1{\epsilon }_2-\frac{c_{max}^2}{4}-\frac{b_{max}{}^2}{4}+{\left({\tilde{c}}_F-\frac{c_{max}}{2}\right)}^2+{\left({\tilde{b}}_F-\frac{b_{max}}{2}\right)}^2\right]\hfill \end{array}$$

(47)

It can be seen that if the selected controller parameter is satisfied:

$$min\left\{{\eta }_2\right\}>max\left\{\left|{\lambda }_1{\epsilon }_1+{\beta }_1{\epsilon }_2+\frac{c_{max}{}^2}{4}+\frac{b_{max}^2}{4}\right|\right\}$$

(48)

Then, from (47), we can deduce that

$$\dot{V}<0$$

(49)

Thus, the closed system is stable according to the Lyapunov criterion.

5. 3WMR’S Navigation and Obstacle Avoidance

The navigation of autonomous vehicles in the workplace is a notable aspect of robotics. To enable automatic navigation, the vehicle must first be provided with environmental map information (mapping) and then locate its own position (localization). Subsequently, the autonomous vehicle must plan its travel path within the work environment (path planning), which involves determining the most efficient route from the current location to the destination. Finally, the challenges of tracking control and obstacle avoidance must be addressed. This section presents how to implement the ABHSMC algorithm for a realistic 3WMR built by the our team, with the detailed hardware structure of the 3WMR system as shown in Figure 6 to tackle these issues. The principal components of the 3WMR system include:

1. The Jetson Nano Developer Kit embedded computer is a high-performance processor that is compact but extremely powerful. It processes information from the Light detecting and ranging (Lidar) A1 sensor directly, calculates the desired trajectory, and sends navigation commands to the Kit Arduino Mega 2560 R3 through the transmission of $v_r,{\omega }_r$ values, where $v_r$ and ${\omega }_r$ are the linear velocities and the reference angular velocity, respectively. Additionally, the coordinate $\mathbf{q}$ of the robot and goal coordinate ${\mathbf{q}}_{goal}$ are also sent to Kit Arduino Mega 2560 R3.

2. The Arduino Mega $2560\mathrm{R}3$ is a low-cost central controller with a good processing speed. It receives navigation commands from the Jetson Nano Developer Kit and transmits control signals to the two bridge circuits ($\mathrm{H}$-driver 1 and $\mathrm{H}$-driver 2), which operate two DC motors $\mathrm{DC}$ (Motor 1 and DC Motor 2). The navigation commands are transmitted through $v_r,{\omega }_r$ values. The proposed ABHSMC control algorithm for 3WMR is installed on this central controller.

3. The ESP32 is a WiFi chip that is programmed to receive data from Kit Arduino Mega 2560 R3 through I2C protocol. It can be connected to Firebase.

4. Lidar A1: this is a lidar sensor produced by Slamtec that uses laser scanning technology to provide 2D data. It is one of the sensor techniques used widely in robot programming because of its less sensitivity to environmental light and determining the exact distance without contact [27]. It serves as a data collector for operational environment maps.

5. The Human-Machine Interface (HMI) employs a touch screen to monitor the complete navigation process of the 3WMRs operating in the field.

6. The website which is connected to Firebase receives the data and displays them in a monitoring interface designed by our research team. The monitoring interface presents a comparison between the actual and reference linear velocity (v and $v_r$, respectively) as well as the actual and references angular velocity ($\omega$ and ${\omega }_r$, respectively). By achieving zero deviation in the linear velocity difference ($e_v=v-v_r=0$) and the angular velocity difference ($e_{\omega }=\omega -{\omega }_r=0$), the 3WMR is capable of following the desired trajectory. Moreover, the interface is also developed to display the tracking errors of the robot’s positions, rotation, velocities, and the distance between the start point and the goal point.

Figure 7 presents the implementation scheme for utilizing the ABHSMC for trajectory tracking control. In the structure of autonomous vehicle navigation, the Hector Slam algorithm is applied to create a map (map); this is a low computational resource SLAM (Simultaneous Localization and Mapping) approach based on laser sensors. The fundamental principle of SLAM is to provide information about the surrounding environment based on sensor systems and build a map of the working space while estimating the position and orientation of the robot [28,29]. From the map data and the desired goal position, the global planner is deployed using the A* algorithm [24,25,30] based on the ROS platform to plan the global path. The global motion plan (global trajectory) will be used as a reference trajectory for the local motion planning (local planner) using the TEB algorithm [24,31]. Finally, the control algorithm ABHSMC is implemented using the local motion plan (local trajectory) and the real-time position information obtained from the sensor to ensure that 3WMRs can move to the destination (goal position) while avoiding obstacles during navigation.

6. Results

6.1. Simulation Results

The proposed ABHSMC controller has been verified for its efficiency through simulation using Matlab/Simulink software. Two preset trajectories were used for the simulation, including sinusoidal orbit and straight-line orbit. The simulation was conducted in two cases: (1) with deterministic model parameters and no disturbances and (2) with uncertain model parameters and disturbances occurring. The RBF neural network has a three-layer structure, as shown in Figure 4, with 30 neurons in the hidden layer. All the simulation results are compared to the BHSMC and the RBFN-SOSMC method studied in [23] and simulate in two cases of the desired trajectories, that is:

• The desired straight-line: ${\mathbf{q}}_{r1}={\left[\begin{array}{ccc}0.25t\hfill & 0.25t\hfill & \pi /4\hfill \end{array}\right]}^T$
• The desired sinusoidal trajectory: ${\mathbf{q}}_{r2}={\left[\begin{array}{ccc}0.2t+1\hfill & 1+0.25sin\left(0.2\pi t\right)\hfill & arctan(0.25\pi cos(0.2\pi t\left)\right)\hfill \end{array}\right]}^T$

The vector of tracking error of this robot is defined as the following:

$$\begin{array}{c}\hfill \mathbf{e}=\mathbf{q}-{\mathbf{q}}_d\end{array}$$

(50)

The parameters of the 3WMR model are provided in

Table 1. Parameters of 3WMR model.

Parameters	m	J	r	L
Value	$10\mathrm{kg}$	$3.818\mathrm{kg}/{\mathrm{m}}^2$	$0.07\mathrm{m}$	$0.19\mathrm{m}$

, and the controller’s parameters are presented in

Table 2. Controller parameters.

Parameters	$a_1$	$a_2$	$c_1$	$c_2$	$k_2$	${\eta }_2$
Value	110	50	25	25	3.1	5
Parameters	${\lambda }_1$	${\beta }_1$	${\gamma }_1$	${\gamma }_2$	${\gamma }_c$	${\gamma }_b$
Value	5.5	0.5	1.5	0.25	0.25	0.25

.

6.1.1. Trajectory Tracking Control without External Disturbances and Unchanged Model Parameter

In case the reference trajectory is a straight line, the output responses of the 3WMR system for ABHSMC, BHSMC and FBFN-SOSMC controllers are described in Figure 8. It can be seen that the positions of the system $(x,y)$ track the reference signal in short time intervals. The setting time when applying our proposed method is faster significantly than using the method introduced in [23]. In detail, the proposed method in this paper makes the positions of the closed-loop system to be stable after only 1.8 s whereas this figure is 4 s for the RBFN-SOSMC method. Additionally, the accuracy of this method is higher than the RBFN-SOSMC technique when the vehicle moves along the y-axis. Although combining the RBF neural network in the closed-loop scheme makes the 3WMR not to be asymptotically stable, its stability is still maintained with an invariant set depicted in (48), and the convergence of the network’s weights ${\hat{\mathbf{W}}}_1,{\hat{\mathbf{W}}}_2$ is guaranteed in spite of the inaccuracy of the mathematical model.

In case one wants to control the 3WMR tracking to the desired trajectory whose shape is as a sinusoidal reference, the proposed method is also acknowledged for this claim. It can be seen that from the Figure 9 when the control diagram is equipped with the ABHSMC controller or RBFN-SOSMC controller, 3WMR’s traveling trajectory closely resembles the desired trajectory. However, with the BHSMC controller, the 3WMR cannot follow the reference trajectory. This highlights the superior performance of the ABHSMC controller over the traditional BHSMC controller. With regard to the property of the closed-loop system’s stability, it can be seen clearly in the three remain simulation results in Figure 9. The asymptotic stability is a beacon for all controllers that want to achieve and the BHSMC entirely guaranteed this stability. Thus, the responses of the tracking error for vehicle’s positions $(x,y)$ and rotation $\theta$ do not have any overshoot values. However, this property is only archived when the mathematical presentation of the system is clear and without uncertain components. Even though the ABHSMC controller, as well as the RBFN-SOSMc controller, cannot bring the system to be asymptotic stable, these methods can be handled with an uncertain system model, the influence of external noises and the closed-loop system is still stable since the invariant set determined in inequality (48) can be shrunk arbitrarily by only selecting parameter ${\eta }_2$.

6.1.2. Trajectory Tracking Control in the Presence of External Disturbances and Changes in Model Parameters

In this case, we assess the effectiveness of the proposed controller under the influence of high-frequency external interference, as depicted in Figure 10. Additionally, we examine the impact of vehicle mass variations by testing scenarios where the mass is 15 kg and 25 kg, as shown in Figure 11 and Figure 12. It is easily seen that when the mass of the robot is changed, the ABHSMC controller still makes the closed-loop system stable at the desired trajectory within a finite time. Despite the non-asymptotic stability of the system, the convergence of the mobile robot’s states still changes to the invariant set (48) even if the high-frequency external disturbances with a large amplitude.

Figure 13 and Figure 14 depict the movement path and tracking errors of 3WMR when the reference path exhibits a sinusoidal pattern while the vehicle’s mass changes to 15 kg and 25 kg. With the ABHSMC controller, the position coordinates, their error, and the error of the navigation angle are pretty small compared to the desired. Meanwhile, the BHSMC controller cannot make the system follow the desired trajectory; in other words, the tracking errors are significant. The reason for this phenomenon is that when the lumped disturbance affects to the closed-loop system. If using only BHSMC, the Lyapunov function:

$$\begin{array}{c}\hfill \dot{V_2}=S_2{\dot{S}}_2=-k_2{S}_2^2-{\eta }_2{S}_{2}sgn\left(S_2\right)+d_m\end{array}$$

(51)

where $d_m=supd\left(t\right)$. Therefore the stability of the closed-loop system is not ensured for the BHSMC controller. By contrast, for the ABHSMC controller, the neural network’s weight has always been changed and updated for compensation for the error affected by lumped disturbance. Consequently, the stability of the closed-loop system is still maintained in a short time interval.

6.2. Experimental Results

In this study, we have developed a web monitoring interface for assessing the output trajectory quality of autonomous vehicles during navigation, as shown in Figure 15. The interface shows any deviations of the angular and linear velocity from their reference values as errors. It can be seen that these errors are gradually approaching zero, which means that 3WMR follows the preset trajectory with the proposed control algorithm.

We next present some results obtained with the ABHSMC algorithm for actual 3WMR as described in the previous section. First, the operating environment data was collected as a 2D map using a Lidar A1 sensor produced by Slamtec, as shown in Figure 16. The main experimental results are depicted in Figure 17, including (1) a web-based monitoring interface that tracks velocity errors in real-time and (2) a 2D map that displays obstacles and navigation as the 3WMR moves. The shaded black area in Figure 17 represents walls or obstacles on the map, and the dark gray area surrounding it shows their spread. The red traces indicate obstacles detected during vehicle operation, which disappear when out of range or no longer detected by the Lidar A1 sensor. The map coordinate frame is the original coordinate frame, while the laser coordinate frame is the coordinate frame attached to the 3WMRs. The green line shows the desired trajectory, and the red line shows the autonomous vehicle’s trajectory.

The navigation process of the 3WMR system by the proposed ABHSMC navigation stack is presented in Figure 17. The process is begun in Figure 17a. In this figure, the ABHSMC navigation stack can be evaluated and a global trajectory can be proposed from the initial position to the goal position, in which the distance between two points is 4.39 m and the initial robot’s velocity is 0.25 (m/s). Notwithstanding the robot’s discovery of the existence of an obstacle (red streak), the initially projected trajectory for the mobile robot is almost the same as the global trajectory (the blue line), since the perception of the 3WMR system for the influence of this obstacle in the planning trajectory has not been completed yet.

Although the initially projected trajectory is not complete, the navigational effectiveness of the proposed stack is demonstrated after a short time interval and when the robot moves nearly to the obstacle. After a short time interval, the robot moves to the position where the distance between it and the goal point is 3.08 m and the existence of the random obstacle is clearly recognised. ABHSMC navigation stack commands changing the actual robot’s trajectory (Figure 17b) to avoid this obstacle. Based on the supplied environmental map and collected data from Lidar A1, the robot’s decision is slowed, and the old path planning is not selected to reach the goal point (Figure 17c). Moreover, the proposed navigational method also helps the 3WMR system to discover many expansions of other obstacle zones, which is depicted by dark grey in Figure 17c. Then, the 3WMR retrogrades (Figure 17d) with a velocity of −0.13 (m/s) and calculated a new, safer projected trajectory. The ABHSMC algorithm helps the 3WMR follow this trajectory, successfully avoiding obstacles (Figure 17e) and reaching the goal point (Figure 17f) with the tracking errors shown in detail in Figure 18.

Figure 18 depicts the tracking error of the robot’s states and its velocities in the navigational process in real time. It can be easily seen that all the amplitudes of the tracking errors are small. In detail, the maximum value of the positional error along to x-axis is only 0.26 cm and the figure for the y-axis is only 0.35 cm. The maximum value of the rotational errors is only from −0.08 degrees to −0.06 degrees. After 26 s from the initial time, the movement of the 3WMR system is finished. All the velocities converge at the origin in a short time interval. However, there exists a distance of about 0.26 m from the robot’s actual finished point to the goal point caused by the inertia constant of the 3WMR system. It is clear that when the robot is at the goal point, it consumes a short time interval so that the input signals are calculated to brake two motors, which are installed on two wheels of the robot. Consequently, during the braking period, the robot has moved beyond the target position by a small distance until it comes to a complete stop.

7. Conclusions

The paper presented a new cooperation regime between a backstepping controller and an RBF neural network-based Hierarchical Sliding Mode Controller (HSMC) for a 3WMR system. The stability of the closed-loop control system was ensured regardless of the influence of external factors such as unpredictable disturbances and uncertain parameters by applying the Lyapunov theorem. The effectiveness of the proposed method was demonstrated through two cases in the simulation results section. In two circumstances, regardless of the change of the robot’s mass or the influences of external disturbances, the proposed ABHSMC controller delivered more significant control performance compared to the traditional BHSMC method and RBFN-SOSMC method introduced in [23]. Furthermore, the ABHSMC navigation stack was first introduced and it helped the navigational process of the 3WMR system more flexible, more optimal with a low computational cost. Due to the robustness of the ABHSMC controller, integrating this controller into the navigational stack not only enhanced the endurance of the closed-loop system with external noises, and the adaption of 3WMR when the change of its parameter but also made the robot avoid the sudden obstacle. The experimental results reflected this contribution to the navigational process. Simultaneously, it validated this control method as highly effective in tasks requiring superior vehicle performance in different environments. Future work will involve integrating computer vision and machine learning technologies, supplementing specialized sensors in order to enhance the autonomous vehicle’s obstacle avoidance and navigation capabilities, and adapting to uncertainties in natural environments.