Graph-Guided Adaptive Formation Control with Prescribed Performance Enhanced by Neural Network-Based Backstepping

1. Introduction

The coordination of multi-autonomous vehicles has become a significant area of research, with a focus on diverse formation control strategies, including behavior-based control [1], virtual structure [2], and decentralized control [3]. Behavior-based formation control emphasizes local interactions among vehicles to form desired patterns, while virtual structure formation control guides the vehicles to maintain their relative positions within an invisible geometric framework. In decentralized control, individual vehicles make decisions based on local information and communication with neighbors. Despite the advantages of these approaches, this paper will investigate leader-follower formation control for multi-autonomous vehicles. This method has been selected due to its simplicity, ease of implementation, and the potential for achieving robust performance under specific conditions [4,5,6].

There has been a significant amount of research conducted on the subject, and a considerable portion of it is of high quality. Yang and Gu et al. [7] realized a non-linear formation keeping and mooring control of multiple autonomous underwater vehicles by combining Lyapunov’s direct method with a smooth feedback control law. A leader-follower framework is utilized to develop a guided formation control scheme through a modular design procedure, drawing inspiration from integrator backstepping and cascade theory concepts [8]. In order to mitigate the negative effects caused by communication delays, a control system that can effectively manage inter-vehicle communication issues is designed [9], which can be particularly challenging in underwater environments. Also, Yan et al. [10] present coordinated control protocols, both with and without time delay, to address the coordination control problem of multiple Autonomous Underwater Vehicles (AUVs) under switching communication topologies using discrete information.

To design stable controllers, many methods are used, among which the sliding mode control (SMC) method is quite popular. SMC is a robust control method that can effectively suppress the effects of external disturbances and modeling uncertainties, making it well-suited for applications with non-linear and uncertain dynamics [11,12,13,14]. Wu et al. [12] combine SMC and backstepping techniques to design a closed-loop control system to deal with uncertainty in formation control and demonstrate the effectiveness of the method. Wang et al. [13] proposed a new method to deal with the uncertainties. The approach involves utilizing sliding mode control, multilayer neural networks, and adaptive robust techniques to design an effective formation controller for underwater vehicles. Also, Su et al. [15] built an adaptive fixed time integral sliding mode disturbance observer to accurately estimate compound disturbance. Nevertheless, the use of sliding mode control (SMC) comes with a notable disadvantage known as chattering [16]. To address this issue and enhance the system’s robustness, various methods have been introduced. Backstepping method is one of them. Backstepping control offers a systematic and recursive approach to designing control laws for complex nonlinear systems, ensuring stability and providing flexibility in handling uncertainties and disturbances, making it an advantageous choice compared to sliding mode control [17,18,19]. The Lyapunov-based backstepping approach is developed and proved to be able to work effectively [17,18]. Also, Wang et al. [20] built a graphtheory-based backstepping controller to deal with the disturbance. Zaidi et al. [21] developed a combined approach of chatter-free sliding mode control and backstepping technique in their design. Yang et al. [22] presented a controller that combines backstepping and sliding mode control techniques to effectively address external disturbances. Also, neural networks have gained popularity in recent years for handling uncertainty disturbances [23,24,25]. Neural networks possess the property of universal function approximation, enabling them to approximate any continuously differentiable function. Using RBFNN to deal with the uncertainties is proved very efficient [26,27]. Using RBFNN and combining LKFs and backstepping techniques, Zhao et al. [28] proposed a control scheme. Besides, guidance laws are widely utilized in trajectory tracking problems. They can effectively improve the robustness and adaptive performance of the system [29,30,31]. Incorporating guidance laws into formation control is a beneficial approach to consider.

However, achieving a stable performance of the system is challenging, despite the inherent difficulty in maintaining stable system performance, a breakthrough was first made with the development of the prescribed performance control (PPC) method. The core concept of achieving predefined transient and steady-state performance of output tracking error is captured by an innovative PPC approach introduced in [32], employing a transformation function that strictly increases or decreases the tracking error. Recently, more PPC methods have been developed [22,33,34,35]. They have achieved good results, but the challenge for formation control remains [36]. Mehdifar et al. [37] presented a graph-based formation control approach for leader-follower multiagent systems in a distributed manner using a prescribed performance strategy, yielding favorable outcomes. However, the method does not account for external disturbances. Dai et al. [36] incorporates barrier Lyapunov functions and an adaptive backstepping procedure to ensure the boundedness of the closed-loop systems while guaranteeing transient performance. However, their method heavily relies on the accuracy of dynamic modeling, which implies that its robustness and adaptive may be limited. Jiang et al. [35] suggests a prescribed-time formation control approach for second-order nonlinear multi-agent systems with a directed graph. However, there is room for further optimization of the system responsiveness of this approach. In a word, the formation members have limited communication capabilities, and complex algorithms may hinder the controller’s performance. Additionally, accurately approximating disturbances is challenging, which can impact the system’s robustness and adaptability [6,38,39].

In this paper, a controller is proposed for autonomous vehicle formation control, combining the leader-follower method and backstepping technique. The leader’s predefined trajectory and desired formation shape guide the generation of desired yaw angles and velocities for the followers using a graph-based guidance law. Nonlinear error handling is achieved through a second-order filter and RBFNN, complemented by an adaptive law. Furthermore, a barrier Lyapunov function is employed to achieve prescribed performance control. In comparison with previously mentioned methods, our main contributions can be summarized as follows: (1) This work combines these approaches, enabling precise trajectory tracking and formation maintenance in autonomous vehicle formation control. (2) The graph-based guidance law generates desired yaw angles and velocities based on the formation members’ graph relationship, ensuring accurate formation control and alignment with the leader’s trajectory. (3) The use of second-order filters and RBFNN effectively handles nonlinear errors, improving control performance by mitigating non-linearities, enhancing stability, and tracking accuracy. The adaptive law further enhances adaptability to changing dynamics.

The rest of this paper is structured as follows: In Section 2, we introduce the vehicle’s kinetic and dynamic model. Section 3 describes the design of the proposed controller, where we employ a Lyapunov function to demonstrate stability. Section 4 presents the simulation results for formation trajectory tracking control in the presence of external disturbances. Finally, Section 5 conclues this work.

2. Kinematics and Dynamics Models

Referring to Figure 1, the kinematics and dynamics of the vehicle is

$$\left\{\begin{array}{cc}\dot{x_i}\hfill & =v_{i}cos\left({\theta }_i\right)\hfill \\ \dot{y_i}\hfill & =v_{i}sin\left({\theta }_i\right)\hfill \\ \dot{{\theta }_i}\hfill & ={\sigma }_i^{-1}{v}_{i}tan\left({\delta }_i\right)+f_{\theta ,i}\left({\theta }_i\right)+d_{\theta ,i}\hfill \\ \dot{v_i}\hfill & =F_i+f_{v,i}\left(v_i\right)+d_{v,i}\hfill \end{array}\right.$$

(1)

where the constant ${\sigma }_i>0$ is the length of each vehicle. Here, $\left(x_i,y_i\right)$ is the reference point of the vehicle with $i\in \{1,\dots ,n\}$ and is placed at the midpoint of the rear axle, with $x_i\in \mathbb{R}$ being the longitudinal position and $y_i\in (-a,a)$ being the lateral position of the vehicle in an inertial frame with Cartesian coordinates $(X,Y)$; $v_i$ is the speed of the vehicle at the point $\left(x_i,y_i\right),{\theta }_i\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ is the angular orientation of the vehicle with respect to the X axis, ${\delta }_i$ is the steering angle of the front wheels relative to the orientation ${\theta }_i$ of the vehicle, and $F_i$ is the acceleration of the vehicle. $f_{\theta ,i}\left({\theta }_i\right)$ and $f_{v,i}\left(v_i\right)$ are unknown nonlinear functions. $d_{\theta ,i}\left(t\right)$ and $d_{v,i}\left(t\right)$ denote unknown and bounded external disturbances.

Lemma 1. 

Nonlinear function $\mathit{g}\left(\mathit{x}\right)$ can be approximated by an $RBFNN$ under given accuracy $\overline{\epsilon }>0$. That is, $\mathit{g}\left(\mathit{x}\right)$ can be written as

$$\mathit{g}\left(\mathit{x}\right)={\mathit{W}}^T\mathit{\phi }\left(\mathit{x}\right)+\epsilon \left(\mathit{x}\right),$$

where $W\in R^l$ is the ideal constant weight vector, $\epsilon \left(\mathit{x}\right)$ is the approximation error satisfying $\left|\epsilon \left(\mathit{x}\right)\right|\le \overline{\epsilon }.\mathit{\phi }\left(\mathit{x}\right)={\left({\phi }_1\left(\mathit{x}\right),{\phi }_2\left(\mathit{x}\right),\dots ,{\phi }_l\left(\mathit{x}\right)\right)}^T$ denotes the Gaussian basis function vector with ${\phi }_j\left(\mathit{x}\right)=exp\left(-\frac{{\parallel \mathit{x}-\varrho \parallel }^2}{2{\sigma }^2}\right),j=1,2,\dots ,l$, where ϱ and σ are the center and width of Gaussian basis function ${\phi }_j\left(\mathit{x}\right)$, respectively. $\mathit{x}={\left(x_1,x_2,\dots ,x_q\right)}^T$. Then we have ${\parallel \mathit{\phi }\left(\mathit{x}\right)\parallel }^2\le \parallel \mathit{\phi }(\dot{\mathit{x}}{)\parallel }^2$, where $\dot{\mathit{x}}={\left(x_1,x_2,\dots ,x_r\right)}^T,r\le q$.

Lemma 2. 

For $\forall a,b\ge 0$, and $p,q>0$, satisfying $1/p+1/q=1$, the inequality holds as $ab\le \frac{a^p}{p}+\frac{b^q}{q}$.

3. Controller Design

In this section, we propose a formation tracking controller for the vehicle formation members. Figure 2 and Figure 3 illustrate the setup, where member i follows leader j as its follower. The whole system is designed with the help of backstepping technique. The leader’s desired trajectory is given, and based on the leader’s kinematics and dynamics, its desired velocity and steering angle are designed and fed into the guidance law. Subsequently, we calculate the desired yaw angle ${\theta }_{d,i}$ and velocity $v_{d,i}$ for the followers, which are then used as inputs to their respective steering angle and velocity controllers. To ensure high robustness against unknown disturbances, we employ a second-order filter to estimate the error and use a Radial Basis Function Neural Network (RBFNN) to approximate the unknown non-linear function. For this purpose, we design an adaptive law. Additionally, we consider a nominal function in the system and introduce another adaptive law to handle it. Finally, we compute the final desired steering angle and velocity to guide the followers effectively.

3.1. Guidance Law Design

The guidance law is specifically designed to enhance the system’s flexibility and adaptability. In this context, the formation’s desired shape is predefined, while the initial positions of the formation members are randomized. To illustrate, let’s consider the scenario involving a leader (denoted as j) and one of its followers (i). The current position of follower i is represented as $p_i$, while $p_{i,d}$ signifies its desired position. The desired position vector ${\mathit{p}}_{i,d}={[x_{i,d},y_{i,d}]}^{\mathrm{T}}={[x_i,y_i]}^{\mathrm{T}}+{[\Delta x,\Delta y]}^{\mathrm{T}}$.

Based on Figure 3, our objective is to control member i effectively, which involves ensuring that $p_i$ converges to the desired position. We define an error term to achieve this and aim to drive it to convergence. This error is described as (2).

$$\left[\begin{array}{c}{x}_{e,i}\hfill \\ y_{e,i}\hfill \end{array}\right]={\left[\begin{array}{cc}cos{\theta }_{i,j,p}& -sin{\theta }_{i,j,p}\\ sin{\theta }_{i,j,p}& cos{\theta }_{i,j,p}\end{array}\right]}^T\left({\mathit{p}}_i-{\mathit{p}}_{\mathit{i},\mathit{d}}\right)$$

(2)

where ${\theta }_{i,j,p}=atan2\left(y_i^{\prime }\left(t\right)+\Delta y^{\prime }\left(t\right),x_i^{\prime }+\Delta x^{\prime }\right)\in \left[-\pi ,\pi \right]$, $y_{e,i}$ is the shortest distance between the real position ${\mathit{p}}_i$ of the vehicle i and the moving line of the desired position ${\mathit{p}}_{i,d}$, $x_{e,i}$ is the distance along the moving line between the real position ${\mathit{p}}_i$ of the vehicle i and the moving line of the desired position ${\mathit{p}}_{i,d}$.

To minimize the error and approach zero, it is essential to calculate the error precisely. We can easily obtain the derivative of $y_{e,i}$ as follows:

$${\dot{y}}_{e,i}=v_{i}sin\left({\theta }_i-{\theta }_{p,i}\right).$$

(3)

The barrier Lyapunov function is employed in the design of the governing law to restrict the error and ensure the controlled member maintains a stable state during motion. Moreover, separate Lyapunov candidates are designed for $y_{e,i}$ and $x_{e,i}$ to significantly reduce interference between different errors. A Lyapunov function for $y_{e,i}$ is presented as follows:

$$V_1=\frac{b_{y,i}^2}{\pi }tan\left(\frac{\pi {y_{e,i}}^2}{2b_{y,i}^2}\right),$$

(4)

where $b_{y,i}$ is the upper boundary of $y_{e,i}$.

By taking the derivative of Eq. (4), one can prove the stability of $V_1$ as follows:

$${\dot{V}}_1=\frac{y_{e,i}\dot{y_{e,i}}}{{cos}^2\left(\frac{\pi {y_{e,i}}^2}{2b_{y,i}^2}\right)}+\frac{2b_{y,i}{\dot{b}}_{y,i}}{\pi }tan\left(\frac{\pi {y_{e,i}}^2}{2b_{y,i}^2}\right)-\left(\frac{{\dot{b}}_{y,i}}{b_{y,i}}\right)\frac{{y_{e,i}}^2}{{cos}^2\left(\frac{\pi {y_{e,i}}^2}{2b_{y,i}^2}\right)}.$$

(5)

Based on the vector field guidance law, the desired yaw angle ${\theta }_{d,i}$ is designed as follows:

$${\theta }_{d,i}={\theta }_{p,i}+arcsin(-\frac{k_y{b}_{y,i}^2}{2v_i\pi y_{e,i}}sin\left(\frac{\pi y_{e,i}^2}{b_{y,i}^2}\right)+\frac{{\dot{b}}_{y,i}}{v_i{b}_{y,i}}{y}_{e,i}),$$

(6)

where $k_y>0$ is the guidance law parameter, which is used to represent the strength of the vector field.

Defining $k_b=sup\left|\frac{{\dot{b}}_{y,i}}{b_{y,i}}\right|$, according to Eqs. (3) and (6), Eq. (5) can be rewritten as follows:

$$\begin{array}{cc}{\dot{V}}_1\hfill & =\frac{y_{e,i}{\dot{y}}_{e,i}}{{cos}^2\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)}+\frac{2b_{y,i}{\dot{b}}_{y,i}}{\pi }tan\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)-\left(\frac{{\dot{b}}_{y,ic}}{b_{y,i}}\right)\frac{y_{e,i}^2}{{cos}^2\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)}\hfill \\ \hfill & \le \frac{y_{e,i}{\dot{y}}_{e,i}}{{cos}^2\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)}-\left(\frac{{\dot{b}}_{y,ic}}{b_{y,i}}\right)\frac{y_{e,i}^2}{{cos}^2\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)}+\frac{2k_b{b}_{y,i}^2}{\pi }tan\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)\hfill \\ \hfill & \le \frac{y_{e,i}{v}_{i}sin\left({\theta }_i-{\theta }_{p,i}\right)}{{cos}^2\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)}-\left(\frac{{\dot{b}}_{y,ic}}{b_{y,i}}\right)\frac{y_{e,i}^2}{{cos}^2\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)}+\frac{2k_b{b}_{y,i}^2}{\pi }tan\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)\hfill \\ \hfill & \le \frac{y_{e,i}\left(-\frac{k_d{b}_{y,i}^2}{2\pi y_{e,i}}sin\left(\frac{\pi y_{e,i}^2}{b_{y,i}^2}\right)+\frac{{\dot{b}}_{y,i}}{b_{y,i}}{y}_{e,i}\right)}{{cos}^2\left(\frac{\pi y_{e,i}^2}{2b_{y,i}}\right)}-\left(\frac{{\dot{b}}_{y,ic}}{b_{y,i}}\right)\frac{y_{e,i}^2}{{cos}^2\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)}+\frac{2k_b{b}_{y,i}^2}{\pi }tan\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)\hfill \\ \hfill & \le -\left(k_d-2k_b\right)\frac{b_{y,i}^2}{\pi }tan\left(\frac{\pi y_{e,i}^2}{2b_{y,i}^2}\right)\hfill \end{array}$$

(7)

By choosing $k_d>2k_b$, the following inequality holds:

$${\dot{V}}_1\le -c_1{V}_1,$$

(8)

where $c_1=k_d-2k_b$.

From Figure 3, the derivative of $x_{e,i}$ can be obtained as follows:

$${\dot{x}}_{e,i}=v_j-v_{i}cos{\beta }_i.$$

(9)

A Lyapunov function is given as:

$$V_2=\frac{b_{x,i}^2}{\pi }tan\left(\frac{\pi {x_{e,i}}^2}{2b_{x,i}^2}\right).$$

(10)

Also, according to the guidance law, the desired velocity is designed as follows:

$$v_{d,i}=\frac{1}{cos{\beta }_i}(\frac{k_v{b}_{x,i}^2}{2\pi x_{e,i}}sin\left(\frac{\pi x_{e,i}^2}{b_{x,i}^2}\right)-\frac{{\dot{b}}_{x,i}}{b_{x,i}}{x}_{e,i}+v_j),$$

(11)

where ${\beta }_i={\theta }_{i,j,p}-{\theta }_i$.

Upon differentiating the equation provided in Eq. (10), the resulting expression for ${\dot{V}}_2$ is given by:

$$\begin{array}{cc}{\dot{V}}_2\hfill & =\frac{x_{e,i}{\dot{x}}_{e,i}}{{cos}^2\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)}+\frac{2b_{x,i}{\dot{b}}_{x,i}}{\pi }tan\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)-\left(\frac{{\dot{b}}_{x,ic}}{b_{x,i}}\right)\frac{x_{e,i}^2}{{cos}^2\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)}\hfill \\ \hfill & \le \frac{x_{e,i}{\dot{x}}_{e,i}}{{cos}^2\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)}-\left(\frac{{\dot{b}}_{x,i}}{b_{x,i}}\right)\frac{x_{e,i}^2}{{cos}^2\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)}+\frac{2k_c{b}_{x,i}^2}{\pi }tan\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)\hfill \\ \hfill & \le \frac{x_{e,i}\left(v_j-v_{i}cos{\beta }_i\right)}{{cos}^2\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)}-\left(\frac{{\dot{b}}_{x,i}}{b_{x,i}}\right)\frac{x_{e,i}^2}{{cos}^2\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)}+\frac{2k_c{b}_{x,i}^2}{\pi }tan\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)\hfill \\ \hfill & \le -\left(k_v-2k_c\right)\frac{b_{x,i}^2}{\pi }tan\left(\frac{\pi x_{e,i}^2}{2b_{x,i}^2}\right)\hfill \end{array}$$

(12)

where $k_c=sup\left|\frac{{\dot{b}}_{x,i}}{b_{x,i}}\right|$.

By selecting $k_v>2k_c$, we can additionally derive:

$${\dot{V}}_2\le -c_2{V}_2,$$

(13)

where $c_2=k_v-2k_c$.

3.2. Orientation Angle Controller Design

Now, with the guidance law supplying the desired yaw angle and velocity, it is imperative to devise controllers for these parameters. The controllers are developed independently, providing various practical advantages, including modularity, specialization, simplicity, robustness, and scalability. For formation member i, defining the angle error as

$$e_{\theta ,i}={\theta }_i-{\theta }_{d,i}$$

(14)

the computations of the derivatives of ${\theta }_{d,i}$ are extremely intricate. To mitigate this complexity, we introduce a second-order filter expressed as:

$$\left\{\begin{array}{cc}\hfill & {\dot{\Phi }}_{10}={\Phi }_{20}\hfill \\ \hfill & {\dot{\Phi }}_{20}=-2{\zeta }_0{\omega }_{n0}{\Phi }_{20}-{\omega }_n^2\left({\Phi }_{10}-{\theta }_{d,i}\right)\hfill \end{array}.\right.$$

(15)

Here, the damping rate ${\zeta }_0$ and frequency ${\omega }_{n0}$ are predetermined constants, ${\theta }_{d,i}$ represents the input, ${\Phi }_{10}$ is the output and an estimation of ${\theta }_{d,i}$, and ${\Phi }_{20}$ can be interpreted as the derivative of ${\theta }_{d,i}$, denoted as ${\widehat{\theta }}_{d,i}$. The estimated error of this second-order filter is defined as follows:

$${\tilde{\dot{\theta }}}_{d,i}={\dot{\theta }}_{d,i}-{\widehat{\dot{\theta }}}_{d,i}.$$

(16)

Consider a Lyapunov candidate

$$V_0=\frac{1}{2}{e}_{\theta ,i}^2$$

(17)

and refer to Eqs. (1) and (16). The derivative of $V_0$ with respect to time can be expressed as:

$$\begin{array}{cc}{\dot{V}}_0\hfill & =e_{\theta ,i}{\dot{e}}_{\theta ,i}\hfill \\ \hfill & =e_{\theta ,i}\left(\dot{{\theta }_i}-{\dot{\theta }}_{d,i}\right)\hfill \\ \hfill & =e_{\theta ,i}\left({\sigma }_i^{-1}{v}_{i}tan\left({\delta }_i\right)+f_{\theta ,i}+d_{\theta ,i}-{\widehat{\dot{\theta }}}_{d,i}-{\tilde{\dot{\theta }}}_{d,i}\right).\hfill \end{array}$$

(18)

RBFNN stands for Radial Basis Function Neural Network, which is a type of artificial neural network that is commonly used for function approximation and pattern recognition tasks. It comprises three layers: an input layer, a hidden layer featuring radial basis functions, and an output layer. In this study, we employ an RBFNN to approximate unknown nonlinear functions. Consequently,

$$\begin{array}{cc}{\dot{V}}_0\hfill & =e_{\theta ,i}\left({\sigma }_i^{-1}{v}_{i}tan\left({\delta }_i\right)+W_{\theta ,i}{\phi }_{\theta ,i}+{\epsilon }_{\theta ,i}+d_{\theta ,i}-{\widehat{\dot{\theta }}}_{d,i}-{\tilde{\dot{\theta }}}_{d,i}\right)\hfill \\ \hfill & =e_{\theta ,i}\left({\sigma }_i^{-1}{v}_{i}tan\left({\delta }_i\right)+W_{\theta ,i}{\phi }_{\theta ,i}-{\widehat{\dot{\theta }}}_{d,i}+{\tilde{d}}_{\theta ,i}\right),\hfill \end{array}$$

(19)

where ${\tilde{d}}_{\theta ,i}={\epsilon }_{\theta ,i}+d_{\theta ,i}-{\tilde{\dot{\theta }}}_{d,i}$ represents total disturbances. It is evident that ${\tilde{d}}_{\theta ,i}$ is bounded, adhering to ${\tilde{d}}_{\theta ,i}\le {\overline{d}}_{\theta ,i}$. Furthermore, ${\tilde{W}}_{\theta ,i}=W_{\theta ,i}-{\widehat{W}}_{\theta ,i}$ and $W_{\theta ,i}\le {\overline{W}}_{\theta ,i}$.

To deal with the unknown nonlinear functions, the adaptive law is designed as follows:

$${\dot{\widehat{W}}}_{\theta ,i}=k_{1,\theta ,i}\left({\phi }_{\theta ,i}{e}_{\theta ,i}-k_{2,\theta ,i}{\widehat{W}}_{\theta ,i}\right)$$

(20)

with positive parameters $k_{1,\theta ,i}$ and $k_{2,\theta ,i}$.

To deal with nominal disturbance, another adaptive law is designed as follows:

$${\dot{\widehat{d}}}_{\theta ,i}=k_{3,\theta ,i}\left(e_{\theta ,i}-k_{4,\theta ,i}{\widehat{d}}_{\theta ,i}\right)$$

(21)

with positive parameters $k_{3,\theta ,i}$ and $k_{4,\theta ,i}$.

For the adaptive tracking controller, the intended steering angle of the front wheels can be formulated as follows:

$${\delta }_{d,i}=arctan\left(v_i^{-1}{\sigma }_i(-k_{5,\theta ,i}{e}_{\theta ,i}-{\widehat{d}}_{\theta ,i}-{\widehat{W}}_{\theta ,i}{\phi }_{\theta ,i}+{\widehat{\dot{\theta }}}_{d,i})\right).$$

(22)

Consider a Lyapunov candidate given by:

$$V_3=\frac{1}{2}{e}_{\theta ,i}^2+\frac{1}{2}{k}_{3,\theta ,i}^{-1}{\widehat{d}}_{\theta ,i}^2+\frac{1}{2}{k}_{1,\theta ,i}^{-1}{\tilde{W}}_{\theta ,i}^2,$$

(23)

where $\tilde{d}=\overline{d}-\widehat{d}$, and combine Eqs. (20), (18), and (22). We can express the time derivative of $V_3$ as follows:

$$\begin{array}{cc}{\dot{V}}_3=\hfill & e_{\theta ,i}{\dot{e}}_{\theta ,i}+{\widehat{d}}_{\theta ,i}{\dot{\widehat{d}}}_{\theta ,i}+{\tilde{W}}_{\theta ,i}{\dot{\tilde{W}}}_{\theta ,i}\hfill \\ =\hfill & e_{\theta ,i}\left({\sigma }_i^{-1}{v}_{i}tan\left({\delta }_i\right)+W_{\theta ,i}{\phi }_{\theta ,i}-{\widehat{\dot{\theta }}}_{d,i}+{\tilde{d}}_{\theta ,i}\right)+{\widehat{d}}_{\theta ,i}\left(e_{\theta ,i}-k_{4,\theta ,i}{\widehat{d}}_{\theta ,i}\right)-{\tilde{W}}_{\theta ,i}{\dot{\widehat{W}}}_{\theta ,i}\hfill \\ =\hfill & e_{\theta ,i}\left(-k_{5,\theta ,i}{e}_{\theta ,i}-{\widehat{d}}_{\theta ,i}-{\widehat{W}}_{\theta ,i}{\phi }_{\theta ,i}+W_{\theta ,i}{\phi }_{\theta ,i}+{\tilde{d}}_{\theta ,i}\right)+{\widehat{d}}_{\theta ,i}\left(e_{\theta ,i}-k_{4,\theta ,i}{\widehat{d}}_{\theta ,i}\right)\hfill \\ \hfill & -{\tilde{W}}_{\theta ,i}\left({\phi }_{\theta ,i}{e}_{\theta ,i}-k_{2,\theta ,i}{\widehat{W}}_{\theta ,i}\right)\hfill \\ \le \hfill & -k_{5,\theta ,i}{e}_{\theta ,i}^2-k_{4,\theta ,i}{\widehat{d}}_{\theta ,i}^2+e_{\theta ,i}{\overline{d}}_{\theta ,i}+k_{2,\theta ,i}{\tilde{W}}_{\theta ,i}{\widehat{W}}_{\theta ,i}\hfill \\ \le \hfill & -k_{5,\theta ,i}{e}_{\theta ,i}^2-k_{4,\theta ,i}{\widehat{d}}_{\theta ,i}^2+e_{\theta ,i}{\overline{d}}_{\theta ,i}+k_{2,\theta ,i}∥{\tilde{W}}_{\theta ,i}∥∥{\widehat{W}}_{\theta ,i}∥\hfill \\ \le \hfill & -k_{5,\theta ,i}{e}_{\theta ,i}^2-k_{4,\theta ,i}{\widehat{d}}_{\theta ,i}^2+e_{\theta ,i}{\overline{d}}_{\theta ,i}+k_{2,\theta ,i}∥{\tilde{W}}_{\theta ,i}∥({\overline{W}}_{\theta ,i}-∥{\tilde{W}}_{\theta ,i}∥)\hfill \end{array}$$

(24)

where $∥W_{\theta ,i}∥\le {\overline{W}}_{\theta ,i}$.

Utilizing Lemma 2, we can derive the following inequalities:

$$∥{\tilde{W}}_{\theta ,i}∥({\overline{W}}_{\theta ,i}-∥{\tilde{W}}_{\theta ,i}∥)\le -\frac{1}{2}{∥{\tilde{W}}_{\theta ,i}∥}^2+\frac{1}{2}{{\overline{W}}_{\theta ,i}}^2$$

(25)

and

$$e_{\theta ,i}{\overline{d}}_{\theta ,i}\le \frac{1}{2}{∥e_{\theta ,i}∥}^2+\frac{1}{2}{{\overline{d}}_{\theta ,i}}^2.$$

(26)

Then, combining these inequalities, we obtain:

$$\begin{array}{cc}{\dot{V}}_3\le \hfill & -k_{5,\theta ,i}{e}_{\theta ,i}^2-k_{4,\theta ,i}{\widehat{d}}_{\theta ,i}^2-\frac{1}{2}{k}_{2,\theta ,i}{∥{\tilde{W}}_{\theta ,i}∥}^2+\frac{1}{2}{∥e_{\theta ,i}∥}^2+\frac{1}{2}{{\overline{d}}_{\theta ,i}}^2+k_{2,\theta ,i}\frac{1}{2}{∥{\overline{W}}_{\theta ,i}∥}^2\hfill \\ \le \hfill & -(k_{5,\theta ,i}-\frac{1}{2})e_{\theta ,i}^2-k_{4,\theta ,i}{\widehat{d}}_{\theta ,i}^2-\frac{1}{2}{k}_{2,\theta ,i}{∥{\tilde{W}}_{\theta ,i}∥}^2+\frac{1}{2}{{\overline{d}}_{\theta ,i}}^2+k_{2,\theta ,i}\frac{1}{2}{∥{\overline{W}}_{\theta ,i}∥}^2\hfill \end{array}$$

(27)

From this, we can conclude that ${\dot{V}}_3\le -{\sigma }_3{V}_3+{\zeta }_1$ where ${\sigma }_3=min\{k_{5,\theta ,i}-\frac{1}{2},k_{4,\theta ,i},k_{2,\theta ,i}\}>0$ and ${\zeta }_1=\frac{1}{2}{{\overline{d}}_{\theta ,i}}^2+k_{2,\theta ,i}\frac{1}{2}{∥{\overline{W}}_{\theta ,i}∥}^2>0$.

3.3. Velocity Controller Design

For formation members, the velocity error is defined as:

$$e_{v,i}=v_i-v_{d,i}$$

(28)

The computations of the derivatives of $v_{d,i}$ are extremely intricate. Therefore, a second-order filter is introduced to solve this problem as

$$\left\{\begin{array}{cc}\hfill & {\dot{\Phi }}_{30}={\Phi }_{40}\hfill \\ \hfill & {\dot{\Phi }}_{40}=-2{\zeta }_0{\omega }_{n0}{\Phi }_{40}-{\omega }_n^2\left({\Phi }_{10}-v_{d,i}\right)\hfill \end{array}\right.$$

(29)

where damp rate ${\zeta }_0$ and frequency ${\omega }_{n0}$ are designed constants, $v_{d,i}$ is the input, ${\Phi }_{30}$ is the output and the estimation of $v_{d,i}$, ${\Phi }_{40}$ can be taken as the derivative of $v_{d,i}$ which is denoted as ${\widehat{v}}_{d,i}$.

The estimate error of this second-order filter is defined as

$${\tilde{\dot{v}}}_{d,i}={\dot{v}}_{d,i}-{\widehat{\dot{v}}}_{d,i}$$

(30)

Differentiating the velocity error,

$$\begin{array}{cc}{\dot{e}}_{v,i}\hfill & =\dot{v_i}-{\dot{v}}_{d,i}\hfill \\ \hfill & =F_i+f_{v,i}\left(v_i\right)+d_{v,i}-{\widehat{\dot{v}}}_{d,i}-{\tilde{\dot{v}}}_{d,i}\hfill \end{array}$$

(31)

In this paper, an RBFNN is employed to approximate unknown nonlinear functions. Then,

$$\begin{array}{cc}{\dot{e}}_{v,i}\hfill & =F_i+W_{v,i}{\phi }_{v,i}+{\epsilon }_{v,i}+d_{v,i}-{\widehat{\dot{v}}}_{d,i}-{\tilde{\dot{v}}}_{d,i}\hfill \\ \hfill & =F_i+W_{v,i}{\phi }_{v,i}+d_{v,i}-{\widehat{\dot{v}}}_{d,i}+{\tilde{d}}_{v,i}\hfill \end{array}$$

(32)

where ${\tilde{d}}_{v,i}={\epsilon }_{v,i}+d_{v,i}-{\tilde{\dot{v}}}_{d,i}$ is total disturbances. Apparently, ${\tilde{d}}_{v,i}$ is bounded, satisfying ${\tilde{d}}_{v,i}\le {\overline{d}}_{v,i}$. ${\tilde{W}}_{v,i}=W_{v,i}-{\widehat{W}}_{v,i}$ and $W_{v,i}\le {\overline{W}}_{v,i}$.

To deal with the unknown nonlinear functions, the adaptive law is designed as

$${\dot{\widehat{W}}}_{v,i}=k_{1,v,i}\left({\phi }_{v,i}{e}_{v,i}-k_{2,v,i}{\widehat{W}}_{v,i}\right)$$

(33)

with positive parameters $k_{1,v,i}$ and $k_{2,v,i}$.

To deal with nominal disturbance, another adaptive law is designed as

$${\dot{\widehat{d}}}_{v,i}=k_{3,v,i}\left(e_{v,i}-k_{4,v,i}{\widehat{d}}_{v,i}\right)$$

(34)

with positive parameters $k_{3,v,i}$ and $k_{4,v,i}$. The adaptive tracking controller for the desired velocity can be formulated as follows:

$$F_{d,i}=-k_{5,v,i}{e}_{v,i}-{\widehat{d}}_{v,i}-{\widehat{W}}_{v,i}{\phi }_{v,i}+{\widehat{\dot{v}}}_{d,i}.$$

(35)

Consider a Lyapunov candidate

$$V_4=\frac{1}{2}{e}_{v,i}^2+\frac{1}{2}{k}_{3,v,i}^{-1}{\widehat{d}}_{v,i}^2+\frac{1}{2}{k}_{1,v,i}^{-1}{\tilde{W}}_{v,i}^2.$$

(36)

Then, by combining Eqs. (31), (34), and (33), we can describe the differential of $V_4$ as follows:

$$\begin{array}{cc}{\dot{V}}_4=\hfill & e_{v,i}{\dot{e}}_{v,i}+{\widehat{d}}_{v,i}{\dot{\widehat{d}}}_{v,i}+{\tilde{W}}_{v,i}{\dot{\tilde{W}}}_{v,i}\hfill \\ =\hfill & e_{v,i}\left(F_i+W_{v,i}{\phi }_{v,i}-{\widehat{\dot{v}}}_{d,i}+{\tilde{d}}_{v,i}\right)+{\widehat{d}}_{v,i}\left(e_{v,i}-k_{4,v,i}{\widehat{d}}_{v,i}\right)-{\tilde{W}}_{v,i}{\dot{\widehat{W}}}_{v,i}\hfill \\ =\hfill & e_{v,i}\left(-k_{5,v,i}{e}_{v,i}-{\widehat{d}}_{v,i}-{\widehat{W}}_{v,i}{\phi }_{v,i}+W_{v,i}{\phi }_{v,i}+{\tilde{d}}_{v,i}\right)+{\widehat{d}}_{v,i}\left(e_{v,i}-k_{4,v,i}{\widehat{d}}_{v,i}\right)\hfill \\ \hfill & -{\tilde{W}}_{v,i}\left({\phi }_{v,i}{e}_{v,i}-k_{2,v,i}{\widehat{W}}_{v,i}\right)\hfill \\ \le \hfill & -k_{5,v,i}{e}_{v,i}^2-k_{4,v,i}{\widehat{d}}_{v,i}^2+e_{v,i}{\overline{d}}_{v,i}+k_{2,v,i}{\tilde{W}}_{v,i}{\widehat{W}}_{v,i}\hfill \\ \le \hfill & -k_{5,v,i}{e}_{v,i}^2-k_{4,v,i}{\widehat{d}}_{v,i}^2+e_{v,i}{\overline{d}}_{v,i}+k_{2,v,i}∥{\tilde{W}}_{v,i}∥∥{\widehat{W}}_{v,i}∥\hfill \\ \le \hfill & -k_{5,v,i}{e}_{v,i}^2-k_{4,v,i}{\widehat{d}}_{v,i}^2+e_{v,i}{\overline{d}}_{v,i}+k_{2,v,i}∥{\tilde{W}}_{v,i}∥({\overline{W}}_{v,i}-∥{\tilde{W}}_{v,i}∥)\hfill \end{array}$$

(37)

where $∥W_{v,i}∥\le {\overline{W}}_{v,i}$.

Utilizing Lemma 2, we can derive the following inequalities:

$$∥{\tilde{W}}_{v,i}∥({\overline{W}}_{v,i}-∥{\tilde{W}}_{v,i}∥)\le -\frac{1}{2}{∥{\tilde{W}}_{v,i}∥}^2+\frac{1}{2}{{\overline{W}}_{v,i}}^2$$

(38)

and

$$e_{v,i}{\overline{d}}_{v,i}\le \frac{1}{2}{∥e_{v,i}∥}^2+\frac{1}{2}{{\overline{d}}_{v,i}}^2.$$

(39)

Combining these inequalities, we obtain:

$$\begin{array}{cc}{\dot{V}}_4\le \hfill & -k_{5,v,i}{e}_{v,i}^2-k_{4,v,i}{\widehat{d}}_{v,i}^2-\frac{1}{2}{k}_{2,v,i}{∥{\tilde{W}}_{v,i}∥}^2+\frac{1}{2}{∥e_{v,i}∥}^2+\frac{1}{2}{{\overline{d}}_{v,i}}^2\hfill \\ \hfill & +k_{2,v,i}\frac{1}{2}{∥{\overline{W}}_{v,i}∥}^2\hfill \\ \le \hfill & -(k_{5,v,i}-\frac{1}{2})e_{v,i}^2-k_{4,v,i}{\widehat{d}}_{v,i}^2-\frac{1}{2}{k}_{2,v,i}{∥{\tilde{W}}_{v,i}∥}^2\hfill \\ \hfill & +\frac{1}{2}{{\overline{d}}_{v,i}}^2+k_{2,v,i}\frac{1}{2}{∥{\overline{W}}_{v,i}∥}^2.\hfill \end{array}$$

(40)

The conclusion can be given that ${\dot{V}}_4\le -{\sigma }_4{V}_4+{\zeta }_2$ with

$$\begin{array}{c}{\sigma }_4=min\{k_{5,v,i}-\frac{1}{2},k_{4,v,i},k_{2,v,i}\}>0\hfill \end{array}$$

(41)

and

$$\begin{array}{c}{\zeta }_2=\frac{1}{2}{{\overline{d}}_{v,i}}^2+k_{2,v,i}\frac{1}{2}{∥{\overline{W}}_{v,i}∥}^2>0.\hfill \end{array}$$

(42)

3.4. Stability Analysis

Consider the Lyapunov candidate represented as:

$$\begin{array}{cc}V\hfill & =V_1+V_2+V_3+V_4\hfill \\ & =\frac{b_{x,i}^2}{\pi }tan\left(\frac{\pi {x_{e,i}}^2}{2b_{x,i}^2}\right)+\frac{b_{y,i}^2}{\pi }tan\left(\frac{\pi {y_{e,i}}^2}{2b_{y,i}^2}\right)+\frac{1}{2}{e}_{\theta ,i}^2+\frac{1}{2}{k}_{3,\theta ,i}^{-1}{\widehat{d}}_{\theta ,i}^2+\frac{1}{2}{k}_{1,\theta ,i}^{-1}{\tilde{W}}_{\theta ,i}^2\hfill \\ & +\frac{1}{2}{e}_{v,i}^2+\frac{1}{2}{k}_{3,v,i}^{-1}{\widehat{d}}_{v,i}^2+\frac{1}{2}{k}_{1,v,i}^{-1}{\tilde{W}}_{v,i}^2\hfill \end{array}$$

(43)

It concludes that $\dot{V}\le -\sigma V+\zeta$ in which

$$\begin{array}{c}\sigma =min\left\{k_d-2k_b,k_v-2k_c,k_{5,\theta ,i}-\frac{1}{2},k_{4,\theta ,i},k_{2,\theta ,i},k_{5,v,i}-\frac{1}{2},k_{4,v,i},k_{2,v,i}\right\}>0\hfill \end{array}$$

(44)

and

$$\begin{array}{c}\zeta =\frac{1}{2}{{\overline{d}}_{v,i}}^2+k_{2,v,i}\frac{1}{2}{∥{\overline{W}}_{v,i}∥}^2+\frac{1}{2}{{\overline{d}}_{\theta ,i}}^2+k_{2,\theta ,i}\frac{1}{2}{∥{\overline{W}}_{\theta ,i}∥}^2>0.\hfill \end{array}$$

(45)

Choosing parameters such that $k_d>2k_b$, $k_v>2k_c$, $k_{5,\theta ,i}>\frac{1}{2}$, and $k_{5,v,i}>\frac{1}{2}$, and integrating Eq. (43), we can derive the inequality $V\le \left(V\left(0\right)-\frac{\zeta }{\sigma }\right)e^{-\sigma t}+\frac{\zeta }{\sigma }$. The conclusion can be drawn that V is bounded. Moreover, $\frac{b_{x,i}^2}{\pi }tan\left(\frac{\pi {x_{e,i}}^2}{2b_{x,i}^2}\right)\le V\le \left(V\left(0\right)-\frac{\zeta }{\sigma }\right)e^{-\sigma t}+\frac{\zeta }{\sigma }$ and $\frac{b_{y,i}^2}{\pi }tan\left(\frac{\pi {y_{e,i}}^2}{2b_{y,i}^2}\right)\le V\le \left(V\left(0\right)-\frac{\zeta }{\sigma }\right)e^{-\sigma t}+\frac{\zeta }{\sigma }$ imply the following constraints hold:

$$\begin{array}{c}{x_{e,i}}^2\le \frac{2b_{x,i}^2}{\pi }{tan}^{-1}\left(\frac{\pi }{b_{x,i}^2}\left(\left(V\left(0\right)-\frac{\zeta }{\sigma }\right)e^{-\sigma t}+\frac{\zeta }{\sigma }\right)\right)<b_{x,i}^2\hfill \end{array}$$

(46)

and

$$\begin{array}{c}{y_{e,i}}^2\le \frac{2b_{y,i}^2}{\pi }{tan}^{-1}\left(\frac{\pi }{b_{y,i}^2}\left(\left(V\left(0\right)-\frac{\zeta }{\sigma }\right)e^{-\sigma t}+\frac{\zeta }{\sigma }\right)\right)<b_{y,i}^2\hfill \end{array}$$

(47)

Therefore, the conclusion can be drawn that $x_{e,i}$ and $y_{e,i}$ are bounded as $\left|x_{e,i}\right|<\left|b_{x,i}\right|$ and $\left|y_{e,i}\right|<\left|b_{y,i}\right|$. Finally, theses terms can be eliminated to a small neighborhood around zero.

4. Simulation Result

In this Section, the simulation results of the formation control with different disturbances are shown. In our simulation, Vehicle 1 serves as the leader, with Vehicles 2, 3, and 4 acting as followers. The leader’s desired trajectory, denoted as $(x_1\left(t\right),y_1\left(t\right))$, follows the equations $x_1\left(t\right)=0.5+5e^{-0.2t}$ and $y_1\left(t\right)=0.5+5e^{-0.2t}$. The vehicles are configured in a quadrilateral formation, with each diagonal measuring 20 meters. Initially, Vehicle 2 is positioned at $(-5.5,-9.9)$, Vehicle 3 at $(-6.5,10)$, and Vehicle 4 at $(-18.5,0.1)$. All vehicles maintain a constant velocity of 1 $m/s$ throughout the simulation.

Table 1. Main parameters of the controller.

Parameter	Value	Parameter	Value
$k_y$	1.01	$k_{4,\theta ,i}$	1.01
$k_v$	1.1	$k_{5,\theta ,i}$	2
$k_{1,\theta ,i}$	0.5	$k_{3,v,i}$	1
$k_{2,\theta ,i}$	8	$k_{4,v,i}$	1
$k_{1,v,i}$	5	$k_{5,v,i}$	0.2
$k_{2,v,i}$	18	${\zeta }_0$	0.8
$k_{3,\theta ,i}$	5	${\omega }_{n0}$	20

Table 1. Main parameters of the controller.

Parameter	Value	Parameter	Value
$k_y$	1.01	$k_{4,\theta ,i}$	1.01
$k_v$	1.1	$k_{5,\theta ,i}$	2
$k_{1,\theta ,i}$	0.5	$k_{3,v,i}$	1
$k_{2,\theta ,i}$	8	$k_{4,v,i}$	1
$k_{1,v,i}$	5	$k_{5,v,i}$	0.2
$k_{2,v,i}$	18	${\zeta }_0$	0.8
$k_{3,\theta ,i}$	5	${\omega }_{n0}$	20

Figure 5 depicts the trajectory of the formation, which maintains its predefined shape effectively even in the presence of various disturbances. The simulation results in Figure 4 and Figure 6 provide a comprehensive analysis of the formation control algorithm’s capability to minimize trajectory tracking errors for the followers.

Initially, at the beginning of the simulation, the position errors along the x and y axes for the followers are approximately 1m, which can be attributed to random initial conditions. However, as the simulation progresses, the formation control algorithm effectively guides the followers towards their target positions, leading to a rapid convergence of position errors. Within a short period, the errors decreased significantly, demonstrating the high precision and effectiveness achieved by the control strategy.

The figures also showcase the error constraints, which define the acceptable range of position errors during operation $\left(0.5m\right)$. The trajectory tracking error consistently stays well within the specified constraints throughout the simulation, affirming the robustness and effectiveness of the proposed formation control algorithm in managing the agents’ motion while adhering to predefined error boundaries.

As depicted in Figure 7, the leader’s angular change exhibits a smooth transition, aligning well with real-world applications. In the initial startup phase (approximately the first 2 seconds), the three followers swiftly adjust their angles within a range of 55 to 140 degrees. Subsequently, they converge to a narrower variation range of 90 to 100 degrees. Around the 5-second mark, Vehicle 4’s angle closely aligns with the leader’s angle, maintaining this alignment throughout the simulation. Similarly, Vehicles 2 and 3 achieve alignment with the leader around the 12-second mark, displaying no visible deviation for the remainder of the simulation. This highlights the efficacy of the formation control algorithm in synchronizing the followers’ angular changes with the leader’s smooth trajectory.

The velocity progression of the formation members is illustrated in Figure 8. Within the initial 3 seconds, Vehicles 2, 3, and 4 rapidly converge their speeds to a range of $1.8-2.5$ m/s and subsequently maintain a speed almost consistent with that of Vehicle 1. Notably, the convergence speed of Vehicles 2 and 3 is significantly faster than that of Vehicle 4. This difference can be attributed to the varying initial positions of the vehicles. The observed behavior indirectly underscores that the proposed method in this study enables the vehicles to swiftly identify their positions and uphold the desired formation alongside the rest of the fleet.

Figure 9–fig:controlinput3 depict the control inputs applied to the followers under various disturbances. The inputs rapidly converge, demonstrating the stability of our approach.

Figure 9. Control inputs of follower1 with different disturbances:(a) Without disturb, (b) $d_{\theta ,i}\left(t\right)$, $d_{v,i}\left(t\right)$=$0.5sin\frac{t}{10}$

Figure 9. Control inputs of follower1 with different disturbances:(a) Without disturb, (b) $d_{\theta ,i}\left(t\right)$, $d_{v,i}\left(t\right)$=$0.5sin\frac{t}{10}$

Figure 10. Control inputs of follower2 with different disturbances:(a) Without disturb, (b) $d_{\theta ,i}\left(t\right)$, $d_{v,i}\left(t\right)$=$0.5sin\frac{t}{10}$

Figure 10. Control inputs of follower2 with different disturbances:(a) Without disturb, (b) $d_{\theta ,i}\left(t\right)$, $d_{v,i}\left(t\right)$=$0.5sin\frac{t}{10}$

Figure 11. Control inputs of follower3 with different disturbances:(a) Without disturb, (b) $d_{\theta ,i}\left(t\right)$, $d_{v,i}\left(t\right)$=$0.5sin\frac{t}{10}$

Figure 11. Control inputs of follower3 with different disturbances:(a) Without disturb, (b) $d_{\theta ,i}\left(t\right)$, $d_{v,i}\left(t\right)$=$0.5sin\frac{t}{10}$

To provide a more detailed depiction of the results, we present the mean and maximum position errors of the followers in Table 2 and Table 3. The tabulated outcomes clearly demonstrate the outstanding performance of the controller, consistent with our expectations.

5. Conclusion

This paper introduces an adaptive leader-follower formation controller with prescribed performance. The guidance law computes the desired velocity and steering angle based on the leader’s trajectory and a predefined formation pattern. To address challenges posed by unknown functions and external disturbances, a second-order filter and an RBFNN, alongside an adaptive law, are employed. Notably, the entire controller adheres to a backstepping method, incorporating distinct velocity and corner controllers to enhance system robustness. Furthermore, the inclusion of a barrier to the Lyapunov function contributes to achieving the prescribed performance. Simulation results illustrate that our proposed controller consistently attains superior performance within the specified limits, even in the presence of various disturbances.