Robust Leader–Follower Formation Control Using Neural Adaptive Prescribed Performance Strategies

Abstract

This paper introduces a novel leader–follower formation control strategy for autonomous vehicles, aimed at achieving precise trajectory tracking in uncertain environments. The approach is based on a graph guidance law that calculates the desired yaw angles and velocities for follower vehicles using the leader’s reference trajectory, improving system stability and predictability. A key innovation is the development of a Neural Adaptive Prescribed Performance Controller (NA-PPC), which incorporates a Radial Basis Function Neural Network (RBFNN) to approximate nonlinear system dynamics and enhances disturbance estimation accuracy. The proposed method enables high-precision trajectory tracking and formation maintenance under random disturbances, which are vital for autonomous vehicle logistics and detection technologies. Leveraging a graph-based guidance law reduces control complexity and improves robustness against external disturbances. The inclusion of second-order filters and adaptive RBFNNs further enhances nonlinear error handling, improving control performance, stability, and accuracy. The integration of guidance laws, leader–follower control strategies, backstepping techniques, and RBFNNs creates a robust formation control system capable of maintaining performance under dynamic conditions. Comprehensive computer simulations validate the effectiveness of this controller, highlighting its potential to advance autonomous vehicle formation control.

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 systems, individual vehicles autonomously make decisions based on locally available information and interactions with neighboring entities. This paper investigates leader–follower formation control for multi-autonomous vehicles, selecting this approach due to its simplicity, ease of implementation, and potential for achieving robust performance under specific conditions [4,5,6].

A significant amount of research has been conducted on the subject, and a considerable portion of it is of high quality. Yang and Gu [7] implemented nonlinear formation alignment and docking control for a fleet of autonomous underwater vehicles by combining Lyapunov’s direct method with a smooth feedback control law. A leader–follower framework has been employed to develop a guided formation control scheme using a modular design approach, incorporating concepts from integrator backstepping and cascade theory [8]. These methods are straightforward and relatively easy to implement; however, they encounter significant challenges when subjected to large disturbances. Essentially, these methods lack the capability to autonomously adjust or effectively resist such disturbances, underscoring a significant limitation in their adaptability and robustness. Numerous approaches have been proposed for designing stable controllers, with the sliding mode control (SMC) method emerging as a particularly favored option. SMC is a robust control strategy that effectively mitigates the effects of external disturbances and modeling uncertainties, making it particularly suitable for applications characterized by nonlinear and unpredictable dynamics [9,10,11,12]. Wu et al. [10] combined SMC and backstepping techniques to design a closed-loop control system to deal with uncertainty in formation control. Wang et al. [11] introduced a method to address uncertainties by integrating SMC, multilayer neural networks, and adaptive robust techniques to develop an effective formation controller for underwater vehicles. Also, Su et al. [13] developed an adaptive fixed-time integral sliding mode observer to precisely estimate compound disturbances. However, the implementation of SMC is associated with a significant drawback known as chattering [14].

To address this issue and enhance system robustness, various methods have been proposed, among which the backstepping method is notable. Backstepping control provides a systematic and recursive approach to designing control laws for complex nonlinear systems, ensuring stability while offering flexibility in managing uncertainties and disturbances, thus presenting advantages over SMC [15,16,17]. The Lyapunov-based backstepping approach has been developed and proven to be able to work effectively [15,16]. Also, Wang et al. [18] built a graph-theory-based backstepping controller to deal with the disturbance. Zaidi et al. [19] combined the chatter-free SMC and backstepping techniques in their design. Yang et al. [20] presented a controller that combines backstepping and SMC techniques to effectively address external disturbances. These combination strategies enhance resistance to perturbations. However, the method imposes a higher computational burden and remains susceptible to instability caused by uncertainty disturbances.

Previous studies have employed Particle Swarm Optimization (PSO), Genetic Algorithm (GA), and Fuzzy Logic Control (FLC) to optimize and effectively address the problem, demonstrating the efficiency of these methods [21,22,23]. Also, neural network technology is continuously developing and has gained popularity in recent years for handling uncertainty disturbances [24,25,26]. Neural networks possess the property of universal function approximation, enabling them to approximate any continuously differentiable function. Utilizing Radial Basis Function Neural Networks (RBFNNs) to address uncertainties has been shown to be highly efficient [27,28]. Zhao et al. [29] proposed using RBFNNs and combining Lyapunov–Krasovskii functionals (LKFs) and backstepping techniques as the control scheme. However, if the output of the RBFNN is not fed back to the control system promptly to adjust the control strategy, the stability of the system will be significantly compromised.

To enhance the robustness and adaptive performance of the system by adjusting control inputs based on the error estimates from the RBFNN and clarifying the control decision-making framework, the guidance law emerges as an effective solution [30,31,32]. Achieving stable performance of the system is challenging; however, a breakthrough was first achieved with the development of the prescribed performance control (PPC) method, despite the inherent difficulties in maintaining stable system performance. The core concept of attaining predetermined levels of transient and steady-state performance in tracking output errors is captured by an innovative PPC approach introduced in [33], employing a transformation function that strictly increases or decreases the tracking error. Recently, more PPC methods have been developed [20,34,35,36]. They have achieved good results, but the challenge for formation control remains [37]. Mehdifar et al. [38] introduced a distributed graph-based formation control method for leader–follower multi-agent systems, employing a prescribed performance strategy that achieved promising results. Nevertheless, the method does not account for external disturbances. Dai et al. [37] incorporates barrier Lyapunov functions and an adaptive backstepping procedure to ensure the stability of the closed-loop systems while maintaining transient performance within specified bounds. The barrier Lyapunov Function is essential when state constraints must be strictly enforced, while a conventional Lyapunov function is sufficient for general stability without explicit state constraints. However, their method heavily relies on the accuracy of dynamic modeling, which implies that its robustness and adaptability may be limited. Jiang et al. [36] suggested a prescribed-time formation control approach for second-order nonlinear multi-agent systems with a directed graph. Nonetheless, potential remains for further optimization of the system responsiveness. In summary, the formation members have limited communication capabilities, and complex algorithms may impede controller performance. Moreover, accurately approximating disturbances poses a challenge, which can affect the robustness and adaptability of a system [6,39,40].

In this paper, a controller is proposed for autonomous vehicle formation control, combining the leader–follower method and backstepping technique. The predefined trajectory of the leader and the desired formation shape direct the generation of desired yaw angles and velocities for the followers through 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 utilized to accomplish controlled performance objectives.

Despite significant progress in vehicle formation control, many existing methods struggle to maintain precise trajectory tracking and formation under random disturbances or unknown nonlinearities. Additionally, these approaches often do not adequately address the balance between control complexity and system robustness in highly dynamic environments. Previous studies have also lacked detailed comparisons with methods using predefined performance constraints (PPCs), which are crucial for evaluating real-world effectiveness.

The key contributions of this work are summarized as follows: (1) This work enables precise trajectory tracking and formation maintenance under random disturbances, addressing a key limitation in current autonomous vehicle formation control methods. This contribution is particularly relevant to future logistics and autonomous vehicle detection technologies. (2) A graph-based guidance law is designed for preemptive input optimization and adjustment, reducing control complexity and significantly enhancing system robustness, allowing PPC systems to handle external variations more effectively and maintain predetermined performance standards. (3) The use of second-order filters and RBFNNs effectively handles nonlinear errors, improving control performance by mitigating nonlinearities, enhancing stability, and tracking accuracy. The adaptive law further enhances adaptability to changing dynamics. (4) This approach successfully integrates the guidance law, leader–follower control, backstepping technique, and RBFNNs to achieve robust formation control under external random disturbances, increasing system robustness while decreasing controller complexity.

The remainder of this study is organized as follows: In Section 2, the kinetic and dynamic model of the vehicle is presented. Section 3 outlines the development of the proposed controller, utilizing a Lyapunov function to establish stability. Section 4 details the simulation outcomes for formation trajectory tracking under external disturbances. Finally, Section 5 provides a summary and the conclusions of this study.

2. Kinematics and Dynamics Models

Referring to Figure 1, the kinematics and dynamics of the i-th vehicle can be expressed as follows:

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

(1)

where the constant ${\sigma }_i>0$ is the length of the i-th vehicle; $\left(x_i,y_i\right)$ denotes the reference point of the i-th vehicle positioned at the midpoint of the rear axle; $x_i\in \mathbb{R}$ represents the longitudinal position; and $y_i\in (-a,a)$ denotes the lateral position of the vehicle in an inertial frame with Cartesian coordinates $(X,Y)$. The speed of the i-th vehicle at point $\left(x_i,y_i\right)$ is denoted by $v_i$, while ${\theta }_i\in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$ represents the angular orientation of the vehicle relative to the X axis. The steering angle of the front wheels relative to the orientation of the i-th vehicle ${\theta }_i$ is denoted by ${\delta }_i$, and the acceleration of the i-th vehicle is denoted by $F_i$. Note that $f_{\theta ,i}\left({\theta }_i\right)$ 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 the unknown and bounded external disturbances, respectively.

Lemma 1. 

The nonlinear function $\mathit{g}\left(\mathit{x}\right)$ can be approximated by an RBFNN with a specified accuracy $\overline{\epsilon }>0$. Specifically, $\mathit{g}\left(\mathit{x}\right)$ is expressed as

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

where $\mathit{W}\in R^{l\times l}$ represents the ideal constant weight matrix, and $\mathit{\epsilon }\left(\mathit{x}\right)\in R^l$ is the approximation error satisfying $\left|\mathit{\epsilon }\left(\mathit{x}\right)\right|\le \overline{\epsilon }$. Here, $\mathit{\phi }\left(\mathit{x}\right)={\left({\phi }_1\left(\mathit{x}\right),{\phi }_2\left(\mathit{x}\right),\dots ,{\phi }_l\left(\mathit{x}\right)\right)}^T$ denotes the vector of Gaussian basis function, where ${\phi }_j\left(\mathit{x}\right)=exp\left(-{\displaystyle \frac{\parallel \mathit{x}-{\varrho }_j{\parallel }^2}{2{\sigma }_j^2}}\right),$ for $j=1,2,\dots ,l$, with ${\varrho }_j$ and ${\sigma }_j$ representing the center and width of the Gaussian basis function ${\phi }_j\left(\mathit{x}\right)$, respectively. Here, $\mathit{x}={\left(x_1,x_2,\dots ,x_q\right)}^T$. Furthermore, there is ${\parallel \mathit{\phi }\left(\mathit{x}\right)\parallel }^2\le \parallel \mathit{\phi }(\dot{\mathit{x}}{)\parallel }^2$, where $\dot{\mathit{x}}={\left({\dot{x}}_1,{\dot{x}}_2,\dots ,{\dot{x}}_r\right)}^T,r\le q$.

Lemma 2. 

For all $a,b\ge 0$, and $p,q>0$, with $1/p+1/q=1$, the inequality $ab\le {\displaystyle \frac{a^p}{p}}+{\displaystyle \frac{b^q}{q}}$ is satisfied.

3. Controller Design

In this section, a formation tracking controller is proposed for the vehicle formation members. Figure 2 describes the structure and the workflow of the proposed system. Figure 3 illustrates the geometric relationship between the leader and the i-th follower and shows how the leader–follower mechanism works, where the i-th member follows the leader as its follower. The whole system is designed using the backstepping technique. Given the desired trajectory of the leader and considering the kinematics and dynamics involved, the desired velocity and steering angle are formulated and subsequently integrated into the guidance law. Subsequently, the desired yaw angle ${\theta }_{d,i}$ and velocity $v_{d,i}$ for the followers are calculated, which are then used as inputs to their respective steering angle and velocity controllers. To ensure high robustness against unknown disturbances, a second-order filter is employed to estimate the error, and an RBFNN is used to approximate the unknown nonlinear function. For this purpose, an adaptive law is designed. Additionally, a nominal function is considered in the system, and another adaptive law is introduced to handle it. Finally, the final desired steering angle and velocity are computed to guide the followers effectively.

3.1. Guidance Law Design

The guidance law is specifically formulated to enhance the flexibility and adaptability of the system. In this paper, the desired shape of the formation is predefined, while the initial positions of the formation members are randomized. For illustration, consider the scenario involving a leader and one of its followers (denoted as i). The current position of follower i is represented as $p_i$, while $p_{i,d}$ signifies its desired position. The desired position vector ${\mathit{p}}_{i,d}={[x_{i,d},y_{i,d}]}^{\mathrm{T}}={[x_i,y_i]}^{\mathrm{T}}+{[\Delta x,\Delta y]}^{\mathrm{T}}$.

Referring to Figure 3, the objective is to effectively control the i-th member, ensuring that $p_i$ converges to the target position. To achieve this, an error term is defined with the objective of driving it to convergence as follows:

$$\left[\begin{array}{c}{x}_{e,i}\hfill \\ y_{e,i}\hfill \end{array}\right]={\left[\begin{array}{cc}cos{\theta }_{i,0,p}& -sin{\theta }_{i,0,p}\\ sin{\theta }_{i,0,p}& cos{\theta }_{i,0,p}\end{array}\right]}^T\left({\mathit{p}}_i-{\mathit{p}}_{\mathit{i},\mathit{d}}\right),$$

(2)

where ${\theta }_{i,0,p}=atan2\left(y_i^{\prime }+\Delta y^{\prime },x_i^{\prime }+\Delta x^{\prime }\right)\in \left[-\pi ,\pi \right]$, $y_{e,i}$ denotes the minimum distance between the current position $\mathit{p}i$ of vehicle i and the trajectory path of its desired position $\mathit{p}i,d$, and $x_{e,i}$ denotes the distance along the trajectory line between the actual position ${\mathit{p}}_i$ of the vehicle i and the moving line of the desired position ${\mathit{p}}_{i,d}$.

Minimizing the error requires precise calculation. The derivative of $y_{e,i}$ is obtained as follows:

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

(3)

The barrier Lyapunov function is used in the design of the control law to constrain the error and ensure the controlled member remains stable during motion. Additionally, distinct Lyapunov candidates are designed for $y_{e,i}$ and $x_{e,i}$ to minimize interference between the different error terms. A Lyapunov function for $y_{e,i}$ is presented as follows:

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

(4)

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

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

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

(5)

Using the vector field guidance principle, the desired yaw angle can be formulated ${\theta }_{d,i}$ as follows:

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

(6)

where $k_y>0$ is the guidance law parameter, representing the strength of the vector field.

According to Equations (3) and (6), Equation (5) can be rewritten as Equation (7).

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

(7)

where $k_b=sup\left|{\displaystyle \frac{{\dot{b}}_{y,i}}{b_{y,i}}}\right|$.

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:

$${\dot{x}}_{e,i}=v_0-v_{i}cos{\beta }_i.$$

(9)

A Lyapunov function for $x_{e,i}$ is presented as follows:

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

(10)

where $b_{x,i}$ is the upper boundary of $x_{e,i}$. Additionally, in accordance with the guidance law, the desired velocity is formulated as follows:

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

(11)

where ${\beta }_i={\theta }_i-{\theta }_{pi}$.

Upon differentiating Equation (10), the resulting expression for ${\dot{V}}_2$ is given by Equation (12).

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

(12)

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

By selecting $k_v>2k_c$, the following can be additionally derived:

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

(13)

where $c_2=k_v-2k_c$.

3.2. Steering Angle Controller Design

With the guidance law providing the desired yaw angle and velocity, it is essential to design controllers for these parameters. These controllers are developed independently, offering several practical advantages, including modularity, specialization, simplicity, robustness, and scalability. For the i-th formation member, the angle error is defined as follows:

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

(14)

Note that the computations of the derivatives of ${\theta }_{d,i}$ are extremely intricate. Therefore, a second-order filter is introduced to mitigate this complexity as follows:

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

(15)

where 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

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

(17)

and refer to Equations (1) and (16). The derivative of $V_0$ can be expressed as follows:

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

(18)

RBFNN is a type of artificial neural network that is commonly used for function approximation and pattern recognition tasks. The network architecture consists of three layers: an input layer, a hidden layer utilizing radial basis functions, and an output layer. In this control system, the RBFNN helps to handle unknown nonlinearities by compensating for steering angle errors. The steering angle error $e_{\theta ,i}$ is fed into the RBFNN, which processes it using radial basis functions. The RBFNN generates a compensation signal by combining the responses from different nodes. The network adjusts its weights in real time using an adaptive law, allowing it to better approximate the nonlinearities in the system as conditions change. Finally, the RBFNN output, along with other control factors, is used to update the steering angle. This ensures the vehicle stays on its desired path, even when facing disturbances or changing conditions. The adaptive nature of the RBFNN makes the system more robust and responsive. Consequently, this process can be expressed as follows:

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

(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 the following:

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

(23)

where $\tilde{d}=\overline{d}-\widehat{d}$. After some straightforward manipulation, the time derivative of $V_3$ can be expressed as Equation (24) as follows:

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

(24)

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

Utilizing Lemma 2, the following inequalities can be described as follows:

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

(25)

and

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

(26)

Then, by combining the inequalities Equations (25) and (26), Equation (24) can be reformulated as follows:

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

(27)

From Equation (27), it can be concluded that ${\dot{V}}_3\le -{\sigma }_3{V}_3+{\zeta }_1$, where ${\sigma }_3=min\{k_{5,\theta ,i}-{\displaystyle \frac{1}{2}},k_{4,\theta ,i},k_{2,\theta ,i}\}>0$ and ${\zeta }_1={\displaystyle \frac{1}{2}}{{\overline{d}}_{\theta ,i}}^2+k_{2,\theta ,i}{\displaystyle \frac{1}{2}}{∥{\overline{W}}_{\theta ,i}∥}^2>0$.

3.3. Velocity Controller Design

For formation members, the velocity error is defined as follows:

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

(28)

Note that the computations of the derivatives of $v_{d,i}$ are extremely intricate. Therefore, a second-order filter is introduced to mitigate this issue as follows:

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

(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 $v_{d,i}$, and ${\Phi }_{40}$ can be taken as the derivative of $v_{d,i}$ which is denoted as ${\widehat{v}}_{d,i}$. The estimation error for this second-order filter is characterized as follows:

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

(30)

By taking the derivative of the velocity error,

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

(31)

Again, the unknown nonlinear functions can be approximated by employing an RBFNN as follows:

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

(32)

where ${\tilde{d}}_{v,i}={\epsilon }_{v,i}+d_{v,i}-{\tilde{\dot{v}}}_{d,i}$ is the total disturbance. 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 follows:

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

(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

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

(36)

Then, by combining Equations (31), (33) and (34), the differential of $V_4$ can be described as Equation (37) as follows:

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

(37)

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

Utilizing Lemma 2, the following inequalities can be derived.

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

(38)

and

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

(39)

Combining inequalities (38) and (39), Equation (37) can be presented as follows:

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

(40)

Thus, it can be concluded that ${\dot{V}}_4\le -{\sigma }_4{V}_4+{\zeta }_2$ with

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

(41)

and

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

(42)

3.4. Stability Analysis

Consider the Lyapunov candidate represented as follows:

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

(43)

It concludes that $\dot{V}\le -\sigma V+\zeta$, where

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

(44)

and

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

(45)

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

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

(46)

and

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

(47)

Hence, it can be concluded that $x_{e,i}$ and $y_{e,i}$ are constrained such that $\left|x_{e,i}\right|<\left|b_{x,i}\right|$ and $\left|y_{e,i}\right|<\left|b_{y,i}\right|$. Ultimately, these terms can be reduced to a narrow vicinity around zero.

4. Simulation Results

In this section, the simulation results are presented to demonstrate formation control under various disturbances. To highlight the performance of the proposed approach clearly, the controller is compared with the conventional controller, which does not employ the PPC method in the simulation results. Main parameters of the controller are listed in

Table 1. Main parameters of the controller.

Parameter	Value	Parameter	Value
$k_y$	0.1	$k_{4,\theta ,i}$	0.01
$k_v$	0.1	$k_{5,\theta ,i}$	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
$k_{3,\theta ,i}$	10	${\omega }_{n0}$	20

. Vehicle 1 serves as the leader, and Vehicles 2 to 6 act as the followers in the simulation. The leader’s desired trajectory, denoted as $(x_1\left(t\right),y_1\left(t\right))$, follows the equations $x_1\left(t\right)=2sin\left({\displaystyle \frac{t}{10}}\right)+2cos\left({\displaystyle \frac{t}{5}}\right)$ and $y_1\left(t\right)=2t$. The vehicles are arranged in a hexagonal formation, with each side measuring 10 m in length. Initially, Vehicles 2 to 6 are positioned at $(2,-9.9)$, $(-6.5,-15)$, $(-6.5,15)$, $(-15.5,0.1)$, and $(-6.5,5.1)$, respectively. The velocity of the leader is set as $v\left(t\right)=\sqrt{{\left({\displaystyle \frac{2}{10}}cos\left({\displaystyle \frac{t}{10}}\right)-{\displaystyle \frac{2}{5}}sin\left({\displaystyle \frac{t}{5}}\right)\right)}^2+2^2}$. The heading angle of the object is set as $\psi \left(t\right)=\mathrm{atan}2\left(2,{\displaystyle \frac{2}{10}}cos\left({\displaystyle \frac{t}{10}}\right)-{\displaystyle \frac{2}{5}}sin\left({\displaystyle \frac{t}{5}}\right)\right)$.

When modeling dynamic systems subject to random perturbations $\chi \left(t\right)$, the stochastic differential equation as follows is utilized:

$$\dot{\chi }\left(t\right)=-2\chi \left(t\right)+u\left(t\right)-0.5,$$

(48)

where $u\left(t\right)$ follows a standard normal distributed random process. This equation characterizes the evolution of external disturbances over time (refer to Figure 4).

Figure 5 shows the vehicle formation trajectories under different controllers, comparing results with and without the proposed PPC method. From the shape of the trajectories, it is evident that both controllers effectively maintained the formation during movement and ensured a smooth motion trajectory. This demonstrates the efficacy of the designed guidance laws, neural networks, and adaptive laws. The simulation results presented in Figure 6 and Figure 7 provide a comprehensive analysis of the effectiveness of the formation control algorithm in minimizing trajectory tracking errors for the followers. Furthermore, these results clearly demonstrate that controllers utilizing the PPC method are more effective at reducing tracking errors than those without the PPC method.

Initially, the position errors along the x and y axes for the followers are approximately 1 m, 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. It is clear that the error curves for controllers without the PPC method converge more slowly and exhibit higher maximum error values compared to those with the PPC method.

The figures also showcase the error constraints, which define the acceptable range of position errors during operation (0.25 m). As shown in Figure 6 and Figure 7, the control errors with the PPC method are confined within the preset constraints throughout the entire simulation process, whereas the errors without the PPC method exceed the constraints before converging and ultimately settle at a noticeably higher level compared to those with the PPC method.

As depicted in Figure 8, the leader’s angular change exhibits a smooth transition. In the initial startup phase (approximately the first 2 s), the five followers swiftly adjust their angles within a range of 58 to 140 degrees. Subsequently, they converge to a narrower variation range of 90 to 100 degrees. Based on the overlap of the curve colors, the convergence speed of the error for controllers with the PPC method is slightly higher than that of controllers without the PPC method.

The velocity progression of the formation members is illustrated in Figure 9. Within the initial three seconds, all followers 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. It is noteworthy that the speed of controllers with the PPC method converges well around 12 s, whereas the speed outputs of controllers without the PPC method only converge well after 15 s. This further demonstrates that controllers incorporating the PPC method are more stable.

Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 depict the control inputs applied to the followers under various disturbances. The inputs rapidly converge, demonstrating the stability of the approach.

To provide a more detailed depiction of the results, the Mean Squared Error and maximum error of the position will be shown to more clearly reflect its performance. For the i-th follower, the mean squared position error $u_i$ is calculated as $u_i={\displaystyle \frac{1}{N}}{\sum }_{t=1}^N{\left(e_i\left(t\right)\right)}^2$, where $e_i\left(t\right)=\sqrt{x_{e,i}^2\left(t\right)+y_{e,i}^2\left(t\right)}$ represents the position error at time step t, and N is the total number of samples. The maximal position error for the i-th follower $m_i$ can be calculated as $m_i={max}_{t=1}^N\left\{e_i\left(t\right)\right\}$.

The results, as shown in

Table 2. The mean square (u) and maximal (m) position errors of vehicles in meters.

Vehicle	u (Without PPC)	u (With PPC)	m (Without PPC)	m (With PPC)
2	0.1870	0.0570	1.2975	1.0334
3	0.0242	0.0114	0.4422	0.5004
4	0.0073	0.0051	0.6607	0.6452
5	0.0111	0.0060	0.6646	0.6470
6	0.1360	0.0604	0.9094	0.8705

, clearly demonstrate the superior performance of the proposed controller compared to the conventional one (without PPC), which aligns with expectations.

5. Conclusions

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 in the Lyapunov function contributes to achieving the prescribed performance. Simulation results illustrate that the proposed controller consistently attains superior performance within the specified limits, even in the presence of various disturbances.