Distributed adaptive event-triggered formation fault-tolerant control for multiple quadrotors

Abstract

In the present paper, addressing the challenges of model uncertainty, actuator faults, and wind disturbances in quadrotor UAV formation systems, this study proposes an adaptive super-twisting integral terminal sliding mode control (ASTITSMC) strategy. This strategy integrates the “leader-follower” formation control approach with adaptive parameter adjustment to optimize the sliding mode controller, effectively estimating the upper bound of unknown disturbances and ensuring system stability. Additionally, based on the leader-follower structure, we design a distributed adaptive event-triggered formation control protocol that does not rely on global network information, thus reducing computational load and conserving communication resources, while strictly proving the avoidance of the Zeno phenomenon. Analysis based on Lyapunov stability theory demonstrates that the proposed algorithm ensures the convergence of the multi-UAV formation tracking error to zero. Simulation results indicate that the proposed control algorithm performs superiorly in terms of fault tolerance compared to two other algorithms and exhibits stronger robustness.

Keywords

Actuator faults wind disturbances quadrotor UAV formation systems adaptive super-twisting integral sliding mode fault-tolerant control distributed adaptive event-triggered formation control

Introduction

In recent years, quadrotor unmanned aerial vehicles (UAVs) have received significant attention due to their advantages of high flexibility, strong maneuverability, and low operational costs, leading to widespread applications in fields such as aerial photography, forest firefighting, and military operations.^1,2 However, with the increasing complexity of application scenarios and the evolving nature of future battlefields, a single UAV often fails to meet the demands of complex missions.^3,4 Consequently, coordinated control of multiple UAV formations, leveraging collective intelligence and collaborative operations, has been highlighted to enhance mission efficiency, improve system, robustness, and adaptability. This approach is suitable for emergency responses, large-scale surveillance, and precise strike missions in complex environments, representing a future development trend.⁵ Currently, prevalent methods for UAV formation control include leader-follower schemes,⁶ behavioral approaches,⁷ virtual structure methods,⁸ artificial potential fields,⁹ and graph theory methods.¹⁰ Despite the critical importance of quadrotor UAV formation systems in both military and civilian domains, several control challenges remain unresolved, with the aim of enhancing real-time collaboration and adaptability in complex environments to ensure efficient and safe mission execution.

Implementing a multi-quadcopter cooperative formation requires addressing two practical issues. First, as quadcopters primarily operate in outdoor environments, they are inevitably affected by airflow, wind speed variations, and other meteorological factors.³ These external disturbances can cause deviations from the intended flight path, impacting the formation’s stability and accuracy. Second, prolonged flight can lead to aging or damage of the quadcopter’s propellers, resulting in actuator faults.¹¹ Such faults not only affect the performance of individual UAVs but can also disrupt the overall stability and coordination of the formation. Addressing both of these issues comprehensively is crucial for the effectiveness of UAV formation systems.

Zhu et al.¹² and Huang et al.¹³ propose a method combining observers and sliding mode control to address the attitude tracking problem of quadcopters, overcoming model parameter uncertainties and external disturbances. Additionally, Song et al.¹⁴ presents a comprehensive sliding mode control scheme that integrates adaptive laws and pole placement control to enhance the robustness and stability of quadcopters. Furthermore, Yu et al.¹⁵ introduces an integral backstepping anti-disturbance control strategy to optimize the tracking accuracy and disturbance rejection performance of quadcopter attitude angles. However, the above literature primarily addresses control issues when quadrotor UAVs are affected by external disturbances and does not consider the impact of actuator faults. In this regard, Baek and Kang¹⁶ employs adaptive dynamic programming and radial basis function neural networks to enhance the tracking control performance of quadcopters under disturbances and actuator faults. Li et al.¹⁷proposes an adaptive fault-tolerant control method based on output feedback to ensure stable flight of quadrotor UAVs in the presence of actuator faults. Gong et al.¹¹ addresses actuator fault detection and compensation issues for quadcopters by proposing a finite-time fault-tolerant control scheme. This approach achieves precise position and attitude tracking while simplifying the control algorithm and pre-allocating stability time.

However, the aforementioned studies only address fault tolerance in single quadrotor UAVs. Further research is needed to achieve safe and stable flight in quadrotor UAV formation control under external disturbances or actuator faults. In this context, Zheng et al.¹⁸ proposes a robust fixed-time method for formation control of underactuated, nonlinear quadrotor UAV groups, achieving convergence in UAV position tracking and attitude stability through distributed fixed-time sliding mode estimators and disturbance observers. Li et al.¹⁹ introduces a consensus flight control method for multiple quadrotor UAV formations considering wind disturbances and time delays, effectively managing the impact of wind disturbances on formation stability. Ullah et al.²⁰ presents a fractional-order adaptive robust control method for multi-UAV formation systems with parameter uncertainties and gust disturbances, ensuring stable formation flight. However, the aforementioned literature does not address actuator fault issues. In response, Wang and Luo²¹ tackles the control problems of multi-quadcopter formations under disturbances, coupling factors, and actuator faults using an extended state observer and a fast terminal sliding mode fault-tolerant controller, ensuring robust fault tolerance performance. Miao et al.²² proposes a solution based on an augmented timed observer and a distributed sliding mode controller to address formation control problems in quadcopters with actuator faults. This approach achieves precise speed tracking and fault compensation, and its effectiveness is validated through hardware testing.

However, the above works are based on traditional time-triggered mechanisms, which involve substantial amounts of redundant and ineffective data processing, leading to inefficient use of computational resources.²³ Therefore, reducing computational burden while ensuring control performance has become a focal point in current research. Event-triggered mechanisms have garnered widespread attention and thorough investigation as a means to alleviate this burden.^24,25 Traditional event-triggered approaches employ fixed thresholds, triggering only when tracking errors exceed predefined thresholds. While effective in reducing trigger occurrences and computational load, fixed thresholds lack flexibility. To address this issue, dynamic threshold-based event-triggered strategies have been extensively explored.²⁶ These dynamically adjust with changes in system states. For example, Yao et al.²⁷ utilizes a robust adaptive event-triggered sliding mode control method to successfully achieve adaptive tracking control for nonlinear multi-agent systems subject to unknown disturbances and limited network bandwidth. Zhou et al.²⁸ introduces dynamic event-triggered mechanisms, successfully achieving stable control and consensus tracking for high-order nonlinear multi-agent systems. Thus, reducing computational burden and conserving communication resources in quadrotor UAV formation control constitutes a key focus of this study.

Although extensive research has been conducted on the control of quadrotor UAVs in complex environments, studies that simultaneously address the impact of model uncertainty, external wind disturbances, and actuator faults on quadrotor systems, and achieve distributed formation control based on adaptive event-triggering, remain relatively scarce. Inspired by the aforementioned literature, and to simultaneously address the comprehensive uncertainties such as model uncertainties, external wind disturbances, and actuator faults in quadcopter formation control while improving the drawbacks of traditional time-triggered computation methods, this paper adopts a leader-follower structure and proposes a distributed adaptive event-triggered formation fault-tolerant control scheme. This approach ensures the safety of UAV formation flight and is validated through comparative simulations to demonstrate the effectiveness of the proposed design. The primary contributions of this paper are as follows:

(1) Breakthrough in Multi-UAV Formation Control: Unlike existing studies^11–17 which focus solely on the tracking of individual quadrotor UAVs, this paper considers the formation control of multiple quadrotor UAVs. A distributed adaptive triggering formation control protocol is utilized, where each UAV makes decisions based on information from neighboring UAVs, updating its position and velocity online. This reduces reliance on global information while achieving stability.

(2) Optimization of Communication Resources and Computational Burden: Although literature Zheng et al.,¹⁸ Li et al.,¹⁹ Ullah et al.,²⁰ Wang and Luo,²¹ and Miao et al.²² discuss formation control of multiple quadrotor UAVs, the distributed adaptive event-triggered formation control protocol designed in this paper can save communication resources and reduce computational burden without compromising system performance, compared to the time-triggered computation methods used in those references.

(3) Proposal of a Comprehensive Fault-Tolerant Control Method: Unlike the existing literature that considers only faults or disturbances individually, this paper addresses the combined impact of model uncertainties, actuator faults, and wind disturbances on quadcopter formations. It designs an adaptive super-twisting integral sliding mode fault-tolerant control method, employing adaptive laws to estimate the upper bound of lumped disturbances, enabling the formation to track the desired trajectory within a finite time. Comparative numerical simulations with the methods proposed in Xiu et al.²⁹ and Eliker and Zhang³⁰ show that this approach exhibits stronger robustness.

The organization of this work is as follows. Section “Preliminaries and problem formulation” introduces some preliminary knowledge and the dynamics model of quadcopters. Section “Distributed adaptive event-triggered formation cooperative control” presents the distributed formation controller and stability analysis. Section “Follower quadrotor UAV ASTITSMC design” focuses on the control algorithm design and stability analysis for follower UAVs. Numerical simulations and experimental results are provided in Section “Analysis of simulation results” to demonstrate the effectiveness of the developed formation control algorithm. Section “Conclusions” concludes the paper.

Preliminaries and problem formulation

Communication topology

Consider an undirected graph $G = (V, E, A)$ that delineates the topology of a master-slave multi-quadrotor system consisting of $N$ nodes. Where, $V = {1, 2, \dots, N}$ denotes the node set, and $E = {(i, j), i, j \in V} \subseteq V \times V$ denotes the edge set. The adjacency matrix $A = [a_{ij}] \in R^{N \times N}$ is defined by non-negative adjacency elements $a_{ij}$ , where element $a_{ij} = 1$ if and only if $(i, j) \in E$ ; otherwise $a_{ij} = 0$ . The set $N_{i} = {j \in V | (i, j) \in E}$ represents the neighbor of node $i$ . The graph $G$ is considered connected if an undirected path exists between any two distinct nodes within $G$ . $L = [l_{ij}] \in R^{N \times N}$ denotes the Laplacian matrix of $G$ , with $l_{ii} = \sum_{j = 1}^{N} a_{ij}$ if and only if $l_{ij} = - a_{ij}, i \neq j$ . Additionally, node 0 represents a virtual leader; if node $i$ can receive relevant information from the leader, the diagonal matrix $B = diag {b_{1}, \dots, b_{N}}$ is defined as $b_{i} > 0$ ; otherwise $b_{i} = 0$ .

Description of the four-rotor UAV dynamics model

The quadrotor UAV is modeled as a rigid body, with the assumption that its geometric center aligns with the center of mass, and the air resistance is ignored. As depicted in Figure 1, the inertial coordinate system $O_{g} (x_{g}, y_{g}, z_{g})$ and the body coordinate system $O (x, y, z)$ are established. In the figure: ${F_{1}, F_{2}, F_{3}, F_{4}}$ represents the lift generated by the four rotors of the UAV; ${ω_{1}, ω_{2}, ω_{3}, ω_{4}}$ represents the speed of the four propellers.

Figure 1.

Schematic diagram of the structure of the quadrotor UAV.

Firstly, the actuator fault model is introduced:

U_{jr} = {U_{jr}}_{a} + f_{jr} (r = 1, 2, 3, 4)

(1)

In the equation, $j$ denotes the j-th quadrotor UAV, $U_{jr}$ represents the actual control input when the r-th actuator of the j-th UAV experiences a failure; ${U_{jr}}_{a}$ denotes the actual control input under normal conditions where all actuators are operational; $f_{jr}$ is the deviation fault factor that causes actuator faults when condition $f_{jr} \neq 0$ is met. Therefore, the full-drive dynamics equation that accounts for uncertainties, wind disturbances, and actuator faults for the j-th quadrotor UAV can be formulated as follows:

\begin{matrix} {\overset{\cdot}{x}}_{j 1} = x_{j 2} + ϖ_{j 1} \\ {\overset{\cdot}{x}}_{j 2} = a_{j 1} x_{j 1} - g_{j} + \frac{\cos x_{j 3} \cos x_{j 5}}{m_{j}} U_{j 1} + ϖ_{j 2} + χ_{j 1} \\ {\overset{\cdot}{x}}_{j 3} = x_{j 4} + ϖ_{j 3} \\ {\overset{\cdot}{x}}_{j 4} = a_{j 2} x_{j 8} x_{j 6} + a_{j 3} {x_{j 4}}^{2} + a_{j 4} {\bar{Ω}}_{j} x_{j 6} + b_{j 1} U_{j 2} + ϖ_{j 4} + χ_{j 2} \\ {\overset{\cdot}{x}}_{j 5} = x_{j 6} + ϖ_{j 5} \\ {\overset{\cdot}{x}}_{j 6} = a_{j 5} x_{j 8} x_{j 4} + a_{j 6} {x_{j 6}}^{2} + a_{j 7} {\bar{Ω}}_{j} x_{j 4} + b_{j 2} U_{j 3} + ϖ_{j 6} + χ_{j 3} \\ {\overset{\cdot}{x}}_{j 7} = x_{j 8} + ϖ_{j 7} \\ {\overset{\cdot}{x}}_{j 8} = a_{j 8} x_{j 4} x_{j 6} + a_{j 9} {x_{j 8}}^{2} + b_{j 3} U_{j 4} + ϖ_{j 8} + χ_{j 4} \\ {\overset{\cdot}{x}}_{j 9} = x_{j 10} + ϖ_{j 9} \\ {\overset{\cdot}{x}}_{j 10} = a_{j 10} x_{j 9} + \frac{U_{j 5}}{m_{j}} U_{j 1} + ϖ_{j 10} + χ_{j 5} \\ {\overset{\cdot}{x}}_{j 11} = x_{j 12} + ϖ_{j 11} \\ {\overset{\cdot}{x}}_{j 12} = a_{j 11} x_{j 11} + \frac{U_{j 6}}{m_{j}} U_{j 1} + ϖ_{j 12} + χ_{j 6} \end{matrix}

(2)

In the equation, $a_{j 1} = \frac{- k_{jz}}{m_{j}}$ , $a_{j 2} = \frac{I_{jy} - I_{jz}}{I_{jx}}$ , $a_{j 3} = \frac{- k_{j ϕ}}{I_{jx}}$ , $a_{j 4} = \frac{- J_{j}}{I_{jx}}$ , $a_{j 5} = \frac{I_{jz} - I_{jx}}{I_{jy}}$ , $a_{j 6} = \frac{- k_{j θ}}{I_{jy}}$ , $a_{j 7} = \frac{- J_{j}}{I_{jy}}$ , $a_{j 8} = \frac{I_{jx} - I_{jy}}{I_{jz}}$ , $a_{j 9} = \frac{- k_{j ψ}}{I_{jz}}$ , $a_{j 10} = \frac{- k_{jx}}{m_{j}}$ , $a_{j 11} = \frac{- k_{jy}}{m_{j}}$ , $b_{j 1} = \frac{d_{j}}{I_{jx}}$ , $b_{j 2} = \frac{d_{j}}{I_{jy}}$ and $b_{j 2} = \frac{C_{jD}}{I_{jz}}$ . $I_{jx}$ , $I_{jy}$ , and $I_{jz}$ represent the moments of inertia of the j-th quadrotor UAV about the $x$ , $y$ , and $z$ axes, respectively. The term $g_{j}$ denotes the gravitational acceleration, while $k_{jx}$ , $k_{jy}$ , and $k_{jz}$ are coefficients of aerodynamic resistance. Additionally, $k_{j ϕ}$ , $k_{j θ}$ , and $k_{j ψ}$ are gyroscopic effect factors. The variable $m_{j}$ stands for the mass of the j-th quadrotor UAV. $d_{j}$ represents the distance between the rotation axis and the center of the quadrotor. The terms $J_{j}$ and $C_{jD}$ denote the inertia and resistance coefficients of the j-th quadrotor UAV motor, respectively. $ϖ_{ji}, (i = 1, 2, . . . ., 12)$ represents model uncertainty, and $χ_{j η} (η = 1, 2, \dots, 6)$ denotes gust disturbances. In addition, there are $x_{j 1} = z_{j}$ , $x_{j 2} = {\overset{\cdot}{z}}_{j}$ , $x_{j 3} = ϕ_{j}$ , $x_{j 4} = {\overset{\cdot}{ϕ}}_{j}$ , $x_{j 5} = θ_{j}$ , $x_{j 6} = {\overset{\cdot}{θ}}_{j}$ , $x_{j 7} = ψ_{j}$ , $x_{j 8} = {\overset{\cdot}{ψ}}_{j}$ , $x_{j 9} = x_{j}$ , $x_{j 10} = {\overset{\cdot}{x}}_{j}$ , $x_{j 11} = y_{j}$ and ${x_{j}}_{12} = {\overset{\cdot}{y}}_{j}$ . $x_{j}$ , $y_{j}$ , and $z_{j}$ denote the position of the j-th quadrotor UAV. The variables $ϕ_{j}$ , $θ_{j}$ , and $ψ_{j}$ represent the three Euler angles of the j-th quadrotor UAV, corresponding to roll, pitch, and yaw angles respectively. $U_{j 1, j 2, j 3, j 4}$ represents the actual control input of the actuators, and $U_{j 5, j 6}$ stands for supplementary control inputs $x_{j}$ and $y_{j}$ .

Additionally, there is $Ω_{j} = ω_{j 1} - ω_{j 2} + ω_{j 3} - ω_{j 4}$ , where $ω_{j 1, j 2, j 3, j 4}$ is the angular velocity satisfying the following relationship with control input $U_{1, 2, 3, 4}$ .

[\begin{matrix} U_{j 1} \\ U_{j 2} \\ U_{j 3} \\ U_{j 4} \end{matrix}] = [\begin{matrix} c_{j} & c_{j} & c_{j} & c_{j} \\ 0 & - d_{j} c_{j} & 0 & d_{j} c_{j} \\ - d_{j} c_{j} & 0 & d_{j} c_{j} & 0 \\ k_{jp} & - k_{jp} & k_{jp} & - k_{jp} \end{matrix}] [\begin{matrix} {ω_{j 1}}^{2} \\ {ω_{j 2}}^{2} \\ {ω_{j 3}}^{2} \\ {ω_{j 4}}^{2} \end{matrix}]

(3)

In the equation, $c_{j}$ represents the lift coefficient of the propeller for the j-th quadrotor UAV, while $k_{jp}$ stands for the torque coefficient of the propeller. For ease of subsequent analysis, denote the combined effects actuator faults, model uncertainties, and wind disturbances in equation (2) be denoted as $Ξ_{j η}$ . Thus, equation (2) can be simplified to as follows:

\begin{matrix} {\overset{\cdot}{x}}_{j 1} = x_{j 2} + ϖ_{j 1} \\ {\overset{\cdot}{x}}_{j 2} = a_{j 1} x_{j 1} - g_{j} + \frac{\cos x_{j 3} \cos x_{j 5}}{m_{j}} U_{j 1 a} + Ξ_{j 1} \\ {\overset{\cdot}{x}}_{j 3} = x_{j 4} + ϖ_{j 3} \\ {\overset{\cdot}{x}}_{j 4} = a_{j 2} x_{j 8} x_{j 6} + a_{j 3} {x_{j 4}}^{2} + a_{j 4} {\bar{Ω}}_{j} x_{j 6} + b_{j 1} U_{j 2 a} + Ξ_{j 2} \\ {\overset{\cdot}{x}}_{j 5} = x_{j 6} + ϖ_{j 5} \\ {\overset{\cdot}{x}}_{j 6} = a_{j 5} x_{j 8} x_{j 4} + {a_{j}}_{6} {x_{j 6}}^{2} + a_{j 7} {\bar{Ω}}_{j} x_{j 4} + b_{j 2} U_{j 3 a} + Ξ_{j 3} \\ {\overset{\cdot}{x}}_{j 7} = x_{j 8} + ϖ_{j 7} \\ {\overset{\cdot}{x}}_{j 8} = a_{j 8} x_{j 4} x_{j 6} + a_{j 9} {x_{j 8}}^{2} + b_{j 3} U_{j 4 a} + Ξ_{j 4} \\ {\overset{\cdot}{x}}_{j 9} = x_{j 10} + ϖ_{j 9} \\ {\overset{\cdot}{x}}_{j 10} = a_{j 10} x_{j 9} + \frac{U_{j 5}}{m_{j}} U_{j 1 a} + Ξ_{j 5} \\ {\overset{\cdot}{x}}_{j 11} = x_{j 12} + ϖ_{j 11} \\ {\overset{\cdot}{x}}_{j 12} = a_{j 11} x_{j 11} + \frac{U_{j 6}}{m_{j}} U_{j 1 a} + Ξ_{j 6} \end{matrix}

(4)

With $Ξ_{j 1} = \frac{\cos x_{j 3} \cos x_{j 5}}{m_{j}} f_{j 1} + ϖ_{j 2} + χ_{j 1}$ , $Ξ_{j 2} = b_{j 1} f_{j 2} + + ϖ_{j 4} + χ_{j 2}$ , $Ξ_{j 3} = b_{j 2} f_{j 3} + ϖ_{j 6} + χ_{j 3}$ , $Ξ_{j 4} = b_{j 3} f_{j 4} + ϖ_{j 8} + χ_{j 4}$ , $Ξ_{j 5} = \frac{U_{j 5}}{m_{j}} f_{j 1} + ϖ_{j 10} + χ_{j 5}$ and $Ξ_{j 6} = \frac{U_{j 6}}{m_{j}} f_{j 1} + ϖ_{j 12} + χ_{j 6}$ .

Assumption 1. Assume that the sum of model uncertainties, actuator faults, and gust disturbances defined in equation (4) is bounded. This bounded sum is denoted as $| Ξ_{j η} | \leq σ_{j η}$ , where $σ_{j η}$ is an unknown positive constant.

Control objectives

In the realm multi-quadrotor UAV formation system operating under an undirected communication topology, this study introduces an advanced fault-tolerant adaptive super-twisting integral terminal sliding mode controller. This innovative control scheme is designed to ensure stable trajectory tracking for follower quadrotor UAVs in the presence of wind disturbances and actuator faults. Follower units within the formation are programmed to derive their requisite positional data from the overall formation structure as well as real-time directives issued by a designated virtual leader UAV. This algorithm empowers each follower UAV to recalibrate its position and velocity autonomously, drawing upon the dynamic state data of adjacent UAVs within the network. This sophisticated mechanism promotes a robust cooperative control environment, meticulously orchestrating the collective aerial maneuvers required to fulfill the stipulated formation flight objectives.

Let $p_{0} (t) = {[x_{0}, y_{0}, z_{0}]}^{T}$ and $v_{0} (t) = {[{\overset{\cdot}{x}}_{0}, {\overset{\cdot}{y}}_{0}, {\overset{\cdot}{z}}_{0}]}^{T}$ denote the position and velocity information of the virtual leader, while $p_{j} (t) = {[x_{j}, y_{j}, z_{j}]}^{T} \in R^{3 \times 1}$ and $v_{j} (t) = {[{\overset{\cdot}{x}}_{j}, {\overset{\cdot}{x}}_{j}, {\overset{\cdot}{x}}_{j}]}^{T} \in R^{3 \times 1}$ represent the desired position and velocity information for the j-th follower UAV. Additionally, Let $δ_{j} (t) = {[δ_{jx}, δ_{jy}, δ_{jz}]}^{T} \in R^{3 \times 1}$ denote the relative position offset between the virtual leader and the j-th follower. Firstly, based on a desired trajectory model, a distributed event-triggered formation coordination law is designed. This law is crucial for ensuring that follower UAVs accurately track the virtual leader’s trajectory, culminating in the formation of the desired spatial configuration $lim_{t \to \infty} ‖ p_{j} - δ_{j} - p_{0} ‖ = 0$ and $lim_{t \to \infty} ‖ {\overset{\cdot}{x}}_{j} - {\overset{\cdot}{x}}_{0} ‖ = 0$ . Secondly, addressing the challenges posed by external wind disturbances and actuator faults, an adaptive super-twisting integral terminal sliding mode fault-tolerant controller is developed. This advanced controller is tailored to ensure that the follower quadrotor UAVs maintain trajectory alignment with the virtual leader, thereby achieving the anticipated formation pattern of the multi-quadrotor system.

Distributed adaptive event-triggered formation cooperative control

Controller design

For clarity and ease of understanding in the forthcoming discussions, several foundational assumptions, and lemmas are stipulated:

Assumption 2. The undirected topology $G$ of the multi-quadrotor UAV formation system is connected, ensuring that at least one follower quadrotor UAV has the capability to receive information from the virtual leader.

Assumption 3. There is a known upper bound for the derivative of the acceleration of the virtual leader quadrotor UAV, ensuring predictability in its dynamic behavior.

Lemma 1. ³¹ For any vector $x, y \in R^{n}$ and constant $ε > 0$ , there exists a $x^{T} y \leq \frac{ε x^{T} x}{2} + \frac{y^{T} y}{2 ε}$ such that (Young’s inequality) holds.

Lemma 2. ³² Given assumption 3, there exists a positive definite matrix $Q = diag {q_{1}, q_{2}, \dots, q_{N}}$ satisfying $H = Q (L + A) + {(L + A)}^{T} Q$ , and there exists $q = {[q_{1}, q_{2}, \dots, q_{N}]}^{T} = {({(L + A)}^{T})}^{- 1} 1_{N}$ . Let $λ_{i}$ be an eigenvalue of $H$ . For analytical convenience, define $λ_{min}^{H} = min_{t \in [0, \infty]} {λ_{i}, i = 1, 2, \dots, N}$ .

The architecture of the distributed adaptive event-triggered cooperative control for multi-quadrotor UAV is depicted in Figure 2.

Figure 2.

Distributed event-triggered formation control block diagram.

In order to realize the synchronized flight of the virtual leader UAV alongside the follower UAV, which subjected to external disturbances and actuator failures, according to the expected formation trajectory, the expected trajectory model is given

{\begin{matrix} {\overset{\cdot}{p}}_{j} = v_{j} \\ {\overset{\cdot}{v}}_{j} = u_{jp} \end{matrix}

(5)

Where, $u_{jp} = {[u_{jx}, u_{jy}, u_{jz}]}^{T} \in R^{3 \times 1}$ represents the control protocol on positions for the x, y, and z, respectively. The model defines the tracking error of the expected trajectory for each follower UAV as follows:

{\begin{matrix} e_{jp} (t) = p_{j} (t) - δ_{j} - p_{0} (t) \\ e_{jv} (t) = v_{j} (t) - {\overset{\cdot}{p}}_{0} (t) \end{matrix}

(6)

To ensure the desired formation flight, a distributed adaptive event-triggered formation control law has been designed

{\begin{matrix} u_{jp} (t) = - M_{j} (t) ℜ_{j} (t_{κ}^{j}) \\ {\overset{\cdot}{M}}_{j} (t) = \frac{1}{Λ} ((1 - \frac{η_{1}}{2}) ℜ_{j}^{T} (t_{κ}^{j}) ℜ_{j} (t_{κ}^{j}) - ε (c_{j} (t) - κ)) \\ ℜ_{j} (t) = μ_{1} [\sum_{i \in N_{j}} a_{ij} ((p_{j} (t) - p_{i} (t)) - (δ_{j} - δ_{i})] \\ + b_{j} (p_{j} (t) - p_{0} (t) - δ_{j}) \\ + μ_{2} [\sum_{i \in N_{j}} a_{ij} (v_{j} (t) - v_{i} (t)) + b_{j} (v_{j} (t) - {\overset{\cdot}{p}}_{0} (t))] \end{matrix}

(7)

Where is the coupling strength of $μ_{1}, μ_{2} > 0$ , and $M_{j} (t)$ represents the time-varying feedback gain of the proposed controller, with the condition $M_{j} (0) > κ$ . $η_{1} \in (0, 2)$ , $κ > 0$ , and $Λ > 0$ are normal numbers. $ε > 0$ is the adjustment parameter, $t_{k}^{j}$ has $t_{0}^{j} = 0 j \in (1, 2, \dots, N)$ , and $t_{k}^{j}$ represents the k-th sampling time of $ℜ_{j} (t)$ .

To facilitate the analysis, define $e_{j} (t) = {\overset{\cdot}{v}}_{j} (t_{κ}^{j}) - {\overset{\cdot}{v}}_{j} (t)$ and $E_{j} (t) = ℜ_{j} (t_{κ}^{j}) - ℜ_{j} (t)$ as sampling errors for all $t \in [\begin{matrix} t_{k}^{j} & t_{k + 1}^{j} \end{matrix}]$ . Subsequently, the event trigger condition is designed as follows:

{\begin{matrix} F_{j} (t) = W_{j 1} E_{j}^{T} (t) E_{j} (t) - W_{j 2} ℜ_{j}^{T} (t_{κ}^{j}) ℜ_{j} (t_{κ}^{j}) - α e^{- β t} \\ W_{j 1} = ς M_{j} (t) + \frac{(2 - η_{1}) (1 - η_{2}) η_{3}}{2 η_{2}} \\ W_{j 2} = \frac{(2 - η_{1}) (1 - η_{3})}{2} \end{matrix}

(8)

The terms $ς > 0$ , $α > 0,$ $β > 0$ , $η_{2} \in (0, 1)$ , and $η_{3} \in (0, 1)$ . To facilitate subsequent proofs, the expected trajectories and dynamic error systems are delineated as follows:

{\overset{\cdot}{e}}_{p} (t) = {\overset{\cdot}{e}}_{v} (t), {\overset{\cdot}{e}}_{v} (t) = - \tilde{M} (t) ℜ (t_{k})

(9)

Where, $ℜ (t_{k}) = {[ℜ_{1}^{T} (t_{κ}^{1}), ℜ_{2}^{T} (t_{κ}^{2}), \dots, ℜ_{N}^{T} (t_{κ}^{N})]}^{T}$ , $e_{p} (t) = [e_{1 p}^{T}, e_{1 p}^{T}, \dots, e_{Np}^{T}]$ , and $e_{v} (t) = [e_{1 v}^{T}, e_{1 v}^{T}, \dots, e_{Nv}^{T}]$ . Let $M (t) = diag {M_{1} (t), . . ., M_{N} (t)}$ . have $\tilde{M} (t) = M (t) \otimes I_{3}$ .

Theorem 1. Considering the expected trajectory model outlined in formula (5), and the distributed event-triggered adaptive formation control law (7). Under the condition that hypothesis 2-3 is true, the expected trajectory, the dynamic error system (9) and the adaptive parameter A all converge within a bounded region. This convergence ensures the convergence of the expected trajectory tracking error of the follower UAV.

Proof: According to lemma 1, the designed Lyapunov function is as follows:

V_{1} (t) = \frac{1}{2} P^{T} (t) Π P (t) + \sum_{j = 1}^{N} \frac{Λ}{2} (M_{j} (t) - \hat{M})^{2}

(10)

Where, $P (t) = {[e_{p}^{T} (t), e_{v}^{T} (t)]}^{T}$ , $Π = [\begin{matrix} γ H^{2} & μ_{1} H \\ μ_{1} H & μ_{2} H \end{matrix}] \otimes 3$ , and $γ >> μ_{1}$ is a standard numerical value. Additionally, $\hat{M}$ is identified as a normal number. According to Li et al.,³³ the matrix $V_{1} (t)$ is positively definite. Taking the derivative of $V_{1} (t)$ proceeds as follows:

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) = γ e_{p}^{T} (t) (H^{2} \otimes 3) e_{v} (t) + μ_{1} e_{v}^{T} (t) (H^{2} \otimes 3) e_{v} (t) \\ - ℜ (t) \tilde{M} (t) ℜ (t_{k}) + \sum_{j = 1}^{N} Λ (M_{j} (t) - \hat{M}) {\overset{\cdot}{M}}_{j} (t) \end{matrix}

(11)

Where $ℜ (t) = {[ℜ_{1}^{T} (t), \dots, ℜ_{N}^{T} (t)]}^{T} = (H^{2} \otimes 3) e_{v} (t) + μ_{2} e_{v} (t)$ , can then be restructured using Young’s inequality, rewrite the derivative of $V_{1} (t)$ in equation (11) as

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) \leq (μ_{1} λ_{max} (H) e_{v}^{T} (t) e_{v} (t) + γ e_{p}^{T} (t) (H^{2} \otimes 3) e_{v} (t) \\ - (1 - \frac{η_{1}}{2}) \hat{M} ℜ^{T} (t_{k}) ℜ (t_{k}) - \sum_{j = 1}^{N} \frac{\bar{σ}}{2} {\tilde{M}}_{j}^{2} (t) \\ + \sum_{j = 1}^{N} \frac{M_{j} (t)}{2 η_{1}} E_{j}^{T} (t) E_{j} (t) + ς) \end{matrix}

(12)

Where, ${\tilde{M}}_{j} (t) = M_{j} (t) - \hat{M}$ and $ς = \frac{N \bar{σ} {(\hat{M} - k)}^{2}}{2}$ , the following relationship is established:

\begin{matrix} - (1 - \frac{η_{1}}{2}) \hat{M} ℜ^{T} (t_{k}) ℜ (t_{k}) \leq - (1 - \frac{η_{1}}{2}) (1 - η_{2}) η_{3} \hat{M} ℜ^{T} (t) ℜ (t) \\ + \sum_{j = 1}^{N} (1 - \frac{η_{1}}{2}) \frac{1 - η_{2}}{η_{2}} η_{3} \hat{M} E_{j} {(t)}^{2} - (1 - \frac{η_{1}}{2}) (1 - η_{3}) \hat{M} ℜ^{T} (t_{k}) ℜ (t_{k}) \end{matrix}

(13)

Additionally,

\begin{matrix} - ℜ^{T} (t) ℜ (t) = - μ_{1}^{2} e_{p}^{T} (t) (H^{2} \otimes 3) e_{p} (t) - 2 μ_{1} μ_{2} e_{p}^{T} (t) \\ (H^{2} \otimes 3) e_{v} (t) - μ_{2}^{2} e_{v}^{T} (t) (H^{2} \otimes 3) e_{v} (t) \end{matrix}

(14)

We choose $γ$ and $\hat{M}$ provided that the following conditions are satisfied:

{\begin{matrix} γ = 2 (1 - \frac{η_{1}}{2}) (1 - η_{2}) η_{3} μ_{1} μ_{2} \hat{M} \\ \hat{M} \geq \frac{1}{(1 - η_{1} / 2) (1 - η_{2}) η_{3} μ_{1}^{2} λ_{min}^{2} (H)} \\ \hat{M} \geq \frac{μ_{1} λ_{max} (H) + 1}{(1 - η_{1} / 2) (1 - η_{2}) η_{3} μ_{2}^{2} λ_{min}^{2} (H)} \end{matrix}

(15)

There is

\begin{array}{l} {\dot{V}}_{1} (t) \leq \hat{M} \sum_{j = 1}^{N} ((\frac{ς M_{j} (t)}{2 η_{1} ς \hat{M}} + \frac{(2 - η_{1}) (1 - η_{2}) η_{3}}{2 η_{2}}) E_{j}^{T} (t) E_{j} (t) \\ - \frac{(2 - η_{1}) (1 - η_{3})}{2} ℜ_{j}^{T} (t_{κ}^{j}) ℜ_{j} (t_{κ}^{j}) - α e^{- β t}) \\ - P^{T} (t) P (t) - \sum_{j = 1}^{N} \frac{\bar{σ}}{2} {\tilde{M}}_{j}^{2} (t) + \hat{M} N α e^{- β t} + ς \end{array}

(16)

Where $\hat{M} \geq 1 / 2 η_{1} ς$ are selected based on the trigger condition (16), ${\overset{\cdot}{V}}_{1} (t)$ in (17) can be restated as follows:

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) \leq - P^{T} (t) P (t) - \sum_{j = 1}^{N} \frac{\bar{σ}}{2} {\tilde{M}}_{j}^{2} (t) + \hat{M} N α e^{- β t} \\ + ς \leq - π V_{1} (t) + \hat{M} N α e^{- β t} + ς \end{matrix}

(17)

There’s $π = \min {(2 / λ \min (Π)), (σ / χ)}$ . According to the comparative lemma in Khalil,³⁴ we obtain:

V_{1} (t) \leq (V_{1} (0) - \frac{ς}{π}) e^{- π t} + \frac{ς}{π} + \hat{M} N Δ (t)

(18)

Where $Δ (t)$ is defined as follows:

\begin{matrix} Δ (t) = {\begin{matrix} α t e^{- π t}, β = π \\ \frac{α}{π - β} (e^{- β t} - e^{- π t}), β \neq π \end{matrix} \end{matrix}

(19)

From this we verify that $P (t)$ and ${\tilde{M}}_{1} (t), \dots, {\tilde{M}}_{N} (t)$ are uniformly and eventually bounded and converge on a bounded set $Z$ . Thus, theorem 1 is proven.

Z \overset{Δ}{=} {P (t), {\tilde{M}}_{1} (t), \dots, {\tilde{M}}_{N} (t) : 0 < V_{1} (t) \leq \frac{ς}{π}}

(20)

Admissibility of triggering condition

The sequence of triggering moments is represented by the monotonically increasing sequence ${t_{κ}^{j}}_{k = 0}^{\infty}$ defined at the triggering moment. Without loss of generality, we assume there exists $t_{κ}^{j} = 0$ . To avoid Zeno behavior, where an infinite number of events could occur in a finite time, it is essential that the interval between events $t_{κ + 1}^{j} - t_{κ}^{j}$ remains positive, facilitated by the presence of $k \in Z \geq 0$ .

Theorem 2. Considering the desired trajectory model outlined in (5), assuming the minimum event interval is positive, the proposed distributed adaptive event-triggered control protocol (7) along with the triggering condition (9) effectively prevent the occurrence of Zeno phenomena.

Proof: For all $t \in [t_{κ}^{j}, t_{κ + 1}^{j}]$ , the right Dini derivative of $‖ {\overset{\cdot}{E}}_{j} (t) ‖$ is denoted as

D^{+} ‖ E_{j} (t) ‖ = \frac{d}{dt} \sqrt{E_{j} (t) E_{j} (t)} = \frac{E_{j}^{T} (t) {\overset{\cdot}{E}}_{j} (t)}{\sqrt{E_{j}^{T} (t) E_{j} (t)}} \leq ‖ {\overset{\cdot}{E}}_{j} (t) ‖

(21)

and we have

\begin{matrix} {\overset{\cdot}{E}}_{j} (t) = - μ_{1} [\sum_{i \in N_{j}} a_{ij} (e_{jv} (t) - e_{iv} (t)) + b_{j} e_{jv} (t)] \\ - μ_{2} [\sum_{i \in N_{j}} a_{ij} ({\overset{\cdot}{e}}_{jv} (t) - {\overset{\cdot}{e}}_{iv} (t)) + b_{j} {\overset{\cdot}{e}}_{jv} (t)] \end{matrix}

(22)

According to Assumptions 2 and 3, for analytical convenience, we assume

ϒ_{k}^{j} = max_{t \in [t_{κ}^{j}, t_{κ + 1}^{j}]} ‖ \begin{matrix} \frac{μ_{1}}{μ_{2}} ℜ_{j} (t_{κ}^{j}) - \frac{{μ_{1}}^{2}}{μ_{2}} (\sum_{i \in N_{j}} a_{ij} (e_{jp} (t) - e_{ip} (t)) + b_{j} e_{jp} (t)) \\ + μ_{2} (\sum_{i \in N_{j}} a_{ij} (e_{jv} (t) - e_{iv} (t)) + b_{j} e_{jv} (t)) \end{matrix} ‖

(23)

Consequently:

D^{+} ‖ E_{j} (t) ‖ \leq \frac{μ_{1}}{μ_{2}} ‖ E_{j} (t) ‖ + ϒ_{k}^{j}, \forall t \in [t_{κ}^{j}, t_{κ + 1}^{j}]

(24)

From the application of the comparative lemma as outlined in Khalil,³⁴ we obtain:

‖ E_{j} (t) ‖ \leq \frac{μ_{1}}{μ_{2}} ϒ_{k}^{j} (e^{\frac{μ_{1}}{μ_{2}} (t - t_{k}^{j})} - 1)

(25)

According to triggering condition (9), it is evident that the boundary of the sample error $‖ E_{j} (t) ‖$ is:

‖ E_{j} (t) ‖ \leq Γ_{k}^{j} (t), \forall t \in [t_{κ}^{j}, t_{κ + 1}^{j}]

(26)

Where there are $Γ_{k}^{j} (t) = \sqrt{(\frac{W_{j 1}}{W_{j 2}}) ℜ_{j}^{T} (t_{κ}^{j}) ℜ_{j} (t_{κ}^{j}) + (\frac{α}{W_{j 1}}) e^{- β t}}$ , $t = t_{k + 1}^{j}$ , and the existence of $E_{j} (t_{k + 1}^{j}) > \sqrt{\frac{α}{W_{j 1}} e^{- β t_{k + 1}^{j}}}$ , according to equation (26) we obtain

t_{κ + 1}^{j} - t_{κ}^{j} \geq \frac{μ_{2}}{μ_{1}} \ln (\frac{μ_{1} \sqrt{\frac{α}{W_{j 1}} e^{- β t_{k + 1}^{j}}}}{μ_{2} ϒ_{k}^{j}} + 1)

(27)

Due to the presence of term $e^{- β t_{k + 1}^{j}}$ , the minimum event interval $t_{κ + 1}^{j} - t_{κ}^{j}$ is guaranteed to be positive over any finite range. Thus, it effectively eliminates Zeno phenomena within the multi-quadrotor UAV formation control process.

Follower quadrotor UAV ASTITSMC design

Super-twisting integral sliding mode surface design

First, we define the tracking error of the j-th follower quadrotor UAV as

Θ_{je} = Θ_{j} - Θ_{jd}

(28)

In equation (28), $Θ_{j} = [\begin{matrix} z_{j} & ϕ_{j} & \begin{matrix} \begin{matrix} θ_{j} & ψ_{j} \end{matrix} & x_{j} & y_{j} \end{matrix} \end{matrix}]$ represents the actual value of the system, and $Θ_{jd} = [\begin{matrix} z_{jd} & ϕ_{jd} & \begin{matrix} \begin{matrix} θ_{jd} & ψ_{jd} \end{matrix} & x_{jd} & y_{jd} \end{matrix} \end{matrix}]$ represents the desired value. Here, the tracking errors and their derivatives for each system component are:

\begin{matrix} {\begin{matrix} e_{jz 1} = z_{j} - z_{jd} \\ e_{jz 2} = {\overset{\cdot}{z}}_{j} - {\overset{\cdot}{z}}_{jd} \end{matrix} {\begin{matrix} e_{j ϕ 1} = ϕ_{j} - ϕ_{jd} \\ e_{j ϕ 2} = {\overset{\cdot}{ϕ}}_{j} - {\overset{\cdot}{ϕ}}_{jd} \end{matrix} \\ {\begin{matrix} e_{j θ 1} = θ_{j} - θ_{jd} \\ e_{j θ 2} = {\overset{\cdot}{θ}}_{j} - {\overset{\cdot}{θ}}_{jd} \end{matrix} {\begin{matrix} e_{j ψ 1} = ψ_{j} - ψ_{jd} \\ e_{j ψ 2} = {\overset{\cdot}{ψ}}_{j} - {\overset{\cdot}{ψ}}_{jd} \end{matrix} \\ {\begin{matrix} e_{jx 1} = x_{j} - x_{jd} \\ e_{jx 2} = {\overset{\cdot}{x}}_{j} - {\overset{\cdot}{x}}_{jd} \end{matrix} {\begin{matrix} e_{jy 1} = y_{j} - y_{jd} \\ e_{jy 2} = {\overset{\cdot}{y}}_{j} - {\overset{\cdot}{y}}_{jd} \end{matrix} \end{matrix}

(29)

To achieve finite-time convergence of the tracking error, the super-twisting integral sliding mode surface for the j-th follower quadrotor UAV is designed as follows:

S_{j η} = S_{jts η} + c_{j η} S_{jst η} (η = 1, 2, \dots, 6)

(30)

Where

s_{jts 1} = e_{jz 2} + l_{j 1} e_{jz 1} + υ_{j 1} \int_{0}^{t} {e_{jz 1} (τ)}^{q_{j 1}} d τ

(31)

s_{jts 2} = e_{j ϕ 2} + l_{j 2} e_{j ϕ 1} + υ_{j 2} \int_{0}^{t} {e_{j ϕ 1} (τ)}^{q_{j 2}} d τ

(32)

s_{jts 3} = e_{j θ 2} + l_{j 3} e_{j θ 1} + υ_{j 3} \int_{0}^{t} {e_{j θ 1} (τ)}^{q_{j 3}} d τ

(33)

s_{jts 4} = e_{j ψ 2} + l_{j 4} e_{j ψ 1} + υ_{j 4} \int_{0}^{t} {e_{j ψ 1} (τ)}^{q_{j 4}} d τ

(34)

s_{jts 5} = e_{jx 2} + l_{j 5} e_{jx 1} + υ_{j 5} \int_{0}^{t} {e_{jx 1} (τ)}^{q_{j 5}} d τ

(35)

s_{jts 6} = e_{jy 2} + l_{j 6} e_{jy 1} + υ_{j 6} \int_{0}^{t} {e_{jy 1} (τ)}^{q_{j 6}} d τ

(36)

Where $c_{j η}$ , $l_{j η}$ , and $ϑ_{j η}$ denote positive constants, and $0 < q_{j η} = \frac{q_{j η 1}}{q_{j η 2}}$ is chosen such that (F) is a positive odd integer satisfying condition (G), and there exists:

Where $c_{j η}$ , $l_{j η}$ , and $ϑ_{j η}$ represent positive constants, and there exists $0 < q_{j η} = \frac{q_{j η 1}}{q_{j η 2}}$ , where $q_{η 1}$ and $q_{η 2}$ are positive odd integers satisfying condition $0 < q_{η 1} < q_{η 2} < 1$ , and there exists

\begin{matrix} s_{jst η} = {\begin{matrix} {s^{n}}_{jts η} & | s_{jts η} | \leq ρ_{j 0} \\ k_{j 1} s_{ts η} + k_{j 2} \tanh (s_{jts η} / Δ_{j η}) {s^{n}}_{jts η} & | s_{jts η} | > ρ_{j 0} \end{matrix} \end{matrix}

(37)

Where there exists $0 < n_{j} = \frac{n_{j 1}}{n_{j 2}} < 1$ , and both $n_{1}$ , and $n_{2}$ are positive odd integers. Additionally, $ρ_{j 0}$ is defined as a positive constant, and $k_{j 1}$ and $k_{j 2}$ are defined as follows:

{\begin{matrix} k_{j 1} = (2 - n_{j}) {ρ_{j 0}}^{n_{j} - 1} \\ k_{j 2} = (n_{j} - 1) {ρ_{j 0}}^{n_{j} - 2} \end{matrix}

(38)

We differentiate equation (30) to obtain

{\overset{\cdot}{S}}_{j η} = {\overset{\cdot}{S}}_{jts η} + c_{j η} {\overset{\cdot}{S}}_{jst η} (η = 1, 2, \dots, 6)

(39)

Based on equation (3), by substituting equations (30), (31) to (36) into equation (39), we obtain

\begin{matrix} {\overset{\cdot}{s}}_{j 1} = a_{j 1} x_{j 1} + l_{j 1} (x_{j 2} - {\overset{\cdot}{z}}_{jd}) - g_{j} + \frac{\cos x_{j 3} \cos x_{j 5}}{m_{j}} U_{j 1} \\ - {\overset{\cdot\cdot}{z}}_{jd} + l_{j 1} ϖ_{j 1} + ϖ_{j 2} + χ_{j 1} + υ_{j 1} {e_{jz 1}}^{q_{j 1}} + c_{j 1} {\overset{\cdot}{s}}_{jst 1} \end{matrix}

(40)

\begin{matrix} {\overset{\cdot}{s}}_{j 2} = a_{j 2} x_{j 8} x_{j 6} + a_{j 3} {x_{j 4}}^{2} + a_{j 4} {\bar{Ω}}_{j} x_{j 6} + b_{j 1} U_{j 2} \\ + l_{j 2} (x_{j 4} - {\overset{\cdot}{ϕ}}_{jd}) - {\overset{\cdot\cdot}{ϕ}}_{jd} + l_{j 2} ϖ_{j 3} + ϖ_{j 4} + χ_{j 2} \\ + υ_{j 2} {e_{j ϕ 1}}^{q_{j 2}} + c_{j 2} {\overset{\cdot}{s}}_{jst 2} \end{matrix}

(41)

\begin{matrix} {\overset{\cdot}{s}}_{j 3} = a_{j 5} x_{j 8} x_{j 4} + a_{j 6} {x_{j 6}}^{2} + a_{j 7} {\bar{Ω}}_{j} x_{j 4} + b_{j 2} U_{j 3} \\ + l_{j 3} (x_{j 6} - {\overset{\cdot}{θ}}_{jd}) - {\overset{\cdot\cdot}{θ}}_{jd} + l_{j 3} ϖ_{j 5} + ϖ_{j 6} + χ_{j 3} \\ + υ_{j 3} {e_{j θ 1}}^{q_{j 3}} + c_{j 3} {\overset{\cdot}{s}}_{jst 3} \end{matrix}

(42)

\begin{matrix} {\overset{\cdot}{s}}_{j 4} = a_{j 8} x_{j 4} x_{j 6} + a_{j 9} {x_{j 8}}^{2} + b_{j 3} U_{j 4} + l_{j 4} (x j_{8} - {\overset{\cdot}{ψ}}_{jd}) \\ - {\overset{\cdot\cdot}{ψ}}_{jd} + l_{j 4} ϖ_{j 7} + ϖ_{j 8} + χ_{j 4} + υ_{j 4} {e_{j ψ 1}}^{q_{j 4}} + c_{j 4} {\overset{\cdot}{s}}_{jst 4} \end{matrix}

(43)

\begin{matrix} {\overset{\cdot}{s}}_{j 5} = a_{j 10} x_{j 9} + \frac{U_{j 5}}{m_{j}} U_{j 1} + l_{j 5} (x_{j 10} - {\overset{\cdot}{x}}_{jd}) - {\overset{\cdot\cdot}{x}}_{jd} \\ + l_{j 5} ϖ_{j 9} + ϖ_{j 10} + χ_{j 5} + υ_{j 5} {e_{jx 1}}^{q_{j 5}} + c_{j 5} {\overset{\cdot}{s}}_{jst 5} \end{matrix}

(44)

\begin{matrix} {\overset{\cdot}{s}}_{j 6} = a_{j 11} x_{j 11} + \frac{U_{j 6}}{m_{j}} U_{j 1} + l_{j 6} (x_{j 12} - {\overset{\cdot}{y}}_{jd}) \\ - {\overset{\cdot\cdot}{y}}_{jd} + l_{j 6} ϖ_{j 11} + ϖ_{j 12} + χ_{j 6} + υ_{j 6} {e_{jy 1}}^{q_{j 6}} + c_{j 6} {\overset{\cdot}{s}}_{jst 6} \end{matrix}

(45)

ASTITSMC design

Let ${\overset{\cdot}{S}}_{j η} = 0$ , neglecting model uncertainties, external wind disturbances, and actuator faults, thus we obtain the corresponding equivalent controller for the j-th follower quadrotor UAV as follows:

\begin{matrix} U_{j 1 a 0} = - \frac{m_{j}}{(\cos x_{j 3} \cos x_{j 5}) σ_{j 1}} (a_{j 1} x_{j 1} - g_{j} + l_{j 1} (x_{j 2} - {\overset{\cdot}{z}}_{jd}) \\ - {\overset{\cdot\cdot}{z}}_{jd} + υ_{j 1} {e_{jz 1}}^{q_{j 1}} + c_{j 1} {\overset{\cdot}{s}}_{jst 1}) \end{matrix}

(46)

\begin{matrix} U_{j 2 a 0} = - \frac{1}{(b_{j 1} σ_{j 2})} (a_{j 2} x_{j 8} x_{j 6} - a_{j 3} {x_{j 4}}^{2} + a_{j 4} {\bar{Ω}}_{j} x_{j 6} \\ + l_{j 2} (x_{j 4} - {\overset{\cdot}{ϕ}}_{jd}) - {\overset{\cdot\cdot}{ϕ}}_{jd} + υ_{j 2} {e_{j ϕ 1}}^{q_{j 2}} + c_{j 2} {\overset{\cdot}{s}}_{jst 2}) \end{matrix}

(47)

\begin{matrix} U_{j 3 a 0} = - \frac{1}{(b_{j 2} σ_{j 3})} (a_{j 5} x_{j 8} x_{j 4} - a_{j 6} {x_{j 6}}^{2} + a_{j 7} {\bar{Ω}}_{j} x_{j 4} \\ + l_{j 3} (x_{j 6} - {\overset{\cdot}{θ}}_{jd}) - {\overset{\cdot\cdot}{θ}}_{jd} + υ_{j 3} {e_{j θ 1}}^{q_{j 3}} + c_{j 3} {\overset{\cdot}{s}}_{jst 3}) \end{matrix}

(48)

\begin{matrix} U_{j 4 a 0} = - \frac{1}{(b_{j 3} σ_{j 4})} (a_{j 8} x_{j 4} x_{j 6} + a_{j 9} {x_{j 8}}^{2} + l_{j 4} (x_{j 8} - {\overset{\cdot}{ψ}}_{jd}) \\ - {\overset{\cdot\cdot}{ψ}}_{jd} + υ_{j 4} {e_{j ψ 1}}^{q_{j 4}} + c_{j 4} {\overset{\cdot}{s}}_{jst 4}) \end{matrix}

(49)

\begin{matrix} U_{j 50} = - \frac{m_{j}}{U_{j 1 a}} (a_{j 10} x_{j 9} + l_{j 5} (x_{j 10} - {\overset{\cdot}{x}}_{jd}) - {\overset{\cdot\cdot}{x}}_{jd} \\ + υ_{j 5} {e_{jx 1}}^{q_{j 5}} + c_{j 5} {\overset{\cdot}{s}}_{jst 5}) \end{matrix}

(50)

\begin{matrix} U_{j 60} = - \frac{m_{j}}{U_{j 1}} (a_{j 11} x_{j 11} + l_{j 6} (x_{j 12} - {\overset{\cdot}{y}}_{jd}) - {\overset{\cdot\cdot}{y}}_{jd} \\ + υ_{j 6} {e_{jy 1}}^{q_{j 6}} + c_{j 6} {\overset{\cdot}{s}}_{jst 6}) \end{matrix}

(51)

In the configuration, constants $σ_{j 1}$ , $σ_{j 2}$ , $σ_{j 3}$ , $σ_{j 4}$ are all defined as positive values.

When applied in practical scenarios, the precise upper limit of parameter $Ξ_{j η}$ , which accounts for modeling uncertainties, wind disturbances, and actuator faults, is typically unknown. To address this challenge, adaptive adjustment methodologies are employed. These methodologies aim to finely approximate the parameter values, initiating with a provisional constant $ξ_{j}$ as a baseline for estimation, with an estimation error is mathematically articulated as:

{\tilde{ξ}}_{j η} = {\hat{ξ}}_{j η} - ξ_{j η}, (\forall η = 1, 2, 3, 4, 5, 6)

(52)

Therefore, we design the adaptive law for the j-th follower UAV as follows:

{\overset{\cdot}{\hat{ξ}}}_{j η} = {γ_{j η}}^{- 1} | s_{j η} |

(53)

Where $γ_{j η}$ is a positive constant, and $s_{j η}$ represents the sliding mode surface for the j-th follower UAV. To mitigate the impact of model uncertainties, wind disturbances, and actuator faults on the system’s performance, we also design a switching control law for the j-th follower UAV, articulated as:

\begin{matrix} U_{j 1 a 1} = ({\hat{ξ}}_{j 1} - ρ_{j 1} ({\sum_{j 1}}^{| s_{j 1} |} - 1) \tanh (\frac{s_{j 1}}{α_{j 1}})) \\ \frac{- m_{j}}{(\cos x_{j 3} \cos x_{j 5}) σ_{j 1}} \end{matrix}

(54)

U_{j 2 a 1} = ({\hat{ξ}}_{j 2} - ρ_{j 2} ({\sum_{j 2}}^{| s_{j 2} |} - 1) \tanh (\frac{s_{j 2}}{α_{j 2}})) \frac{- 1}{(b_{j 1} σ_{j 2})}

(55)

U_{j 3 a 1} = ({\hat{ξ}}_{j 3} - ρ_{j 3} ({\sum_{j 3}}^{| s_{j 3} |} - 1) \tanh (\frac{s_{j 3}}{α_{j 3}})) \frac{- 1}{(b_{j 2} σ_{j 3})}

(56)

U_{j 4 a 1} = ({\hat{ξ}}_{j 4} - ρ_{j 4} ({\sum_{j 4}}^{| s_{j 4} |} - 1) \tanh (\frac{s_{j 4}}{α_{j 4}})) \frac{- 1}{(b_{j 3} σ_{j 4})}

(57)

U_{j 5 a 1} = ({\hat{ξ}}_{j 5} - ρ_{j 5} ({\sum_{j 5}}^{| s_{j 5} |} - 1) \tanh (\frac{s_{j 5}}{α_{j 5}})) \frac{- m_{j}}{U_{j 1 a}}

(58)

U_{j 6 a 1} = ({\hat{ξ}}_{j 6} - ρ_{j 6} ({\sum_{j 5}}^{| s_{j 6} |} - 1) \tanh (\frac{s_{j 6}}{α_{j 6}})) \frac{- m_{j}}{U_{j 1 a 0}}

(59)

Here, $ρ_{j η} > 0$ and $0 < \sum_{j η} < 1 (\forall η = 1, \dots, 6)$ .

Therefore, combining equations (46) to (51) and equations (54) through (59), we obtain the adaptive super-twisting integral terminal sliding mode controller for the j-th follower UAV.

\begin{array}{l} U_{j 1 a} = \frac{m_{j}}{(\cos x_{j 3} \cos x_{j 5}) σ_{j 1}} (a_{j 1} x_{j 1} - g_{j} + l_{j 1} (x_{j 2} - {\dot{z}}_{j d}) - {\overset{\cdot\cdot}{z}}_{j d} \\ + υ_{j 1} {e_{j z 1}}^{q_{j 1}} + c_{j 1} {\dot{s}}_{j s t 1} + {\hat{ξ}}_{j 1} - ρ_{j 1} ({\sum_{j 1}}^{| s_{j 1} |} - 1) \tanh (\frac{s_{j 1}}{α_{j 1}})) \end{array}

(60)

\begin{matrix} U_{j 2 a} = - \frac{1}{(b_{j 1} σ_{j 2})} (a_{j 2} x_{j 8} x_{j 6} - a_{j 3} {x_{j 4}}^{2} + a_{j 4} {\bar{Ω}}_{j} x_{j 6} \\ + l_{j 2} (x_{j 4} - {\overset{\cdot}{ϕ}}_{jd}) - {\overset{\cdot\cdot}{ϕ}}_{jd} + υ_{j 2} {e_{j ϕ 1}}^{q_{j 2}} + c_{j 2} {\overset{\cdot}{s}}_{jst 2} \\ + ({\hat{ξ}}_{j 2} - ρ_{j 2} ({\sum_{j 2}}^{| s_{j 2} |} - 1) \tanh (\frac{s_{j 2}}{α_{j 2}})) \end{matrix}

(61)

\begin{matrix} U_{j 3 a} = - \frac{1}{(b_{j 2} σ_{j 3})} (a_{j 5} x_{j 8} x_{j 4} - a_{j 6} {x_{j 6}}^{2} + a_{j 7} {\bar{Ω}}_{j} x_{j 4} \\ + l_{j 3} (x_{j 6} - {\overset{\cdot}{θ}}_{jd}) - {\overset{\cdot\cdot}{θ}}_{jd} + υ_{j 3} {e_{j θ 1}}^{q_{j 3}} + c_{j 3} {\overset{\cdot}{s}}_{jst 3} \\ + {\hat{ξ}}_{j 3} - ρ_{j 3} ({\sum_{j 3}}^{| s_{j 3} |} - 1) \tanh (\frac{s_{j 3}}{α_{j 3}})) \end{matrix}

(62)

\begin{matrix} U_{j 4 a} = - \frac{1}{(b_{j 3} σ_{j 4})} (a_{j 8} x_{j 4} x_{j 6} + a_{j 9} {x_{j 8}}^{2} + l_{j 4} (x_{j 8} - {\overset{\cdot}{ψ}}_{jd}) - {\overset{\cdot\cdot}{ψ}}_{jd} \\ + υ_{j 4} {e_{j ψ 1}}^{q_{j 4}} + c_{j 4} {\overset{\cdot}{s}}_{jst 4} + {\hat{ξ}}_{j 4} - ρ_{j 4} ({\sum_{j 4}}^{| s_{j 4} |} - 1) \tanh (\frac{s_{j 4}}{α_{j 4}})) \end{matrix}

(63)

\begin{matrix} U_{j 5} = - \frac{m_{j}}{U_{j 1 a}} (a_{j 10} x_{j 9} + l_{j 5} (x_{j 10} - {\overset{\cdot}{x}}_{jd}) - {\overset{\cdot\cdot}{x}}_{jd} + υ_{j 5} {e_{jx 1}}^{q_{j 5}} \\ + c_{j 5} {\overset{\cdot}{s}}_{jst 5} + {\hat{ξ}}_{j 5} - ρ_{j 5} ({\sum_{j 5}}^{| s_{j 5} |} - 1) \tanh (\frac{s_{j 5}}{α_{j 5}})) \end{matrix}

(64)

\begin{array}{l} U_{j 6} = - \frac{m_{j}}{U_{j 1 a}} (a_{j 11} x_{j 11} + l_{j 6} (x_{j 12} - {\dot{y}}_{j d}) - {\overset{\cdot\cdot}{y}}_{j d} + υ_{j 6} {e_{j y 1}}^{q_{j 6}} \\ + c_{j 6} {\dot{s}}_{j s t 6} + {\hat{ξ}}_{j 6} - ρ_{j 6} ({\sum_{j 5}}^{| s_{j 6} |} - 1) \tanh (\frac{s_{j 6}}{α_{j 6}})) \end{array}

(65)

Where $σ_{1, 2, 3, 4}$ and $α_{η}$ are established as known constants.

Theorem 3. Considering the model uncertainties, wind disturbances, and actuator faults of the j-th follower quadrotor UAV system as defined in (3). Assuming that the super-twisting sliding mode surface and adaptive law are delineated by (30) and (53) respectively, and the control input is crafted according to equations (60) to (65). Under Assumption 1, it is posited that the tracking error of the system will asymptotically converge.

Proof: The controller designed for the j-th follower UAV is confirmed to stable. For the demonstration of stability, we choose the Lyapunov function as

V_{j η} (s_{j η}, {\tilde{ξ}}_{j η}) = \frac{1}{2} {s_{j η}}^{2} + \frac{1}{2} γ_{j η} {\tilde{ξ}}_{j η} {\overset{\cdot}{\hat{ξ}}}_{j η}

(66)

We differentiate the above equation, and given the existence of ${\overset{\cdot}{\tilde{ξ}}}_{j η} = {\overset{\cdot}{\hat{ξ}}}_{j η}$ , we obtain

{\overset{\cdot}{V}}_{j η} (s_{η}, {\tilde{ξ}}_{η}) = {\overset{\cdot}{s}}_{j η} s_{j η} + {\tilde{ξ}}_{j η} {\overset{\cdot}{\hat{ξ}}}_{j η}

(67)

Upon substituting equations (39) through (44), as well as equations (52) and (53), into equation (67), we derive:

\begin{matrix} {\overset{\cdot}{V}}_{j 1} = s_{j 1} (a_{j 1} x_{j 1} + l_{j 1} (x_{j 2} - {\overset{\cdot}{z}}_{jd}) - g_{j} + \frac{\cos x_{j 3} \cos x_{j 5}}{m_{j}} U_{j 1 a} \\ - {\overset{\cdot\cdot}{z}}_{jd} + l_{j 1} ϖ_{j 1} + ϖ_{j 2} + υ_{j 1} {e_{jz 1}}^{{q_{j}}_{1}} + c_{j 1} {\overset{\cdot}{s}}_{jst 1}) + {\tilde{ξ}}_{j} | s_{j 1} | \end{matrix}

(68)

\begin{matrix} {\overset{\cdot}{V}}_{j 2} = s_{j 2} (a_{j 2} x_{j 8} x_{j 6} + a_{j 3} {x_{j 4}}^{2} + a_{j 4} {\bar{Ω}}_{j} x_{j 6} + b_{j 1} σ_{j 2} U_{j 2 a} \\ + l_{j 2} (x_{j 4} - {\overset{\cdot}{ϕ}}_{jd}) - {\overset{\cdot\cdot}{ϕ}}_{jd} + l_{j 2} ϖ_{j 3} + ϖ_{j 4} + υ_{j 2} {e_{j ϕ 1}}^{q_{j 2}} \\ + c_{j 2} {\overset{\cdot}{s}}_{jst 2}) + {\tilde{ξ}}_{j 2} | s_{j 2} | \end{matrix}

(69)

\begin{matrix} {\overset{\cdot}{V}}_{j 3} = s_{j 3} (a_{j 5} x_{j 8} x_{j 4} + a_{j 6} {x_{j 6}}^{2} + a_{j 7} {\bar{Ω}}_{j} x_{j 4} + b_{j 2} U_{j 3 a} \\ + l_{j 3} (x_{j 6} - {\overset{\cdot}{θ}}_{jd}) - {\overset{\cdot\cdot}{θ}}_{jd} + l_{j 3} ϖ_{j 5} + ϖ_{j 6} + υ_{j 3} {e_{j θ 1}}^{q_{j 3}} \\ + c_{j 3} {\overset{\cdot}{s}}_{jst 3}) + {\tilde{ξ}}_{j 3} | s_{j 3} | \end{matrix}

(70)

\begin{matrix} {\overset{\cdot}{V}}_{j 4} = s_{j 4} (a_{j 8} x_{j 4} x_{j 6} + a_{j 9} {x_{j 8}}^{2} + b_{j 3} U_{j 4 a} + l_{j 4} (x_{j 8} - {\overset{\cdot}{ψ}}_{jd}) \\ - {\overset{\cdot\cdot}{ψ}}_{jd} + l_{j 4} ϖ_{j 7} + ϖ_{j 8} + υ_{j 4} {e_{j ψ 1}}^{q_{j 4}} + c_{j 4} {\overset{\cdot}{s}}_{jst 4}) + {\tilde{ξ}}_{j 4} | s_{j 4} | \end{matrix}

(71)

\begin{matrix} {\overset{\cdot}{V}}_{j 5} = s_{j 5} (a_{j 10} x_{j 9} + \frac{U_{j 5}}{m_{j}} U_{j 1 a} + l_{j 5} (x_{j 10} - {\overset{\cdot}{x}}_{jd}) \\ - {\overset{\cdot\cdot}{x}}_{jd} + l_{j 5} ϖ_{j 9} + ϖ_{j 10} + υ_{j 5} {e_{jx 1}}^{q_{j 5}} + c_{j 5} {\overset{\cdot}{s}}_{jst 5}) + {\tilde{ξ}}_{j 5} | s_{j 5} | \end{matrix}

(72)

\begin{matrix} {\overset{\cdot}{V}}_{j 6} = s_{j 6} (a_{j 11} x_{j 11} + \frac{U_{j 6}}{m_{j}} U_{j 1 a} + l_{j 6} (x_{j 12} - {\overset{\cdot}{y}}_{jd}) \\ - {\overset{\cdot\cdot}{y}}_{jd} + l_{j 6} ϖ_{j 11} + ϖ_{j 12} + υ_{j 6} {e_{jy 1}}^{η 6} + c_{j 6} {\overset{\cdot}{s}}_{jst 6}) + {\tilde{ξ}}_{j 6} | s_{j 6} | \end{matrix}

(73)

By substituting equations (60) through (65) into equations (68) through (73) and conducting a subsequent simplification, we derive:

\begin{matrix} {\overset{\cdot}{V}}_{j η} (s_{j η}, {\tilde{ξ}}_{j η}) = s_{j η} (N_{j η} - ({\hat{ξ}}_{j} - ρ_{j η} ({Σ_{j η}}^{| s_{j η} |} - 1)) \tanh (\frac{s_{j η}}{α_{j η}})) \\ + {\tilde{ξ}}_{j η} | s_{j η} | \\ \leq - ρ_{j η} ({Σ_{j η}}^{| s_{j η} |} - 1) | s_{j η} | + {\tilde{ξ}}_{j η} | s_{j η} | + ξ_{j η} | s_{j η} | \\ - {\hat{ξ}}_{j η} | s_{j η} | + ξ_{j η} | s_{j η} | - ξ_{j η} | s_{j η} | \end{matrix}

(74)

From equation (52) and through detailed simplification, we obtain:

\begin{matrix} {\overset{\cdot}{V}}_{j η} (s_{j η}, {\tilde{ξ}}_{j η}) \leq - ρ_{j η} ({Σ_{j η}}^{| s_{j η} |} - 1) | s_{j η} | + ({\hat{ξ}}_{j η} - ξ_{j η}) | s_{j η} | \\ - ({\hat{ξ}}_{j η} - ξ_{j η}) | s_{j η} | + ξ_{j η} | s_{j η} | - ξ_{j η} | s_{j η} | \\ = - ρ_{j η} ({Σ_{j η}}^{| s_{j η} |} - 1) | s_{j η} | \leq 0 \end{matrix}

(75)

According to the Lyapunov stability condition, it is confirmed that the designed controller is stable. This implies that the tracking error of the system can converge to the verification. The overall control block diagram illustrating this process is depicted in Figure 3.

Figure 3.

Block diagram of adaptive super-twisting integrating terminal sliding mode control for multi-quadrotor UAV based on leader-follower configuration.

Analysis of simulation results

Simulation parameters

To validate the effectiveness of the newly designed control algorithm designed, a detailed simulation was conducted on a MATLAB platform, establishing a multi-quadcopter UAV formation system. Uniformity in system parameters across all UAVs in the formation ensures consistency in the simulation environment, thereby enabling a focused evaluation of the control strategy’s performance. The structural parameters, which are critical to the dynamics and behavior of the UAVs, are meticulously listed in Table 1.

Table 1.

Structural parameters of four-rotor UAV.

Parameters	Value
$m_{j}$	$2 kg$
$I_{jx}$	$5.745 \times 1 0^{- 3} kg \cdot m^{2}$
$I_{jy}$	$5.745 \times 1 0^{- 3} kg \cdot m^{2}$
$I_{jz}$	$1.175 \times 1 0^{- 2} kg \cdot m^{2}$
$k_{Ti}$	$1.116 \times 10^{- 5} kg \cdot m^{2}$
$k_{mi}$	$1.489 \times 1 0^{- 7} kg \cdot m^{2}$
$k_{jx} = k_{jy} = k_{jz}$	$0.01$
$k_{j ϕ} = k_{j θ} = k_{j ψ}$	$0.012$
$g$	$9.81 m \cdot s^{- 2}$
$l_{i}$	$0.225 m$

The fault-tolerant control simulation described in this paper involves a configuration comprising one virtual leader UAV and three follower UAVs. Figure 4 depicts the communication topology utilized among the UAV within the formation.

Figure 4.

Topology structure of UAV formation communication.

In this simulation scenario, Follower UAV 1 is specifically designated to encounter both wind disturbance and actuator faults simultaneously at 10 s, collectively termed as “aggregate disturbance.” The specific parameters are outlined as follows.

\begin{matrix} f_{1 x} = {\begin{matrix} 0 (t < 10) \\ \cos (2 t) + 0.16 (10 \leq t \leq 20) \end{matrix} \\ f_{1 y} = {\begin{matrix} 0 (t < 10) \\ \cos (2 t) + 0.16 (10 \leq t \leq 0) \end{matrix} \end{matrix}

\begin{matrix} f_{1 z} = {\begin{matrix} 0 (t < 10) \\ \cos (2 t) + 0.25 (10 \leq t \leq 20) \end{matrix} \\ f_{1 ϕ} = {\begin{matrix} 0 (t < 10) \\ \cos (2 t) + 0.25 (10 \leq t \leq 20) \end{matrix} \end{matrix}

\begin{matrix} f_{1 θ} = {\begin{matrix} 0 (t < 10) \\ \cos (2 t) + 0.25 (10 \leq t \leq 20) \end{matrix} \\ f_{1 ψ} = {\begin{matrix} 0 (t < 10) \\ \cos (2 t) + 0.25 (10 \leq t \leq 20) \end{matrix} \end{matrix}

The fault-tolerant control algorithm for the multi-UAV formation designed in this paper is divided into two primarily parts: a distributed self-adaptive event-triggered cooperative controller and an adaptive terminal sliding mode controller. The specific parameters designated for the formation cooperative controller are $ε = 0.2$ , $μ_{1} = μ_{2} = 10$ , $Λ = 4.1$ , $k = 1$ , $η_{1} = 0.2$ , $η_{2} = η_{3} = 0.1$ , $ς = 0.5$ , $α = 0.1$ , and $β = 0.01$ .

For the adaptive integral terminal sliding mode control of the follower UAVs, the parameters are precisely set as $c_{η} = 0.5$ , $l_{η} = 20$ , $υ_{η} = 0.1$ , $q_{η 1} = 7$ , $q_{η 2} = 9$ , $n = \frac{3}{5}$ , $ρ_{0} = 0.1$ , $λ_{η} = 0.2$ , $\sum_{η} = \frac{3}{5}$ , $σ_{1} = 1$ and $σ_{2} = σ_{3} = σ_{4} = 0.1$ .

It is noteworthy that the parameters for the aforementioned controllers are selected through a trial-and-error method, where different choices significantly influence the control performance. For instance, within the distributed formation coordination controller, parameters $μ_{1}$ and $μ_{2}$ primarily impact the systems response amplitude. Conversely, in the adaptive super-twisting integral terminal controller, parameters $c_{η}$ , $q_{η 1}$ , and $q_{η 2}$ are critical in dictating the system’s convergence rate. Moreover, the interaction among various controller parameters plays a pivotal role in shaping the overall system performance. These interactions define the response characteristics of the system, underscoring the complexity of parameter optimization. Therefore, when optimizing one parameter, it is usually necessary to consider its interdependencies with other parameters to ensure the achievement of optimal control performance across the system.

Numerical simulation results of distributed adaptive event-triggered formation cooperative control

We set the relative position offsets as $δ_{1} (t) = {[0, - 2, 0]}^{T}$ m, $δ_{2} (t) = {[2, - 2, 0]}^{T}$ m, and $δ_{3} (t) = {[2, 0, 0]}^{T}$ m, and the desired trajectory for the virtual leader as $p_{0} (t) = {[3 \cos t, 3 sint, 2 + 0.5 t]}^{T}$ m and $v_{0} (t) = {[- 3 \sin t, 3 \cos t, 0.5]}^{T}$ m/s.

Figure 5 depicts the three-dimensional desired trajectory curve for formation cooperative control. Figures 6 to 8 show the relative distance error curves between the virtual leader and the three followers. Figures 9 to 11 display the relative velocity error curves between the virtual leader and the three followers. Figure 12 illustrates the event-triggering time intervals. From Figures 5 to 12, it can be observed that the designed distributed cooperative control protocol can achieve the generation of desired trajectories for quadcopter UAV formations, and Figure 12 indicates that this can be achieved with fewer computational resources.

Figure 5.

Formation cooperative control of 3D desired trajectory curve.

Figure 6.

Virtual leader and follower 1 relative distance error.

Figure 7.

Virtual leader and follower 2 relative distance error.

Figure 8.

Virtual leader and follower 3 relative distance error.

Figure 9.

Virtual leader and follower 1 relative speed error.

Figure 10.

Virtual leader and follower 2 relative speed error.

Figure 11.

Virtual leader and follower 3 relative speed error.

Figure 12.

Event triggering interval.

Numerical simulation results of fault-tolerant control of the follower quad-rotor UAV

To validate the performance advantages of the follower UAV adaptive super-twisting integral sliding mode control (ASTITSMC) algorithm, numerical simulations were conducted comparing Algorithm 1 from Xiu et al.²⁹ and Algorithm 2 from Eliker and Zhang.³⁰ The position tracking of follower UAV 1 is shown in Figures 13 to 15. From the figures, it can be observed that initially, there is no significant difference in convergence speed between ASTITSMC and the comparative algorithms. However, at time t, when the UAV is simultaneously affected by gust disturbances and actuator faults, although all three control methods converge to the desired values, zooming into the graphs reveals that ASTITSMC exhibits smaller overshoot and errors compared to the other two algorithms. It achieves faster convergence and precise tracking of the desired trajectory, demonstrating strong robustness. Figure 16 illustrates the three-dimensional trajectory tracking curves, where it can be seen that ASTITSMC outperforms the other two algorithms in terms of higher tracking accuracy and stronger disturbance rejection capabilities.

Figure 13.

Position x tracking curve.

Figure 14.

Position y tracking curve.

Figure 15.

Position z tracking curve.

Figure 16.

3D tracking curve.

The attitude angle tracking of follower UAV 1 is shown in Figures 17 to 19. From the graphs, it is evident that initially, although the two comparative algorithms converge quickly, they exhibit noticeable overshoot. In contrast, ASTITSMC shows smaller overshoot. At 10 s, when follower UAV 1 experiences gust disturbances and actuator faults simultaneously, ASTITSMC demonstrates stronger robustness compared to the two comparative algorithms. It effectively compensates for disturbances and faults, achieving faster convergence and smaller errors, thereby enabling precise tracking of the desired trajectory.

Figure 17.

Tracking curve of $ϕ$ system.

Figure 18.

Tracking curve of $θ$ system.

Figure 19.

Tracking curve of $ψ$ system.

The adaptive law parameter estimation for the position and attitude of follower UAV 1 is shown in Figure 20. From the figure, we observe that initially, the parameter estimation values converge rapidly to a constant, ensuring that the UAV effectively tracks the desired trajectory. When follower UAV 1 encounters external disturbances and actuator faults simultaneously, the adaptive law adjusts the parameter estimation values to another constant through active adaptation. This allows estimation of the upper bound of aggregate disturbances, thereby enhancing system robustness and maintaining good tracking performance of the UAV.

Figure 20.

Adaptive parameter estimation curve: (a) estimate of parameter $ξ_{5, 6}$ , (b) estimate of parameter $ξ_{1, 2}$ , and (c) estimate of parameter $ξ_{3, 4}$ .

The UAV formation flight tracking trajectory is depicted in Figure 21. From the figure, it can be observed that the designed leader-follower distributed event-triggered formation coordination control protocol provides real-time updates of positions to the follower UAVs. Additionally, the designed follower UAV adaptive super-twisting integral terminal sliding mode control algorithm achieves precise tracking of desired trajectories, enabling formation coordination control.

Figure 21.

Formation flight trajectory curve.

Conclusions

This paper addresses the problem of tracking control for multi-quadcopter formations in the presence of model uncertainties, wind disturbances, and actuator faults. We have designed an adaptive super-twisting integral terminal sliding mode controller (ASTITSMC), which estimates the upper bound of aggregate disturbances through an adaptive law, enabling the quadcopters to effectively track the desired trajectory within a finite time despite actuator faults and external disturbances. The stability of the controller is thoroughly analyzed, and the performance of ASTITSMC is compared with two other control algorithms in actual flight scenarios. Simulation results demonstrate that the ASTITSMC controller exhibits significant advantages in both performance and adaptability, effectively guiding quadcopter formations through complex flight tasks. Additionally, a distributed adaptive event-triggered formation coordination control protocol is proposed. This protocol does not rely on global information and significantly improves convergence speed and reduces unnecessary energy consumption compared to existing methods. Through proof, the constructed event-triggered conditions effectively eliminate Zeno behavior. Finally, numerical simulations validate the effectiveness of this control protocol in formation flight.

Despite the demonstrated performance of the proposed control scheme and protocol, there remain several issues and areas for further research and improvement. For instance, this study primarily focuses on formation control in static environments, while robustness and adaptability in dynamic or unknown environments still need further investigation. Additionally, sensor errors and communication delays in practical applications present challenges to the distributed control protocol. Future work should involve experimental validation on real UAVs and explore more efficient formation control in complex environments.

Footnotes

Author contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by the first author and second author. The first draft of the manuscript was written by the first author and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project has received funding from the National Natural Science Foundation of China. (Fund Project Number :61463030)

ORCID iD

Ming-Min Luo

Data availability statement

The data underlying this study cannot be made publicly available due to confidentiality agreements. All data were handled in accordance with strict privacy and security protocols to ensure accuracy and integrity. While the specific data cannot be shared, we affirm that the research findings and conclusions are based on thoroughly validated information.

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