Bearing-Based Formation Control of Multi-UAV Systems with Conditional Wind Disturbance Utilization

Abstract

This paper investigates bearing-based formation control of multiple unmanned aerial vehicles (UAVs) flying in low-altitude wind fields. In such environments, time-varying wind disturbances can distort the formation geometry, enlarge bearing errors, and even induce potential collisions among neighboring UAVs, yet they also contain components that can be beneficial for the formation motion. Conventional disturbance compensation methods treat wind as a purely harmful factor and aim to reject it completely, which may sacrifice responsiveness and energy efficiency. To address this issue, we propose a pure bearing-based formation control framework with Conditional Disturbance Utilization (CDU). First, a real-time disturbance observer is designed to estimate the wind-induced disturbances in both translational and rotational channels. Then, based on the estimated disturbances and the bearing-dependent potential function, CDU indicators are constructed to judge whether the current disturbance component is beneficial or detrimental with respect to the formation control objective. These indicators are embedded into the bearing-based formation controller so that favorable wind components are exploited to accelerate formation convergence, whereas adverse components are compensated. Using an angle-rigid formation topology and a Lyapunov-based analysis, we prove that the proposed CDU-based controller guarantees global asymptotic stability of the desired formation. Simulation results on triangular and hexagonal formations under complex wind disturbances show that the proposed method achieves faster convergence and improved responsiveness compared with traditional disturbance observer-based control, while preserving formation stability and safety.

1. Introduction

In recent years, under the backdrop of interdisciplinary integration, research on collaborative control of unmanned aerial vehicles (UAVs) has emerged as a key direction in the field of intelligent systems, drawing sustained attention from researchers worldwide for both theoretical exploration and engineering validation [1,2,3,4,5]. Yuan et al. [6] developed a flight formation control framework based on a six-degree-of-freedom dynamics model, proposing a distributed model predictive control architecture with a leader–follower topology. Pan et al. [7] established a theoretical framework for collaborative control in three-dimensional space for multi-UAV systems under non-holonomic constraints. Compared to other multi-rotor UAVs, quadrotor UAVs are widely utilized in flight control research due to their compact size and agile maneuverability. In UAV swarms, each UAV can acquire pose information from neighboring units, enabling highly consistent swarm behavior through control theory. Liu et al. [8] proposed a distributed fault-tolerant control method that integrates topology-switching mechanisms with adaptive compensation algorithms.

In the evolution of formation control theory, the research focus has shifted from traditional position and velocity control to angle-constrained control [9,10]. Unlike conventional formation control methods that rely on distance and relative position information [11], angle-based rigid formation control requires only easily obtainable neighboring angle measurements [12]. This approach simplifies controller design while meeting complex control requirements and precision demands. Moreover, pure bearing-based formation control avoids target ambiguity, effectively maintaining geometric identifiability of relative poses in three-dimensional space. Zhang et al. [13] introduced an information association algorithm based solely on minimal angular distance, while Bao et al. [14] developed a simple mathematical model to adjust UAV positions using pure bearing-based passive localization. However, these studies assume ideal environments and overlook the impact of multi-source time-varying disturbances on system robustness. A critical factor in achieving UAV formation tasks lies in handling environmental disturbances. Disturbances affecting UAVs are closely tied to their operating environment, including electromagnetic disturbance [15], persistent wind disturbances, and time-varying turbulent wind fields [16]. The coupled effects of these disturbances lead to nonlinear accumulation [17] of formation pose errors, significantly increasing the technical complexity of formation control. Wind disturbances, characterized by time-varying directions and random intensities (e.g., frequent gusts and spatially non-uniform turbulence) [18], are particularly prone to causing angular deviations in formations, making them a complex and critical challenge in UAV formation control. Zhang et al. [19] designed a disturbance observer to achieve online estimation and compensation of wind disturbances, with experiments demonstrating a significant reduction in heading angle errors. Meng et al. [20] modeled stochastic wind fields to derive the minimum energy consumption boundary for UAV wind-resistant control, providing a theoretical basis for optimizing control strategies.

However, most of the aforementioned studies treat disturbances as purely harmful factors that must be fully suppressed, addressing disturbance compensation independently from other control objectives [21,22]. This overlooks the potential utility of disturbances under specific conditions. Emerging research has begun to challenge the conventional view that disturbances are solely detrimental [23,24,25]. In certain scenarios, particularly when disturbance directions align with control objectives, these signals can be transformed into auxiliary control inputs. Inspired by this, in wind field environments, favorable winds can enhance system propulsion and efficiency, enabling UAVs to exhibit behaviors that surpass traditional compensation methods. This insight is one of the primary motivations for this study. Despite significant progress in UAV formation control, challenges remain when it comes to interacting with the environment and achieving robust flight control under wind disturbances. The main contributions of this study are as follows:

(1) This paper proposes a novel pure bearing-based formation control scheme for multiple UAVs that leverages CDU to address the challenge of maintaining formation in complex wind disturbance environments, enabling the system to exploit favorable disturbances to improve convergence performance.

(2) A bearing-dependent potential energy function is introduced into the CDU-based formation control strategy to ensure safety during disturbance utilization, including collision risk reduction and safe formation maintenance under wind disturbances.

The remainder of this paper is organized as follows. Section 2 formulates the wind disturbance model, the multi-UAV dynamics, and the bearing-based formation control objective. Section 3 presents the proposed CDU-based bearing formation control scheme. Section 4 provides the stability analysis and Section 5 reports the simulation results. Section 6 concludes the paper.

2. Problem Statement

2.1. Wind Disturbance Modeling

In low-altitude environments, UAVs face mean wind, gusts, and wind shear. The wind field is modeled as follows:

$$v_w(h,q)=v_a\left(h\right)+v_b\left(q\right)+v_c\left(h\right),$$

(1)

decomposed into the body coordinate axes:

$$v_{wi}\left(t\right)=\left[\begin{array}{c}{v}_{wxi}\left(t\right)\\ v_{wyi}\left(t\right)\\ v_{wzi}\left(t\right)\end{array}\right].$$

The wind speed components are

$$\begin{array}{cc}\hfill v_a\left(h\right)& =v_{6.096}{\displaystyle \frac{ln(h/z_0)}{ln(6.096/z_0)}},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill v_b\left(q\right)& =\left\{\begin{array}{cc}0,\hfill & q<0,\hfill \\ {\displaystyle \frac{v_{bmax}}{2}}\left(1-cos{\displaystyle \frac{\pi q}{d}}\right),\hfill & 0\le q\le d,\hfill \\ v_{bmax},\hfill & q>d,\hfill \end{array}\right.\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill v_c\left(h\right)& ={\displaystyle \frac{v_f}{k}}ln\left({\displaystyle \frac{h}{H_c}}\right).\hfill \end{array}$$

(4)

Here, h is the flight altitude, $z_0$ is the surface roughness length, $v_{6.096}$ is the mean wind speed at 6.096 m; q is the distance to the gust center, d is the gust scale, $v_{bmax}$ is the peak gust speed; $H_c$ is thw zero-wind height, k is the Kármán constant, and $v_f$ is the friction velocity. These wind disturbances affect UAV position and attitude.

2.2. Mathematical Model

Fixed-wing UAV motion includes translation (position change, constant attitude) and rotation (Euler angle changes) [26]. In wind-disturbed environments, assuming zero roll and pitch, the UAV system decouples into position and angle subsystems. The model for UAV i is

$$\begin{array}{cc}\hfill {\dot{r}}_i& =R_i(v_i+d_{v_i}\left(t\right)),\hfill \\ \hfill {\dot{\theta }}_i& ={\omega }_i+d_{{\omega }_i}\left(t\right),\hfill \end{array}$$

(5)

where $r_i={\left[\begin{array}{ccc}{x}_i& y_i& z_i\end{array}\right]}^T$ is the position, ${\dot{r}}_i$ is the velocity, $i=1,\dots ,n$; $R_i=\left[\begin{array}{ccc}cos{\theta }_i& -sin{\theta }_i& 0\\ sin{\theta }_i& cos{\theta }_i& 0\\ 0& 0& 1\end{array}\right]$ is the rotation matrix, ${\theta }_i$ is the yaw angle, $d_{v_i}\left(t\right)$ is the wind disturbance with regard to velocity, $d_{{\omega }_i}\left(t\right)$ is the wind disturbance on yaw, ${\omega }_i$ is the yaw angular velocity, and $v_i={\left[\begin{array}{ccc}{V}_{ix}& V_{iy}& V_{iz}\end{array}\right]}^T$ is the heading velocity. The actual velocity is $u_i=v_i+d_{v_i}$, with $d_{v_i}={\left[\begin{array}{ccc}{d}_{v_{ix}}& d_{v_{iy}}& d_{v_{iz}}\end{array}\right]}^T$, $d_{{\omega }_i}={\left[\begin{array}{ccc}{d}_{{\omega }_{ix}}& d_{{\omega }_{iy}}& d_{{\omega }_{iz}}\end{array}\right]}^T$.

2.3. Bearing Rigidity and Control Goal

The distance and bearing vectors are written as follows:

$$r_{ij}=r_i-r_j,g_{ij}={\displaystyle \frac{r_{ij}}{\parallel r_{ij}\parallel }},$$

(6)

where $\parallel r_{ij}\parallel$ is the Euclidean distance and $g_{ij}$ is the relative bearing. For undirected topology, $r_{ij}=-r_{ji}$, $g_{ij}=-g_{ji}$. The position vector is $r={\left[\begin{array}{c}{r}_1^T,\dots ,r_n^T\end{array}\right]}^T$. The desired bearing is $g_{ij}^{*}$. The topology graph $G=(V,E)$ has n nodes ($V=\{1,\dots ,n\}$) and the edge set $E=\{(i,j)\in V\times V\mid j\in N_i\}$, where $N_i$ is the neighbor set of node i.

Figure 1 shows a five-UAV planar topology, with edges as bidirectional communication links. This supports an algebraic graph model for adaptive bearing formation control.

For formation structure, the position set is ${\left\{r_i\right\}}_{i=1}^n$ and the position vector is $r={\left[\begin{array}{c}{r}_1^T,\dots ,r_n^T\end{array}\right]}^T\in {\mathbb{R}}^{3n}$. Graph G has m edges, numbered 1 to m. Edge $(i,j)$ corresponds to edge k, with edge vector $e_k=r_{ij}=r_i-r_j$ and bearing vector $g_k={\displaystyle \frac{e_k}{\parallel e_k\parallel }}$. Define $e={\left[\begin{array}{c}{e}_1^T,\dots ,e_m^T\end{array}\right]}^T$, $g={\left[\begin{array}{c}{g}_1^T,\dots ,g_m^T\end{array}\right]}^T$. The incidence matrix gives $e=H\otimes I_{d}r=\overline{H}r$. The desired bearing is $g_k^{*}=g_{ij}^{*}$, with $g^{*}={\left[\begin{array}{c}{g}_1^{*T},\dots ,g_m^{*T}\end{array}\right]}^T$. The topology is infinitesimally bearing-rigid, ensuring unique formation from relative bearing measurements, scaling, and translation coefficients.

2.4. Formation Control Objective

Design a bearing-only control strategy to achieve desired bearing-rigid formation from any initial state:

$$\underset{t\to \infty }{lim}(g_{ij}\left(t\right)-g_{ij}^{*}\left(t\right))=0,\underset{t\to \infty }{lim}({\theta }_i\left(t\right)-{\theta }_i^{*})=0,$$

where $r_i\left(0\right)\in {\mathbb{R}}^3$, ${\theta }_i^{*}$ is a positive constant.

3. Bearing Formation Control Scheme

3.1. Disturbance Observer

External disturbances, such as wind, can impair system performance. A disturbance observer is proposed to estimate and compensate for these disturbances in real time, ensuring robust control [27,28,29,30]. The observer includes two decoupled components for linear and angular velocity channels:

$$\begin{array}{cc}& \left\{\begin{array}{c}{\dot{z}}_{vi}=-L_{vi}{R}_i(z_{vi}+L_{vi}{R}_i)-L_{vi}{R}_i{v}_i,\hfill \\ {\widehat{d}}_{vi}=z_{vi}+L_{vi}{R}_i,\hfill \end{array}\right.\hfill \end{array}$$

(7)

$$\begin{array}{cc}& \left\{\begin{array}{c}{\dot{z}}_{\omega i}=-l_{\omega i}(z_{\omega i}+l_{\omega i}{\theta }_i)-l_{\omega i}{\omega }_i,\hfill \\ {\widehat{d}}_{\omega i}=z_{\omega i}+l_{\omega i}{\theta }_i,\hfill \end{array}\right.\hfill \end{array}$$

(8)

where $z_{vi}$ and $z_{\omega i}$ are auxiliary variables, $L_{vi}$ and $l_{\omega i}$ are observer gain vectors with $L_{vi}\ge {\displaystyle \frac{1}{4}}{R_i}^T$, $l_{\omega i}\ge {\displaystyle \frac{1}{4}}$, and ${\widehat{d}}_{vi}$, ${\widehat{d}}_{\omega i}$ are estimated disturbances for linear and angular velocity channels, respectively.

Assumption 1. 

Disturbances $d_{vi}$ and $d_{\omega i}$ in system (5) are bounded, with ${lim}_{t\to \infty }{\dot{d}}_{vi}\left(t\right)=0$ and ${lim}_{t\to \infty }{\dot{d}}_{\omega i}\left(t\right)=0$.

Remark 1. 

Assumption 1 is in line with typical low-altitude wind conditions and is a standard hypothesis in the design of disturbance observers. In many UAV applications, the main wind components (mean wind, slowly varying gusts and shear) evolve on a time scale slower than the outer-loop formation dynamics and can be regarded as bounded signals with slowly changing derivatives. This boundedness requirement also facilitates stable numerical implementation of the observer. In addition, the inertia and attitude control bandwidth of fixed-wing UAVs attenuate very fast disturbance variations, so Assumption 1 is appropriate for the multi-UAV formation scenario considered in this work.

3.2. Bearing-Rigid Formation Controller

Safety is also essential in formation control [31]. The potential energy function between UAV i and its neighboring UAV j is defined as follows:

$$P_{ij}=\parallel g_{ij}-g_{ij}^{*}{\parallel }^2\parallel r_{ij}\parallel .$$

Define

$${\beta }_{ij}={\displaystyle \frac{\partial P_{ij}}{\partial r_{ij}}}=2(g_{ij}-g_{ij}^{*}){\displaystyle \frac{I-g_{ij}^{\top }{g}_{ij}}{\parallel r_{ij}\parallel }}\parallel r_{ij}\parallel +{(g_{ij}-g_{ij}^{*})}^{\top }(g_{ij}-g_{ij}^{*}){\displaystyle \frac{r_{ij}}{\parallel r_{ij}\parallel }}=2(g_{ij}-g_{ij}^{*}).$$

(9)

The total potential field for UAV i is

$$P_i=\sum _{j\in N_i}{P}_{ij}\left(r_{ij}\right).$$

A formation control law based on bearing measurements is designed to ensure that the UAV system, starting from any initial position $r_i\left(0\right)\in {\mathbb{R}}^3$, $i=1,2,\dots ,n$, forms and maintains the desired globally stable formation. The potential function $P_{ij}$ is smooth and equals zero when the relative bearing between UAVs i and j reaches the desired value $g_{ij}^{*}$.

3.3. Conditional Disturbance Utilization

A criterion for CDU is designed, considering desired angular error and external disturbances:

$$\begin{array}{cc}\hfill {\eta }_i& ={\displaystyle \frac{R_i^T{\nabla }_{r_i}{P}_i{\widehat{d}}_{vi}^T}{|R_i^T{\nabla }_{r_i}{P}_i{\widehat{d}}_{vi}^T|+{\epsilon }_i}},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\rho }_i& ={\displaystyle \frac{({\theta }_i-{\theta }_i^{*}){\widehat{d}}_{\omega i}^T}{|({\theta }_i-{\theta }_i^{*}){\widehat{d}}_{\omega i}^T|+{\varsigma }_i}},\hfill \end{array}$$

(11)

where $R_i$ is the rotation matrix, ${\theta }_i^{*}$ is the desired angle, ${\epsilon }_i$ and ${\varsigma }_i$ are small positive constants, and ${\nabla }_{r_i}{P}_i={\left[\begin{array}{c}{\nabla }_{p_i}{P}_{x_i},{\nabla }_{p_i}{P}_{y_i},{\nabla }_{p_i}{P}_{z_i}\end{array}\right]}^T$.

Equations (10) and (11) define the CDU strategy, assessing whether disturbances are beneficial or detrimental. If ${\eta }_i\le 0$ and ${\rho }_i\le 0$, the disturbance is beneficial; otherwise, if ${\eta }_i>0$ or ${\rho }_i>0$, it is detrimental [32].

Remark 2. 

Whether an external signal should be exploited or rejected depends on its direction relative to the formation error. If the signal acts against the error (i.e., ${\eta }_i\le 0$ in (10) or ${\rho }_i\le 0$ in (11)), it plays a damping-like role: the Lyapunov function derivative $\dot{W}$ decreases faster, convergence is accelerated, and control effort can be reduced. On the other hand, when it points in the same direction as the error (i.e., ${\eta }_i>0$ or ${\rho }_i>0$), it enlarges formation deviation and needs to be actively compensated using the observer-based estimates in (7) and (8). In this way, the CDU mechanism keeps and utilizes favorable components while suppressing those that undermine formation stability.

3.4. Control Law Design

To achieve the control objectives, the bearing formation control law is designed as follows:

$$\begin{array}{cc}\hfill v_i& =-k_i{R}_i^T\left({\nabla }_{r_i}{P}_i\right)-{\eta }_i{\widehat{d}}_{vi},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill w_i& =-l_i({\theta }_i-{\theta }_i^{*})-{\rho }_i{\widehat{d}}_{\omega i},\hfill \end{array}$$

(13)

where $k_i\ge 1$, $l_i\ge 1$, ${\eta }_i$ is given by (10), and ${\rho }_i$ is given by (11).

Remark 3. 

The proposed controller employs a disturbance characterization indicator built upon the estimates provided by the disturbance observer, so that disturbance components that hinder formation regulation are compensated, whereas those aligned with the control objective are actively utilized. At the same time, a bearing-dependent potential function is included to enforce collision avoidance among neighboring UAVs, thereby improving the overall safety of the formation during disturbance utilization.

4. Stability Analysis

4.1. Stability of the Disturbance Observer

Theorem 1. 

Consider the disturbance observer (7)–(8) and define the estimation errors

$$e_{vi}=d_{vi}-{\widehat{d}}_{vi},e_{\omega i}=d_{\omega i}-{\widehat{d}}_{\omega i}.$$

(14)

Under Assumption 1 and for $L_{vi}\ge {\displaystyle \frac{1}{4}}{R}_i^T$, $l_{\omega i}\ge {\displaystyle \frac{1}{4}}$, the disturbance estimation error system is asymptotically stable, i.e.,

$$\underset{t\to \infty }{lim}{e}_{vi}\left(t\right)=0,\underset{t\to \infty }{lim}{e}_{\omega i}\left(t\right)=0.$$

Proof. 

From the disturbance observer (7)–(8), the estimation error dynamics can be written as

$$\begin{array}{cc}\hfill {\dot{e}}_{vi}& ={\dot{d}}_{vi}-{\dot{\widehat{d}}}_{vi}={\dot{d}}_{vi}-{\dot{z}}_{vi}={\dot{d}}_{vi}+L_{vi}{R}_i\left(z_{vi}+L_{vi}{R}_i\right)+L_{vi}{R}_i{v}_i-L_{vi}{R}_i(v_i+d_{vi})\hfill \\ \hfill & ={\dot{d}}_{vi}+L_{vi}{R}_i{\widehat{d}}_{vi}-L_{vi}{R}_i{d}_{vi}={\dot{d}}_{vi}-L_{vi}{R}_i{e}_{vi},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\dot{e}}_{\omega i}& ={\dot{d}}_{\omega i}-{\dot{\widehat{d}}}_{\omega i}={\dot{d}}_{\omega i}-{\dot{z}}_{\omega i}={\dot{d}}_{\omega i}+l_{\omega i}\left(z_{\omega i}+l_{\omega i}{\theta }_i\right)+l_{\omega i}{\omega }_i-l_{\omega i}({\omega }_i+d_{\omega i})\hfill \\ \hfill & ={\dot{d}}_{\omega i}+l_{\omega i}{\widehat{d}}_{\omega i}-l_{\omega i}{d}_{\omega i}={\dot{d}}_{\omega i}-l_{\omega i}{e}_{\omega i}.\hfill \end{array}$$

(16)

Choose the Lyapunov functions

$$V_{vi}={\displaystyle \frac{1}{2}}{e}_{vi}^2,V_{\omega i}={\displaystyle \frac{1}{2}}{e}_{\omega i}^2.$$

(17)

Taking their time derivatives along (15)–(16) yields

$$\begin{array}{cc}\hfill {\dot{V}}_{vi}& =e_{vi}{\dot{e}}_{vi}=e_{vi}\left({\dot{d}}_{vi}-L_{vi}{R}_i{e}_{vi}\right)=-L_{vi}{R}_i{e}_{vi}^2+e_{vi}{\dot{d}}_{vi},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\dot{V}}_{\omega i}& =e_{\omega i}{\dot{e}}_{\omega i}=e_{\omega i}\left({\dot{d}}_{\omega i}-l_{\omega i}{e}_{\omega i}\right)=-l_{\omega i}{e}_{\omega i}^2+e_{\omega i}{\dot{d}}_{\omega i}.\hfill \end{array}$$

(19)

Under Assumption 1, $d_{vi}$ and $d_{\omega i}$ are bounded and satisfy ${lim}_{t\to \infty }{\dot{d}}_{vi}\left(t\right)=0$ and ${lim}_{t\to \infty }{\dot{d}}_{\omega i}\left(t\right)=0$. Since $L_{vi}\ge {\displaystyle \frac{1}{4}}{R}_i^T$, $l_{\omega i}\ge {\displaystyle \frac{1}{4}}$ and $R_i$ is a rotation matrix, there exist positive constants ${\underline{L}}_i,{\underline{l}}_i>0$ such that

$$\begin{array}{cc}\hfill {\dot{V}}_{vi}& \le -{\underline{L}}_i{e}_{vi}^2,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\dot{V}}_{\omega i}& \le -{\underline{l}}_i{e}_{\omega i}^2.\hfill \end{array}$$

(21)

Therefore, $e_{vi}$ and $e_{\omega i}$ converge to zero asymptotically. □

4.2. Formation System Stability Analysis

Theorem 2. 

Consider the closed-loop multi-UAV system (5) under the bearing-based control laws (12)–(13) with the CDU strategy (10)–(11). Under Assumption 1, for sufficiently large gains $k_i\ge 1$, $l_i\ge 1$, $L_{vi}\ge {\displaystyle \frac{1}{4}}{R}_i^T$, and $l_{\omega i}\ge {\displaystyle \frac{1}{4}}$, the desired bearing-rigid formation is asymptotically achieved, i.e.,

$$\underset{t\to \infty }{lim}\left(g_{ij}\left(t\right)-g_{ij}^{*}\left(t\right)\right)=0,\underset{t\to \infty }{lim}\left({\theta }_i\left(t\right)-{\theta }_i^{*}\right)=0.$$

Proof. 

The stability of the translational formation subsystem is analyzed by integrating the control laws with the potential energy functions. Define the Lyapunov function

$$W_i=\sum _{i=1}^n{P}_i+\sum _{i=1}^n{e}_{vi}^2.$$

(22)

For undirected communication topology, ${\displaystyle \frac{\partial P_{ij}}{\partial r_{ij}}}={\displaystyle \frac{\partial P_{ij}}{\partial r_i}}=-{\displaystyle \frac{\partial P_{ij}}{\partial r_j}}$, $j\in N_i$, yielding

$${\displaystyle \frac{d}{dt}}\sum _{i=1}^n{P}_i=\sum _{i=1}^n\sum _{j\in N_i}{\dot{P}}_{ij}^T{\nabla }_{r_{ij}}{P}_{ij}=2\sum _{i=1}^n{\dot{P}}_i^T{\nabla }_{r_i}{P}_i.$$

The derivative of $W_i$ is

$${\dot{W}}_i=2\sum _{i=1}^n{\dot{P}}_i^T{\nabla }_{r_i}{P}_i+2\sum _{i=1}^n{e}_{vi}^T{\dot{e}}_{vi}.$$

(23)

Considering the CDU, two cases are analyzed:

Case 1: When ${\eta }_i>0$,

$$\begin{array}{cc}\hfill {\dot{W}}_i& =2\sum _{i=1}^n{\left({\nabla }_{r_i}{P}_i\right)}^T{R}_i\left[-k_i{R}_i^T\left({\nabla }_{r_i}{P}_i\right)-\left(1-{\displaystyle \frac{{\epsilon }_i}{|R_i^T{\nabla }_{r_i}{P}_i{\widehat{d}}_{vi}|}}\right){\widehat{d}}_{vi}+d_{vi}\right]-2\sum _{i=1}^n{L}_{vi}{R}_i{e}_{vi}^2\hfill \\ \hfill & \le -2k_i\sum _{i=1}^n{\left|{\nabla }_{r_i}{P}_i\right|}^2+2\sum _{i=1}^n{R}_i^T{\nabla }_{r_i}{P}_i(d_{vi}-{\widehat{d}}_{vi})-2\sum _{i=1}^n{L}_{vi}{R}_i{e}_{vi}^2\hfill \\ \hfill & \le -2k_i\sum _{i=1}^n|{\nabla }_{r_i}{P}_i{|}^2+2{∥\sum _{i=1}^n{\nabla }_{r_i}{P}_i∥}^2+{\displaystyle \frac{1}{2}}\sum _{i=1}^n{\parallel R_i{e}_{vi}\parallel }^2-2\sum _{i=1}^n{L}_{vi}{R}_i{e}_{vi}^2.\hfill \end{array}$$

(24)

Case 2: When ${\eta }_i\le 0$,

$$\begin{array}{cc}\hfill {\dot{W}}_i& =2\sum _{i=1}^n{\left({\nabla }_{r_i}{P}_i\right)}^T{R}_i\left[-k_i{R}_i^T\left({\nabla }_{r_i}{P}_i\right)-{\eta }_i{\widehat{d}}_{vi}+d_{vi}\right]-2\sum _{i=1}^n{L}_{vi}{R}_i{e}_{vi}^2\hfill \\ \hfill & \le -2k_i\sum _{i=1}^n{\left|{\nabla }_{r_i}{P}_i\right|}^2+2\sum _{i=1}^n{R}_i^T{\nabla }_{r_i}{P}_i(d_{vi}-{\widehat{d}}_{vi})-2\sum _{i=1}^n{L}_{vi}{R}_i{e}_{vi}^2\hfill \\ \hfill & \le -2k_i\sum _{i=1}^n|{\nabla }_{r_i}{P}_i{|}^2+2{∥\sum _{i=1}^n{\nabla }_{r_i}{P}_i∥}^2+{\displaystyle \frac{1}{2}}\sum _{i=1}^n{\parallel R_i{e}_{vi}\parallel }^2-2\sum _{i=1}^n{L}_{vi}{R}_i{e}_{vi}^2.\hfill \end{array}$$

(25)

When $L_{vi}\ge {\displaystyle \frac{1}{4}}$, both cases yield

$$\begin{array}{cc}\hfill {\dot{W}}_i& \le -{\overline{k}}_i{∥\sum _{i=1}^n{\nabla }_{r_i}{P}_i∥}^2-{\overline{\lambda }}_i\sum _{i=1}^n{\parallel R_i{e}_{vi}\parallel }^2\le 0,\hfill \end{array}$$

(26)

where ${\overline{k}}_i>0$, ${\overline{\lambda }}_i>0$. Since ${\dot{W}}_i\le 0$ and $W_i$ is bounded below by zero, ${lim}_{t\to \infty }{W}_i\left(t\right)$ exists. Define $T\left(g_{ij}\left(t\right)\right)=-2{\overline{k}}_i{\sum }_{i=1}^n{\parallel {\nabla }_{r_i}{P}_i\parallel }^2$. Integrating gives

$$\underset{t\to \infty }{lim}{\int }_0^{t}T\left(g_{ij}\left(\tau \right)\right)d\tau \le W_i\left(r_{ij}\left(0\right)\right)-W_i\left(r_{ij}(\infty )\right).$$

(27)

Since ${\int }_0^{t}T\left(g_{ij}\left(\tau \right)\right)d\tau$ is bounded, $T\left(g_{ij}\left(t\right)\right)$ is continuous for all $g_{ij}\left(t\right)\in W$, $t\in [0,+\infty )$. By uniform continuity and Barbalat’s lemma, ${lim}_{t\to \infty }T\left(g_{ij}\left(t\right)\right)=0$, implying ${\sum }_{i=1}^n{\sum }_{j\in N_i}(g_{ij}-g_{ij}^{*})=0$. In matrix form,

$$r^T{\overline{H}}^T(g-g^{*})=e^T(g-g^{*})={\displaystyle \frac{1}{2}}\sum _{k=1}^m\parallel e_k\parallel \parallel g_k-g_k^{*}{\parallel }^2=0.$$

For the rotational angle system, choose the Lyapunov function

$$W_{\theta i}=\sum _{i=1}^n{({\theta }_i-{\theta }_i^{*})}^2+\sum _{i=1}^n{e}_{\omega i}^2.$$

(28)

Its derivative is

$${\dot{W}}_{\theta i}=2\sum _{i=1}^n({\theta }_i-{\theta }_i^{*}){\dot{\theta }}_i+2\sum _{i=1}^n{e}_{\omega i}^T{\dot{e}}_{\omega i}.$$

(29)

Considering (13), two cases are analyzed:

Case 1: When ${\rho }_i>0$,

$$\begin{array}{cc}\hfill {\dot{W}}_{\theta i}& =-2\sum _{i=1}^n{l}_i{({\theta }_i-{\theta }_i^{*})}^2+2\sum _{i=1}^n({\theta }_i-{\theta }_i^{*})e_{\omega i}-2\sum _{i=1}^n{l}_{\omega i}{e}_{\omega i}^2\hfill \\ \hfill & \le -2\sum _{i=1}^n{l}_i{({\theta }_i-{\theta }_i^{*})}^2+2\sum _{i=1}^n\parallel {\theta }_i-{\theta }_i^{*}{\parallel }^2+{\displaystyle \frac{1}{2}}\sum _{i=1}^n{\parallel e_{\omega i}\parallel }^2-2\sum _{i=1}^n{l}_{\omega i}{e}_{\omega i}^2.\hfill \end{array}$$

(30)

Case 2: When ${\rho }_i\le 0$,

$$\begin{array}{cc}\hfill {\dot{W}}_{\theta i}& =-2\sum _{i=1}^n{l}_i{({\theta }_i-{\theta }_i^{*})}^2-2\sum _{i=1}^n({\theta }_i-{\theta }_i^{*}){\rho }_i{\widehat{d}}_{\omega i}+2\sum _{i=1}^n({\theta }_i-{\theta }_i^{*})e_{\omega i}-2\sum _{i=1}^n{l}_{\omega i}{e}_{\omega i}^2\hfill \\ \hfill & \le -2\sum _{i=1}^n{l}_i{({\theta }_i-{\theta }_i^{*})}^2+2\sum _{i=1}^n\parallel {\theta }_i-{\theta }_i^{*}{\parallel }^2+{\displaystyle \frac{1}{2}}\sum _{i=1}^n{\parallel e_{\omega i}\parallel }^2-2\sum _{i=1}^n{l}_{\omega i}{e}_{\omega i}^2.\hfill \end{array}$$

(31)

When $l_{\omega i}\ge {\displaystyle \frac{1}{4}}$, both cases yield

$${\dot{W}}_{\theta i}\le -{\overline{\tau }}_i\sum _{i=1}^n\parallel {\theta }_i-{\theta }_i^{*}{\parallel }^2-{\overline{\gamma }}_i\sum _{i=1}^n{\parallel e_{\omega i}\parallel }^2\le 0,$$

(32)

where ${\overline{\tau }}_i>0$, ${\overline{\gamma }}_i>0$. Thus, ${\theta }_i\left(t\right)$ converges to ${\theta }_i^{*}$, i.e., ${lim}_{t\to \infty }({\theta }_i\left(t\right)-{\theta }_i^{*})=0$.

By uniform continuity, ${lim}_{t\to \infty }(g_{ij}\left(t\right)-g_{ij}^{*}\left(t\right))=0$ and ${lim}_{t\to \infty }({\theta }_i\left(t\right)-{\theta }_i^{*})=0$, ensuring all angle errors converge to zero. This confirms the global asymptotic stability of the bearing-rigid formation, achieving the desired formation shape. □

5. Simulation

5.1. Wind Disturbance Simulation

Wind disturbances are modeled as the sum of mean wind, gusts, and wind shear, as defined in the wind field model:

$$v_w(h,q)=v_a\left(h\right)+v_b\left(q\right)+v_c\left(h\right),$$

(33)

The components are given by

$$\begin{array}{cc}\hfill v_a\left(h\right)& =v_{6.096}{\displaystyle \frac{ln(h/z_0)}{ln(6.096/z_0)}},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill v_b\left(q\right)& =\left\{\begin{array}{cc}0,\hfill & q<0,\hfill \\ {\displaystyle \frac{v_{bmax}}{2}}\left(1-cos{\displaystyle \frac{\pi q}{d}}\right),\hfill & 0\le q\le d,\hfill \\ v_{bmax},\hfill & q>d,\hfill \end{array}\right.\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill v_c\left(h\right)& ={\displaystyle \frac{v_f}{k}}ln\left({\displaystyle \frac{h}{H_c}}\right),\hfill \end{array}$$

(36)

where h is the UAV altitude, $z_0=0.1$ m is the surface roughness length, $v_{6.096}=5$ m/s is the mean wind speed at 6.096 m; q is the distance to the gust center, $d=10$ m is the gust scale, $v_{bmax}=2$ m/s is the peak gust speed; $H_c=0.01$ m is the zero-wind height, $k=0.4$ is the Kármán constant, and $v_f=0.5$ m/s is the friction velocity. For simulation, the wind disturbance is incorporated into the UAV dynamics as $d_{v_i}\left(t\right)=v_{wi}\left(t\right)$, with $d_{{\omega }_i}\left(t\right)=0.1cos{\theta }_i$.

MATLAB (2023b) simulations validate the proposed algorithm’s effectiveness. The UAV system dynamics are modeled as

$$\left\{\begin{array}{c}{\dot{r}}_i=R_i(v_i+1.2cos{v}_i),\hfill \\ {\dot{\theta }}_i={\omega }_i+0.1cos{\theta }_i,\hfill \end{array}\right.$$

with the initial positions, velocity constraints, and communication radius configured based on real UAV performance and mission scenarios.

5.2. Case 1

Six UAVs have initial coordinates: $r_1=[1,0.2,0]$, $r_2=[1,0.4,0]$, $r_3=[1,0.6,0]$, $r_4=[1,0.8,0]$, $r_5=[1,1,0]$, $r_6=[1,1.2,0]$.

In 3D space, six UAVs start from initial positions and form a regular triangular formation after 6 s of attitude adjustment, as illustrated in Figure 2. The simulation results in Figure 3 and

Table 1. Comparison of CDU and DOBC in convergence speed and formation distance.

Metric	CDU	DOBC	Improvement
Convergence time $T_{\mathrm{settle}}$ (s)	3.8	6.0	36.7% faster
Formation distance (m)	19.8	17.1	15.8% farther

show that the CDU method achieves zero tracking error within 4 s with high formation precision, demonstrating its superiority in rapid convergence and accuracy. In contrast, the traditional Disturbance Observer-Based Control (DOBC) method converges more slowly, highlighting CDU’s advantage; this is also reflected in the bearing error convergence curves in Figure 4 and Figure 5. Moreover, under favorable wind conditions, the CDU strategy enables the formation to travel further along the desired direction within the same time horizon, as seen from the trajectories in Figure 3.

5.3. Case 2

Six UAVs have initial coordinates: $r_1=[-1,-1,0]$, $r_2=[-2,-2,0]$, $r_3=[-3,-3,0]$, $r_4=[-4,-4,0]$, $r_5=[-5,-5,0]$, $r_6=[-6,-6,0]$.

In 3D space, six UAVs take off from initial positions and form a regular hexagonal formation after 8 s of adjustment, as illustrated in Figure 6. Simulation results in Figure 7 and

Table 2. Comparison of CDU and DOBC in convergence speed and formation distance.

Metric	CDU	DOBC	Improvement
Convergence time $T_{\mathrm{settle}}$ (s)	6.9	8.0	13.8% faster
Formation distance (m)	12.8	10.6	26.4% farther

indicate that the CDU algorithm reduces tracking error to zero within 6 s while maintaining high formation accuracy, confirming its effectiveness in fast convergence and precise control. Conversely, the DOBC method exhibits slower convergence, underscoring CDU’s superior performance; this trend is also reflected in the bearing error convergence shown in Figure 8 and Figure 9. Moreover, under the same simulation duration, the CDU-based formation moves farther than the DOBC-based one while still preserving the prescribed hexagonal geometry.

6. Conclusions

In this paper, a method based on CDU is proposed to solve the formation control problem of UAVs in complex wind disturbance environments. Differing from the traditional disturbance compensation methods, the CDU framework actively identifies and exploits the beneficial components in the external disturbance to improve the performance and stability of the system. By estimating the disturbance dynamics with a real-time disturbance observer, the CDU mechanism is able to dynamically decide whether to exploit these disturbances according to the control objectives. Combined with the angle-rigid formation topology, the proposed method ensures global stability and fast convergence, and the angle-based potential energy function is used to reduce the collision risk caused by disturbance. Simulation and experimental results verify the significant advantages of the proposed method in improving UAV formation stability, response speed, and accuracy. Compared with the traditional methods, the CDU framework can deal with the disturbance problem in the dynamic environment more effectively, which provides a new idea for UAV formation control. In future work, we will explore proportional–integral disturbance observers and barrier Lyapunov-based constraint handling to further enhance the robustness and safety of the proposed CDU formation control framework under wind disturbances.