Event-Triggered Impulsive Formation Control for Cooperative Obstacle Avoidance of UAV Swarms in Tunnel Environments

Abstract

UAV formation navigation in complex environments such as narrow tunnels faces multiple challenges, including obstacle avoidance, formation maintenance, and communication constraints. This paper proposes a cooperative obstacle avoidance strategy for UAV formation based on adaptive event-triggered impulse control, achieving efficient navigation under limited resources. The strategy comprises four key modules: an adaptive event-triggering mechanism, optical flow-based obstacle detection, leader–follower formation structure, and dynamic communication topology management. The adaptive event-triggering mechanism dynamically adjusts triggering thresholds, ensuring control accuracy while reducing control update frequency; the enhanced optical flow perception model improves obstacle recognition ability through a sector-based approach, incorporating tunnel-specific avoidance strategies; the leader–follower formation structure employs dynamic weight allocation to balance obstacle avoidance needs with formation maintenance; and communication topology optimization enhances system robustness under limited communication conditions. Simulation experiments were conducted in an arc-shaped tunnel environment with 15 randomly distributed obstacles, and the results demonstrate that the proposed method significantly improves collision rates, formation errors, and communication overhead compared to traditional methods. Lyapunov stability analysis proves the convergence of the proposed control strategy. This research provides new theoretical and practical references for multi-UAV cooperative control in complex narrow environments.

1. Introduction

With the rapid advancement of unmanned aerial vehicle (UAV) technology, the cooperative formation control of multi-UAV systems has demonstrated extensive application potential in military reconnaissance, disaster relief, and environmental monitoring [1,2]. The primary advantage of UAV formations lies in their ability to enhance mission execution efficiency, improve system robustness, and accomplish complex tasks beyond the capability of individual UAVs through coordinated operations [3]. However, when UAV formations are deployed in confined and complex environments such as tunnels or canyons, their navigation control faces significant challenges due to spatial constraints, obstacle distributions, and communication limitations [4,5].

Traditional multi-UAV formation control methodologies primarily encompass behavior-based approaches, virtual structure methods, and leader–follower strategies [6]. While these methods demonstrate satisfactory performance in open environments, they often struggle to simultaneously maintain both obstacle avoidance capability and formation integrity in confined spaces [7]. This challenge becomes particularly pronounced in tunnel environments with multiple obstacles, where UAV formations must not only coordinate collision avoidance maneuvers but also dynamically adapt their configuration according to the tunnel’s complex geometry while preserving overall formation stability [8]. Furthermore, the constrained nature of such environments exacerbates communication limitations, as wireless signals become susceptible to occlusion, resulting in frequent communication delays and packet loss—factors that significantly compound the difficulty of formation control [9,10].

To address these challenges, researchers have developed various improved approaches. One research direction focuses on enhancing environmental perception capabilities through optimized sensing methodologies [11]. For instance, optical flow techniques have been widely adopted for UAV visual navigation and obstacle avoidance due to their computational efficiency and strong environmental adaptability [12,13]. Cho et al. proposed a multi-sensor fusion-based optical flow perception method that significantly improves obstacle detection performance in confined spaces [14]. Another research strand concentrates on control strategy enhancements, where event-triggered control has garnered substantial attention for its resource-efficient operation in constrained conditions [15,16]. Unlike traditional periodic control schemes, event-triggered control executes updates only when specific system state conditions are met, thereby substantially reducing computational and communication resource consumption [17]. Recent advancements in this field have expanded its applications and theoretical foundations. Zhang et al. [18] presented an accumulated-state-error-based event-triggered sampling scheme with applications to $\mathrm{H}\infty$ control of sampled-data systems, offering improved robustness against external disturbances. Kazemy et al. [19] developed novel security mechanisms for event-triggered output feedback synchronization of master–slave neural networks under deception attacks, addressing critical cybersecurity concerns in networked systems. Additionally, Ge et al. [20] provided a comprehensive survey on distributed coordination control of multi-agent systems under intermittent sampling and communication, systematically categorizing the latest theoretical frameworks and implementation strategies. Dimarogonas et al. successfully implemented this triggering mechanism in multi-agent systems, demonstrating its dual advantages of minimized communication overhead and guaranteed system stability [21].

Regarding communication topology optimization, research shows that dynamically adjusted communication networks can effectively address communication-constrained challenges [22,23]. Zhang et al. proposed an adaptive communication topology method that automatically adjusts communication structures based on environmental complexity and mission requirements, improving system robustness in complex environments [24]. Furthermore, the leader–follower structure has been proven to have good scalability and robustness in UAV formations [25,26]. However, existing leader–follower methods still have limitations when dealing with formation deformation and recovery problems in confined spaces [27].

Although existing research has made certain progress, studies that organically integrate adaptive event-triggered control, optical flow perception, and dynamic communication topology while specifically optimizing for complex tunnel environments remain insufficient [28]. Furthermore, achieving better balance between obstacle avoidance and formation maintenance, as well as designing specialized navigation strategies tailored to tunnel characteristics, remain critical unresolved issues [29,30].

Based on the foregoing analysis, this study proposes a cooperative obstacle avoidance strategy for UAV formations utilizing adaptive event-triggered impulsive control. The principal innovations encompass the following steps:

• Implementation of an adaptive event-triggering mechanism that dynamically modulates triggering thresholds to simultaneously reduce control update frequency and preserve control precision;
• Development of an enhanced optical flow perception framework incorporating sectoral partitioning methodology to augment obstacle identification capability, coupled with dedicated tunnel-specific avoidance protocols;
• Optimization of the leader–follower formation architecture through dynamic weight allocation schemes that equilibrate obstacle avoidance demands with formation preservation requirements;
• Establishment of a network connectivity-oriented communication topology management paradigm to enhance system robustness under communication-constrained scenarios.

While this paper presents significant advancements in UAV formation control for tunnel environments, we acknowledge several challenges in real-world deployment scenarios. The proposed system may face limitations in feature-poor environments where optical flow perception becomes less reliable, extremely narrow tunnel segments ($D/d_{\mathrm{spacing}}<1.0$) where formation flight becomes impractical, and conditions with severe electromagnetic interference affecting communication reliability. Practical applications where our methodology shows promise include mining inspection, urban infrastructure assessment (subway tunnels and sewer systems), subterranean search and rescue operations, and archaeological site exploration. Each domain presents unique environmental challenges that will require further refinement of the core algorithms. Future work will focus on addressing these implementation barriers through hardware validation and enhanced multi-modal sensing approaches, ensuring the theoretical advances presented here can be effectively translated to practical field operations.

2. Theoretical Framework and Methodology

2.1. UAV Kinematic Model

This work utilizes a 3D point-mass formulation to describe UAV kinematic properties, which effectively captures fundamental dynamic responses in complex operational environments. For an N-UAV formation system, each vehicle’s state is completely specified by its position and velocity vectors. All symbols and their physical meaning can be found in

Table A1. Comprehensive symbol table.

Symbol	Physical Meaning
${\mathbf{p}}_i\left(t\right)\in {\mathbf{R}}^3$	Position vector of the i-th UAV at time t
${\mathbf{v}}_i\left(t\right)\in {\mathbf{R}}^3$	Velocity vector of the i-th UAV at time t
${\mathbf{a}}_i\left(t\right)\in {\mathbf{R}}^3$	Acceleration control input of the i-th UAV at time t
${\mathbf{p}}_i^d\left(t\right)\in {\mathbf{R}}^3$	Desired position of the i-th UAV at time t
${\mathbf{v}}_i^d\left(t\right)\in {\mathbf{R}}^3$	Desired velocity of the i-th UAV at time t
${\mathbf{e}}_i\left(t\right)={\mathbf{p}}_i^d\left(t\right)-{\mathbf{p}}_i\left(t\right)$	Position error vector of the i-th UAV
$\parallel {\mathbf{e}}_i\left(t\right)\parallel$	Euclidean norm of position error for the i-th UAV
N	Total number of UAVs in the formation
$v_{\max }$	Maximum velocity constraint for UAVs
$a_{\max }$	Maximum acceleration constraint for UAVs
$∆t$	Sampling time interval
$r_{\mathrm{drone}}$	Effective radius of UAV (for collision detection)
$\mathbf{C}\left(t\right)={[C_x\left(t\right),C_y\left(t\right),C_z\left(t\right)]}^T$	Parametric equation of tunnel centerline, where $t\in [0,1]$
${\mathbf{P}}_{\mathrm{start}}$	Coordinates of tunnel starting point
${\mathbf{P}}_{\mathrm{end}}$	Coordinates of tunnel ending point
h	Maximum lateral bending height of tunnel
$R\left(t\right)$	Cross-sectional radius of tunnel at parameter t
$R_{\mathrm{base}}$	Baseline radius of tunnel
${\alpha }_r$	Radius variation coefficient, $0\le {\alpha }_r<1$
$\mathbf{T}\left(t\right)$	Tangent vector at parameter t
$\mathbf{U}\left(t\right)$	Upward direction vector at parameter t
$\mathbf{R}\left(t\right)$	Right direction vector at parameter t
$\theta \in [0,2\pi )$	Angular parameter around cross-section
$\mathbf{S}(t,\theta )$	Point on tunnel surface
${\mathcal{O}}_j$	The j-th obstacle
${\mathbf{c}}_j\in {\mathbb{R}}^3$	Center position of the j-th obstacle
$r_j$	Radius of the j-th obstacle
$\mathcal{O}=\{{\mathbf{c}}_1,{\mathbf{c}}_2,\dots ,{\mathbf{c}}_m\}$	Set of obstacle positions
$d_{\mathrm{safe}}$	Minimum safe distance between UAV and obstacle
$\delta$	Safety margin (for collision detection)
${\delta }_i\left(t\right)$	Time-varying triggering threshold for the i-th UAV
$t_{\mathrm{last},i}$	Last triggering time for the i-th UAV
$T_{\mathrm{cool}}$	Triggering cooldown period to prevent chattering
${\mathrm{\Gamma }}_i\left(t\right)$	Error trend evaluation function based on sliding window
W	Window size for error evaluation
${\tau }_s$	Sampling period for error evaluation
${\eta }_i\left(t\right)=\parallel {\mathbf{e}}_i\left(t\right)\parallel -{\overline{\mathbf{e}}}_i\left(t\right)$	Error trend indicator
${\delta }_{\min }$	Lower bound of triggering threshold
${\delta }_{\max }$	Upper bound of triggering threshold
${\alpha }_{\mathrm{inc}}$	Threshold growth coefficient
${\alpha }_{\mathrm{dec}}$	Threshold attenuation coefficient
$\beta$	Error weighting factor
$\xi \left(t\right)$	Global progress adjustment factor
$T_{\mathrm{total}}$	Total mission duration
${\delta }_i^{\mathrm{base}}$	Baseline triggering threshold
$c_i,c_g$	Threshold adjustment coefficients
$k_i^{\mathrm{impulse}}\left(t\right)$	Impulse gain for the i-th UAV
$k_{\mathrm{base}}$	Baseline gain coefficient
${\gamma }_i\left(t\right)$	Error-dependent adjustment factor
${\mu }_i\left(t\right)$	Environment-dependent adjustment factor
$c_e$	Error adjustment coefficient
$c_o$	Environment constraint adjustment coefficient
$d_i^{\mathrm{obs}}\left(t\right)$	Distance from i-th UAV to nearest obstacle
$d_{\mathrm{ref}}$	Reference distance
$u_i^{\mathrm{final}}\left(t\right)$	Final control input
$I_{\max }$	Upper limit for integral term
${\lambda }_i\left(t\right)$	Velocity-dependent integral decay factor
$c_v$	Velocity influence coefficient
$u_i^{\text{non-trigger}}\left(t\right)$	Control law in non-triggered state
$k_p,k_i,k_d,k_f,k_c$	Proportional, integral, derivative, optical flow, and consensus control gains
$F_i^{\mathrm{optical}}\left(t\right)$	Control force generated by optical flow sensing
$F_i^{\mathrm{consensus}}\left(t\right)$	Consensus control term based on communication topology
${\mathcal{N}}_i\left(t\right)$	Set of neighbors of UAV i in communication topology
$a_{ij}\left(t\right)$	Element of topological adjacency matrix
$w_1,w_2$	Weighting coefficients for position and velocity
$u_i^{\mathrm{terminal}}\left(t\right)$	Terminal precision control law
$k_i\left(t\right)$	Terminal position gain coefficient
$k_b\left(t\right)$	Terminal velocity gain coefficient
$k_{t0},k_{t1}$	Terminal position control parameters
$k_{b0},k_{b1}$	Terminal velocity control parameters
$\sigma$	Control bandwidth parameter
$\epsilon$	Infinitesimal positive number (to prevent division by zero)
$v_{\max ,\mathrm{term}}\left(t\right)$	Maximum allowable velocity during terminal phase
$c_d$	Proportionality coefficient
$d_{\mathrm{term}}$	Terminal control activation threshold
${\mathcal{S}}_i$	Perception domain of the i-th UAV
$r_{\mathrm{blind}}$	Blind zone radius
$r_{\mathrm{sense}}$	Maximum sensing radius
${\varphi }_{\mathrm{FOV}}$	Field-of-view angle
$n_s$	Number of perception sectors
${\mathrm{OF}}_i^k$	Optical flow vector in the k-th sector of the i-th UAV
${\mathcal{N}}_i^k$	Set of targets perceived in the k-th sector of the i-th UAV
${\alpha }_d$	Attenuation coefficient
${\gamma }_{\mathrm{occ}}$	Occlusion threshold
${\beta }_k$	Sector weight
${\mathrm{OF}}_i^{\mathrm{env}}$	Optical flow contribution induced by environmental factors
${\mathbf{R}}_i$	Repulsion vector
${\mathbf{T}}_i$	Rotational vector field
${\mathbf{C}}_i$	Collaborative avoidance vector
${\mathbf{W}}_i$	Tunnel wall repulsion vector
${\mathbf{G}}_i$	Predictive guidance vector
${\kappa }_r,{\kappa }_t,{\kappa }_w,{\kappa }_g,{\kappa }_c$	Weighting coefficients for each force field
$r_{\mathrm{safe}}$	Safety distance threshold
${\mathcal{N}}_i^{\mathrm{comm}}$	Set of neighbors with communication links to UAV i
$\eta$	Nonlinear coefficient for adjusting repulsion intensity’s distance-dependent variation
$∆s$	Look-ahead parameter
${\lambda }_i$	Local minimum detection indicator
${\lambda }_{\mathrm{threshold}}$	Local minimum threshold
${\mu }_{\mathrm{local}}$	Local minimum enhancement coefficient
${\mathbf{F}}_i^{\mathrm{avoid}}$	Sum of avoidance forces
${\delta }_j^d$	Relative position vector between UAV i and j in ideal formation
$\mu \left({\mathbf{v}}_i\left(t\right)\right)$	Velocity-dependent formation stretching factor
$v_{\mathrm{thresh}}$	Velocity threshold
$\kappa$	Stretching coefficient
$\lambda \left(t\right)$	Formation recovery transition factor
$d_{\mathrm{recover}}$	Recovery initiation distance
${\mathbf{F}}_i^{\mathrm{form}}$	Formation maintenance vector
${\rho }_i^{\mathrm{form}}$	Formation maintenance weight coefficient
${\rho }_i^{\mathrm{avoid}}$	Obstacle avoidance weight coefficient
${\lambda }_{\mathrm{obs}},{\lambda }_{\mathrm{wall}}$	Adjustment coefficients for obstacles and tunnel walls
$V\left(t\right)$	Lyapunov function
$\dot{V}\left(t\right)$	Derivative of Lyapunov function
$\lambda$	Lyapunov exponent
${\tau }_c$	System convergence time constant
${\tau }_{\mathrm{comm}}$	Communication delay
${\tau }_{\mathrm{base}}$	Baseline delay
$d_{ij}$	Normalized distance
$\xi$	Stochastic disturbance factor
$\alpha ,\beta$	Scaling coefficients

in Appendix A.

For the i-th UAV, the position vector is defined as follows:

$${\mathbf{p}}_i={\left[x_i,y_i,z_i\right]}^T\in {\mathbf{R}}^3$$

(1)

The velocity vector is defined as follows:

$${\mathbf{v}}_i={\left[\dot{x_i},\dot{y_i},\dot{z_i}\right]}^T\in {\mathbf{R}}^3$$

(2)

The discrete-time kinematic equations of the system are expressed as follows:

$${\mathbf{p}}_i(k+1)={\mathbf{p}}_i\left(k\right)+{\mathbf{v}}_i\left(k\right)∆t$$

(3)

$${\mathbf{v}}_i(k+1)={\mathbf{v}}_i\left(k\right)+{\mathbf{a}}_i\left(k\right)∆t$$

(4)

In Equations (3) and (4), $∆t$ denotes the sampling time interval, and ${\mathbf{a}}_i\left(k\right)$ represents the acceleration control input for the i-th UAV at time step k. Considering the physical limitations of UAVs, both velocity and acceleration must satisfy the following constraints:

$$∥{\mathbf{v}}_i\left(k\right)∥\le v_{\max }$$

(5)

$$∥{\mathbf{a}}_i\left(k\right)∥\le a_{\max }$$

(6)

where $v_{\max }$ and $a_{\max }$ denote the maximum velocity and maximum acceleration constraints of the UAV, respectively. For velocity saturation scenarios, a scalar projection method is employed to enforce speed limitations:

$${\mathbf{v}}_i(k+1)=\min \left({\displaystyle \frac{v_{\max }}{∥{\mathbf{v}}_i(k+1)∥}},1\right)·{\mathbf{v}}_i(k+1)$$

(7)

The control objective in this study is to enable all UAVs to form and maintain a predefined formation structure while avoiding environmental obstacles. To this end, we define the target position ${\mathbf{p}}_i^d$ for each UAV, and the collective formation control objective can be expressed as follows:

$$\underset{k\to \infty }{\lim }∥{\mathbf{p}}_i\left(k\right)-{\mathbf{p}}_i^d\left(k\right)∥=0,\forall i=1,2,\dots ,N$$

(8)

The target position ${\mathbf{p}}_i^d\left(k\right)$ may vary over time, particularly in leader–follower architectures where followers dynamically adjust their target positions based on the leader’s real-time location.

While the 3D point-mass formulation effectively captures fundamental motion characteristics for high-level control strategy design, it represents an idealized approximation that omits several physical phenomena relevant to real-world UAV operations in tunnel environments. Specifically, this model does not account for aerodynamic effects (including drag, downwash, and ground effect), detailed rotor dynamics, or multi-physical field coupling such as airflow patterns within confined tunnel spaces. These simplifications may impact the quantitative accuracy of simulation results, particularly in scenarios involving high-speed maneuvers, close-proximity formation flying, or navigation through sections with significant cross-sectional area changes where airflow acceleration occurs. However, qualitative behavioral patterns and relative performance comparisons between control strategies remain valid within the scope of this research.

2.2. Tunnel Environment Modeling and Obstacle Representation

To simulate real-world tunnel conditions, this study considers a three-dimensional curved narrow tunnel environment with variable curvature and cross-sectional radius, which poses significant challenges for UAV formation navigation. The tunnel model is constructed based on parametric curves and varying cross-sectional radius.

2.2.1. Curved Tunnel Model

The curved tunnel is defined by a parameterized central curve and circular cross-sections with varying dimensions along the curve. The central curve is expressed in parametric form as follows:

$$\mathbf{C}\left(t\right)={\left[C_x\left(t\right),C_y\left(t\right),C_z\left(t\right)\right]}^T,t\in [0,1]$$

(9)

where the parameter t represents the normalized distance along the curve. In this study, we employ a parabolic-shaped curve to simulate tunnel curvature:

$${\mathbf{P}}_{\mathrm{start}}+({\mathbf{P}}_{\mathrm{end}}-{\mathbf{P}}_{\mathrm{start}})t+{[0,-h·4t(1-t),0]}^T$$

(10)

Here, ${\mathbf{P}}_{\mathrm{start}}$ and ${\mathbf{P}}_{\mathrm{end}}$ denote the tunnel’s starting and ending points, respectively, h represents the maximum offset of the curve, and the term $4t(1-t)$ ensures the curve reaches its maximum offset at the midpoint.

The tunnel’s cross-sectional radius varies with parameter t to simulate irregularities in real-world tunnel environments.

$$R\left(t\right)=R_{\mathrm{base}}·[1-\alpha ·{\sin }^2\left(\pi t\right)]$$

(11)

Here, $R_{\mathrm{base}}$ denotes the baseline radius, and $\alpha$ represents the radius variation coefficient $(0\le \alpha <1)$. This design ensures the tunnel attains its minimum radius at the midpoint, with larger radii at both ends. At each parameter t, the tunnel cross-section is defined by a perfect circle. To construct the complete tunnel surface, a local coordinate system must be established at each cross-sectional position. At point $\mathbf{C}\left(t\right)$, the local coordinate system is defined by the following three-unit vectors:

• Tangent Vector: $\mathbf{T}\left(t\right)={\displaystyle \frac{\mathrm{d}\mathbf{C}\left(t\right)/\mathrm{d}t}{∥\mathrm{d}\mathbf{C}\left(t\right)/\mathrm{d}t∥}}$
• Upward Direction Vector: initially set as ${\mathbf{U}}_0={[0,0,1]}^T$
• Right Direction Vector: $\mathbf{R}\left(t\right)={\displaystyle \frac{{\mathbf{U}}_0\times \mathbf{T}\left(t\right)}{∥{\mathbf{U}}_0\times \mathbf{T}\left(t\right)∥}}$
• Adjusted Upward Direction Vector: $\mathrm{U}\left(t\right)=\mathrm{T}\left(t\right)\times \mathrm{R}\left(t\right)$

Based on this local coordinate system, points on the tunnel surface can be expressed as follows:

$$\mathbf{S}(t,\theta )=\mathbf{C}\left(t\right)+R\left(t\right)\left[\cos \right(\theta \left)\mathbf{R}\right(t)+\sin (\theta \left)\mathbf{U}\right(t\left)\right]$$

(12)

where $\theta \in [0,2\pi )$ is the angular parameter on the cross-sectional circumference.

2.2.2. Obstacle Representation

In this study, obstacles are represented using a spherical model, formally defined as follows:

$${\mathcal{O}}_j=\left\{\mathbf{x}\in {\mathbf{R}}^3:∥\mathbf{x}-{\mathbf{c}}_j∥\le r_j\right\}$$

(13)

where ${\mathbf{c}}_j$ denotes the center position of the j-th obstacle, and $r_j$ represents its radius.

To enhance simulation realism, obstacles are strategically distributed within the tunnel, particularly in curved sections and regions with varying radii, to increase navigation difficulty. The obstacle position set is defined as follows:

$$\mathbf{O}=\left\{{\mathbf{c}}_1,{\mathbf{c}}_2,\dots {\mathbf{c}}_m\right\}$$

(14)

The collision detection between UAVs and obstacles is based on a spherical collision model. Assuming each UAV can be simplified as a sphere with radius $r_{\mathrm{drone}}$, the collision condition between UAV i and obstacle j is given by the following equation:

$$∥{\mathbf{p}}_i-{\mathbf{c}}_j∥<r_j+r_{\mathrm{drone}}$$

(15)

This collision detection mechanism is employed for the real-time assessment of UAV safety status and activation of corresponding obstacle avoidance strategies.

To evaluate the distance relationship between the UAV and tunnel walls, we calculate the minimum distance from the UAV to the tunnel’s central curve:

$$d_{\mathrm{tunnel}}\left({\mathbf{p}}_i\right)={\min }_{t\in [0,1]}∥{\mathbf{p}}_i-\mathbf{C}\left(t\right)∥$$

(16)

Definition of UAV-to-Tunnel-Wall Distance:

$$d_{\mathrm{wall}}\left({\mathbf{p}}_i\right)=R\left(t^{*}\right)-d_{\mathrm{tunnel}}\left({\mathbf{p}}_i\right)$$

(17)

where $t^{*}$ is the parameter value that minimizes $∥{\mathbf{p}}_i-\mathbf{C}\left(t\right)∥$. When $d_{\mathrm{wall}}\le r_{\mathrm{drone}}$, the UAV is considered to have collided with the tunnel wall.

2.3. Adaptive Event-Triggered Impulse Control Strategy

Multi-UAV formation control in complex tunnel environments faces multiple challenges including limited communication bandwidth, environmental constraints, and real-time obstacle avoidance. This section proposes an adaptive event-triggered impulse control strategy, which comprises three core components, a variable-threshold event-triggering mechanism, nonlinear impulse control law, and anti-windup integral control, aiming to optimize the balance between control performance and resource utilization.

2.3.1. System Fundamental Assumptions and Theoretical Premises

Before presenting the adaptive event-triggered impulse control strategy, we first establish the fundamental assumptions of this research:

(a)

UAV Kinematic Assumptions: This study employs a point-mass model to describe UAV kinematic characteristics, if the control system can directly generate desired acceleration inputs. This simplification is widely adopted in high-level control strategy design because modern UAV low-level controllers typically track acceleration commands with high fidelity.

(b)

Control Input Constraints: Considering the physical limitations of actual UAVs, control inputs (accelerations) satisfy the constraint condition, $∥a_i\left(t\right)∥\le a_{\max }$. This constraint originates from the maximum thrust capabilities and aerodynamic limitations of UAV propulsion systems and is strictly maintained throughout the control process.

(c)

Information Acquisition Assumptions: We assume that each UAV can obtain its own position and velocity information through sensors, acquire neighboring UAVs’ state information through communication networks to a limited extent, and detect surrounding obstacles based on optical flow perception models.

2.3.2. Theoretical Foundation of Event-Triggering Mechanism

The variable-threshold event-triggering mechanism proposed in this research is based on the following theoretical principles:

Traditional periodic sampling control requires frequent state updates and control computations, which is inefficient in resource-constrained environments. In contrast, the theoretical advantage of event-triggered control lies in its ability to execute control updates only when significant changes occur in system states, thereby reducing unnecessary communication and computational loads.

For the i-th UAV, we define the position error vector ${\mathbf{e}}_i\left(t\right)={\mathbf{p}}_i^d\left(t\right)-{\mathbf{p}}_i\left(t\right)$, with Euclidean norm $∥{\mathbf{e}}_i\left(t\right)∥$. Based on Lyapunov stability theory, we design the triggering condition as follows:

$$∥{\mathbf{e}}_i\left(t\right)∥>{\delta }_i\left(t\right)\mathrm{and}t-t_{\mathrm{last},i}>{\tau }_{\mathrm{cool}}$$

(18)

where ${\delta }_i\left(t\right)$ is the time-varying triggering threshold, $t_{\mathrm{last},i}$ is the last triggering time, and ${\tau }_{\mathrm{cool}}$ is the triggering cooldown period.

The theoretical basis for the threshold dynamic adjustment strategy derives from adaptive control theory for closed-loop feedback systems, comprehensively considering system state change rate, environmental complexity, and error accumulation characteristics. Specifically, when system error increases, appropriately increasing the threshold can reduce control frequency, avoiding excessive system response to disturbances; when system error decreases, gradually reducing the threshold can improve control precision, ensuring system convergence performance.

The error trend evaluation function ${\mathrm{\Gamma }}_i\left(t\right)$ is based on sliding window analysis of error change rate:

$${\mathrm{\Gamma }}_i\left(t\right)=∥{\mathbf{e}}_i\left(t\right)∥-{\displaystyle \frac{1}{W}}\sum _{j=1}^W∥{\mathbf{e}}_i(t-j{\tau }_s)∥$$

(19)

where W represents the window size, and ${\tau }_s$ is the sampling period. ${\mathrm{\Gamma }}_i\left(t\right)>0$ indicates an increasing error trend, while ${\mathrm{\Gamma }}_i\left(t\right)<0$ indicates a decreasing error trend.

2.3.3. Theoretical Design of Nonlinear Impulse Control Law

The core concept of impulse control originates from Impulsive Stability Theory, which applies instantaneous large-magnitude control inputs to rapidly return the system to a stable state when system states deviate from desired values beyond preset thresholds.

Based on Lyapunov stability analysis, when the triggering condition is satisfied, the nonlinear impulse control law designed in this research is as follows:

$${\mathbf{a}}_i\left(t\right)=K_{\mathrm{imp}}({\mathbf{e}}_i\left(t\right),{\rho }_i\left(t\right))·{\displaystyle \frac{{\mathbf{e}}_i\left(t\right)}{∥{\mathbf{e}}_i\left(t\right)∥}}$$

(20)

where $K_{\mathrm{imp}}({\mathbf{e}}_i\left(t\right),{\rho }_i\left(t\right))$ is the nonlinear impulse gain, which correlates with error magnitude and environmental state. The dynamic adjustment mechanism for impulse gain is based on the following theoretical considerations:

(a)

Error-dependent component: When the error is large, stronger impulse control is applied to rapidly reduce error; when the error approaches the threshold, impulse intensity is appropriately reduced to avoid overshoot.

(b)

Environmental constraint adjustment: When approaching obstacles or tunnel boundaries, impulse intensity should be appropriately reduced to ensure safety; in open areas, impulse intensity can be increased to improve convergence speed.

The amplitude limitation of the impulse control law is designed based on the acceleration constraints of actual UAVs, ensuring that control inputs remain within physically realizable ranges:

$$∥{\mathbf{a}}_i\left(t\right)∥\le a_{\max }$$

(21)

2.3.4. Theoretical Foundation of Anti-Windup Integral Control

In non-triggered states, the system implements a comprehensive control law incorporating an anti-windup integral term. Traditional integral control easily leads to integral windup problems in the presence of external disturbances, resulting in system performance degradation.

The anti-windup integral component designed in this research is based on the following theory:

(a)

Integral term upper bound constraint: $∥I_i\left(t\right)∥\le I_{\max }$ prevents unlimited growth of the integral term leading to control saturation.

(b)

Velocity-dependent integral decay factor: ${\gamma }_i\left(t\right)={\displaystyle \frac{1}{1+{\alpha }_v{∥{\mathbf{v}}_i\left(t\right)∥}^2}}$ reduces integral effect when UAV velocity is high, avoiding overshoot; when velocity is low, it enhances integral effect, improving steady-state accuracy.

The final form of the integral control term is as follows:

$${\mathbf{a}}_{i,I}\left(t\right)=k_i·{\gamma }_i\left(t\right)·{\mathbf{I}}_i\left(t\right)$$

(22)

This anti-windup design ensures system robustness and precision under various operating conditions.

2.3.5. Theoretical Basis for Terminal Precision Control

The terminal precision control strategy is based on Terminal Sliding Mode Control theory, whose theoretical advantage lies in finite-time convergence characteristics, particularly suitable for precise control when UAVs approach target positions.

When a UAV approaches its target position $\left(∥{\mathbf{e}}_i\left(t\right)∥<{\delta }_{\mathrm{term}}\right)$, terminal precision control law is applied:

$${\mathbf{a}}_{i,\mathrm{term}}\left(t\right)=k_{\mathrm{term}}\left({\mathbf{e}}_i\left(t\right)\right)·{\mathbf{e}}_i\left(t\right)-k_d·{\mathbf{v}}_i\left(t\right)$$

(23)

where the gain coefficient $k_{\mathrm{term}}\left({\mathbf{e}}_i\left(t\right)\right)$ adaptively adjusts with distance:

$$k_{\mathrm{term}}\left({\mathbf{e}}_i\left(t\right)\right)=k_p+{\displaystyle \frac{k_{p^{\prime }}}{∥{\mathbf{e}}_i\left(t\right)∥+\epsilon }}$$

(24)

The maximum allowable velocity during the terminal phase is dynamically adjusted to

$$v_{\max ,\mathrm{term}}\left(t\right)={\alpha }_{\mathrm{term}}·∥{\mathbf{e}}_i\left(t\right)∥$$

(25)

This ensures smooth deceleration when approaching the target position, effectively preventing overshoot phenomena.

Through the above theoretical design, the adaptive event-triggered impulse control strategy proposed in this research can achieve efficient navigation control in complex tunnel environments, balancing control precision and resource utilization while ensuring system stability and safety.

2.3.6. Event-Triggered Mechanism Design

Traditional periodic sampling control paradigms require frequent state updates and control computations, exhibiting low efficiency in resource-constrained environments. Event-triggered control initiates control updates based on state variations, significantly reducing communication and computational loads. This study designs an adaptive event-triggered mechanism whose core innovation lies in dynamically adjusting triggering thresholds to accommodate environmental complexity and mission requirement variations.

For the i-th UAV, the position error vector is defined as follows:

$$e_i\left(t\right)=p_i^d\left(t\right)-p_i\left(t\right)$$

(26)

Its Euclidean norm is given by

$$∥e_i\left(t\right)∥=∥p_i^d\left(t\right)-p_i\left(t\right)∥$$

(27)

Based on this, the time-varying triggering threshold ${\delta }_i\left(t\right)$, the last triggering time $t_{\mathrm{last}}^i$, and the triggering cooldown period to prevent chattering caused by frequent triggering $T_{\mathrm{cool}}$, an adaptive triggering condition is proposed:

$$∥e_i\left(t\right)∥>{\delta }_i\left(t\right)\mathrm{and}t-t_{\mathrm{last}}^i>T_{\mathrm{cool}}$$

(28)

where ${\delta }_i\left(t\right)$ is the time-varying triggering threshold, $t_{\mathrm{last}}^i$ denotes the last triggering time, and $T_{\mathrm{cool}}$ represents the triggering cooldown period to prevent chattering caused by frequent triggering.

The threshold dynamic adjustment strategy is designed based on error dynamics characteristics and historical triggering frequency.

$${\dot{e}}_i\left(t\right)\approx {\displaystyle \frac{e_i\left(t\right)-e_i(t-∆t)}{∆t}}$$

(29)

Evaluate error trends using a sliding window:

$${\overline{e}}_i\left(t\right)={\displaystyle \frac{1}{W}}\sum _{j=t-W+1}^t∥e_i\left(j\right)∥$$

(30)

$${\eta }_i\left(t\right)=∥e_i\left(t\right)∥-{\overline{e}}_i\left(t\right)$$

(31)

where W denotes the window size, and ${\eta }_i\left(t\right)$ characterizes the increasing/decreasing trend of the error.

Accordingly, the threshold update strategy is

$${\delta }_i(t+∆t)=\left\{\begin{array}{cc}\min \left\{{\delta }_{\max },{\delta }_i\left(t\right)(1+{\alpha }_{\mathrm{inc}})\right\},\hfill & {\eta }_i\left(t\right)>0\hfill \\ \max \left\{{\delta }_{\min },{\delta }_i\left(t\right)(1-{\alpha }_{\mathrm{dec}}\exp (-\beta ∥e_i\left(t\right)∥))\right\},\hfill & {\eta }_i\left(t\right)\le 0\hfill \end{array}\right.$$

(32)

where ${\delta }_{\min }$ and ${\delta }_{\max }$ denote the lower threshold limits, ${\alpha }_{\mathrm{inc}}$ and ${\alpha }_{\mathrm{dec}}$ represent the increasing and decreasing coefficients, respectively, and $\beta$ is the error weighting factor. This design causes the threshold to gradually decrease during system stabilization to improve accuracy and rapidly increase during system disturbances to reduce communication frequency.

Considering the effects of obstacle density and mission phase, a global progress adjustment factor is introduced:

$$\min \left\{{\displaystyle \frac{1}{T_{\mathrm{total}}·c_g}}\right\}$$

(33)

The final threshold is

$${\delta }_i^{\mathrm{final}}\left(t\right)=(1-\xi \left(t\right)){\delta }_i\left(t\right)+\xi \left(t\right){\delta }_{\mathrm{base}}(1-c_t\xi \left(t\right))$$

(34)

where ${\delta }_{\mathrm{base}}$ is the baseline threshold, $c_g$ and $c_t$ are adjustment coefficients, and $T_{\mathrm{total}}$ is the total mission duration.

2.3.7. Nonlinear Impulse Control Law

When a triggering event occurs, the system applies an impulse control law to deliver an instantaneously larger control input, rapidly adjusting the UAV’s state. Based on Lyapunov stability theory and the system’s overall convergence, the nonlinear impulse control law is designed as follows:

$${\mathbf{a}}_i\left(t\right)=k_{\mathrm{impulse}}^t\left(t\right)·{\displaystyle \frac{e_i\left(t\right)}{∥e_i\left(t\right)∥}}$$

(35)

The dynamic adjustment mechanism for pulse gain is

$$k_{\mathrm{impulse}}^t\left(t\right)=k_{\mathrm{base}}·{\gamma }_i\left(t\right)·{\mu }_i\left(t\right)$$

(36)

where $k_{\mathrm{base}}$ is the baseline gain, ${\gamma }_i\left(t\right)$ is the error-dependent adjustment factor, and ${\mu }_i\left(t\right)$ is the environment-dependent adjustment factor.

The error-dependent adjustment factor is designed as follows:

$${\gamma }_i\left(t\right)=\max \left\{0.5,\min \left\{1.0,c_e·∥e_i\left(t\right)∥\right\}\right\}$$

(37)

The environment-constrained adjustment factor is

$${\mu }_i\left(t\right)={\displaystyle \frac{1}{1+c_o\exp (-d_i^{\mathrm{obs}}\left(t\right)/d_{\mathrm{ref}})}}$$

(38)

where $d_i^{\mathrm{obs}}\left(t\right)$ denotes the distance between the UAV and the nearest obstacle, $d_{\mathrm{ref}}$ is the reference distance, and $c_e$ and $c_o$ are adjustment coefficients.

To prevent control saturation, limit the amplitude of impulse control inputs:

$$u_i^{\mathrm{final}}=\left\{\begin{array}{cc}{u}_i^{\mathrm{impulse}}\left(t\right),\hfill & ∥u_i^{\mathrm{impulse}}\left(t\right)∥\le a_{\max },\hfill \\ a_{\max }{\displaystyle \frac{u_i^{\mathrm{impulse}}\left(t\right)}{∥u_i^{\mathrm{impulse}}\left(t\right)∥}},\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(39)

2.3.8. Anti-Windup Integral Control and Non-Triggered State Control Law

During non-triggered intervals, the system implements an anti-windup integral control scheme that holistically accounts for

$$u_i^{\text{non-trigger}}\left(t\right)=k_p{e}_i\left(t\right)+k_i{I}_i\left(t\right)+k_d(-v_i\left(t\right))+k_f{F}_{\mathrm{optical}}^i\left(t\right)+k_c{F}_{\mathrm{consenuse}}^i\left(t\right)$$

(40)

where $k_p$, $k_i$, $k_d$, $k_f$, and $k_c$ represent the proportional, integral, derivative, optical flow, and consensus control gains respectively, $F_{\mathrm{optical}}^i\left(t\right)$ denotes the control force generated by optical flow sensing, and $F_{\mathrm{consenuse}}^i\left(t\right)$ is the consensus control term based on the communication topology.

To prevent integral windup, an anti-windup integral term is designed:

$$\Rightarrow I_i\left(t\right)=\left\{\begin{array}{cc}{I}_i(t-∆t)+e_i\left(t\right)∆t,\hfill & \mathrm{if}\parallel I_i(t-∆t)+e_i\left(t\right)∆t\parallel \le I_{\max },\hfill \\ I_{\max }{\displaystyle \frac{I_i(t-∆t)+e_i\left(t\right)∆t}{\parallel I_i(t-∆t)+e_i\left(t\right)∆t\parallel }},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(41)

Introduce a velocity-dependent integral decay factor:

$${\lambda }_i\left(t\right)={\displaystyle \frac{1}{1+c_v{\parallel v_i\left(t\right)\parallel }^2}}$$

(42)

The integral term should be amended as follows:

$$I_i^{\mathrm{final}}\left(t\right)=I_i\left(t\right)·{\lambda }_i\left(t\right)$$

(43)

The consensus control term utilizes network topology information, incorporating the relative states of neighboring UAVs:

$$F_{\mathrm{consensus}}^i\left(t\right)=\sum _{j\in {\mathcal{N}}_i\left(t\right)}{a}_{ij}\left(t\right)\left[w_1(p_j\left(t\right)-p_i\left(t\right)\right)+w_2\left(v_j\left(t\right)-v_i\left(t\right))\right]$$

(44)

where ${\mathcal{N}}_i\left(t\right)$ denotes the neighbor set of UAV i in the communication topology, $a_{ij}\left(t\right)$ is the element of the topological adjacency matrix, and $w_1$ and $w_2$ are weighting coefficients.

2.3.9. Terminal Precision Control Strategy

To enhance the positioning accuracy when UAVs approach the target location, a terminal precision control law is designed:

$$u_i^{\mathrm{terminal}}\left(f\right)=k_i\left(f\right)e_i\left(f\right)-k_b\left(f\right)v_i\left(f\right)$$

(45)

where the terminal gain coefficient adaptively adjusts with distance:

$$k_b\left(t\right)=k_{t0}+{\displaystyle \frac{k_{t1}}{\parallel e_i\left(t\right)\parallel +ϵ}}$$

(46)

$$k_b\left(t\right)=k_{b0}+k_{b1}\exp \left(-{\displaystyle \frac{\parallel e_i{\left(t\right)\parallel }^2}{{\sigma }^2}}\right)$$

(47)

where $k_{t0}$, $k_{t1}$, $k_{b0}$, and $k_{b1}$ are control parameters, $ϵ$ is an infinitesimal positive number to prevent division by zero, $\sigma$ is the control bandwidth parameter, and the maximum allowable velocity during the terminal phase dynamically adjusts with distance.

$${\mathcal{V}}_{{\max }^{\mathrm{term}}}\left(t\right)=\min \left\{{\mathcal{V}}_{\max },c_d·\parallel e_i\left(t\right)\parallel \right\}$$

(48)

where $c_d$ is the proportionality coefficient.

The composite control law is

$$u_i\left(t\right)=\left\{\begin{array}{cc}{u}_i^{\mathrm{final}}\left(t\right),\hfill & \mathrm{if}\mathrm{trigger}\mathrm{condition}\mathrm{is}\mathrm{satisfied},\hfill \\ u_i^{\mathrm{terminal}}\left(t\right),\hfill & \mathrm{if}\parallel e_j\left(t\right)\parallel <d_{\mathrm{term}},\hfill \\ u_i^{\text{non-trigger}}\left(t\right),\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(49)

where $d_{\mathrm{term}}$ is the terminal control activation threshold.

The adaptive event-triggered impulse control strategy proposed in this section significantly reduces communication requirements through a meticulously designed variable-threshold triggering mechanism, leverages nonlinear impulse control to provide rapid response capability, and ensures system stability and precision by integrating anti-windup integral control with terminal precision control.

2.3.10. Analysis of Non-Zeno Behavior

In event-triggered control systems, Zeno behavior refers to the occurrence of an infinite number of triggering events within a finite time interval. This phenomenon is undesirable as it compromises the practical implementation of control algorithms and negates the resource-saving benefits of event-triggered mechanisms.

The temporal properties and Zeno behavior in event-triggered systems have been thoroughly investigated in several studies [31,32]. The reviewer correctly points out that while our system includes a cooldown period (Equation (28)) to mitigate chattering, a formal analysis proving the absence of Zeno behavior would strengthen the theoretical foundations of our approach.

Theorem 1.

The proposed adaptive event-triggered mechanism guarantees a positive lower bound on inter-execution times, thereby ensuring the absence of Zeno behavior.

Proof. 

We begin by analyzing the triggering condition in Equation (28). The mandatory cooldown period $T_{\mathrm{cool}}$ immediately ensures a minimum separation of $T_{\mathrm{cool}}$ between consecutive triggering events. However, this alone is insufficient for a formal proof, as we must also establish that the system does not reach a state where triggering occurs exactly every $T_{\mathrm{cool}}$ seconds indefinitely. □

To establish a comprehensive proof, we analyze the error dynamics between triggering events. At time when a triggering occurs for the i-th UAV, the control input changes discontinuously due to the impulse control law. From Equation (35). After applying this control, the error decreases rapidly. Let us define $\tau =t-t_k$ as the time elapsed since the last triggering. The error dynamics can be expressed as follows:

$${\displaystyle \frac{d∥{\mathbf{e}}_i(t_k+\tau )∥}{d\tau }}={\displaystyle \frac{d∥{\mathbf{p}}_i^d(t_k+\tau )-{\mathbf{p}}_i(t_k+\tau )∥}{d\tau }}$$

(50)

Given the kinematic model in Equations (3) and (4), and assuming bounded desired trajectory derivatives, this can be bounded by

$${\displaystyle \frac{d∥{\mathbf{e}}_i(t_k+\tau )∥}{d\tau }}\le ∥{\mathbf{p}}_i(t_k+\tau )-{\mathbf{v}}_i(t_k+\tau )∥\le M_v$$

(51)

where $M_v$ is a positive constant representing the maximum relative velocity.

For the error to grow from below the minimum threshold ${\delta }_{\min }$ to above it, a minimum time ${\tau }_{\min }$ is required:

$${\tau }_{\min }\ge {\displaystyle \frac{{\delta }_{\min }}{M_v}}$$

(52)

The minimum time between consecutive triggering events is bounded by

$$∆t_{\min }=\max \left\{T_{\mathrm{cool}},{\tau }_{\min }\right\}=\max \left\{T_{\mathrm{cool}},{\displaystyle \frac{{\delta }_{\min }}{M_v}}\right\}$$

(53)

Since both $T_{\mathrm{cool}}$ and ${\displaystyle \frac{{\delta }_{\min }}{M_v}}$ are positive constants, $∆t_{\min }>0$. This establishes a positive lower bound on inter-execution times, guaranteeing the absence of Zeno behavior.

Furthermore, our adaptive threshold design reinforces the non-Zeno property. From Equation (32), when errors increase $({\eta }_i\left(t\right)>0)$, the threshold also increases, which reduces triggering frequency. This adaptive mechanism creates a self-regulating effect that inherently counteracts potential Zeno behavior.

The lower bound constraint on the threshold is

$${\delta }_i\left(t\right)\ge {\delta }_{\min }$$

(54)

2.4. Optical Flow Perception and Obstacle Avoidance Strategy

2.4.1. Enhanced Optical Flow Perception Model

Optical flow perception is a biologically inspired sensing method derived from insect visual navigation, offering advantages of low computational complexity and strong environmental adaptability. In complex environments such as tunnels, traditional optical flow models face challenges including perception blind spots, occlusion effects, and inaccurate distance estimation. To address these issues, this paper proposes a sector-based enhanced optical flow perception model. The model simulation diagram is as follows:

In Figure 1, the UAV formation is arranged in a cross-shaped configuration, with green spheres representing the formation’s optical flow perception range, and red spheres indicating the centroid locations and physical presence of obstacles.

For the i-th UAV, its perception domain is defined as a spherical region centered at its current position:

$${\mathcal{S}}_i=\{q\in {\mathbf{R}}^3\mid r_{\mathrm{bind}}<\Vert q-p_i\Vert \le r_{\mathrm{sense}}\}$$

(55)

where $r_{\mathrm{bind}}$ denotes the blind zone radius, and $r_{\mathrm{sense}}$ represents the maximum sensing radius.

Accounting for the field-of-view limitations of unmanned aerial vehicles, the effective perception domain is characterized as follows:

$$S_i^{\mathrm{eff}}=\left\{q\in S_i\mid \arccos \left({\displaystyle \frac{(q-p_i)·v_i}{\Vert q-p_i\Vert ·\Vert v_i\Vert }}\right)\le {\displaystyle \frac{{\varphi }_{\mathrm{FOV}}}{2}}\right\}$$

(56)

Here, ${\varphi }_{\mathrm{FOV}}$ denotes the field-of-view (FOV) angle of the unmanned aerial vehicle.

To enhance spatial perception accuracy, the perception domain is partitioned into $n_s$ sectors:

$${\mathcal{S}}_i^k=\left\{q\in {\mathcal{S}}_i^{\mathrm{eff}}\mid {\theta }_k\le \mathrm{atan}2(q_y-p_{i,y},q_x-p_{i,x})<{\theta }_{k+1}\right\}$$

(57)

where $k\in \left\{1,2,\dots ,n_s\right\}$, and ${\theta }_k=2\pi (k-1)/n_s$.

Within each sector, define the optical flow vector based on relative motion as follows:

$${\mathbf{OF}}_i^k=\sum _{j\in {\mathcal{N}}_i^k}{w}_{ij}·{\displaystyle \frac{({\mathbf{v}}_j-{\mathbf{v}}_i)·D_{ij}}{\Vert p_j-p_i{\Vert }^2}}$$

(58)

Here, ${\mathcal{N}}_i^k$ denotes all perceived targets (including other UAVs and obstacles) within the k-th sector of the i-th UAV, where $w_{ij}$ represents the weighting coefficient and $D_{ij}$ is the distance-dependent attenuation factor.

$$D_{ij}=\exp \left(-{\alpha }_d·\Vert p_j-p_i\Vert \right)$$

(59)

Here, ${\alpha }_d$ denotes the attenuation coefficient.

Considering occlusion effects, an occlusion detection mechanism is introduced. For any two targets j and l within the sector, the following condition is satisfied:

$${\displaystyle \frac{(p_j-p_i)·(p_l-p_i)}{\Vert p_j-p_i\Vert \Vert p_l-p_i\Vert }}>{\gamma }_{\mathrm{occ}} \mathrm{and} \Vert p_l-p_i\Vert <\Vert p_j-p_i\Vert$$

(60)

If target j is occluded by target l, its weight is modified as follows:

$$w_{\overrightarrow{y}}=w_{\overrightarrow{y}}·(1-{\gamma }_{occ})$$

(61)

Here, ${\gamma }_{occ}$ denotes the occlusion threshold. By integrating optical flow from all sectors and incorporating sector weights ${\beta }_k$, the overall optical flow perception vector is obtained as follows:

$${\mathbf{OF}}_i=\sum _{k=1}^{n_s}{\beta }_k·{\mathbf{OF}}_i^k+{\mathbf{OF}}_i^{\mathrm{env}}$$

(62)

Here, ${\mathbf{OF}}_i^{\mathrm{env}}$ denotes the optical flow contribution induced by environmental factors.

The selection of sector count ($n_s=8$) represents a critical design decision balancing perceptual resolution against computational efficiency. Our parameter optimization revealed that while $n_s<6$ resulted in significant directional blind spots leading to 23.5% higher obstacle detection failure rates, increasing ns beyond 8 yielded diminishing returns in perception accuracy ($+2.7\%$ for $n_s=12$) at substantially higher computational cost ($+73\%$). The eight-sector design provides optimal 45° directional resolution, aligning with UAV maneuverability constraints in confined tunnel environments. The weighting coefficients ($w_{ij}$, ${\beta }_k$) were determined through sensitivity analysis to maximize obstacle detection reliability while minimizing false positives. For target weighting, the distance-dependent coefficient $w_{ij}$ follows an inverse quadratic relationship (Equation (59)) with attenuation parameter ${\alpha }_d=0.35$, creating a detection priority sphere extending approximately 3.5 m from each UAV. The sector weighting coefficients ${\beta }_k$ implement a priority distribution where forward-facing sectors (k = 4.5) receive higher weights (${\beta }_k=0.18$), side sectors (k = 3.6) receive moderate weights (${\beta }_k=0.15$), and rear sectors (k = 1.8) receive lower weights (${\beta }_k=0.12$), reflecting directional collision risk probabilities while maintaining 360° situational awareness. This asymmetric weighting scheme improves critical obstacle detection rates by 31.7% compared to uniform sectoral distribution while maintaining acceptable peripheral awareness.

2.4.2. Sector-Based Obstacle Avoidance Decision-Making

Based on sectorized optical flow perception, a multi-level obstacle avoidance decision-making mechanism is designed, comprising three tiers: emergency avoidance, coordinated avoidance, and path optimization.

At the emergency avoidance level, when potential collision risks are detected, a repulsion vector is constructed:

$${\mathbf{R}}_i=\sum _{j\in {\mathcal{O}}_{\mathrm{close}}}{\kappa }_r·{\left(1-{\displaystyle \frac{\parallel P_j-P_i\parallel }{r_{\mathrm{safe}}}}\right)}^2·{\displaystyle \frac{P_i-P_j}{\parallel P_i-P_j\parallel }}$$

(63)

Here, ${\mathcal{O}}_{\mathrm{close}}$ denotes the set of obstacles near the UAV, satisfying $\parallel P_j-P_i\parallel <r_{\mathrm{safe}}$, where ${\kappa }_r$ is the repulsion gain and $r_{\mathrm{safe}}$ is the safety distance threshold.

To avoid local minimum points, a rotational vector field is introduced:

$${\mathbf{T}}_i={\kappa }_t·\sum _{j\in {\mathcal{O}}_{\mathrm{close}}}\exp \left(-{\displaystyle \frac{\parallel p_j-p_i\parallel }{r_{\mathrm{rot}}}}\right)·{\displaystyle \frac{(p_i-p_j)\times v_i}{\parallel (p_i-p_j)\times v_i\parallel }}$$

(64)

Here, ${\kappa }_t$ denotes the rotational gain and $r_{\mathrm{rot}}$ represents the radius of rotational influence.

In the coordinated avoidance layer, a cooperative obstacle avoidance strategy is constructed by utilizing shared information from multiple UAVs. The collaborative avoidance vector is defined as follows:

$${\mathbf{C}}_i=\sum _{m\in {\mathcal{N}}_i^{\mathrm{comm}}}{A}_{im}·{\displaystyle \frac{{\mathbf{R}}_m+{\mathbf{T}}_m}{|{\mathcal{N}}_i^{\mathrm{comm}}|}}$$

(65)

Here, ${\mathcal{N}}_i^{\mathrm{comm}}$ denotes the set of neighbors with communication links to UAV i, and $A_{im}$ represents the adjacency matrix element.

2.4.3. A Tunnel-Specific Obstacle Avoidance Strategy

A specialized obstacle avoidance strategy for tunnel environments is designed, incorporating tunnel wall repulsion, curved section prediction, and dynamic formation adjustment.

For tunnel wall avoidance, a distance-dependent repulsive vector field is established:

$${\mathbf{W}}_i={\kappa }_w·{\left({\displaystyle \frac{r_t\left(s^{*}\right)-d_{i,\mathrm{wall}}}{r_t\left(s^{*}\right)}}\right)}^{\eta }·{\displaystyle \frac{p_i-c\left(s^{*}\right)}{\parallel p_i-c\left(s^{*}\right)\parallel }}$$

(66)

Here, ${\kappa }_w$ denotes the wall repulsion gain, $r_t\left(s^{*}\right)$ represents the tunnel radius corresponding to the UAV’s current position (where $s^{*}$ is the parameter of the closest point on the parameterized path), $d_{i,\mathrm{wall}}$ is the distance between the UAV and tunnel wall, and $\eta$ serves as the nonlinear coefficient for adjusting the repulsion intensity’s distance-dependent variation rate.

In curved tunnel sections, a predictive guidance vector is introduced:

$${\mathbf{G}}_i={\kappa }_g·{\displaystyle \frac{c(s^{*}+∆s)-p_i}{\parallel c(s^{*}+∆s)-p_i\parallel }}$$

(67)

Here, ${\kappa }_g$ denotes the guidance gain, $∆s$ represents the look-ahead parameter, and $c(s^{*}+∆s)$ indicates the coordinate of the look-ahead point.

The detection and resolution of local minima in tunnel environments is

$${\lambda }_i={\displaystyle \frac{\parallel {\mathbf{R}}_i+{\mathbf{W}}_i\parallel ·\parallel v_i\parallel }{({\mathbf{R}}_i+{\mathbf{W}}_i)·v_i+ϵ}}$$

(68)

When ${\lambda }_i>{\lambda }_{\mathrm{threshold}}$, a local minimum is identified and the rotational vector field is enhanced:

$${\mathbf{T}}_i^{\mathrm{enhanced}}={\mu }_{\mathrm{local}}·{\mathbf{T}}_i$$

(69)

where ${\mu }_{\mathrm{local}}$ denotes the local minimum enhancement coefficient.

2.4.4. Hybrid Force Field Optimization

By integrating the obstacle avoidance strategies, a hybrid force field is constructed to achieve multi-objective optimization:

$${\mathbf{F}}_l^{\mathrm{avoid}}={\alpha }_r{\mathbf{R}}_i+{\alpha }_t{\mathbf{T}}_i+{\alpha }_w{\mathbf{W}}_i+{\alpha }_g{\mathbf{G}}_i+{\alpha }_c{\mathbf{C}}_i$$

(70)

Here, ${\alpha }_r$, ${\alpha }_t$, ${\alpha }_w$, ${\alpha }_g$, and ${\alpha }_c$ represent the weighting coefficients for each force field, dynamically adjusted based on environmental complexity and mission phase:

$${\alpha }_x={\alpha }_x^{\mathrm{base}}·f_{\mathrm{env}}(d_{i,\mathrm{obs}},d_{i,\mathrm{wall}})·f_{\mathrm{task}}(t/T_{\mathrm{total}})$$

(71)

Here, $x\in \left\{r,t,w,g,c\right\}$, $f_{\mathrm{env}}$, and $f_{\mathrm{task}}$ denote the adjustment functions for environmental complexity and mission phases, respectively.

The resultant control force generated by optical flow perception is expressed as follow:

$${\mathbf{F}}_i^{\mathrm{OF}}=k_{\mathrm{OF}}·{\mathbf{OF}}_i+{\mathbf{F}}_i^{\mathrm{avoid}}$$

(72)

where $k_{\mathrm{OF}}$ denotes the optical flow gain coefficient.

The proposed multi-level obstacle avoidance framework, integrating optical-flow-based perception with tunnel-optimized hybrid force fields, demonstrates robust performance in complex tunnel environments, ensuring both safety and formation stability for UAV swarms.

2.5. Leader–Follower Formation Maintenance Mechanism

For formation maintenance in complex tunnel environments, this section proposes an adaptive leader–follower mechanism that dynamically balances formation preservation and obstacle avoidance through weighted optimization and topological adaptation, achieving unified flexibility and stability.

2.5.1. Dynamic Formation Structure Design

In the standard formation structure, one UAV (designated as l) serves as the leader while the remaining UAVs act as followers. The leader assumes navigation responsibilities, whereas followers maintain relative positional relationships. The ideal formation structure is defined as follows:

$${\delta }_{ij}^d=p_j^d-p_i^d$$

(73)

where ${\delta }_{ij}^d$ denotes the relative position vector between UAV i and UAV j in the ideal formation, while $p_i^d$ and $p_j^d$ represent the reference positions of UAV i and UAV j, respectively.

A dynamic formation adjustment strategy is proposed to address tunnel-specific constraints. For follower i, its target position is dynamically updated based on the leader’s position:

$$p_i^d\left(t\right)=p_l\left(t\right)+R\left({\theta }_l\left(t\right)\right)·{\delta }_{il}^d$$

(74)

where $R\left({\theta }_l\left(t\right)\right)$ represents the rotation matrix computed based on the leader’s heading angle ${\theta }_l\left(t\right)$:

$$R\left({\theta }_l\left(t\right)\right)=\left[\begin{array}{ccc}\cos {\theta }_l\left(t\right)& -\sin {\theta }_l\left(t\right)& 0\\ \sin {\theta }_l\left(t\right)& \cos {\theta }_l\left(t\right)& 0\\ 0& 0& 1\end{array}\right]$$

(75)

An elastic formation deformation mechanism is introduced to accommodate tunnel geometric constraints:

$${\delta }_{il}^d\left(t\right)=\sigma \left(t\right)·{\delta }_{il}^{d,0}$$

(76)

where ${\delta }_{il}^{d,0}$ denotes the relative position in the standard formation, and $\sigma \left(t\right)$ represents the elastic coefficient dynamically adjusted according to local tunnel characteristics:

$$\sigma \left(t\right)={\sigma }_{\mathrm{base}}·\min \left\{1,{\displaystyle \frac{r_t\left(s_l^{*}\right)}{r_{\mathrm{ref}}}}\right\}·\gamma \left(v_l\right)$$

(77)

where ${\sigma }_{\mathrm{base}}$ denotes the baseline elasticity coefficient, $r_{\mathrm{ref}}$ represents the reference tunnel radius, and $\gamma \left(v_l\right)$ signifies the velocity-dependent adjustment factor:

$$\gamma \left(v_l\right)=1+\left({\displaystyle \frac{\Vert v_l\Vert }{v_{\max }}}-0.5\right)$$

(78)

where ${\beta }_v$ denotes the velocity influence coefficient.

2.5.2. Formation Maintenance and Obstacle Avoidance Balancing

In complex tunnel environments, UAVs must achieve equilibrium between formation maintenance and obstacle avoidance. The formation maintenance vector is defined as follows:

$${\mathbf{F}}_i^{\mathrm{form}}=k_p(p_i^d-p_i)+k_d(v_i^d-v_i)$$

(79)

where $k_p$ and $k_d$ denote the position and velocity gain coefficients, respectively, and $v_i^d$ represents the target velocity.

To achieve dynamic equilibrium between formation maintenance and obstacle avoidance, an adaptive weight adjustment mechanism is designed:

$${\mathbf{F}}_i^{\mathrm{total}}={\rho }_i^{\mathrm{form}}·{\mathbf{F}}_i^{\mathrm{form}}+{\rho }_i^{\mathrm{avoid}}·{\mathbf{F}}_i^{\mathrm{OF}}$$

(80)

where ${\rho }_i^{\mathrm{form}}$ and ${\rho }_i^{\mathrm{avoid}}$ denote the weight coefficients for formation maintenance and obstacle avoidance, respectively, with the constraint ${\rho }_i^{\mathrm{form}}+{\rho }_i^{\mathrm{avoid}}=1$.

The weight coefficients are dynamically adjusted based on environmental complexity and mission phase:

$${\rho }_i^{\mathrm{avoid}}=\min \{1,\max \{{\rho }_{\min }^{\mathrm{avoid}},\varphi (d_{i,\mathrm{obs}},d_{i,\mathrm{wall}})\}\}$$

(81)

where $\varphi (d_{i,\mathrm{obs}},d_{i,\mathrm{wall}})$ denotes the adjustment function based on obstacle distance and tunnel wall proximity:

$$\varphi (d_{i,\mathrm{obs}},d_{i,\mathrm{wall}})=1-\exp \left(-{\lambda }_{\mathrm{obs}}·{\displaystyle \frac{1}{d_{i,\mathrm{obs}}^2}}·-{\lambda }_{\mathrm{wall}}·{\displaystyle \frac{1}{d_{i,\mathrm{wall}}^2}}\right)$$

(82)

where ${\lambda }_{\mathrm{obs}}$ and ${\lambda }_{\mathrm{wall}}$ denote the adjustment coefficients.

The integrated leader–follower formation maintenance mechanism, incorporating dynamic formation adjustment and adaptive weight allocation, simultaneously ensures formation stability and effective obstacle avoidance in complex tunnel environments, enabling safe and efficient swarm navigation.

2.5.3. Parameter Tuning for Dynamic Weight Allocation in Practical Scenarios

The dynamic weight allocation mechanism described in Equations (79)–(82) requires appropriate parameter tuning to achieve optimal performance across different operational environments. While the theoretical formulation provides the framework for balancing formation maintenance and obstacle avoidance priorities, practical implementation demands systematic calibration approaches. This section outlines the key factors influencing weight parameters and presents methodologies for their adjustment in real-world scenarios.

Environmental Factors Influencing Optimal Weight Settings:

• Several environmental characteristics significantly impact the optimal weight configuration:

Obstacle Density: In environments with high obstacle density (>$0.08$ obstacles/m³), increasing the obstacle avoidance weight coefficient (${\alpha }_i^{\mathrm{avoid}}$) by 25–35% improves collision avoidance while maintaining acceptable formation integrity. Our experimental results indicate that for every 0.02 increase in volumetric obstacle density, the optimal ${\alpha }_i^{\mathrm{avoid}}$ value increases by approximately 0.07.

2.

Tunnel Geometry Complexity:

The ratio of tunnel curvature to tunnel width represents a critical factor. For high-complexity sections (curvature/width ratio > 0.15), increasing formation maintenance weights (${\alpha }_i^{\mathrm{form}}$) by 15–20% enhances formation stability during navigation through curved segments while still allowing sufficient flexibility for obstacle avoidance.

3.

Spatial Constraints:

In narrow tunnel sections where the ratio of tunnel diameter to formation diameter falls below 2.0, reducing the formation maintenance weight by 30–40% enables greater deformation capability, preventing deadlock situations. The adjustment factor can be approximated as ${\gamma }_{\mathrm{spatial}}=\min \{1.0,0.5+0.25·(D_{\mathrm{travel}}/D_{\mathrm{formation}})\}$

4.

Environmental Uncertainty:

In scenarios with perceptual uncertainty (e.g., poor lighting, dust, smoke), increasing the safety margins by reducing the threshold parameters (${\beta }_{\mathrm{obs}}$, ${\beta }_{\mathrm{wall}}$ in Equation (82) by 20–30% provides more conservative behavior. The adjustment correlates with the standard deviation of perception errors.

3. Experimental Design and Results Analysis

3.1. Experimental Setup and Parameter Configuration

To better demonstrate the simulation environment, the key experimental parameters are specified in

Table 1. Experimental environment and parameter configuration.

Parametric Class	Parameter Name	Methodology of This Paper	APF-A* Physical Method	Unit
Software environment	Simulation software	MATLAB R2023a		–
Drone formation configuration	Drones count	5		N/A
	Formation structure	cruciform		–
	Max velocity	12.0		m/s
	Max acceleration	10.0		m/s2
	Follower spacing	8.0		m
Tunnel environmental parameters	Tunnel length	90		m
	Tunnel base radius	20		m
	Bend height	20		m
	Radius variation coefficient	0.3		–
	Tunnel curve path points	80		N/A
	Tunnel min radius	15		m
Selection parameters	Obstacles count	15		N/A
	Obstacles radius	1		m
Feature dimension classification parameters	Base trigger threshold	0.5	–	m
	Max trigger threshold	3.0		m
	Min trigger threshold	0.15		m
	Threshold growth factor	0.04		m
	Threshold attenuation factor	0.25		–
	Trigger cooldown	0.6		–
	Pulse gain factor	6.5		–
	Error window size	10		–
	Position proportional gain ($k_p$)	2.0		sample
	Velocity differential gain ($k_v$)	1.5		2.0
	Integral gain ($k_i$)	0.15		1.5
	Optical flow control gain	0.6		–
APF-A*	Route planning interval	–	5.0	s
	Gravitational coefficient of artificial potential field		1.0	–
	Obstacle repulsion factor		15.0	–
	Drone repulsion factor		10.0	–
	Follow leader factor		2.0	–
	Tunnel guidance factor		12.0	–
	Obstacle impact distance		10.0	m
	Drone impact distance		15.0	m
	Path weight		1.5	–
	APF weight		2.5	–
	Mesh size		2.0	m
Communication parameters	Communication range	20		m
	Packet loss probability	15		%
	Communication delay	0.01		s
	Topology change interval	10		s
Safety parameters	Sensing radius	25		m
	Proximity blind spot radius	3		m
	Field of view	270		°
	Sensed sectors count	8		N/A

:

As evidenced by the preceding table, this study adopts the APF-A* method as the benchmark, with scientifically configured parameters to ensure equitable experimental performance comparison.

3.1.1. Tunnel Environment Modeling

To accurately represent real-world tunnel environments, we constructed a sophisticated 3D model that captures key characteristics including variable cross-sections, curved pathways, and complex geometry. The tunnel environment was designed to challenge formation control algorithms with realistic navigational constraints.

The tunnel is defined by a parameterized central curve and circular cross-sections with varying dimensions along the curve. The central curve follows a parabolic path described by

$$\gamma \left(t\right)=P_s+(P_e-P_s)t+[0,-h_c·4t(1-t),0]$$

(83)

where $P_s=[10,0,20]$ and $P_e=[90,0,20]$ represent the starting and ending points of the tunnel (in meters), respectively, $h_c=20$ m denotes the maximum lateral bending height, and $t\in [0,1]$ is the normalized distance parameter along the tunnel length. This parabolic formulation creates a realistic arc-shaped tunnel with maximum curvature at the midpoint, challenging the formation control system with non-uniform directional changes.

The tunnel’s cross-sectional dimensions vary along its length according to

$$r\left(t\right)=r_b·(1-{\alpha }_r·{\sin }^2\left(\pi t\right))$$

(84)

where $r_b=20$ m is the baseline radius, ${\alpha }_r=0.3$ is the radius variation coefficient, and $r\left(t\right)$ is constrained to a minimum value of $r_{\min }=15$ m. This radius variation models the constrictions and expansions commonly found in natural tunnel formations, creating sections where the formation must adapt its configuration to navigate safely. The tunnel geometry is discretized into $N_{\mathrm{path}}=80$ curve points along its length with $N_{\mathrm{seg}}=30$ segments per cross-section to generate a high-fidelity 3D mesh representation.

For each point along the tunnel path, a local coordinate system was established to accurately model the tunnel’s curved geometry. At each parameter value t, the local coordinate basis consists of the following:

• Tangent Vector: $\overline{T}\left(t\right)={\displaystyle \frac{d\gamma \left(t\right)}{dt}}/∥{\displaystyle \frac{d\gamma \left(t\right)}{dt}}∥$
• Initial Upward Direction: ${\overrightarrow{U}}_0=[0,0,1]$
• Right Direction: $\overrightarrow{R}\left(t\right)=\overrightarrow{T}\left(t\right)\times {\overrightarrow{U}}_0/∥\overrightarrow{T}\left(t\right)\times {\overrightarrow{U}}_0∥$
• Adjusted Upward Direction: $\overrightarrow{U}\left(t\right)=\overrightarrow{R}\left(t\right)\times \overrightarrow{T}\left(t\right)$

Using this local coordinate system, points on the tunnel surface are expressed as

$$S(t,\theta )=\gamma \left(t\right)+r\left(t\right)(\cos \theta ·\overrightarrow{R}\left(t\right)+\sin \theta ·\overrightarrow{U}\left(t\right))$$

(85)

where $\theta \in [0,2\pi ]$ is the angular parameter defining points around the cross-sectional circumference. The resulting tunnel structure exhibits several key characteristics that challenge UAV formation control:

• Variable width profile: The tunnel radius reaches its minimum value at the midpoint (t = 0.5), requiring formation compression in this region.
• Continuous curvature: The parabolic centerline creates a smooth but continuously changing direction of travel.
• Compound complexity: The combination of variable radius and curvature creates regions where both horizontal and vertical formation adjustments are simultaneously required.

To accurately represent the tunnel structure in the simulation environment, the mesh representation was constructed by sampling both parameters (t and $\theta$) at regular intervals, generating a mesh with ($N_{\mathrm{path}}\times N_{\mathrm{seg}}$) vertices. The triangulation of this mesh creates a realistic surface for collision detection and visualization purposes.

Figure 2 illustrates the tunnel’s geometry, highlighting the variable radius sections and curvature characteristics. The figure shows (a) the complete 3D mesh representation of the tunnel with color-coded radius values, (b) a top-down view showing the centerline path and radius variations, and (c) cross-sectional profiles at different positions along the tunnel length, demonstrating the narrowing at the midpoint.

This tunnel modeling approach creates a sophisticated environment that closely resembles real-world tunnel structures, providing a challenging yet realistic testbed for evaluating UAV formation control algorithms under complex spatial constraints.

3.1.2. Obstacle Distribution and Complexity

To evaluate formation control performance under challenging navigational conditions, we strategically distributed 15 spherical obstacles throughout the tunnel environment, with concentration in critical areas. Each obstacle is formally defined as follows:

$$O_j=\{x\in R^3: \parallel x-c_j\parallel \le r_o\}$$

(86)

where $c_j\in R^3$ denotes the center position of the j-th obstacle, and $r_o=1$ m represents its radius.

The obstacles were strategically arranged into four functional groups:

• Tunnel entrance region (3 obstacles): positioned near the entrance (15–25 m) to test formation initialization and early coordination patterns.
• Maximum curvature section (5 obstacles): concentrated in the region of highest tunnel curvature and minimum radius (40–60 m from entrance), creating combined navigation challenges requiring simultaneous handling of tight turns and constricted passages.
• Variable radius sections (4 obstacles): placed at transitions between different tunnel radius zones (30–36 m and 62–66 m) to force formation reconfiguration as the tunnel dimensions vary.
• Exit approach region (3 obstacles): positioned in the final segment (72–83 m) to challenge terminal phase stability as the formation transitions to its final configuration.

The obstacle density function $\rho \left(x\right)$ at any point $x\in R^3$ within the tunnel can be quantified as follows:

$$\rho \left(x\right)=\sum _{j=1}^{15}\exp \left(-{\displaystyle \frac{\parallel x-c_j{\parallel }^2}{2{\sigma }^2}}\right)$$

(87)

where $\sigma =5$ m determines the influence range of each obstacle.

This obstacle arrangement creates several particularly challenging navigation points:

• Narrow corridors where passage width is less than 1.5× the formation diameter;
• Slalom configurations requiring sequential avoidance maneuvers;
• Combined vertical–horizontal challenges necessitating 3D formation adjustments.

For collision detection purposes, the minimum safety distance between a UAV and an obstacle is defined as follows:

$$d_{\mathrm{safe}}=r_o+r_{\mathrm{uav}}+\delta$$

(88)

where $r_{\mathrm{uav}}=0.5$ m is the UAV’s effective radius, and $\delta =0.2$ m is an additional safety margin.

This obstacle distribution was specifically designed to test collision avoidance capabilities while maintaining formation integrity, representing realistic challenges encountered in tunnel exploration scenarios.

3.1.3. Baseline Method Implementation (APF-A*)

The artificial potential field with A* algorithm (APF-A*) serves as our benchmark, representing a state-of-the-art approach widely used in robotic navigation. Our implementation integrates global path planning (A*) with local reactive control (APF) through a hybrid architecture designed to balance global efficiency with local obstacle avoidance.

A*-APF Integration Mechanism

The hybrid control strategy updates UAV acceleration commands $a_i\left(t\right)$ for the i-th UAV at time t according to $a_i\left(t\right)=w_{\mathrm{path}}·F_{\mathrm{path}}\left(p_i\left(t\right)\right)+w_{\mathrm{apf}}·F_{\mathrm{apf}}(p_i\left(t\right),v_i\left(t\right))+w_{\mathrm{damp}}·F_{\mathrm{damp}}\left(v_i\left(t\right)\right)$ where $w_{\mathrm{path}}=1.5$ is the path tracking weight, $w_{\mathrm{apf}}=2.5$ is the APF force weight, and $w_{\mathrm{damp}}=0.6$ is the damping coefficient. The position and velocity of the UAV are denoted by $p_i\left(t\right)$ and $v_i\left(t\right)$, respectively.

A* Path Planning Component

The A* path planning executes at intervals of $T_{\mathrm{plan}}=0.5$ s, generating a sequence of waypoints $\{w_1,w_2,\dots ,w_n\}$ from the current position to the goal. The environment is discretized into a 3D grid with resolution ${∆}_{\mathrm{grid}}=2.0$ m, and path planning incorporates a safety margin of $1.5\times r_o$ around obstacles.

The A* algorithm employs a heuristic function $h\left(n\right)$ for node n defined as follows:

$$h\left(n\right)=\parallel p_n-p_{\mathrm{goal}}\parallel$$

(89)

where $p_n$ is the position corresponding to node n and $p_{\mathrm{goal}}$ is the goal position.

The cost function $g\left(n\right)$ represents the path cost from the start node to node, and the total evaluation function $f\left(n\right)$ is

$$f\left(n\right)=g\left(n\right)+h\left(n\right)$$

(90)

The generated path is smoothed using B-spline techniques to ensure dynamically feasible trajectories. For a path with waypoints $\{w_1,w_2,\dots ,w_n\}$, the smoothed path points $s\left(u\right)$ at parameter $u\in [0,1]$ are computed as follows:

$$s\left(u\right)=\sum _{i=1}^n{w}_i·B_{i,\widehat{k}}\left(u\right)$$

(91)

where $B_{i,\widehat{k}}\left(u\right)$ are B-spline basis functions of degree $k=3$. The path following force $F_{\mathrm{path}}\left(p_i\left(t\right)\right)$ is calculated as follows:

$$F_{\mathrm{path}}\left(p_i\left(t\right)\right)={\displaystyle \frac{w_{\mathrm{next}}-p_i\left(t\right)}{\parallel w_{\mathrm{next}}-p_i\left(t\right)\parallel }}$$

(92)

where $w_{\mathrm{next}}$ is the next waypoint in the path.

Artificial Potential Field Component

The APF force $F_{\mathrm{apf}}(p_i\left(t\right),v_i\left(t\right))$ comprises several components:

• Goal attraction force: $F_{\mathrm{att}}\left(p_i\right)=k_{\mathrm{att}}·{\displaystyle \frac{p_{\mathrm{goal}}-p_i}{\parallel p_{\mathrm{goal}}-p_i\parallel }}$ with coefficient $k_{\mathrm{att}}=1.0$.
• Obstacle repulsion force: $F_{\mathrm{rep},\mathrm{obs}}\left(p_i\right)={\sum }_{j=0}^W{k}_{\mathrm{rep},\mathrm{obs}}·{\left({\displaystyle \frac{1}{d_{ij}}}-{\displaystyle \frac{1}{d_0^{\mathrm{obs}}}}\right)}^2·{\displaystyle \frac{p_i-c_j}{\parallel p_i-c_j\parallel }}$ where $d_{ij}=\parallel p_i-c_j\parallel$ is the distance to obstacle surface, coefficient $k_{\mathrm{rep},\mathrm{obs}}=15.0$, and influence distance $d_0^{\mathrm{obs}}=10.0$ m.
• Inter-UAV repulsion force: $F_{\mathrm{rep},\mathrm{uav}}\left(p_i\right)={\sum }_{j\ne 1}{k}_{\mathrm{rep},\mathrm{uav}}·{\left({\displaystyle \frac{1}{d_{ij}^{uav}}}-{\displaystyle \frac{1}{d_0^{\mathrm{uav}}}}\right)}^2·{\displaystyle \frac{p_i-p_j}{\parallel p_i-p_j\parallel }}$ where $d_{ij}^{uav}=\parallel p_i-c_j\parallel$ is the distance to obstacle surface, coefficient $k_{\mathrm{rep},\mathrm{uav}}=10.0$, and influence distance $d_0^{uav}=5.0$ m.
• Leader following force (for non-leader UAVs):

  $$F_{\mathrm{follow}}\left(p_i\right)=k_{\mathrm{follow}}·{\displaystyle \frac{p_l-p_i}{\parallel p_l-p_i\parallel }}$$

  (93)

  where $p_l$ is the leader position and coefficient $k_{\mathrm{follow}}=2.0$.
• Tunnel guidance force: $F_{\mathrm{tunnel}}\left(p_i\right)=k_{\mathrm{tunnel}}·{\left({\displaystyle \frac{d_i^{\mathrm{wall}}}{r\left(t_i\right)}}\right)}^2·{\displaystyle \frac{P_{\mathrm{center}}\left(t_i\right)-p_i}{\parallel P_{\mathrm{center}}\left(t_i\right)-p_i\parallel }}$ where $d_i^{\mathrm{wall}}$ is the distance to tunnel wall, $P_{\mathrm{center}}\left(t_i\right)$ is the closest point on tunnel centerline, and coefficient $k_{\mathrm{tunnel}}=12.0$.
• Velocity damping force: $F_{\mathrm{damp}}\left(v_i\right)=-v_i$.
  The total APF force is the weighted sum of these components:

  $$F_{\mathrm{apf}}(p_i,v_i)=F_{\mathrm{att}}\left(p_i\right)+F_{\mathrm{rep},\mathrm{obs}}\left(p_i\right)+F_{\mathrm{rep},\mathrm{nav}}\left(p_i\right)+{\beta }_i·F_{\mathrm{follow}}\left(p_i\right)+F_{\mathrm{tunnel}}\left(p_i\right)$$

  (94)

  where ${\beta }_i=0$ for the leader UAV and ${\beta }_i=1.0$ for followers.

Dynamic Force Balancing

The force balancing mechanism dynamically adjusts weights based on environmental conditions:

$$w_{\mathrm{path}}\left(t\right)=w_{\mathrm{path}}^0·(1-{\alpha }_{\mathrm{obs}}·{\rho }_i\left(t\right))$$

(95)

$$w_{\mathrm{apf}}\left(t\right)=w_{\mathrm{apf}}^0·(1+{\alpha }_{\mathrm{obs}}·{\rho }_i\left(t\right))$$

(96)

where $w_{\mathrm{path}}^0=1.5$ and $w_{\mathrm{apf}}^0=2.5$ are baseline weights, ${\alpha }_{\mathrm{obs}}=0.5$ is the obstacle influence coefficient, and ${\rho }_i\left(t\right)$ is the local obstacle density at the UAV’s position.

This force balancing ensures smooth transitions between global path following and local obstacle avoidance behaviors, with APF dominance increasing in high-risk situations while path following dominates in obstacle-free regions.

3.1.4. UAV Formation Configuration

The UAV formation is configured in a cross-shaped pattern with five UAVs: one leader (UAV 3, centrally positioned) and four followers arranged at cardinal positions (UAV 1: upper, UAV 4: lower, UAV 2: right, and UAV 5: left). The designed inter-UAV spacing is $d_{\mathrm{spacing}}=8$ m, selected based on the tunnel dimensions and obstacle sizes to provide adequate separation while maintaining formation cohesion.

Formation Structure and Dynamics

The ideal formation structure is defined by the set of desired relative positions between UAVs:

$${∆}_{ij}^{\mathrm{ref}}=p_j^{\mathrm{ref}}-p_i^{\mathrm{ref}}$$

(97)

where $p_i^{\mathrm{ref}}$ and $p_j^{\mathrm{ref}}$ are the reference positions of UAVs i and j, respectively.

For a follower UAV i, its target position $p_i^{\mathrm{target}}\left(t\right)$ at time t is dynamically updated based on the leader’s position:

$$p_i^{\mathrm{target}}\left(t\right)=p_l\left(t\right)+R\left({\theta }_l\left(t\right)\right)·{∆}_{il}^{\mathrm{ref}}·\mu \left(v_l\left(t\right)\right)$$

(98)

where $p_l\left(t\right)$ is the leader’s position, and $R\left({\theta }_l\left(t\right)\right)$ is the rotation matrix computed based on the leader’s heading angle ${\theta }_l\left(t\right)$:

$$R\left({\theta }_l\left(t\right)\right)=\left[\begin{array}{ccc}\cos {\theta }_l\left(t\right)& -\sin {\theta }_l\left(t\right)& 0\\ \sin {\theta }_l\left(t\right)& \cos {\theta }_l\left(t\right)& 0\\ 0& 0& 1\end{array}\right]$$

(99)

The formation stretching factor is

$$\mu \left(v_l\left(t\right)\right)=\left\{\begin{array}{cc}1.0,\hfill & \mathrm{if}\parallel v_l\left(t\right)\parallel \le v_{\mathrm{thresh}}\hfill \\ 1.0+\kappa ·(\parallel v_l\left(t\right)\parallel - v_{\mathrm{thresh}}),\hfill & \mathrm{if}\parallel v_l\left(t\right)\parallel >v_{\mathrm{thresh}}\hfill \end{array}\right.$$

(100)

where $v_{\mathrm{thresh}}=5.0$ m/s is the threshold speed and $\mathcal{K}=0.02$ s/m is the stretching coefficient, allowing the formation to elongate by up to 20% along the direction of travel at high speeds to reduce aerodynamic interference.

Formation Recovery Mechanism

When approaching the destination, the formation gradually transitions back to the standard configuration. The transition factor $\lambda \left(t\right)$ is computed as follows:

$$\lambda \left(t\right)=\max (0,\min (1.0,1-{\displaystyle \frac{\parallel p_i\left(t\right)-p_{goal}\parallel }{d_{recovery}}}))$$

(101)

where $d_{recovery}=35.0$ m is the recovery initiation distance.

The target position incorporating formation recovery is

$$p_i^{target}\left(t\right)=(1-\lambda \left(t\right))·p_i^{dynamic}\left(t\right)+\lambda \left(t\right)·p_i^{standard}\left(t\right)$$

(102)

where $p_i^{dynamic}\left(t\right)$ is the dynamically adjusted position and $p_i^{standard}\left(t\right)$ is the standard formation position.

Leader Slowdown Adaptation

The leader implements a slowdown mechanism when followers lag behind, with the slowdown factor proportional to the maximum formation error:

$$\eta \left(t\right)=\max (0.5,\min (1.0,{\displaystyle \frac{d_{spacing}·{\alpha }_{slowdown}}{{\max }_{i\ne l}\left(e_i\left(t\right)\right)}}))$$

(103)

where $e_i\left(t\right)=\parallel p_i\left(t\right)-p_i^{target}\left(t\right)\parallel$ is the position error of follower i, and ${\alpha }_{slowdown}=1.2$ is the slowdown threshold coefficient.

The leader’s acceleration is then scaled by this factor:

$$a_l\left(t\right)=\eta \left(t\right)·a_l^{original}\left(t\right)$$

(104)

This adaptive slowdown ensures that the leader does not outpace followers, maintaining formation cohesion throughout the navigation task.

The formation configuration and adaptation mechanisms are designed to balance individual UAV maneuverability with collective formation integrity, providing robust performance in complex and dynamic environments while ensuring safe and coordinated navigation.

3.2. Evaluation Metrics

To comprehensively evaluate the performance of the proposed method, four categories of evaluation metrics are designed:

• Safety Metrics: including collision counts/rates with other UAVs/obstacles and minimum safety distance.
• Formation Accuracy Metrics: including formation shape error $E_{\mathrm{form}}$ and formation recovery time $T_{\mathrm{recorvery}}$, defined as follows:

  $$E_{\mathrm{form}}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\displaystyle \frac{1}{T}}{\int }_0^T∥p_i\left(t\right)-p_i^d\left(t\right)∥dt$$

  (105)
• Communication Efficiency Metrics: including average communication latency, communication success rate, and network connectivity degree.
• Stability Metrics: including Lyapunov function value variation and convergence rate.

Based on the aforementioned evaluation metrics, the performance of the algorithms proposed in this study is comprehensively assessed to provide an objective evaluation.

3.3. Comparative Experiments

To validate the effectiveness of the proposed method, the representative conventional approach combining artificial potential field with A* algorithm (APF-A*) is selected as the benchmark. This hybrid method, widely adopted in robotic obstacle avoidance and path planning, employs A* for global path planning and artificial potential field for local obstacle avoidance. Both methods were tested under identical tunnel environments and obstacle configurations, with Monte Carlo simulations conducted for multiple trials to eliminate stochastic effects.

The following figure group compares 3D results generated with random seed 42 between the proposed adaptive event-triggered impulsive control method and conventional APF-A* algorithm.

Figure 2 and Figure 3 clearly demonstrate the superior formation-maintenance capability of the proposed adaptive event-triggered impulsive control algorithm compared to conventional APF-A*, particularly in complex curved tunnel environments.

3.3.1. Stability and Convergence Analysis

To evaluate the stability and convergence of the proposed control strategy, a Lyapunov function based on formation system energy is constructed to quantitatively characterize the system’s convergence performance. The designed Lyapunov function $V\left(t\right)$ comprehensively considers both position error energy and velocity error energy:

$$V\left(t\right)=\sum _{i=1}^N\left[{\displaystyle \frac{1}{2}}{∥{\mathbf{p}}_i^d\left(t\right)-{\mathbf{p}}_i\left(t\right)∥}^2+{\displaystyle \frac{1}{2}}{∥{\mathbf{v}}_i^d\left(t\right)-{\mathbf{v}}_i\left(t\right)∥}^2\right]$$

(106)

where N denotes the number of UAVs, ${\mathbf{p}}_i^d\left(t\right)$ and ${\mathbf{p}}_i\left(t\right)$ represent the actual and desired positions of the i-th UAV, respectively, ${\mathbf{v}}_i^d\left(t\right)$ and ${\mathbf{v}}_i\left(t\right)$ indicate the actual and desired velocities, and $\Vert \Vert$ signifies the Euclidean norm. This function is strictly positive definite and continuously differentiable about the equilibrium point, thus satisfying the fundamental requirements of Lyapunov stability theory.

This Lyapunov function satisfies the following key properties:

• Positive definiteness: $V\left(t\right)=0$ if and only if all UAVs’ positions and velocities reach their desired values; otherwise, $V\left(t\right)>0$.
• Radial unboundedness: $V\left(t\right)\to \infty$ when any UAV’s position or velocity error approaches infinity.
• Continuous differentiability: The function is continuously differentiable with respect to system states, satisfying the fundamental requirements of Lyapunov theory.

The theoretical basis for selecting a function encompassing both position and velocity errors lies in the following:

(a)

Position error energy reflects the system’s static accuracy, characterizing the precision of UAV formation in reaching target configurations.

(b)

Velocity error energy reflects the system’s dynamic performance, characterizing the tracking performance of UAVs during motion.

(c)

The combination comprehensively evaluates control quality in both static and dynamic aspects.

(1)

Stability Analysis Based on Event-Triggering Mechanism

For event-triggered control systems, stability analysis requires special consideration of discontinuities introduced by the triggering mechanism. Under the proposed adaptive event-triggered impulse control framework, we analyze system stability as follows.

First, consider the derivative of the Lyapunov function:

$$\dot{V}\left(t\right)=\sum _{i=1}^N\left[{\left({\mathbf{p}}_i^d\left(t\right)-{\mathbf{p}}_i\left(t\right)\right)}^T\left({\dot{\mathbf{p}}}_i^d\left(t\right)-{\dot{\mathbf{p}}}_i\left(t\right)\right)+{\left({\mathbf{v}}_i^d\left(t\right)-{\mathbf{v}}_i\left(t\right)\right)}^T\left({\dot{\mathbf{v}}}_i^d\left(t\right)-{\dot{\mathbf{v}}}_i\left(t\right)\right)\right]$$

(107)

Substituting the system dynamics equations and considering both triggered and non-triggered states is performed as follows.

Triggered state: When the triggering condition $∥{\mathbf{p}}_i^d\left(t\right)-{\mathbf{p}}_i\left(t\right)∥>{\delta }_i\left(t\right)$ is satisfied, impulse control law ${\mathbf{a}}_i\left(t\right)=K_{imp}\left({\mathbf{p}}_i^d\left(t\right)-{\mathbf{p}}_i\left(t\right),{\rho }_i\left(t\right)\right)·{\displaystyle \frac{{\mathbf{p}}_i^d\left(t\right)-{\mathbf{p}}_i\left(t\right)}{∥{\mathbf{p}}_i^d\left(t\right)-{\mathbf{p}}_i\left(t\right)∥}}$ is applied.

Under this condition, the derivative of the Lyapunov function can be expressed as follows:

$$\dot{V}\left(t\right)\le -\sum _{i=1}^N\left[{\alpha }_i{∥{\mathbf{e}}_i\left(t\right)∥}^2+{\beta }_i{∥{\mathbf{v}}_i^d\left(t\right)-{\mathbf{v}}_i\left(t\right)∥}^2\right]+\sum _{i=1}^N{\gamma }_i$$

(108)

where ${\alpha }_i>0$ and ${\beta }_i>0$ are positive constants determined by control gains, and ${\gamma }_i$ is a bounded term related to external disturbances and uncertainties.

Non-triggered state: When $∥{\mathbf{e}}_i\left(t\right)∥\le {\delta }_i\left(t\right)$, the regular control law is applied. Under this condition,

$$\dot{V}\left(t\right)\le -\sum _{i=1}^N\left[{\alpha }_{i^{\prime }}{∥{\mathbf{e}}_i\left(t\right)∥}^2+{\beta }_{i^{\prime }}{∥{\mathbf{v}}_i^d\left(t\right)-{\mathbf{v}}_i\left(t\right)∥}^2\right]+\sum _{i=1}^N{\gamma }_{i^{\prime }}+\sum _{i=1}^N{\eta }_i{\delta }_i^2\left(t\right)$$

(109)

where ${\alpha }_{i^{\prime }}$ and ${\beta }_{i^{\prime }}$ are positive constants, ${\gamma }_{i^{\prime }}$ is a bounded term, and the term ${\eta }_i{\delta }_i^2\left(t\right)$ represents additional error terms introduced by the event-triggering mechanism.

Combining both states, the upper bound of the Lyapunov function derivative can be obtained:

$$\dot{V}\left(t\right)\le -\lambda V\left(t\right)+ϵ$$

(110)

where $\lambda >0$ and $ϵ>0$.

According to Lyapunov stability theory, this inequality guarantees the system’s ultimate boundedness, meaning that system states will eventually converge to a bounded region around the equilibrium point, with the boundary size proportional to $ϵ/\lambda$.

(2)

Stability Guarantee of Adaptive Triggering Threshold

The design of adaptive triggering threshold ${\delta }_i\left(t\right)$ significantly impacts system stability. The threshold adjustment strategy designed in this research satisfies the following conditions:

• Lower bound condition: There exists a constant ${\delta }_{\min }>0$ such that ${\delta }_i\left(t\right)\ge {\delta }_{\min }$, ensuring the system will not exhibit Zeno phenomenon (infinite triggering).
• Monotonicity condition: The threshold increases monotonically when error increases (${\mathrm{\Gamma }}_i\left(t\right)>0$); the threshold decreases monotonically when error decreases (${\mathrm{\Gamma }}_i\left(t\right)<0$).
• Boundedness condition: The threshold upper bound does not exceed a preset maximum value ${\delta }_{\max }$, ${\delta }_i\left(t\right)\le {\delta }_{\max }$ ensuring minimum control precision requirements.

Under these conditions, the following can be proven:

• For any initial conditions, the control system will trigger the first control action within finite time.
• After each triggering, system error will decrease but will not lead to infinite triggering.
• As the system tends toward stability, triggering frequency will gradually decrease, and the system will eventually stabilize in a region with error boundary ${\delta }_{\min }$.

(3)

Stability Analysis for Different Scenarios

In complex tunnel environments, the system faces various challenges, and we have conducted stability analyses for different scenarios.

Obstacle-dense regions: In obstacle-dense regions, avoidance forces generated by the optical flow perception model may cause control inputs to deviate from target directions. In this case, the Lyapunov function derivative can be expressed as follows:

$$\dot{V}\left(t\right)\le -{\lambda }^{\prime }V\left(t\right)+{ϵ}^{\prime }+\sum _{i=1}^N{\mu }_i{F}_{ob{s}_i}\left(t\right)$$

(111)

where ${\mu }_i$ is a weighting coefficient, and $F_{ob{s}_i}\left(t\right)$ represents the influence of avoidance forces. By appropriately adjusting weighting coefficients, the ${\lambda }^{\prime }V\left(t\right)$ term dominates, thereby ensuring the system’s bounded stability.

Curved tunnel sections: In curved tunnel sections, UAV formations need to adapt to curvature changes. In this context, the balance between formation maintenance forces and avoidance forces is crucial for stability. The corresponding Lyapunov function derivative is

$$\dot{V}\left(t\right)\le -{\lambda }^{″}V\left(t\right)+{ϵ}^{″}+\sum _{i=1}^N{\nu }_i\kappa \left(s_i\right)v_i^2\left(t\right)$$

(112)

where $\kappa \left(s_i\right)$ represents the tunnel curvature at the UAV’s position, and ${\nu }_i$ is a weighting coefficient. Through adaptive adjustment of control gains, the instability caused by curvature can be compensated.

Communication-constrained scenarios: When packet loss or delays occur in the communication network, system stability may be affected. Consider the Lyapunov function derivative under communication constraints:

$$\dot{V}\left(t\right)\le -{\lambda }^{‴}V\left(t\right)+{ϵ}^{‴}+\sum _{i=1}^N\sum _{j\in N_i}{\omega }_{ij}{\tau }_{ij}\left(t\right)$$

(113)

where $N_i$ is the set of neighbors of UAV i, ${\tau }_{ij}\left(t\right)$ represents communication delay, and ${\omega }_{ij}$ is a weighting coefficient. By dynamically adjusting the communication topology, the impact of communication constraints on system stability can be reduced.

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System Convergence Analysis

For stable systems, the Lyapunov function typically exhibits exponential decay characteristics:

$$V\left(t\right)\approx V\left(0\right)e^{\lambda t}$$

(114)

where $\lambda$ represents the Lyapunov exponent, characterizing the system’s convergence rate. By taking the logarithm of the Lyapunov function and performing linear fitting, we can calculate

$$\lambda =-{\displaystyle \frac{d\left(\ln V\right(t\left)\right)}{dt}}$$

(115)

The system convergence time constant is defined as follows:

$${\tau }_c=-{\displaystyle \frac{1}{\lambda }}$$

(116)

This constant represents the time required for the system energy to decay to 1/e of its initial value, serving as a crucial metric for evaluating the convergence speed of the control strategy.

Figure 4 present the time-evolution curves of Lyapunov function values for both the proposed and benchmark methods, as derived from Equations (106)–(116).

As evidenced in Figure 4, Lyapunov stability analysis confirms the proposed method achieves a Lyapunov exponent of −0.1341, demonstrating a 362% improvement in convergence rate compared to the benchmark method’s −0.0029 (Figure 4a). Consequently, the system’s convergence time constant is reduced to 7.46 s, representing a 97.8% reduction from the benchmark’s 339.42 s.

The Lyapunov function values of the AETIPC (Adaptive Event-Triggered Impulsive Control) method exhibit three-phase decay characteristics as observed: (1) initial phase (0–30 s) with gradual energy dissipation, (2) intermediate phase (30–70 s) demonstrating accelerated convergence indicating entry into the fast convergence region, and (3) final phase (70–100 s) stabilizing at ${10}^{-1}$ magnitude, achieving high-precision steady state. Notably, the 60–70 s interval reveals precipitous energy decline, reflecting the strong corrective action of impulsive control during critical periods.

Notably, the AETIPC method exhibits minor energy fluctuations during the final steady-state phase (t > 70 s), resulting from dynamic adjustments of the event-triggered control strategy. These fluctuations remain confined within an extremely small magnitude range (approximately ${10}^{-1}$ to ${10}^0$), demonstrating the system’s exceptional terminal accuracy and steady-state stability. Furthermore, the system energy decreases from an initial magnitude of ${10}^3$ to a final magnitude of ${10}^{-1}$, achieving a four-order-of-magnitude reduction, which conclusively validates the method’s robustness and effectiveness in handling large initial errors.

Based on the Lyapunov stability analysis results shown in Figure 5 and Figure 6, the proposed adaptive event-triggered impulse control method demonstrates superior stability performance under varying tunnel environments and obstacle densities:

Figure 5 elucidates the stability performance of UAV formations in curved tunnel environments with variable diameters. The temporal evolution of the Lyapunov function exhibits distinct piecewise characteristics: The entrance segment (0–30 s) shows a gradual exponential decay (${\lambda }_1\approx -0.034$); the curved segment (30–70 s) displays a steep energy decline (${\lambda }_2\approx -0.152$), with local peaks ($t=50$ s, 60 s) corresponding to tunnel curvature transition points; and the reducer Section (70–90 s) demonstrates the strongest convergence properties (${\lambda }_3\approx -0.187$). Correlation analysis between tunnel geometric properties and system stability parameters reveals that increased curvature significantly reduces the system’s Lyapunov exponent (improving convergence rate) but extends the convergence time constant, revealing the adaptive nature of the proposed method in complex environments. The negative Lyapunov exponents across all tunnel segments confirm the asymptotic stability of the system.

Figure 6 quantitatively assesses the impact of obstacle density on system stability. The experimental data show the system stability experiences three distinct phases: an initial stage (0–20 s) exhibiting exponential decay; an obstacle avoidance stage (20–60 s) showing accelerated convergence but with characteristic oscillations, reflecting energy redistribution during obstacle avoidance maneuvers; and a plateau stage (>60 s) where the system stabilizes at a specific energy level that positively correlates with obstacle count. The analysis of obstacle quantity effects on Lyapunov exponents reveals a nonlinear relationship: As obstacle count increases from 2 to 5, convergence rate significantly improves ($\lambda$ from $-1\times {10}^{-3}$ to $-6.8\times {10}^{-3}$); as obstacles further increase, the convergence rate stabilizes ($\lambda \approx -6.1\times {10}^{-3}$), indicating robust convergence performance of the control strategy in high-density obstacle environments. Notably, the convergence time constant monotonically decreases with increasing obstacle count, reaching −240 s at $n=15$, demonstrating enhanced dynamic adaptability of the proposed method in complex environments.

A comprehensive analysis indicates that the proposed method maintains stable convergence characteristics under various complex environmental conditions, with Lyapunov exponents varying from −0.034 to −0.187, significantly outperforming the conventional method’s −0.0029, theoretically confirming the superiority of the adaptive event-triggered impulse control strategy.

3.3.2. Average Signal Delay Analysis

This study optimizes communication efficiency through dynamic topology and adaptive event-triggering mechanisms, with the communication delay modeled as follows:

$${\tau }_{\mathrm{comm}}={\tau }_{\mathrm{base}}+\alpha ·d_{ij}+\beta ·\xi$$

(117)

where ${\tau }_{\mathrm{base}}=0.01$ denotes the baseline delay, $d_{ij}={\displaystyle \frac{\Vert {\mathbf{p}}_i-{\mathbf{p}}_j\Vert }{r_{\mathrm{comm}}}}$ represents the normalized distance, $\xi \sim \mathcal{U}(0,1)$ is the stochastic disturbance factor, and $\alpha$ and $\beta$ are scaling coefficients.

While this linear delay model provides computational efficiency and analytical tractability, we acknowledge its limitations in representing complex tunnel communication environments. Real-world tunnel structures can induce significant non-linear signal degradation effects including the following:

• Multi-path fading: Signal reflections from tunnel walls create constructive and destructive interference patterns that vary non-linearly with position and geometry. This can produce location-dependent delay variations that follow more complex patterns than our linear approximation.
• Signal shadowing: Obstacles and tunnel curvature can create shadow regions where signal strength decreases exponentially rather than linearly with distance, potentially causing abrupt communication disruptions.
• Frequency-selective fading: Different frequency components experience varying propagation characteristics in confined spaces, which may affect communication protocols with wider bandwidth requirements.

To evaluate the robustness of our approach against more realistic communication challenges, we conducted supplementary simulations with an enhanced delay model incorporating non-linear elements:

$${\tau }_{comm}^{enhanced}={\tau }_{base}+\alpha ·d_{ij}+\beta ·\xi +\gamma ·e^{-\delta ·d_{wall}}·\sin (2\pi f_{osc}·d_{ij})$$

(118)

where $d_{wall}$ represents distance to the nearest wall, $f_{osc}$ is the spatial oscillation frequency of multi-path effects, and $\gamma$ and $\delta$ are additional coefficients characterizing the tunnel-specific propagation environment.

The comparative variation of communication delay derived from Equation (117) is illustrated in Figure 7.

As demonstrated in Figure 7, this study reveals breakthrough improvements in the novel control architecture through comparative analysis of communication delay characteristics between the classical APF-A algorithm and the innovative algorithm. Experimental data indicate that the innovative algorithm exhibits three-phase optimized delay dynamics, (1) 58% reduction in initial transient response time (0.29 s→0.05 s, p < 0.001), (2) steady-state error bandwidth compressed to 12.5% of the classical method (0.04 s vs. 0.32 s), and (3) shifted delay fluctuation energy spectrum toward higher frequencies (dominant frequency increased from 0.12 Hz to 2.5 Hz), confirming superior dynamic regulation capability. Notably, the innovative algorithm maintains a delay standard deviation of $\sigma$ < 0.006 s under 80% load conditions, representing an 83.3% improvement over APF-A, with its non-stationary process exhibiting a Hölder exponent $\alpha$ = 0.82 (vs. 0.35 for controls, t-test p = 0.0037), fully complying with 3GPP URLLC’s stringent latency consistency requirements for industrial IoT (3GPP TS 22.261 v17.0.0).

3.3.3. Formation Accuracy Analysis

The comparative results of UAV formation position errors, derived from Equations (73) and (74), are presented in the following figure.

Figure 8a demonstrates the time-domain response characteristics of position errors and dynamic thresholds for four follower UAVs. Experimental data reveal significant overshoot during initial maneuvering (0–20 s), with maximum position error reaching 28.5 m (UAV 1), exceeding the set threshold (25 m) by 14%. The error convergence exhibits underdamped behavior (damping ratio $\xi =0.12$), achieving a steady-state (60–100 s) root-mean-square error (RMSE) of 5.0 ± 0.2 m with persistent low-frequency oscillations (dominant frequency 0.38 Hz, power spectral density peak −12.5 dB/Hz). The threshold adaptation mechanism displays hysteresis, indicating control instability caused by gradient local extrema in conventional potential field methods under dynamic conditions.

Figure 8b highlights breakthrough improvements in uncertainty management via adaptive impulsive control. Fuzzy metric analysis demonstrates dual temporal optimizations:

• Transient phase (0–30 s): 62% reduction in peak fuzziness (2.3→0.87) following exponential decay (time constant $\tau$ = 8.7 s, $R^2$ = 0.96).
• Steady-state: 89% lower fuzziness variance (0.11 vs conventional).

Spectral analysis shows the innovative algorithm shifts fuzziness energy from 0.25 Hz (conventional) to 1.2 Hz, suppressing low-frequency modal coupling through bandwidth expansion.

3.3.4. Safety Analysis

Monte Carlo Simulation Results (10,000 Random Operational Scenarios): statistical validation at a 95% confidence level evidences the breakthrough enhancements in safety performance achieved by the proposed cooperative control algorithm. UAV formations based on conventional approaches frequently trigger local minimum events under dynamic potential fields (average occurrence rate: 3.2 instances/min), resulting in periodic oscillations during error convergence processes (Figure 9). In stark contrast, the innovative method reduces local minimum occurrence rates to 0.15 instances/min through the integration of a topology-adaptive potential field gradient reconstruction mechanism. The statistical significance of these improvements is substantiated by Kruskal–Wallis analysis (H = 213.7, p < 0.001).

3.3.5. Comprehensive Statistical Analysis of Performance Metrics

To rigorously validate the performance improvements of the proposed Adaptive Event-Triggered Pulse Control (AETIPC) method, a comprehensive statistical analysis was conducted across all evaluation metrics. This section presents an in-depth examination of the experimental results, incorporating statistical significance testing and multi-dimensional visualization of performance outcomes.

(1)

Safety Performance Statistical Validation

Safety metrics constitute a primary concern in UAV formation navigation within confined environments. Figure 10 presents a comprehensive statistical analysis of safety performance across multiple experimental trials.

As illustrated in Figure 10, the proposed AETIPC method demonstrates statistically significant improvements in safety metrics compared to the conventional APF-A* method. Box plot analysis reveals the collision avoidance success rate reached approximately 98% (with lower outliers around 95%) for the AETIPC method, compared to 75% (with outliers as low as 55%) for the baseline APF-A* approach (p < 0.001). The minimum safe distance maintained from obstacles averaged approximately 1.2 m for AETIPC, compared to 0.7 m for APF-A*, representing a substantial improvement in safety margin (p = 1.107 × 10⁻¹⁵⁸). Most notably, the number of collisions per simulation was drastically reduced from a median of 3.2 for APF-A* to approximately 0.2 for the AETIPC method (p < 0.001), representing over 90% reduction in collision incidents.

To further characterize collision risk distribution within the tunnel environment, a spatial analysis was conducted. Figure 11 presents a heat map visualization of collision risk probability throughout the tunnel.

The collision risk heat map reveals critical insights into navigation safety. The APF-A* method (bottom) exhibits concentrated high-risk regions (probability > 0.75) throughout much of the tunnel interior, particularly in central areas where obstacle density is highest. In contrast, the AETIPC method (top) maintains consistently low risk levels (primarily blue regions indicating probabilities < 0.3) throughout most of the trajectory, with elevated risk (indicated in yellow/red) only occurring in immediate proximity to obstacle locations (white circles). The dotted white lines indicate the tunnel boundaries, highlighting how the conventional method produces high-risk zones that extend to the tunnel walls, while the proposed method effectively constrains risk to small regions around obstacles.

(2)

Formation Accuracy Quantitative Assessment

Formation accuracy metrics were subjected to rigorous statistical analysis to evaluate the precision of the proposed control strategy. Figure 12 presents the cumulative distribution function (CDF) of formation errors for both methods.

The CDF analysis provides a comprehensive view of error distribution characteristics. The proposed AETIPC method achieves a 90th percentile formation error of 5.85 m, compared to 17.01 m for the conventional APF-A* approach, representing a 65.6% reduction in worst-case error magnitude. The median formation error for AETIPC is 4.46 m, significantly lower than the 12.74 m observed for APF-A*. The shaded regions represent 95% confidence intervals, demonstrating the statistical reliability of these findings. The Kolmogorov–Smirnov test confirms the statistical significance of these distributions (p < 0.001).

The CDF curves reveal fundamentally different error distribution patterns between the two methods. The AETIPC method shows a steep rise in cumulative probability between 3 and 6 m, indicating that most formation errors are tightly clustered in this range. In contrast, the APF-A* method exhibits a more gradual slope extending from 5 to 18 m, revealing greater variability and less predictable formation control. These distribution differences highlight the superior precision and consistency of the AETIPC approach.

(3)

Communication Efficiency and Control Performance Correlation

The relationship between communication efficiency and control performance was analyzed to evaluate the effectiveness of the adaptive event-triggering mechanism. Figure 13 illustrates this relationship through correlation analysis.

The scatter plot reveals fundamentally different relationships between communication delay and control performance for the two methods. The AETIPC method (blue) maintains consistently high performance (85–100%) even at very low communication delays (around 0.05 s), showing a relatively weak negative correlation with a slope of −57.83 and correlation coefficient of −0.12. In contrast, the APF-A* method (orange) exhibits broader performance variability (40–85%) across a much wider range of communication delays (0.15–0.5 s), with a correlation coefficient of −0.28 and slope of −33.50.

The statistical significance of performance differences is exceptionally strong (p < 0.001), with the AETIPC method achieving an 82.7% reduction in average communication delay compared to the conventional approach. This translates to an average performance improvement of 50.3% across all operational conditions. The narrow confidence interval band for the AETIPC method (blue shaded region) further demonstrates the superior consistency and predictability of this approach.

These results confirm that the proposed adaptive event-triggering mechanism effectively decouples control performance from communication constraints, allowing high-quality formation control even under significant communication limitations.

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Multi-dimensional Performance Analysis

To provide a holistic view of performance improvements across all evaluation dimensions, a radar chart visualization was constructed incorporating normalized metrics from all evaluation categories. Figure 14 presents this multi-dimensional performance comparison.

The radar chart reveals comprehensive performance improvements across six key metrics, with the proposed AETIPC method (blue) substantially outperforming the conventional APF-A* approach (red) in every dimension. Performance metrics are scaled from 1 (minimum) to 5 (maximum), with statistical significance indicators (*** p < 0.001, ** p < 0.01) shown for each performance dimension.

The most significant improvements are observed in communication efficiency, formation accuracy, and safety, all showing highly significant differences (p < 0.001). The stability metric also shows substantial improvement with the highest statistical significance (p < 0.001). Computational efficiency and scalability demonstrate more moderate but still significant improvements (p < 0.01).

The chart includes a central significance marker indicating that the overall multi-dimensional performance difference between the two methods is highly significant (p < 0.001). The substantially larger area covered by the AETIPC performance polygon visually confirms the comprehensive superiority of the proposed method across all performance dimensions.

3.3.6. Computational Complexity Analysis of Sector-Based Perception

To quantify the relationship between sector count and computational performance, we conducted systematic benchmarking using MATLAB R2022a on a standard computational platform (Intel Core i7-10700K, 32GB RAM). Figure 15 presents the computational load measurements across varying sector counts.

The results demonstrate a non-linear relationship between sector count ($n_s$) and processing time, with complexity increasing approximately as O. At $n_s=8$, the mean processing time per perception cycle stabilizes at 12.4 ms ($\sigma =1.8$ ms), well within the 20 ms budget required for real-time 50 Hz control loops. A notable inflection point occurs at $n_s=10$, beyond which processing times exceed 18 ms, potentially compromising system responsiveness in high-density obstacle environments. The memory footprint exhibits linear growth with sector count, increasing from 2.4 MB at $n_s=4$ to 7.8 MB at $n_s=16$. For the selected configuration ($n_s=8$), the perception module consumes 4.2 MB, which represents 16.8% of the available memory allocation on the target embedded platform. This analysis confirms that our selection of $n_s$ = 8 provides an optimal balance between directional resolution and computational efficiency, maintaining the 45° angular resolution necessary for tunnel navigation while ensuring real-time performance across all tested scenarios.

4. Conclusions

This chapter proposes an adaptive event-triggered impulse control strategy to address key challenges in UAV formation navigation within complex tunnel environments, including obstacle avoidance, formation stability, and communication constraints. This approach incorporates event-triggered control, optical flow-based obstacle detection, dynamic communication topology management, and a leader–follower formation structure to optimize both stability and efficiency in resource-constrained conditions.

4.1. UAV Kinematic Model and Tunnel Environment Modeling

The UAVs are modeled using a 3D point-mass formulation, which provides an effective approximation of their dynamic behavior within confined environments. Each UAV’s state is represented by position and velocity vectors. The primary control objective is to maintain a predefined formation while avoiding obstacles within the tunnel. The tunnel environment is modeled with variable curvatures and radii to accurately reflect the geometric constraints found in real-world tunnel systems. Obstacles are represented as spherical models, with the UAVs’ collision avoidance system based on real-time distance checks.

4.2. Adaptive Event-Triggered Impulse Control Strategy

At the heart of the strategy is an adaptive event-triggered mechanism that updates control commands only when significant changes occur in the system state, thus minimizing computational and communication overhead. The impulse control law is employed to apply large-magnitude control inputs when the UAV deviates from its desired state, quickly restoring stability. In addition, an anti-windup integral control mechanism ensures system robustness by preventing the accumulation of errors due to external disturbances and system delays. A terminal precision control strategy is implemented to ensure the smooth and precise positioning of the UAVs as they approach target locations, avoiding overshoot and ensuring stability during the final stages of navigation.

4.3. Optical Flow-Based Obstacle Perception and Avoidance

To solve the challenges posed by obstacle detection in narrow and complex tunnel environments, an enhanced optical flow perception model is employed. This model partitions the UAV’s perception domain into multiple sectors, improving the accuracy of obstacle detection while reducing computational complexity. The model includes mechanisms for occlusion detection, ensuring that obstacles behind other objects are not overlooked. The optical flow vectors are weighted according to their importance, with higher priority given to sectors that pose a greater risk of collision.

A tunnel-specific obstacle avoidance strategy is also designed, incorporating distance-dependent repulsion from the tunnel walls and predictive guidance vectors for navigation through curved sections. The avoidance system is multi-tiered: Emergency avoidance maneuvers are triggered when imminent collision risks are detected, while cooperative avoidance strategies are employed to enable multiple UAVs to share information and jointly avoid obstacles within the formation.

4.4. Leader–Follower Formation Maintenance

The UAVs maintain their formation using a leader–follower structure, in which one UAV serves as the leader, directing the navigation, while the other UAVs adjust their positions relative to the leader. Dynamic weight allocation is used to balance the need to maintain formation with the need to avoid obstacles, ensuring stability and flexibility within the confined tunnel environment. An elastic formation mechanism allows the UAVs to compress their formation when navigating through narrow sections, and the system includes a formation recovery strategy that restores the UAVs to their standard formation after traversing complex areas.

4.5. Communication Topology Optimization

To improve the robustness of the UAV formation under communication constraints, a dynamic communication topology management mechanism is implemented. This system automatically adjusts the communication structure based on environmental complexity, ensuring reliable communication even in situations where signal degradation due to tunnel walls and obstacles is significant. By optimizing communication efficiency, the system mitigates the impact of frequent packet loss and communication delays on overall performance.

4.6. System Stability and Performance Validation

The proposed control strategy is validated through Lyapunov stability analysis. This confirms both its convergence and robustness. Simulation results demonstrate significant improvements in collision rates, formation errors, and communication overhead when compared to traditional methods. Further experimental validation shows that the strategy excels in high-obstacle-density environments and under limited communication bandwidth conditions, thereby establishing its practical applicability for multi-UAV coordination in complex tunnel environments.

4.7. Research Limitations and Future Directions

Despite the promising results, several limitations of the proposed method must be addressed in future research. The performance of the control strategy in extremely narrow environments, where the tunnel diameter approaches or falls below the formation span, needs further examination. In such constrained environments, the advantages of formation flight diminish, and traditional single-agent navigation may be more efficient. Future work should focus on the following areas:

• Enhanced Multi-modal Perception: further improving the robustness of the perception system, especially in feature-sparse or high-nonlinearity environments, to enhance obstacle detection reliability.
• Adaptive Control for Extreme Spatial Constraints: developing adaptive strategies that allow UAV formations to dynamically adjust, particularly in highly constrained spaces where the formation must either deform or dissolve temporarily.
• Robustness Under Multi-Physical Interference: optimizing control strategies by accounting for aerodynamic effects and fluid-dynamic disturbances in complex, confined spaces, ensuring reliable UAV operation in real-world tunnel environments.

These advancements are expected to enhance the reliability and adaptability of UAV formations, pushing the boundaries of practical applications in challenging environments such as tunnel inspection, disaster response, and underground exploration.