Online Synchronous Coordinated Assignment and Planning for Heterogeneous Fixed-Wing UAVs

Abstract

This paper addresses the Multi-Target Reconnaissance (MTR) problem for heterogeneous Fixed-Wing Unmanned Aerial Vehicles (FW-UAVs), focusing on synchronized and time-optimal mission execution under stringent constraints. A two-stage coordinated assignment and planning framework is proposed. First, a time-balanced clustering algorithm is designed to minimize the overall mission duration while balancing individual UAV workloads by jointly employing a target reallocation strategy and an improved Genetic Algorithm (GA). Subsequently, an online trajectory planning method based on differential flatness is developed, integrating a robust replanning and flight-time synchronization strategy to ensure coordinated execution. Simulation results unequivocally demonstrate that the proposed approach enhances time optimality and temporal coordination in complex scenarios.

1. Introduction

Multi-Target Reconnaissance (MTR) is a critical mission in both military and civilian domains, with applications ranging from battlefield target engagement to disaster-site data acquisition [1,2]. Generally, MTR efforts relied on single Unmanned Aerial Vehicle (UAV) performing sequential reconnaissance of multiple targets [3,4]. However, the single UAV proves inefficient, particularly in time-sensitive scenarios. To enhance task efficiency, the deployment of Fixed-Wing Unmanned Aerial Vehicles (FW-UAVs) has gained increasing traction. FW-UAVs offer distinct advantages over single UAVs, due to their superior cooperative sensing and reconnaissance capabilities [5,6,7].

Nevertheless, in practical emergency MTR scenarios, FW-UAVs are typically heterogeneous, with different cruise speeds [8]. Furthermore, to maximize the operational capacity of each UAV while ensuring mission continuity, it is imperative that all subtasks reach saturation and are completed synchronously. MTR inherently involves multiple interrelated processes, including Multi-Target Assignment (MTA), path planning, and robust control. Consequently, two critical challenges emerge during the above processes:

• Considering the varying optimal cruising speeds of heterogeneous UAVs, how can a set of targets be optimally assigned to ensure each FW-UAV completes its MTR within an equal and minimized mission duration?
• Subsequent to MTA, how can kinematically feasible trajectories be generated for each FW-UAV under dynamic environmental constraints, while simultaneously guaranteeing temporal synchronization among FW-UAVs throughout the whole process?

MTA constitutes the initial phase in tackling the MTR problem. By judiciously distributing targets among specific FW-UAVs, the inherently complex multi-UAV planning task can be decomposed into simpler, independent single-UAV problems. A diverse array of algorithms has been proposed for MTA, broadly categorized into traditional heuristic methods and metaheuristic optimization approaches. Traditional methods include Genetic Algorithms (GAs) [9], Particle Swarm Optimization (PSO) [10], Ant Colony Optimization (ACO) [11], Self-Organizing Maps (SOMs), K-means, and so on [12,13]. For example, a decentralized GA-based strategy for cooperative search, where each agent independently optimizes its task, was introduced in [14]. Choi proposed a two-stage GA framework enabling individual agents to optimize local task sequences in [15]. Furthermore, a modified PSO algorithm was presented in [16] to address uncertain time constraints in rescue missions. To surmount the limitations in modeling complexity in conventional methods, Li et al. developed a deep reinforcement learning model specifically tailored for simultaneous MTA in [17]. However, a limitation of most aforementioned studies is their oversight of the inherent heterogeneity among UAVs.

The aforementioned approaches commonly fail to fully consider the diverse operational characteristics of heterogeneous UAVs. To address this issue, several algorithms have been explored. For example, the K-means method, employed in [18], tends to produce clusters for multi-robot MTA. Similarly, the SOMs method, initially developed for high-dimensional data visualization [19], was adapted in [20] for MTA and path planning through a neural mapping approach. However, these methods primarily focus on basic optimization goals and remain inadequate for complex missions that require precise temporal coordination and optimal performance under UAV heterogeneity.

Following MTA, it becomes imperative to synchronously plan kinematically feasible and safe trajectories for each UAV. A diverse range of path planning algorithms has been developed, broadly categorized into geometry-based, sampling-based, optimization-based, and intelligent approaches [21]. Geometry-based approaches, including Dijkstra, A, D, and D* Lite, are widely employed for path planning in static environments. The A* algorithm was successfully applied in [22] to address path planning under wind disturbances. Optimization-based methods, such as GA, have been extensively utilized to fulfill specific mission requirements [23,24]. Ref. [25] proposed a Generative AI (GAN) algorithm combined with traditional RRT and BFOA to predict UAV paths in a dynamic environment. Nevertheless, prior works predominantly concentrate on single-UAV path planning and neglect crucial aspects of time synchronization. Furthermore, most existing path planning approaches, though effective for single-UAV missions or static environments, remain inadequate for coordinated multi-UAV operations that demand synchronization, compliance with kinematic constraints, and obstacle avoidance in dynamic and uncertain settings.

To facilitate synchronized execution, various Trajectory Planning (TP) methods have been developed. A representative example is the cooperative penetration strategy proposed by Luo et al. [26], which utilizes a deep deterministic policy gradient (DDPG) algorithm to enable multiple UAVs to penetrate defenses. Liu et al. introduced a spatiotemporal-refined voting mechanism for PSO to address the cooperative path planning problem in [27]. Their objective function accounted for obstacles, threat regions, and arrival time constraints. However, the iterative nature of such methods poses significant challenges for large-scale real-time deployment, particularly when maintaining precise temporal coordination across heterogeneous UAVs.

Alternatively, Differential Flatness (DF) theory [28] has been applied to reduce the state dimensionality of TP problems, while consensus-based approaches adjust replanning intervals to achieve synchronization. For example, ref. [29] introduced consensus variables to align replanning intervals across UAVs, but without explicit mechanisms to match them to the global mission timeline. Although DF offers computational efficiency and consensus algorithms enhance coordination, most DF-based methods assume fully known environments. Many consensus approaches neglect heterogeneous kinematic constraints and the real-time requirement for dynamic replanning. Developing a framework that integrates the above advantages with online replanning under dynamic constraints and environmental uncertainty remains an important open challenge.

In this paper, we address the MTR problem for heterogeneous FW-UAVs by proposing a synchronized, coordinated, and online framework that integrates MTA and TP for temporally coordinated reconnaissance missions. A global optimization problem is formulated to minimize the overall mission duration and control effort while balancing individual UAV flight times, and is decomposed into two coupled sub-problems. The first sub-problem focuses on time-balanced task allocation, where K-means-based spatial initialization is combined with an iterative target reallocation procedure to equalize flight times, followed by an improved GA to optimize the intra-cluster visiting sequence. The second sub-problem addresses flight-time-consistent trajectory planning by leveraging the DF property of fixed-wing UAV dynamics, enabling the real-time generation of kinematically feasible trajectories with online replanning for collision avoidance, explicit flight-time synchronization, and robustness to dynamic and unforeseen environmental changes.

Compared with existing studies, the main contributions of this work are summarized as follows.

• A new practical time-balanced clustering algorithm is proposed for heterogeneous FW-UAVs. This method minimizes the overall mission duration and balances individual UAV flight durations by strategically reallocating targets and optimizing the intra-cluster visiting sequence. By decoupling temporal coordination from route optimization, the proposed approach achieves significantly improved computational efficiency for time-coordinated MTA problems, which is further validated through theoretical time-complexity analysis and extensive numerical simulations.
• A practical replanning flight-time synchronization mechanism is proposed, which adaptively adjusts the replanning duration for each UAV. Inspired by consensus-based coordination principles, this mechanism enables the synchronization of flight times, and a rigorous convergence proof is provided to guarantee persistent synchronization.
• An online trajectory planning algorithm is developed using the DF property of FW-UAVs. This planner operates under stringent kinematic constraints, ensures collision avoidance in unknown and dynamic environments, and rigorously respects terminal time constraints. The applicability of DF to fixed-wing UAVs is explicitly derived and discussed, extending its use beyond conventional rotary-wing platforms.

The remainder of this paper is organized as follows. Section 2 outlines the preliminaries, including FW-UAV dynamics, graph theory, and the problem formulation. Section 3 describes the time-balanced clustering algorithm, and Section 4 details the DF-based trajectory planner with the flight-time synchronization mechanism. Section 5 presents the simulation results, and Section 6 concludes the paper.

2. Preliminaries

2.1. FW-UAV Model

In this paper, the term UAVs specifically refers to FW-UAVs. Let $p_i^e\in {\mathcal{R}}^{3\times 1}$ represent the position vector of UAV i in the Earth-fixed coordinate frame, and let ${\mathrm{\Theta }}_i={[{\varphi }_i,{\theta }_i,{\psi }_i]}^T\in {\mathbb{R}}^{3\times 1}$ denote its Euler angles. The rotation matrix $R_{b,i}^e\in SO\left(3\right)$, mapping vectors from the body-fixed frame to the Earth-fixed frame, is defined accordingly. $R_{a,i}^b$ denotes the rotation matrix from the air-relative frame to the body-fixed frame. The coordinate frames are illustrated in Figure 1.

${\alpha }_i$ is the angle of attack and ${\beta }_i$ is the sideslip angle. The dynamics of UAV i are described by the following  [29,30]:

$$\begin{array}{cc}\hfill {\dot{p}}_i^e& =v_{a,i}^e,\hfill \\ \hfill m_i{\dot{v}}_{a,i}^e& =R_{b,i}^e\left[R_{a,i}^b\left(T{h}_i+D_i+L_i\right)\right]+G_i,\hfill \\ \hfill {\dot{R}}_{b,i}^e& =R_{b,i}^e{⌊{\mathrm{\Omega }}_i⌋}_{\times },\hfill \\ \hfill {\mathrm{\Omega }}_i& ={\mathrm{\Omega }}_{c,i},\hfill \end{array}$$

(1)

where $m_i$ is the mass of UAV i, and $G_i$ denotes the gravitational force vector, with $G_i=-m_{i}g{e}_2$. $e_i$ is the unit vector in the i-direction in Earth-fixed coordinate frame, and g denotes the gravitational constant. ${\mathrm{\Omega }}_i={\left[{\omega }_{x,i},{\omega }_{y,i},{\omega }_{z,i}\right]}^T$, with ${\omega }_{\mu ,i},\mu =x,y,z$ representing the angular velocities, and ${⌊{\mathrm{\Omega }}_i⌋}_{\times }$ denotes the skew-symmetric matrix of ${\mathrm{\Omega }}_i$ as

$${⌊{\mathrm{\Omega }}_i⌋}_{\times }=\left[\begin{array}{ccc}0& -{\omega }_{z,i}& {\omega }_{y,i}\\ {\omega }_{z,i}& 0& -{\omega }_{x,i}\\ -{\omega }_{y,i}& {\omega }_{x,i}& 0\end{array}\right].$$

(2)

$L_i$ is the lift force of UAV i along the $y_a$ air axis and $D_i$ is the drag force along the negative $x_a$ air axis.

Assumption 1. 

1. 

The wind speed is below 3 m/s, so sideslip ${\beta }_i$ can be neglected;

2. 

The aerodynamic forces generated by control-surface deflections ${\delta }_i$ are negligible due to the imposed limits on control-surface ranges.

Under Assumption 1, the aerodynamic forces can be expressed as

$$\begin{array}{cc}\hfill L_i& \approx {\displaystyle \frac{1}{2}}\rho V_{a,i}^2{S}_i{C}_{L,i}\left({\alpha }_i\right)a_2,\hfill \\ \hfill D_i& \approx -{\displaystyle \frac{1}{2}}\rho V_{a,i}^2{S}_i{C}_{D,i}\left({\alpha }_i\right)a_1,\hfill \end{array}$$

(3)

where $\rho$ is the air density, $V_{a,i}=\left|\right|v_{a,i}\left|\right|$ is the airspeed of UAV i, and $S_i$ is the wing reference area. $C_{L,i}$ and $C_{D,i}$ are the lift and drag aerodynamic coefficients, respectively. Let ${\mathrm{\Omega }}_{c,i}={\left[{\omega }_{c,x,i},{\omega }_{c,y,i},{\omega }_{c,z,i}\right]}^T$ be the commanded angular velocities and $T{h}_i=T{e}_i\cos \left({\alpha }_i\right)a_1$ be the commanded thrust along the $x_a$ air axis, where the attack angle is small. $a_i$ is the unit vector in the i-direction in the air-relative coordinate frame. In this paper, the control input for UAV i is denoted by $U_i$:

$$U_i=\left\{{\omega }_{c,x,i},{\omega }_{c,y,i},{\omega }_{c,z,i},T{e}_i\right\}.$$

(4)

To simplify notation, the position vector $p_i^e$ is subsequently written as $p_i$. The main symbols used in this article are shown in

Table 1. Symbol and Meaning.

Symbol	Meaning
N	Number of UAVs
M	Number of reconnaissance targets
Q	Number of no-fly zones
$P_{target}$	Set of target positions
A	Target index set
$wP{t}_i$	Ordered target position set of UAV i
$A_i$	Ordered target index set of UAV i
$P_{obs}$	Set of no-fly zone positions
$K_i$	Number of targets of UAV i
$O_{A,i}$	Center of the target cluster of UAV i
$V_{cruise}$	Set of optimal cruising speeds for UAVs
$P_i$	Post-optimization trajectory of UAV i
$p_i$	Current position of UAV i
$L{o}_i$	Local replanning region of UAV i
$wP{t}_{i,local}$	Ordered local target set in the replanning region of UAV i
$N_{l,i}$	Number of waypoints in the replanning region of UAV i
$N_p$	Number of Bézier curve segments

.

2.2. Graph Theory

Consider a formation of N FW-UAVs, indexed from $i=1$ to N. The interactions among them are modeled by a directed graph $G=(V,E,W)$, where $V=\{v_1,\dots ,v_N\}$ is the set of nodes, and $E\subset V\times V$ is the set of directed edges. $W=\left[w_{i,j}\right]\in {\mathcal{R}}^{N\times N}$ indicates the weighted adjacency matrix with $w_{i,j}>0$, if and only if $(v_i,v_j)\in E$. For node $v_i$, the weighted in-degree is defined as ${\mu }_i={\sum }_{j=1}^N{w}_{i,j}$, where $N_i=\left\{j\right|(v_i,v_j)\in E\}$ is the set of neighbors of node $v_i$. Define the Laplacian matrix as $L=D-W$, where $D=diag\left\{{\mu }_i\right\}\in {\mathbb{R}}^{N\times N}$. A directed path from node $v_i$ to $v_j$ is a sequence of directed edges connecting them. If a node has a directed path to every other node in the graph through a subset of edges G, G is said to have a spanning tree and the node is called the root.

2.3. Problem Formulation

In MTR, a swarm of $N\in \mathbb{N}$ heterogeneous UAVs is considered, where each UAV i has a distinct optimal cruising speed $V_{cruise,i}$. Let the target set be $A=\left\{j|j=1,\dots ,M\right\}$, with their corresponding position set being $P_{target}=\left\{p_{target,j}\subseteq {\mathbb{R}}^{3\times 1}|j\in A\right\}$. For each UAV i, a subset of assigned targets is denoted by $A_i=\left\{k\right|k\in A\}$, with its associated ordered position set represented by $wP{t}_i\subseteq P_{target}$. Both $A_i$ and $wP{t}_i$ are treated as ordered sequences. Simultaneously, there are Q no-fly zones randomly distributed over the mission area $\mathcal{M}$, with their positions represented by the set $P_{obs}$. Each UAV is required to visit all assigned targets in its respective set $A_i$ and return to its initial departure point.

Given the randomly distributed target set A and a set of heterogeneous FW-UAVs with differing optimal cruising speeds $V_{cruise,i}$, it is necessary to (1) assign each UAV i an ordered target subset $A_i$ such that all UAVs complete their missions in equal and minimized mission durations T; and (2) in response to environmental uncertainty and limited sensing range, plan a trajectory for each UAV online to minimize control effort, while ensuring compliance with the same flight time $T_i=T$, obstacle avoidance Q, and kinematic constraints.

This problem is formally modeled as the following constrained optimization problem:

$$\begin{array}{cc}\hfill & \underset{A_i,w{P}_{t_i}}{\min }\max T_i+\underset{P_i}{\min }\sum _{i=1}^N{\int }_0^{T_i}{(p_i^{\left(3\right)}\left(t\right))}^{2}dt,\hfill \\ \hfill & s.t.\hfill \\ \hfill & S{1}_1:\bigcup _{i=1}^N{A}_i=A,{\cap }_{i=1}^N{A}_i=⌀,\hfill \\ \hfill & S{1}_2:\max \left(T_i\right)-\min \left(T_j\right)<{\epsilon }_1,\hfill \\ \hfill & S{2}_1:w{P}_{t_i}\subseteq P_i,\hfill \\ \hfill & S{2}_2:\parallel T_i-T\parallel \le {\epsilon }_T,\hfill \\ \hfill & S{2}_3:\parallel p_i\left(t\right)-p_{obs,q}\parallel >R_{obs,q}+R_{UAV},\forall q\in Q,\hfill \\ \hfill & S{2}_4:\parallel p_i\left(t\right)-p_j\left(t\right)\parallel >2R_{UAV},i,j\in N,j\ne i,\hfill \\ \hfill & S{2}_5:V_{a,i}\left(t\right)\in [V_{a,i,\min },V_{a,i,\max }],\hfill \\ \hfill & S{2}_6:\parallel n_{i,\mu }\parallel \le n_{i,\mu \_max},\parallel n_{i,z}\parallel \le v_i^2/\left(g{R}_{turn}\right),\hfill \\ \hfill & S{2}_7:{\varphi }_i\left(t\right)\in [-{\varphi }_{i,\max },{\varphi }_{i,\max }],{\theta }_i\left(t\right)\in [-{\theta }_{i,\max },{\theta }_{i,\max }].\hfill \end{array}$$

(5)

where $\mu \in \{x,y,z\}$. $S1$ represents the synchronized coordination constraints among UAVs, and $S2$ includes flight time, obstacle avoidance, and dynamic feasibility constraints for individual UAV i.

For constraints in $S2$, $p_i\left(t\right)\in P_i$ represents the position of UAV i at time t. In the individual trajectory planning process, each UAV must traverse its assigned targets in sequence order, adhering to $S{2}_1$. To satisfy $S{2}_2$, the actual flight time $T_i$ must converge to the desired mission duration $T\in {\cap }_{i=1}^N\left[T_{i,min},T_{i,max}\right]$, where $T_{i,min},T_{i,max}$ represent the permitted bounded durations for UAV i. When flying in unknown environments, such as battlefield scenarios, UAVs must dynamically avoid time-varying obstacles and maintain safe distances from other UAVs in real time. Specifically, $S{2}_3$ ensures that UAVs maintain a safe distance from obstacles, where $p_{obs,q},q\in Q$ is the center position of the q-th obstacle, $R_{obs,q}$ is the required safety radius, and $R_{UAV}$ is the safe radius of the UAV. Concurrently, $S{2}_4$ ensures that UAVs do not conflict with one another during the simultaneous execution of their tasks. The velocity constraint $S{2}_5$ guarantees that UAVs operate within their feasible speed range. The UAV’s overloads in all directions, denoted by $n_{i,\mu }=a_{i,\mu }\left(t\right)/m_{i}g,\mu \in x,y,z$, are constrained by structural strength and turning radius limitations, specifically $S{2}_6$, where $n_{i,\mu \_max}$ denotes the maximum allowable overload in direction $\mu$, and $R_{turn}$ is the minimum turning radius. Furthermore, each UAV must satisfy specific attitude constraints $S{2}_7$, where ${\varphi }_i$ and ${\theta }_i$ represent the roll angle and pitch angle of UAV i, respectively. To constrain the lateral acceleration, ${\varphi }_i$ needs to satisfy $\left|{\varphi }_i\right|\le {45}^{\circ }$.

To address this constrained optimization problem, a divide-and-conquer strategy is adopted. The original problem is decomposed into two sub-problems: a time-balanced clustering-based MTA problem and a DF-based TP problem. The overall solution framework is illustrated in Figure 2.

3. Time-Balanced Clustering Algorithm for MTA

This section presents a time-balanced clustering algorithm designed for heterogeneous UAVs. This algorithm addresses the MTA problem under synchronized task execution requirements, formulated as follows:

Sub-problem 1:

$$\begin{array}{cc}\hfill & \underset{A_i,wP{t}_i}{\min }\max T_i,\hfill \\ \hfill & s.t.\hfill \\ \hfill & S{1}_1:\bigcup _{i=1}^N{A}_i=A,\bigcap _{i=1}^N{A}_i=⌀,\hfill \\ \hfill & S{1}_2:\max \left(T_i\right)-\min \left(T_j\right)<{\epsilon }_1.\hfill \end{array}$$

(6)

This sub-problem aims to minimize the maximum flight time among all UAVs by optimizing the assigned target sets $A_i$ and their visiting sequences $wp{t}_i$, subject to two constraints. $S{1}_1$ ensures that the target sets assigned to different UAVs are mutually exclusive and that all targets are allocated. $S{1}_2$ enforces approximate equality of flight times across all UAVs.

To expedite the initial target allocation process, a standard K-means clustering algorithm is employed. Based on the spatial distribution of the target positions $p_{target}\in P_{target}$ and the number N of UAVs, an initial division of targets into N clusters is performed. Then, an improved GA is utilized to optimize the visiting order among the targets within that cluster, forming an ordered set $A_i$ and its corresponding ordered position set $wP{t}_i$. For notational simplicity, define $wPt=\left\{wP{t}_1,\dots ,wP{t}_N\right\}$.

In practical scenarios, the varying number of targets M and the random distribution of targets $P_{target}$, coupled with the differences in UAV optimal cruising speeds $V_{cruise,i}\in V_{cruise}$, result in significant discrepancies in individual UAV flight time $T_i$ without coordination. $T_i$ can be approximated as

$$T_i={\gamma }_{turn,i}\sum _{j\in A_i}\left(\left|\right|p_{target,j+1}-p_{target,j}\left|\right|/V_{cruise,i}\right).$$

(7)

where ${\gamma }_{turn,i}$ is a scaling coefficient derived from experimental data, which corrects flight-time estimation bias according to each UAV’s minimum turning radius.

3.1. Cluster-Based Target Reallocation Strategy

To address the aforementioned limitations, a target reallocation strategy based on dynamic cluster adjustments is proposed in Algorithm 1.

Algorithm 1 Time-Balanced Clustering Algorithm for MTA

Input: Number of UAVs N, optimal cruising speeds $V_{cruise}$, and target positions $P_{target}$.

Output: Adjusted target clusters $A_i$, waypoint sets $wP{t}_i$, and minimized mission duration T.

1. Initialization

$wPt\leftarrow K\_MEANS\left(P_{target},N\right)$

For  $i=1,\dots ,N$  do

$wP{t}_i\leftarrow GA\left(wP{t}_i\right)$

End For

$k_2=1$

2. Iterative Refinement

While $\max \left(T_j\right)-\min \left(T_i\right)>threshold$ and $k_2<L_2$ do

$k_1=1$

2.1. Calculate the center position $O_{A,i}$ and flight time $T_i$ for each cluster:

$O_{A,i}={\sum }_{i=1}^{K_i}wP{t}_i/K_i$

$T_i\leftarrow {\gamma }_{turn,i}{\displaystyle \sum _{j\in A_i}}\left(\left|\right|p_{target,j+1}-p_{target,j}\left|\right|/V_{cruise,i}\right),i=1,\dots ,N$

$T\leftarrow TIME\_MEAN\left(T_1,\dots ,T_N,N\right)$

2.2. Sort. Sort UAVs in descending order of flight time $T_i$.

2.3. Iterative Refinement

For each UAV $i=1,\dots ,N-1$ in the sorted order do

While $\left|\right|T_i-T\left|\right|>threshold$ and $k_1<L_1$ do

If $T_i<T$ do

$\left(p_{tmp},j\right)\leftarrow FIND\_POINT\_j\left(wP{t}_i,O_{A,i},P_{target}\right)$

$wP{t}_j\leftarrow REMOVE\_POINT\left(wP{t}_j,p_{tmp}\right)$

$wP{t}_i\leftarrow ADD\_POINT\left(wP{t}_i,p_{tmp}\right)$

Else

$p_{tmp}\leftarrow FIND\_POINT\_i\left(wP{t}_i,O_{A,i}\right)$

$wP{t}_i\leftarrow REMOVE\_POINT\left(wP{t}_i,p_{tmp}\right)$

$j\leftarrow FIND\_CLUSTER\left(i,p_{tmp},O_A\right)$

$wP{t}_j\leftarrow ADD\_POINT\left(wP{t}_j,p_{tmp}\right)$

calculate $T_i,i=1,\dots ,N$ and update T

$k_1\leftarrow k_1+1$

End If

End While

End For

2.4. Optimize inter-order

For $i=1,\dots ,N$ do

$wP{t}_i\leftarrow GA\left(wP{t}_i\right)$

End For

$k_2\leftarrow k_2+1$

pruning

End While

Algorithm 1 is executed to perform adaptive adjustments of the clusters. In Step 1, the initial target clusters $A_i$ are obtained. Step 2 is the core part of this algorithm, where the visiting order within each set $A_i$ and $wp{t}_i$ is optimized.

In Step 2.1, the mission duration $T={\displaystyle \sum _{i=1}^N}{T}_i/N$ is calculated. In Step 2.2, to ensure adjustment of target clusters in each iteration, the cluster adjustment order is determined. In Step 2.3, to minimize the impact of target point selection on the existing target clusters, $p_{tmp}$ is defined as the target point that is closest to the center point $O_{A,i}$ of the set $A_i$, whose corresponding cluster is $j,j\ne i$, i.e.,:

$$\begin{array}{c}\hfill \mathrm{FIND}\_\mathrm{POINT}\_\mathrm{j}\left(wPt,O_{A,i},P_{target}\right):p_{tmp}=\underset{p\in wP{t}_j,j>i}{\arg \min }\left(\left|\right|p-O_{A,i}\left|\right|\right).\end{array}$$

(8)

Conversely, $\mathrm{FIND}\_\mathrm{POINT}\_\mathrm{i}\left(wP{t}_i,O_{A,i}\right)$ is defined as finding the point $p_{tmp}$ within set $A_i$ that is furthest from $O_{A,i}$:

$$\begin{array}{c}\hfill \mathrm{FIND}\_\mathrm{POINT}\_\mathrm{i}\left(wP{t}_i,O_{A,i}\right):p_{tmp}=\underset{p\in wP{t}_i}{\arg \max }\left(\left|\right|p-O_{A,i}\left|\right|\right).\end{array}$$

(9)

Furthermore, $\mathrm{FIND}\_\mathrm{CLUSTER}\left(p_{tmp},O_A\right)$ is defined as finding the unadjusted cluster j such that the distance between its center point $O_{A,j}\left(j>i\right)$ and $p_{tmp}$ is minimized.

When $p_{tmp}$ is added to the ordered set $wP{t}_i$, the point $p_{mid,j}=\underset{p\in wP{t}_i}{\arg \min }\left(\left|\right|p-p_{tmp}\left|\right|\right)$ closest to $p_{tmp}$ and its corresponding subsequent point $p_{mid,j+1}$ are identified within $wP{t}_i$ in $\mathrm{ADD}\_\mathrm{POINT}\left(wP{t}_i,p_{tmp}\right)$. Subsequently, $p_{tmp}$ is inserted between these two points to ensure that target cluster $A_i$ and $wP{t}_i$ remain ordered.

Step 2.4 ensures that the flight time $T_i$ for each target cluster is minimized. This algorithm incorporates a 2-opt local search operator, which eliminates path crossings and reduces redundant segments within the generated routes. The 2-opt local search operator is applied to the worst 50% of individuals. The core of the improved GA comprises three main components: a selection operator, a crossover operator, and a mutation operator. In the selection phase, a portion of highly fit individuals is chosen using a roulette wheel strategy; another portion is randomly generated to maintain diversity; and the remaining individuals are constructed based on the visiting order derived from Algorithm 1. In the crossover operator, parent individuals are crossed using a random slicing method, which inherently avoids conflict detection. In the mutation operator, a randomly selected segment of an offspring’s sequence is reordered to achieve rapid exploration of the solution space. The main parameters of Algorithm 1 are constructed in Table 2. Threshold ${\tau }_{\mathrm{time}}$ depends on the spatial distribution of UAVs and the scale of the environment. A practical selection strategy is as follows: assign an initially small value (typically below 50); then, execute Algorithm 1, and check whether the algorithm terminates because the maximum iteration limits $L_1$ or $L_2$ are reached. If this occurs, ${\tau }_{\mathrm{time}}$ should be further reduced.

Algorithm 1 is heuristic in nature but admits a well-defined convergence behavior. Specifically, the MTA in Algorithm 1 has a finite search space since both the number of UAVs N and the number of targets M are finite. The algorithm is designed to accept a new assignment state only if it yields a strict reduction in the mission-duration objective while satisfying the feasibility constraints. Step 2.3 preserves constraint satisfaction and monotonically reduces flight-time imbalance, while Step 2.4 ensures a non-increasing objective value for the visiting sequence optimization [31], consistent with established convergence properties of evolutionary algorithms. To prevent cycling in the finite state space, a pruning mechanism is introduced to discard previously visited states. With explicit termination conditions, Algorithm 1 is therefore guaranteed to converge to a locally optimal solution within a finite number of iterations, which provides a practical improvement guarantee for the MTA problem.

Table 2. Main parameters of Algorithm 1.

Simulation Parameter	Value
Iteration in K_MEANS	$25\times \min (M,10\times N)$ [32]
Threshold	${\tau }_{\mathrm{time}}$
$L_2$	N
$L_1$	$10\times L_2$
Max Generations in GA	35 × $(10-k_2)$
Population Size	80
Roulette Wheel Probability in Selection Operator	$0.81$
Random Probability in Selection Operator	$0.09$
Unchanged Probability in Selection Operator	$0.1$
Crossover Operator Rate	$0.65$
Mutation Rate	$0.25$

Table 2. Main parameters of Algorithm 1.

Simulation Parameter	Value
Iteration in K_MEANS	$25\times \min (M,10\times N)$ [32]
Threshold	${\tau }_{\mathrm{time}}$
$L_2$	N
$L_1$	$10\times L_2$
Max Generations in GA	35 × $(10-k_2)$
Population Size	80
Roulette Wheel Probability in Selection Operator	$0.81$
Random Probability in Selection Operator	$0.09$
Unchanged Probability in Selection Operator	$0.1$
Crossover Operator Rate	$0.65$
Mutation Rate	$0.25$

3.2. Time Complexity Analysis

Assuming that the average number of targets in each cluster is $M/N$, the overall time complexity of Step 2.3 is $O(L_1(N-1)M/N)$, where $L_1$ is the number of iterations. Considering that Step 2 needs to iterate $L_2$ times, and the GA’s sub-generation iterates $L_3$ times, the total time complexity of Algorithm 1 is

$$\begin{array}{cc}\hfill O\left(L_2\left\{\left[L_1\left(N-1\right)M/N\right]+NZ{L}_3{(M/N)}^2\right\}\right)\approx & O\left(L_1{L}_{2}M+L_2{L}_{3}Z{M}^2/N\right)\hfill \\ \hfill \propto & O\left(max\left(M,{\displaystyle \frac{M^2}{N}}\right)\right),\hfill \end{array}$$

(10)

where Z is the number of sub-generations. In contrast, if this problem were solved as a MTSP, the time complexity would be $O(M!)$.

Algorithm 1 achieves flight time $T_i$ balance among heterogeneous UAVs, and T is selected as the mission duration for all subsequent UAVs in the subsequent planning phases.

4. Flight-Time-Consistent Algorithm Based on DF for TP

In actual flight, various constraints, including kinematic limitations and environmental factors, must be strictly satisfied. In this section, these challenges are addressed through a flight-time-consistent TP algorithm by generating executable trajectories $P_i$ that ensure satisfy mission duration T for all UAVs $i\in N$.

Sub-problem 2: This sub-problem aims to minimize the integral of ${\left({p_i}^{\left(3\right)}\left(t\right)\right)}^2$ for all UAVs, subject to waypoint visit constraints, strict mission duration consistency T, obstacle avoidance, inter-UAV collision avoidance, velocity limits, overload constraints, and attitude limitations. Note that if $S{2}_7$ in (5) was transformed into a hard constraint, the closed-form solution would be prohibitively complex. Therefore, for practical implementation, these constraints are incorporated as soft constraints using penalty functions $J_{e,i}$ within the optimization framework. Mathematically, the sub-problem ${\displaystyle \underset{P_i}{\min }}{\displaystyle \sum _{i=1}^N}{\int }_{t=0}^{T_i}{\left({p_i}^{\left(3\right)}\left(t\right)\right)}^{2}dt$ subject to $S2$ in (5) is reformulated as

$$\begin{array}{c}\hfill \min \sum _{i=1}^N\left[\sum _{\mu \in \left\{x_i,y_i,z_i\right\}}\left(J_{\mu ,i}\right)+J_{e,i}\right]=\min \sum _{i=1}^N\left[\sum _{\mu \in \left\{x_i,y_i,z_i\right\}}\left({\int }_{t=0}^{T_i}{\left({p_{i,\mu }}^{\left(3\right)}\left(t\right)\right)}^{2}dt\right)+{\int }_{t=0}^{T_i}{J}_{e,i}^{2}dt\right]\end{array}$$

s.t.

$$\begin{array}{cc}\hfill & {\mathrm{S}2}_1:wP{t}_i\subseteq P_i,\hfill \\ \hfill & {\mathrm{S}2}_2:\left|\right|T_i-T\left|\right|\le {\epsilon }_T,\hfill \\ \hfill & {\mathrm{S}2}_3:\left|\right|p_i\left(t\right)-p_{obs,q}\left|\right|>R_{obs,q}+R_{UAV},\forall q\in Q,\hfill \\ \hfill & {\mathrm{S}2}_4:\left|\right|p_i\left(t\right)-p_j\left(t\right)\left|\right|>2R_{UAV},i,j\in N,j\ne i,\hfill \\ \hfill & {\mathrm{S}2}_5:V_{a,i}\left(t\right)\in \left[V_{a,i,min},{V_{a,i}}_{,max}\right],\hfill \\ \hfill & {\mathrm{S}2}_6:\left|\right|n_{i,\mu }\left|\right|\le n_{i,\mu \_max},\left|\right|n_{i,z}\left|\right|\le v_i^2/g{R}_{turn}.\hfill \end{array}$$

(11)

The algorithm is structured into two main steps:

Step 1: Initial Trajectory Generation: Initially, neglecting the obstacle avoidance constraints $S{2}_3\wedge S{2}_4$, an initial reference trajectory $P_i^0$ is generated for each UAV. This trajectory passes through all assigned waypoints and adheres to the desired total mission duration, thereby providing a preliminary solution to sub-problem 2 by leveraging the properties of flat outputs. However, in real-world flight operations, constraints $S{2}_3\wedge S{2}_4$ cannot be disregarded. Furthermore, due to unforeseen obstacles and strict motion constraints, a UAV’s actual position deviate from its desired reference, leading to uneven remaining flight times among UAVs, which in turn violates constraints $\mathrm{S}{2}_2\wedge S{2}_5\wedge S{2}_6$.

Step 2: Flight-Time-Consistent Replanning: A flight-time-consistent replanning feedback framework is proposed based on the initial reference trajectory $P_i^0$. This framework can continuously perform spatiotemporal replanning over a replanning interval $L{o}_i$ through remaining flight time feedback to satisfy all constraints. By adaptively adjusting the local flight time $T_{d,i}$ of $L{o}_i$, all UAVs are guided to fly in accordance with T, further satisfying constraint $S{2}_2$. Simultaneously, within $L{o}_i$, a safe corridor satisfying environmental constraints $S{2}_3\wedge S{2}_4$ is rapidly generated. Subsequently, optimal local trajectories satisfying $S{2}_5\wedge S{2}_6$ are obtained within this safe corridor through DF optimization.

4.1. Flight-Time-Consistent Replanning Strategy

Given the limited sensing range of UAVs and the imperative to maintain precise flight-time consistency, a dynamic replanning strategy is proposed. The starting point of the replanning interval $L{o}_i$ is the UAV’s current position $p_i\left(t\right)$, and the end point is the next waypoint position $p_{i,nextPoint}\in wP{t}_i$.

Assuming that all waypoints $wP{t}_i$ are reachable, and their surrounding areas are guaranteed to be free of obstacles, i.e., $\left|\right|wP{t}_i\left(m\right)-P_{obs}\left(q\right)\left|\right|>R_{obs,q}+R_{UAV},\forall m,q$. To satisfy constraint $S{2}_2$, define the desired local flight time for the UAV i as $T_{d,i}$ and the replanning time interval as ${\tau }_{planning}$. To ensure flight-time consistency among all UAVs, a variable ${\epsilon }_i$ for UAV i is defined as follows:

$${\epsilon }_i\left[k\right]={\displaystyle \frac{t_{i,nextPoint}-\left(T_{l,i}\left[k\right]-{\tau }_{control}\right)}{T}},$$

(12)

where ${\tau }_{control}$ is the single-step control time interval, $T_{l,i}$ is the actual local flight time of current replanning interval $L{o}_i$, $t_{i,nextPoint}$ is the desired reference time to reach the next waypoint obtained from $P_i^0$, and ${\epsilon }_i\in \left[0,1\right]$. Due to physical motion constraints, $T_{l,i}\ne T_{d,i}$. Define the error variable between $T_{d,i}\left[k\right]$ and $T_{l,i}\left[k\right]$ as ${\delta }_i\left[k\right]=T_{d,i}\left[k\right]-T_{l,i}\left[k\right]$, which characterizes flight process for the UAV i. Based on ${\epsilon }_i$, $T_{d,i}$ is designed as

$$\begin{array}{cc}\hfill & T_{d,i}\left[k\right]={\displaystyle \frac{\mathrm{\Gamma }\left(L{o}_i\right)}{V_{cruise,i}}}\left[\sum a_{i,j}{w}_{i,j}\left({\epsilon }_i\left[k\right]-{\epsilon }_j\left[k\right]\right)\right.\left.+b_{i,N+1}{w}_{i,N+1}\left({\epsilon }_i\left[k\right]-\epsilon \left[k\right]\right)+1\right],\epsilon =t/T,\hfill \end{array}$$

(13)

where $\epsilon$ is a global absolute time reference, $\epsilon \in \left[0,1\right]$, $a_{i,j},b_{i,j}$ are positive coordination coefficients, and $w_{i,N+1}$ is the adjacency value from UAV i to $\epsilon$ in the communication graph, here $w_{i,N+1}=1,\forall i$. $\mathrm{\Gamma }\left(·\right)$ is the estimated path length for $L{o}_i$. After specifying the $L{o}_i$, the A* algorithm is adopted to generate a sequence of sub-waypoints $wP{t}_{i,local}$ within $L{o}_i$, then

$$\begin{array}{c}\hfill \mathrm{\Gamma }\left(wP{t}_{i,local}\right)=\sum _{q=1}^{N_{l,i}-1}\left|\right|wP{t}_{i,local}\left(q\right)-wP{t}_{i,local}\left(q+1\right)\left|\right|,\end{array}$$

(14)

where $N_{l,i}$ is the number of waypoints in $L{o}_i$.

Based on the definition above, a flight-time-consistent replanning strategy is proposed, as detailed in Algorithm 2. To handle delays arising from acceleration limits or obstacle avoidance, as well as turn restrictions induced by turning constraints, a verification module check() is incorporated into Algorithm 2, which systematically examines the satisfaction of critical dynamic constraints. If either the velocity constraint or the $S{2}_7$ constraint is violated, the framework automatically triggers Algorithm 1 to reallocate the targets and restore feasibility. Each UAV i obtains ${\epsilon }_j$ of neighboring UAVs and $\epsilon$, then continuously adjusts its desired local flight time $T_{d,i}$ for $L{o}_i$ to ensure that all UAVs maintain temporal consistency. Specifically, consider a scenario where UAV i encounters an unforeseen obstacle at the k-th planning step; this encounter will cause an increase in $\mathrm{\Gamma }\left(wP{t}_{i,local}\right)$, leading to an increase in $T_{d,i}\left[k\right]$, which will further cause ${\epsilon }_i\left[k\right]$ to decrease, indicating that the UAV’s time progression is lagging. When updating at step $k+1$, the consistency algorithm will adjust $T_{d,i}\left[k+1\right]$ to decrease, implicitly instructing UAV i to speed up during the next replanning step. Concurrently, neighboring UAVs will adaptively increase their $T_{d,j}\left[k+1\right]$ to accommodate the lagging UAV i, achieving flight-time consistency. Thus, constraint $S{2}_2$ is satisfied.

Algorithm 2 Flight-Time-Consistent Algorithm

Input: current time t, environmental map $\mathcal{M}$, communication graph matrix L, and so on.

1. Initialization

To satisfy constraints $\mathrm{S}{2}_1\wedge S{2}_5\wedge S{2}_6$ when neglecting the obstacle avoidance

constraints $S{2}_3\wedge S{2}_4$.

$\epsilon ={\epsilon }_i=0,i=1,2,\dots ,N$, $k=1$

$\begin{array}{c}\left(P_i^0,T_i\right)\leftarrow GLOBAL\_PLANNING\left(\mathcal{M},wP{t}_i,T_i\right)\hfill \end{array}$

2. Replanning strategy

To satisfy all constraints $\mathrm{S}{2}_1\wedge \mathrm{S}{2}_2\wedge \mathrm{S}{2}_3\wedge \mathrm{S}{2}_4\wedge S{2}_5\wedge S{2}_6$.

While  $p_i\ne wP{t}_i\left(end\right)$  do

$wP{t}_{i,local}\leftarrow A^{*}\left(\mathcal{M},p_i,p_{i,nextPoint}\right)$

Calculate $\mathrm{\Gamma }\left(wP{t}_{i,local}\right)$

Calculate $T_{d,i}\left[k\right]$

$\begin{array}{c}\left(P_i^k,T_{l,i}\left[k\right]\right)\leftarrow LOCAL\_PLANNING\left(\mathcal{M},wP{t}_{i,local},T_{d,i}\left[k\right]\right)\hfill \end{array}$

While $t<t\left[k\right]+{\tau }_{planning}$ do

$\begin{array}{c}{U}_i\leftarrow CONTROL\left(P_i^k\right)\hfill \\ \left(X_i,\mathcal{M}\right)\leftarrow UAVSYSTEM\left(U_i\right)\hfill \end{array}$

End While

Update $\epsilon$, ${\epsilon }_i$ and $p_{i,nextPoint}$.

$k=k+1$

check()

End While

Theorem 1.

Considering sub-problem 2 (11) with the initial trajectory obtained from Step 1, and incorporating the consistency variable (12) together with the desired flight time (13), the flight-time coordination framework can be established through the flight-time-consistent algorithm in Algorithm 2. Within this framework, the constraint $S{2}_2$ in (11) is guaranteed, and the overall solution of sub-problem 2 can be ultimately achieved under Assumption 2 and appropriate selection of design parameters.

Assumption 2.

1. 

In Equations (12) and (13), $\parallel \delta \left[k\right]\parallel \le \Delta <\infty$ is uniformly bounded. This assumption is ensured by the local A*-based trajectory planner. In our scenarios, A* can always find a feasible path within finite time [33].

2. 

In Equation (15), $H_{1,i}$ and $H_{2,i}$ are both bounded as $0<{\underline{H}}_1\le {\lambda }_{\min }\left(H_1\left[k\right]\right)\le {\lambda }_{\max }\left(H_1\left[k\right]\right)\le {\overline{H}}_1,0<{\underline{H}}_2\le {\lambda }_{\min }\left(H_2\left[k\right]\right)\le {\lambda }_{\max }\left(H_2\left[k\right]\right)\le {\overline{H}}_2,\forall k$. Due to the definitions of $H_{1,i}$ and $H_{2,i}$, this bound holds naturally.

3. 

The communication topology L contains at least one spanning tree.

Proof. 

Considering Equation (12), we have

$$\begin{array}{c}\hfill {\epsilon }_i\left[k+1\right]=-H_{1,i}\left[k\right]\left[a_{i,j}{w}_{i,j}\left({\epsilon }_i\left[k\right]-{\epsilon }_j\left[k\right]\right)+b_{i,N+1}{w}_{i,N+1}\left({\epsilon }_i\left[k\right]-\epsilon \left[k\right]\right)\right]-H_{1,i}\left[k\right]+H_{2,i}\left[k\right],\end{array}$$

(15)

where $H_{2,i}=t_{i,nextPoint}+{\delta }_i\left[k\right]+{\tau }_{control}/T$, $H_{1,i}=\mathrm{\Gamma }\left(wP{t}_{i,local}\right)/\left(V_{cruise,i}T\right)$.

Under the Assumption 2, rewrite Equation (15) in matrix form:

$$\epsilon \left[k+1\right]=-H_1\left[k\right]\overline{L}\epsilon \left[k\right]-H_1\left[k\right]+H_2\left[k\right],$$

(16)

where $\overline{L}=\left[l_{i,j}\right]\in {\mathbb{R}}^{\left(N+1\right)\times \left(N+1\right)}$ is the augmented Laplacian matrix that incorporates the absolute time reference $\epsilon$. Define $a_{i,N+1}\triangleq b_{i,N+1}$ and $w_{N+1,j}=0,\forall j$. Then, $l_{i,j}=-a_{i,j}{w}_{i,j}$, $l_{i,i}={\sum }_{i\ne j}{a}_{i,j}{w}_{i,j}$, $l_{N+1,j}=0$, $\forall i\in \{1,\dots ,N\},\forall j\in \{1,\dots ,N+1\}$. Referring to Proposition 1 in [29], a transformation matrix $I\in {\mathbb{R}}^{\left(N-1\right)\times N}$ for the complementary consensus subspace is introduced such that

$$\vartheta \left[k\right]=I\epsilon \left[k\right].$$

(17)

I satisfies $I{1}_n=0$. This implies that $\vartheta =0$ is equivalent to $\epsilon \in \mathrm{span}\left\{1_n\right\}$, meaning the UAV achieves flight time coordination at step k. Choosing $I^{T}I={\mathbb{I}}_{n\times n}-{\displaystyle \frac{1}{n}}{1}_n{1}_n^T$, then, $I^T\overline{L}I=\overline{L}$ can be further derived based on $\overline{L}{1}_n=0$. Furthermore, the error system is

$$\begin{array}{cc}\hfill \vartheta \left[0\right]& =0,\hfill \\ \hfill \vartheta \left[k+1\right]& =-I{H}_1\left[k\right]\overline{L}{I}^T\vartheta \left[k\right]-I\left(H_1\left[k\right]+H_2\left[k\right]\right).\hfill \end{array}$$

(18)

Equation (18) can be written as the discrete-time system as

$$\begin{array}{c}\hfill \vartheta [k+1]=\mathrm{\Phi }\left[k\right]\vartheta \left[k\right]+d\left[k\right],k\in {\mathbb{Z}}_{\ge 0},\end{array}$$

(19)

where $\mathrm{\Phi }\left[k\right]=-I{H}_1\left[k\right]\overline{L}{I}^T,d\left[k\right]=-I\left(H_1\left[k\right]+H_2\left[k\right]\right)$. Based on Assumption 2, we have

$$\begin{array}{c}\hfill \parallel \mathrm{\Phi }\left[k\right]\parallel \le {\overline{H}}_1{\lambda }_{\max }\left(\overline{L}\right)=\mathrm{\Lambda }.\end{array}$$

(20)

By choosing parameters $\gamma$ and adjusting $\overline{L}$ such that $\mathrm{\Lambda }<\gamma <1$. Therefore, there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \parallel \mathrm{\Phi }[k-1]\cdots \mathrm{\Phi }\left[0\right]\parallel \le C{\gamma }^k.\end{array}$$

(21)

The iteration solution of Equation (19) is

$$\begin{array}{c}\hfill \vartheta \left[k\right]=\mathrm{\Phi }[k-1]\cdots \mathrm{\Phi }\left[0\right]\vartheta \left[0\right]+\sum _{j=0}^{k-1}\mathrm{\Phi }[k-1]\cdots \mathrm{\Phi }[j+1]d\left[j\right].\end{array}$$

(22)

Since ${\sum }_{\ell =0}^{k-1}{\gamma }^{\ell }\le {\sum }_{\ell =0}^{\infty }{\gamma }^{\ell }={\displaystyle \frac{1}{1-\gamma }}$, we obtain $\parallel \vartheta \left[k\right]\parallel \le C{\gamma }^k\parallel \vartheta \left[0\right]\parallel +{\displaystyle \frac{C}{1-\gamma }}{\sup }_{0\le t\le k}\parallel d\left[t\right]\parallel .$ Hence $\vartheta \left[k\right]$ converges exponentially to a bounded neighborhood related to $\sup \parallel d\left[t\right]\parallel$.

If the case of time variations is considered, the synchronization law (18) becomes a linear system of time variations. The system remains uniformly asymptotically stable if $\mathrm{\Gamma }\left(wP{t}_{i,local}\right)$ satisfies certain conditions. Similarly, when the effect of communication delays is taken into account, (15) introduces a time delay into the system. Convergence can still be guaranteed if the maximum delay does not exceed a specified threshold. □

4.2. DF Based Planning Algorithm for TP

In this section, the DF-based planning method implements GLOBAL_PLANNING and LOCAL_PLANNING mentioned in Algorithm 2.

A nonlinear system $\dot{X}=f\left(X,U\right)$ is said to be differentially flat if there exists a vector $Y=f_1\left(X,\dots ,X^{\left(m_1\right)},\dots ,U,\dots ,U^{\left(m_2\right)}\right)$ that satisfies the following properties:

• Y is differentiable;
• The system states X and control inputs U can be expressed as functions of Y and its finite-order derivatives $X=g_1\left(Y,\dots ,Y^{\left(n_1\right)}\right)$ and $U=g_2\left(Y,\dots ,Y^{\left(n_2\right)}\right).$

Such systems are termed flat systems, and Y is the differential flat output. Through DF, the original system can be mapped to a lower-dimensional, linear flat space. By planning the flat outputs, algebraic solutions for X and U can be directly obtained.

Theorem 2.

Considering (1), a DF output for FW-UAVs is selected as $Y_i=\left\{p_i,{\alpha }_i\right\}$. From this flat output, all other state variables $X_i=\left\{p_i,v_{a,i}^e,V_{a,i},{\mathrm{\Theta }}_i,{\mathrm{\Omega }}_i\right\}$ and control inputs $U_i=\left\{{\mathrm{\Omega }}_{c,i},T{e}_i\right\}$ can be uniquely derived. The applicability of this theorem is restricted to the case of translational-flatness parametrization [34,35].

Proof. 

First, based on the definition of ${\dot{p}}_i=v_{a,i}^e$, $V_{a,i}=\left|\right|v_{a,i}\left|\right|$, it can be readily shown that $p_i,v_{a,i}^e,V_{a,i}$ can be characterized by $Y_i$ as follows:

$$V_{a,i}=g_1\left(p_i\right).$$

(23)

Then, it can be derived that

$$R_{a,i}^b=g_2\left({\alpha }_i\right).$$

(24)

Furthermore, the transformation matrix $R_{a,i}^e$ can be further expressed as

$$R_{a,i}^e=\left[x_{a,i},y_{a,i},z_{a,i}\right],$$

(25)

where $x_{a,i}$ is the projection of the $x_a$ axis of air coordinates in the inertial system. Given that the velocity vector is aligned with the $x_a$ axis of the air-relative frame, we can obtain

$$x_{a,i}^e={\displaystyle \frac{v_{a,i}^e}{\left|\right|v_{a,i}^e\left|\right|}}.$$

(26)

Based on (1), we can obtain

$$m_i{\dot{v}}_{a,i}^e+m_{i}g{e}_2=\left(T{e}_i\cos \left({\alpha }_i\right)-D_i\right)x_{a,i}^e+L_i{y}_{a,i}^e.$$

(27)

Dot multiplying both sides by $x_{a,i}^e$ yields

$${x_{a,i}^e}^T\left(m_i{\dot{v}}_{a,i}^e+m_{i}g{e}_2\right)=T{e}_i\cos \left({\alpha }_i\right)-D_i.$$

(28)

Substituting the above equation into (1), we get

$$L_i{y}_{a,i}^e=m_i{\dot{v}}_{a,i}^e+m_{i}g{e}_2-{x_{a,i}^e}^T\left(m_i{\dot{v}}_{a,i}^e+m_{i}g{e}_2\right)x_{a,i}^e.$$

(29)

Considering $\left|\right|y_{a,i}^e\left|\right|=1$, then

$$y_{a,i}^e={\displaystyle \frac{{\dot{v}}_{a,i}^e+g{e}_2-{x_{a,i}^e}^T\left({\dot{v}}_{a,i}^e+g{e}_2\right)x_{a,i}^e}{\left|\right|{\dot{v}}_{a,i}^e+g{e}_2-{x_{a,i}^e}^T\left({\dot{v}}_{a,i}^e+g{e}_2\right)x_{a,i}^e\left|\right|}}.$$

(30)

According to the orthogonality of coordinate, we can obtain

$$z_{a,i}^e=x_{a,i}^e\times y_{a,i}^e.$$

(31)

Thus

$${\mathrm{\Theta }}_i=g_3\left(R_{b,i}^e\right)=g_3\left(R_{a,i}^e{R_{a,i}^b}^T\right)=g_4\left(Y_i\right).$$

(32)

Considering ${⌊{\mathrm{\Omega }}_i⌋}_{\times }={R_{b,i}^e}^T{\dot{R}}_{b,i}^e$, it is derived that

$${\mathrm{\Omega }}_i=g_5\left(Y_i\right).$$

(33)

Thus, ${\mathrm{\Omega }}_{c,i}={\mathrm{\Omega }}_i=g_5\left(Y_i\right)$.

According to (28), $T{e}_i$ can be expressed as

$$T{e}_i={\displaystyle \frac{{x_{a,i}^e}^T\left(m_i{\dot{v}}_{a,i}^e+m_{i}g{e}_2\right)+D_i}{\cos \left({\alpha }_i\right)}}.$$

(34)

Considering $D_i=g_6\left(Y_i\right)$, then

$$T{e}_i=g_7\left(Y_i\right).$$

(35)

Therefore, all system states $X_i$ and control inputs $U_i$ are explicitly expressed as functions of $Y_i$ and its finite-order time derivatives. □

During the planning process, constraints $S2$ defined in the state space can be mapped to constraints $Y_i$ in the flat output space. Subsequently, all desired $X_i$ and $U_i$ can be inversely mapped from the planned flat outputs. However, if the real-time state of the UAV is available, the control inputs $U_i$ can be derived by introducing a feedback controller to achieve rapid tracking of the desired state variables, as shown in Figure 3. This feedback control eliminates the need for explicit inverse mapping calculations from flat outputs to desired controls, thereby significantly improving control efficiency and responsiveness.

When applying the DF-based approach, the TP problem is reformulated as an optimization problem, in which the following aspects must be taken into account: trajectory representation, flight time allocation, cost function definition, and constraint definition.

A. Trajectory representation

For $Y_i$, Bézier curves are employed for fitting. Further specific details on Bézier curves can be found in [36]. To simplify, the UAV’s cruising altitude $y_i\left(t\right)=H_c$ is assumed to be fixed. The angle of attack ${\alpha }_i\left(t\right)$ can then be determined based on the lift–gravity balance equation:

$${\alpha }_i\left(t\right)=C_{L,i}^{-1}\left({\displaystyle \frac{2m_{i}g}{\rho V_{a,i}^2{S}_i}}\right),$$

(36)

where $C_{L,i}^{-1}\left(·\right)$ denotes the inverse mapping from the lift coefficient to ${\alpha }_i\left(t\right)$. Consequently, only the horizontal positions $x_i,z_i$ are planned. The Bézier curve parameterization for a dimension $\mu =x,z$ of one segment is given by the following:

$$\begin{array}{c}\hfill x_i=\left\{\begin{array}{c}{B}_1\left(t\right)=s_{B_1}{\sum }_{h=0}^{N_p}{c}_{B_1}^h{b}_{N_p}^h\left(\frac{t-T_{i,1}}{s_{B1}}\right),T_{i,1}\le t\le T_{i,2},\hfill \\ ⋮\hfill \\ B_{N_{l,i}-1}\left(t\right)=s_{B_{N_{l,i}-1}}{\sum }_{h=0}^{N_p}{c}_{B_{N_{l,i}-1}}^h{b}_{N_p}^h\left(\frac{t-T_{i,N_{l,i}-1}}{s_{B_{N_{l,i}-1}}}\right),T_{i,N_{l,i}-1}\le t\le T_{i,N_{l,i}}.\hfill \end{array}\right.\end{array}$$

(37)

where $b_{N_p}^h\left(t_B\right)=C\left({\displaystyle \genfrac{}{}{0pt}{}{N_p}{h}}\right){t_B}^h{\left(1-t_B\right)}^{N_p-h},t_B\in \left[0,1\right]$ are the Bernstein basis polynomials, $N_p$ is the degree of the curve, and $C\left(·\right)$ is the binomial coefficient. For the first piecewise segment $B_1\left(t\right)$ from the beginning point to the next waypoint, $c_{B_1}^h$ denotes the h-th control point for this specific curve segment. The parameters $s_d,d=B_1,\dots ,B_{N_{l,i}-1}$ are scaling factors used to scale the time interval of each segment, where $N_{l,i}$ is the total number of waypoints $wP{t}_i$ or $wP{t}_{i,local}$ in the segment.

B. Flight Time Allocation

The Bézier desired flight time for each waypoint within the trajectory is allocated using the following formula:

$$T_{i,d}=T_{i,d-1}+T_i\left|\right|p_{i,d+1}-p_{i,d}\left|\right|/\sum _{\begin{array}{c}d=1,\dots ,N_{l,i}-1\end{array}}\left(\left|\right|p_{i,d+1}-p_{i,d}\left|\right|\right).$$

(38)

C. Cost Function Definition

Based on the aforementioned definitions, for a single UAV, the cost function to be optimized is reformed to minimize the motion jerk and additional cost:

$$\begin{array}{c}\hfill \min J_i=\min \left(\sum _{\mu \in \left\{x_i,z_i\right\}}\left(J_{\mu ,i}\right)+J_{e,i}\right)=\min \left(\sum _{\mu \in \left\{x_i,z_i\right\}}{\int }_{t=0}^{T_i}{\left({p_{i,\mu }}^{\left(3\right)}\left(t\right)\right)}^{2}dt+{\int }_{t=0}^{T_i}{j}_{e,i}^{2}dt\right),\end{array}$$

(39)

where $J_{\mu ,i}$ represents the minimization of the UAV’s motion jerk. Taking the $x_i$-direction as an example, the cost function can be expressed as $J_{x_i,i}={c_{x,i}}^T{Q}_{x,i}{c}_{x,i}$, where $c_i=\left\{c_{B_1}^0,\dots ,c_{B_{N_{l,i}-1}}^{N_p}\right\}$ is the vector of all control points in the $x_i$-direction, which are the variables to be optimized. $Q_{x,i}$ is the positive semi-definite Hessian matrix of the cost function.

For the symmetric attitude soft constraint $J_{e,i}$, a barrier function $\mathrm{\Psi }\left(x,x_{\max },x_{smooth}\right)$ is defined to enforce it as a soft constraint:

$$\mathrm{\Psi }\left(x,x_{\max },x_{smooth}\right)=K_{sig}{\displaystyle \frac{Sig\left(x,x_{\max },x_{smooth}\right)\left|\right|x\left|\right|}{{\epsilon }_{sig}}},$$

(40)

where ${\epsilon }_{sig}$ is a very small positive number to prevent a logarithm of zero or negative values, and $K_{sig}$ is a scaling factor. $Sig\left(x,x_{\max },x_{smooth}\right)=1/\left(1+\exp \left(10\left(x_{\max }-\left|\right|x\left|\right|\right)/x_{smooth}\right)\right)$. $\mathrm{\Psi }\left(x,x_{\max }.x_{smooth}\right)$ has the following properties:

• $\mathrm{\Psi }\left(x,x_{\max },x_{smooth}\right)\ge 0$;
• As $\left|\right|x\left|\right|\le x_{\max }-x_{smooth}$, $\mathrm{\Psi }\to 0$;
• When $\left|\right|x\left|\right|>x_{\max }$, $\mathrm{\Psi }\ge K_{sig}(1-\epsilon )\left|\right|x\left|\right|/{\epsilon }_{sig}$, which reaches a maximum value.

By minimizing this barrier function $\mathrm{\Psi }\to 0$, it can be guaranteed that $\left|\right|x\left|\right|\le x_{\max }-x_{smooth}$. Thus, constraint $\mathrm{S}{2}_7$ is transformed into

$$j_{e,i}=\mathrm{\Psi }\left({\varphi }_i\left(t\right),{\varphi }_{i,max},{\varphi }_{i,smooth}\right)+\mathrm{\Psi }\left({\theta }_i\left(t\right),{\theta }_{i,max}{\theta }_{i,smooth}\right).$$

(41)

D. Constraint Definition

Waypoint Constraints: For both the initial reference trajectory $P_i^0$ and subsequent local replanning trajectories $P_i^k$, $S{2}_1$ must be satisfied. In the initial global planning phase, $P_i^0$ is required to pass through all assigned waypoints $wP{t}_i$ and satisfy initial and terminal position, velocity, and acceleration constraints. In the online replanning phase, $P_i^k$ only needs to satisfy the start and end point constraints of the current replanning segment, as intermediate waypoints are dynamically handled by the safe flight corridors. Considering the properties of Bézier curves, these waypoint constraints can be transformed into constraints on the curve control points $c_d^h,d=B_1,\dots ,B_{N_{l,i}-1},h=0/N_p$ and their derivatives. Taking the start point constraint of the first curve segment for $x_i$-direction as an example, we have

$$\begin{array}{cc}\hfill c_{B_1}^0{s}_{B_1}& =x_{i,start},p_{i,start}=wP{t}_i\left(1\right),\hfill \\ \hfill c_{B_1}^{0,\left(1\right)}& ={\dot{x}}_{i,start},\hfill \\ \hfill c_{B_1}^{0,\left(2\right)}{s_{B_1}}^{-1}& ={\ddot{x}}_{i,start},\hfill \end{array}$$

(42)

where $c_{B_1}^{h,\left(l\right)},h=0,\dots N_p,l=0,\dots 2$ represents the l-th order derivative of the h-th control point of this curve segment, $c_{B_1}^{h,\left(l\right)}={\displaystyle \frac{N_p!}{\left(N_p-l\right)!}}\left(c_{B_1}^{h+1,\left(l-1\right)}-c_{B_1}^{h,\left(l-1\right)}\right)$.

Continuity Constraints: To ensure smooth and continuous UAV motion, the connections between adjacent Bézier curve segments must be multi-order continuity connections. Taking the first two curve segments for $x_i$-direction as an example, this constraint can be expressed as equality constraints on control points $c_{B_1}^{N_p}$, $c_{B_2}^0$ and their derivatives, i.e.,

$$c_{B_1}^{N_p,\left(l\right)}{s}_{B_1}^{1-l}=c_{B_2}^{0,\left(l\right)}{s}_{B_2}^{1-l},l=0,1,2.$$

(43)

Safe Flight Corridors Constraint: During the online replanning phase, to ensure obstacle avoidance constraints $S{2}_3\wedge S{2}_4$, safe flight corridor constraints are introduced. Taking the $x_i$-direction as an example, considering the convex hull property of Bézier curves, ensuring that all control points $c_d^h,d=B_1,\dots ,B_{N_{l,i}-1},h=0,\dots ,N_p$ of a curve segment lie within a defined convex hull guarantees that the entire trajectory segment also remains within that hull. This property also applies to the derivatives of the curve.

First, the local replanning area is discretized into a grid. Then, an A* algorithm is utilized to generate a sequence of central path points $wP{t}_{i,local}$ that satisfy the obstacle avoidance constraints $S{2}_3\wedge S{2}_4$. Subsequently, each sub-waypoint is expanded to form a rectangular safe corridor ${\mathrm{\Xi }}_d=\left\{p|\forall p\in {\mathrm{\Xi }}_d,psatisfiesS{2}_3\wedge S{2}_4\right\}$, ultimately forming the total safe corridor for the local planning interval $\mathrm{\Xi }=\cup {\mathrm{\Xi }}_d,{\mathrm{\Xi }}_d\cap {\mathrm{\Xi }}_{d+1}\ne ⌀$. For a certain curve segment d within the interval, it is only necessary to apply constraints to all control points of the curve, ensuring that they satisfy

$$x_{d,\min }\le c_d^h{s}_d\le x_{d,\max },$$

(44)

where $x_{d,\min },x_{d,\max }$ are the lower and upper bounds of the safe corridor for the d-th curve segment.

Kinematic Constraints: Since the derivatives of control points also possess the convex hull property, velocity and acceleration constraints can be directly imposed on the control point derivatives. Considering constraints $S{2}_5\wedge S{2}_6$, taking a curve segment for the $x_i$-direction as an example, we have

$$\begin{array}{cc}\hfill -{v_i}_{,max}& \le c_d^{h,\left(1\right)}\le {v_i}_{,max},\hfill \\ \hfill -{a_i}_{,max}{m}_{i}g& \le c_d^{h,\left(2\right)}{s_d}^{-1}\le {a_i}_{,max}{m}_{i}g.\hfill \end{array}$$

(45)

For $\mu =z_i$, turning constraints also need to be considered.

In summary, the TP problem can be formulated as a constrained optimization problem. The waypoint visit and continuity constraints are transformed into linear equality constraints on the control points. Safe corridor constraints and kinematic limits are represented as linear inequality constraints on control points and their derivatives. Finally, attitude constraints are incorporated as soft constraints through barrier functions. Thus, this planning problem can be formally described as

$$\begin{array}{c}\min {c_i}^T{Q}_i{c}_i+J_{e,i}\hfill \\ s.t.\hfill \\ A_{eq,i}{c}_i=b_{eq,i},\hfill \\ A_{ie,i}{c}_i\le b_{ie,i},\hfill \end{array}$$

(46)

where $c_i=\left[c_{{B_1}_{,x}}^0,\dots ,c_{{B_1}_{,x}}^{N_p},\dots ,c_{{B_{N_{l,i}-1}}_{,x}}^0,\dots ,c_{{B_{N_{l,i}-1}}_{,x}}^{N_p},c_{{B_1}_{,z}}^0,\dots ,c_{{B_{N_{l,i}-1}}_{,z}}^{N_p}\right]$. When $J_{e,i}$ is not considered, this problem reduces to a convex quadratic programming problem, which can be efficiently solved using a QP solver to obtain an initial feasible trajectory. Subsequently, $J_{e,i}$ is considered for further optimization to obtain the final trajectory.

5. Simulation and Analysis

The performance of the algorithm is evaluated and validated in this section. All algorithms are executed on the same hardware platform: Intel i5-9300H CPU, 32 GB RAM, and NVIDIA GeForce GTX 1650 GPU. Due to the fixed cruising altitude assumption for UAVs, the simulation environment is simplified to a two-dimensional scenario. Target positions $P_{target}$ and no-fly zone positions $P_{obs}$ are randomly generated within a 2.5 km × 2.5 km area. All UAVs are assumed to depart from the origin $\left(0,0\right)$ m, and return to $\left(0,0\right)$ m after completing the MTR mission. A precise aerodynamic force model and 6-DOF kinematic motion are constructed, while attitude control is provided by a simulated PX4 controller. Additionally, the UAV’s detectable range is set to 200 m. The MOSEK optimizer is employed to solve the convex QP problems arising from the trajectory generation. The main parameters for the UAVs and environment are constructed in

Table 3. Main parameters of the UAVs and environment.

Simulation Parameter	Value
Number of UAVs N	4
Number of targets M	40
Number of no-fly zones Q	8
Optimal cruise speed $V_{cruise}$	$\left\{80,60,50,30\right\}$ m/s
Bézier curve degree $N_p$	8
Safe radius of UAV $R_{UAV}$	10 m
Radius of static no-fly zones $R_{obs,q}$	$\left[10,100\right]$ m
Smooth parameters ${\varphi }_{i,smooth}$ and ${\theta }_{i,smooth}$	$5^{\circ }$
A* grid length	5 m
Corridor width	30 m
Detection range	200 m
Forward field of view	$\pm {30}^{\circ }$
Dual feasibility tolerance in MOSEK	${10}^{-4}$
Primal feasibility tolerance in MOSEK	${10}^{-4}$
Infeasibility tolerance in MOSEK	${10}^{-4}$

.

5.1. Time-Balanced Clustering Simulation

The simulation result of Algorithm 1 is shown in Figure 4, Figure 5 and Figure 6. Figure 4 illustrates the convergence of flight times during the optimization process: the green-shaded region indicates the execution of the target reallocation, while the red-shaded region denotes the application of the intra-cluster optimization. As the optimization progresses, the individual UAV flight times gradually converge, with the flight times of the four UAVs being $T_1=180\mathrm{s}$, $T_2=185\mathrm{s}$, $T_3=185\mathrm{s},T_4=186\mathrm{s}$. The average flight time decreases from 218 s to 184 s. Figure 5 shows the target allocation and visiting order. Each target position is represented by a solid dot, where lighter colors indicate earlier reconnaissance and darker colors indicate later reconnaissance within a UAV’s sequence. The different colored lines represent the distinct routes planned for each UAV. Specifically, Figure 5a shows the initial target routes $wPt$, where targets belonging to each UAV are spatially grouped with minimal differences in the number of assigned targets. Figure 5b illustrates the final adjusted target allocation and optimized target routes $wPt$. Notably, since $UA{V}_1$ has the highest optimal cruising speed, its optimized route is consequently the longest, followed by $UA{V}_2$, while $UA{V}_1$ has the shortest route, demonstrating the algorithm’s ability to balance mission durations despite heterogeneity.

Figure 6a shows the scalability of Algorithm 1 under various configurations $(M,N)$, presenting runtime, time variance, and constraint-violation ratios (percentage of optimizations completed within the threshold). The computational efficiency of the time-balanced clustering is further evaluated against representative MTA algorithms ([29,37]), as illustrated in Figure 6b with $M=5N$.

5.2. Flight-Time-Consistent Planning Simulation

In this section, the simulation result of Algorithm 2 is shown in Figure 7, Figure 8 and Figure 9. The allowed speeds for each UAV are $V_{a,i,min}=0.5$ × $V_{cruise,i}$ and $V_{a,i,\max }=2$ × $V_{cruise,i}$. The desired mission duration for all UAVs is selected as $T=185$ s. All UAVs are subject to specific kinematic constraints: $n_{i,x\_max}=2$, ${\varphi }_{i,max}={30}^{\circ }, {\theta }_{i,max}={45}^{\circ }$, $R_{turn}=100\mathrm{m}$. The replanning time interval ${\tau }_{planning}=5\mathrm{s}$, the control time interval ${\tau }_{control}=0.05\mathrm{s}$, the coordination coefficients $a_{i,j}=b_{i,j}=1$, and the adjacency value $w_{i,i+1}=1,i=1,2,3$. The initial and terminal velocities for all UAVs are set to $V_{a,i}\left(0\right)=V_{a,i}\left(T\right)=50\mathrm{m}/\mathrm{s},\forall i$. During fixed-wing flight, the Bank-to-Turn coordinated turning method is employed to generate ${\mathrm{\Omega }}_{c,i}$, and attitude control is achieved through PX4 inner-loop attitude control. To achieve inner–outer loop decoupling and reduce system complexity, the inner-loop bandwidth is designed to be ten times larger than that of the outer loop.

Figure 7 illustrates both the global and local planning processes for the UAVs. In Figure 7a, the colored solid curves depict the optimized initial reference trajectory $P_i^0$ that initially disregard no-fly zones, with lighter colors indicating lower speeds. Blue circular regions represent the no-fly zones, and for clarity, only those impacting the trajectories are shown. The dashed curves represent the actual flight trajectories obtained after local online replanning by each UAV. Figure 7b provides a magnified local view, clearly demonstrating that when a global reference trajectory conflicts with a no-fly zone, a safe local trajectory can be planned through the dynamically generated safe flight corridor, enabling the UAV to circumvent the obstacle and satisfy safety constraints.

Figure 8a,b collectively demonstrate the efficacy of the proposed constraint enforcement framework. Figure 8a presents the velocity profiles of all UAVs. The straight horizontal lines indicate the upper and lower velocity constraint bounds. The dashed curves in the green-shaded region on the left represent the UAV velocities planned without explicit velocity constraints, while the solid curves on the right show the UAV velocities when constraints are actively enforced. Figure 8b displays the overload curve for UAV1. The dashed curve in the green-shaded region on the left indicates the axial overload of UAV1 when overload constraints are not considered, whereas the solid curve on the right represents the UAV’s overload when these constraints are actively applied.

Figure 9 illustrates the temporal evolution of the consistency variable ${\epsilon }_i$ and the global absolute time reference $\epsilon$ for all UAVs. The green-shaded region on the left shows the variable changes when the flight-time-consistent strategy is not introduced, highlighting significant deviations. In contrast, the right side demonstrates the changes after introducing the proposed algorithm. This comparison clearly shows that all UAVs can strictly adhere to the absolute time reference for real-time planning, thereby successfully achieving precise flight time coordination.

6. Conclusions

This article investigated the synchronous and coordinated MTA and TP problem for heterogeneous FW-UAVs under dynamic environments and kinematic constraints. A time-balanced clustering-based assignment algorithm that minimizes the overall mission completion time while simultaneously ensuring equitable flight-time balance among heterogeneous UAVs is developed. Furthermore, a robust DF-based online planning algorithm, which guarantees the generation of kinematically feasible, collision-free, and time-synchronized trajectories, is proposed. The proposed framework decomposes the original large-scale and tightly coupled optimization problem into two sub-problems, thereby enabling scalable and real-time implementation. Rigorous theoretical analysis and extensive simulations demonstrated the feasibility and effectiveness of the proposed approach in achieving synchronized MTR missions.

The hierarchical MTA–TP framework proposed in this paper is compatible with the onboard computational architecture of most existing UAV platforms. In addition, this framework can be further integrated with AI-driven autonomous systems. For example, generative AI algorithms [38] could be incorporated into the upper-layer decision-making module to provide high-quality initial cluster partitions, thereby reducing computational time. Such integration represents a promising direction for future development.