Physical property-aware multi-UAV spatial coordination for energy-efficient mobile edge computing

Abstract

In recent years, Unmanned Aerial Vehicle (UAV)-assisted Mobile Edge Computing (MEC) systems have emerged as innovative solutions for delivering efficient communication and computing services to Internet of Things (IoT) devices. However, the three-dimensional deployment and trajectory decision of UAVs remain challenging due to their highly non-convex and complex process characteristics. Existing methods often face scalability limitations, hindering their applicability to collaborative tasks as the number of UAVs increases. Furthermore, many approaches rely on simplified UAV models, neglecting the complexities of real-world physical dynamics. To address these issues, we propose a joint optimization framework designed to simultaneously minimize real-world UAV system overhead and enhance Air-to-Ground (A2G) communication capabilities. Our approach incorporates a deployment and trajectory design strategy that captures the comprehensive kinematic and dynamic properties of UAVs. In light of the problem’s inherent nonconvex structure and computational intractability, we introduce a collaborative multi-operator Differential Evolution (DE) variant algorithm with a semi-adaptive strategy, termed CSADE. This algorithm utilizes three distinct mutation strategies and integrates an external archiving mechanism to optimize both the number and locations of UAV Task Points (TPs). Additionally, we present an end-to-end dynamic UAV allocation and integrated flight path optimization method to ensure efficient route planning. The proposed method is evaluated through experiments on four data instances and compared with two related algorithms. Results demonstrate that our approach significantly reduces system operating costs while maintaining effectiveness and stability, highlighting its potential for large-scale UAV-assisted MEC applications.

Introduction

In recent years, the rapid advancement of Internet of Things (IoT) and communication technologies has led to a surge in data-intensive IoT devices [1, 2–3]. To manage the massive data generated by user terminals, Mobile Edge Computing (MEC) has emerged as a promising solution [4]. By deploying computing resources at the network edge, MEC enables task offloading, reducing service delays and enhancing Quality of Service (QoS) [5]. However, traditional terrestrial networks rely on fixed-location edge servers, limiting their coverage and signal quality, especially in disaster scenarios or temporary high-density areas [6, 7–8]. To address these limitations, unmanned aerial vehicles (UAVs) have been explored as wireless relays or mobile base stations (BSs) [9, 10]. With their high mobility and flexible deployment [11], UAVs integrated with MEC provide efficient task offloading for IoT devices. Compared to conventional methods, UAV-assisted MEC systems enhance line-of-sight (LoS) communication [12], improve coverage [13], and reduce transmission delays [14].

In UAV-assisted MEC systems, the primary research goal is to minimize energy consumption while ensuring data throughput, coverage, and adherence to UAV physical constraints. Factors such as UAV locations, deployment numbers, and flight trajectories significantly influence the system’s energy efficiency. Early studies primarily focused on single UAV scenarios [15, 16], with Li et al. [17] proposing an energy-efficient framework to address challenges in fully or partially collecting data in sparse IoT networks. Similarly, Yu et al. [18] optimized UAV deployment, communication, and resource allocation to reduce delay and energy consumption, while Han et al. [19] developed a two-layer optimization algorithm for UAV deployment and flight trajectory optimization to minimize energy usage. The application of multiple UAVs has gained attention due to increasing user density and the growing coverage demands of IoT devices, necessitating improved communication efficiency and energy optimization [20, 21]. For example, Wang et al. [22] optimized user associations and UAV trajectories to reduce IoT energy consumption, Khalid et al. [23] introduced energy harvesting networks to support energy-constrained IoT devices while maximizing computational efficiency, and Zhang et al. [24] incorporated QoS and service capacity considerations to minimize UAV usage and enhance coverage rates. Despite these advancements, most existing studies simplify UAV models, overlooking their real-world physical constraints, which can hinder practical mission execution and affect the continuity and efficiency of UAV operations in engineering applications.

Determining the optimal number and locations of UAV high-altitude task points (TPs) is a critical step in enhancing overall system performance. This decision directly influences the quality of subsequent trajectory planning, thereby improving both network performance and energy efficiency [25]. Several studies focus on refining UAV positioning when the number of UAVs is predefined [26, 27]. For instance, Zhong et al. [28] proposed a network enhancement method using opportunistic UAVs to enable real-time relay assignment and channel allocation. Similarly, Gao et al. [29] employed a potential game-based multi-agent deep deterministic policy gradient approach, demonstrating that the service assignment problem converges to a Nash equilibrium, allowing for optimized positioning and trajectory planning. Wang et al. [30] modeled the three-dimensional (3D) UAV deployment problem as a real-time single-step Markov decision process, aiming to minimize energy consumption under conditional value-at-risk constraints.

UAV deployment should be dynamically adjusted based on mission requirements and the distribution of device nodes, rather than relying on a fixed number of UAVs. Such an approach enhances the flexibility and adaptability of deployment, reducing potential errors. Furthermore, the time-varying nature of UAV positions highlights the significant benefits of joint trajectory design for improving system performance. When dealing with a variable number of TPs, the association between UAVs and IoT devices becomes critical. Deployment and scheduling strategies addressing this challenge have been explored in studies such as [14, 22, 31]. For instance, Pervez et al. [32] focused on reducing the weighted cost of energy consumption and latency, and El-Emary et al. [33] aimed to minimize user equipment energy consumption. Despite these advancements, most existing studies that explore the simultaneous optimization of TPs and trajectory planning treat these as submodular problems. Typically, to optimize both the number and locations of TPs, UAV speed is assumed to be constant, and the motion energy consumption model is simplified. Additionally, under 3D assumptions, human intervention is often introduced beforehand, limiting the number of UAVs available for task assignment and constraining the system’s adaptability.

This problem is typically formulated as a mixed-integer programming problem with NP-hard complexity, making it challenging to solve. Most approaches rely on heuristic algorithms [34, 35, 36–37], approximation and transformation methods [32, 38, 39], or reinforcement learning (RL) techniques [34, 40, 41]. Among these, heuristic algorithms have demonstrated remarkable effectiveness, with differential evolution (DE) emerging as a mainstream solution. For instance, Zhang et al. [42] proposed two DE algorithm variants to minimize UAV flight time during data collection in large-scale systems. Similarly, Huang et al. [43] developed a path planning model with three coupled subproblems to reduce energy consumption, employing a variable population size DE algorithm.

To address complex, high-dimensional problems, numerous effective variants of DE algorithms have been developed. While dynamic population sizing and parameter adaptation are well-established features in many DE algorithms, current parameter adjustment methodologies predominantly rely on leveraging historical records of successful parameter configurations [44, 45–46]. Zhan et al. [44] proposed to introduce evolutionary state estimation and to adaptively adjust the population size using feedback information. Wang et al. [47] developed a Multi-niche sampling strategy to support a diverse search in early evolutionary stages and an intensive search in later stages. Sui et al. [48] focused on leveraging the best and worst individuals within the population to maintain diversity and prevent premature convergence. Wang et al. [49] investigated a variant based on accompanying population and piecewise evolution strategy. Furthermore, the design of novel and effective mutation strategies also presents a significant avenue for enhancement. In [37], a DE variant with a differentiated update strategy, distinguishing between elite and disadvantaged individuals, was used to optimize UAV trajectory in mountainous environment.

Drawing inspiration from [50], it is evident that no singular mutation strategy can satisfactorily address all emergent problems. Consequently, the application of diverse mutation strategies, each with distinct control parameter configurations, is likely to yield superior problem-solving capabilities. Zhou et al. [51] divided the population into three subpopulations based on overall fitness values and applied distinct mutation strategies to each. Sun et al. [52] proposed an improved DE algorithm for traffic flow optimization, which is characterized by two sets of mutation and restart strategies that can be dynamically adjusted according to traffic flow. However, while some studies adopted adaptive mechanisms to update algorithm parameters, they often failed to fully utilize historical feedback. To accelerate convergence towards the global optimum, algorithms should dynamically adjust based on the current evolutionary state, incorporating historical parameter successes to improve adaptability. Additionally, although variable population sizes are employed, information exchange between populations remains inefficient. As evolution progresses, diversification of sub-populations through multiple mutation strategies and adaptive size adjustments can enhance solution diversity, balance exploration and exploitation, and prevent stagnation in later stages.

In a multi-stage decision problem, the decision outcome at each stage inherently introduces a degree of systematic misdirection towards the final decision scheme [53]. This is because the locally optimal number of decision points does not necessarily guarantee the globally minimal fitness of the resulting trajectory, known as inductive bias. To this end, we need to formulate a global optimization method to reduce the impact of relying on staged or local decisions on the global decision. In accordance with the above discussions, we aim to employ a model with realistic physical characteristics to replace the simplified UAV model, enabling more comprehensive performance connectivity. Considering the coupled effects of TP locations and quantity, UAV flight speed, and multi-UAV allocation strategies on overall energy consumption, this paper proposes an end-to-end dynamic approach that automates the entire process, from deployment to flight path planning. To address the NP-hard nature of the problem, we propose a modified dynamic population DE variant algorithm with a semi-adaptive multi-strategy. The main contributions of this paper are as follows:

• We develop a multi-UAV-assisted MEC network, deploying multiple UAVs to support task offloading for IoT devices. A kinematic energy model for electric quadrotor UAVs is introduced, incorporating kinetic energy parameters, physical properties, and other practical characteristics for position deployment and 3D trajectory planning.

• A collaborative multi-operator DE variant algorithm with a semi-adaptive strategy is proposed. Its primary theoretical advantage lies in the integration of dynamic parallel exploration and hierarchical semi-adaptive parameter control, which mitigates the issue of erratic search directions due to extreme parameter values. In contrast to static allocation methods, the algorithm dynamically assigns operators to subpopulations based on ongoing performance.

• We design an end-to-end dynamic framework that combines base selection, UAV assignment, and integrated flight path planning to enhance system flexibility. The effectiveness of the proposed coordination method is validated through four numerical simulations, demonstrating significant reductions in system operating costs while maintaining stability and effectiveness compared to existing algorithms.

The subsequent sections of this paper are structured as follows: Section II presents the system model and problem formulation. In Section III, we detail the proposed algorithm. The experimental setup, experimental results and analysis are outlined in Section IV. Finally, Section V concludes this paper.

System model and problem formulation

As shown in Fig. 1, we consider a multi-UAV-assisted system in IoT networks. There are $M$ UAVs, represented by the set $\mathcal{U}=\{u_1,u_2,\dots ,u_M\}$, and $K$ ground device nodes (DNs), represented by the set $\mathcal{G}=\{g_1,g_2,\dots ,g_K\}$. Each UAV departs from the ground BS as the starting point, flies to designated TPs to collect data, and returns to the original BS upon completion of the mission.

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Fig. 1

The schematic diagram of multi-UAV-assisted MEC system

We assume there are $N$ TPs, represented by the set $\mathcal{T}$, where $N$ is an undetermined variable. The location of the n-th TP is denoted by $X_n=(x_n,y_n,h_n)$, $n\in \mathcal{N}=\{1,2,\dots ,N\}$. The position of the m-th UAV at time slot $t$ is given by $X_m(t)=(x_m(t),y_m(t),h_m(t))$ with $m\in \mathcal{M}=\{1,2,\dots ,M\}$, $t\in [0,T]$, where T denotes the mission duration of the UAV. Additionally, DNs are randomly distributed with coordinates $Y_k=(x_k,y_k,0)$, $k\in \mathcal{K}=\{1,2,\dots ,K\}$. We define the set of TPs assigned to the m-th UAV as ${\mathcal{L}}_m=\{1,2,\dots ,l_m\}$, where $l_m$ denotes the number of hovering TPs of the m-th UAV. Furthermore, the flight path of the m-th UAV is represented as a sequence of TPs: ${\Lambda }_m=\left\{X_{m,1},,,X_{m,2},,,\dots ,,,X_{m,l_m}\right\}$, where $X_{m,l_m}$ denotes the location of the $l_m$-th TP of the m-th UAV.

Air-to-ground communication model

We assume that TPs are at the same height. Thus, the Euclidean distance between the n-th TP and the k-th DN is expressed as:

1

$$\begin{array}{c}\hfill d_{n,k}={\Vert X_n-Y_k\Vert }_2.\end{array}$$In relatively open terrain, the Air-to-Ground (A2G) channel is dominated by LoS propagation [54]. Considering complex and dynamic nature of UAV communication environments, we adopt a probabilistic path loss model in which LoS and Non-Line-of-Sight (NLoS) coexist. Signal attenuation varies with the degree of obstruction, and NLoS typically experiences higher path loss than LoS due to shadowing effects and signal reflection [55]. The path loss for both LoS and NLoS links between the n-th TP and k-th DN can be expressed as:

2

$$\begin{array}{cc}& L_{n,k}^{\text{LoS}}=L_{n,k}^{\text{FS}}+{\eta }_{\text{LoS}},\hfill \end{array}$$

3

$$\begin{array}{cc}& L_{n,k}^{\text{NLoS}}=L_{n,k}^{\text{FS}}+{\eta }_{\text{NLoS}},\hfill \end{array}$$where ${\eta }_{\text{LoS}}$ and ${\eta }_{\text{NLoS}}$ denote the shadow fading of the corresponding links. $L_{n,k}^{\text{FS}}$ represents the free space path loss and is expressed as:

4

$$\begin{array}{c}\hfill L_{n,k}^{\text{FS}}=20log\left(d_{n,k}\right)+20log(f_c)+20log\left(\frac{4\pi }{v_c}\right),\end{array}$$where $f_c$ represents the signal frequency, and $v_c$ is the speed of light in a vacuum.

The probabilities of the LoS and NLoS links between the n-th TP and the k-th DN are given by:

5

$$\begin{array}{cc}& P_{n,k}^{\text{LoS}}={\left(1,+,{\xi }_a,exp,\left(-,{\xi }_b,\left(arcsin,\left(\frac{h_n}{d_{n,k}}\right),-,{\xi }_a\right)\right)\right)}^{-1},\hfill \end{array}$$

6

$$\begin{array}{cc}& P_{n,k}^{\text{NLoS}}=1-P_{n,k}^{\text{LoS}},\hfill \end{array}$$where ${\xi }_a$ and ${\xi }_b$ are fixed values determined by the specific environment.

Therefore, the total path loss between the n-th TP and the k-th DN can be expressed as:

7

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L_{n,k}& =P_{n,k}^{\text{LoS}}·L_{n,k}^{\text{LoS}}+P_{n,k}^{\text{NLoS}}·L_{n,k}^{\text{NLoS}}\hfill \\ \hfill & =\left({\eta }_{\text{LoS}},-,{\eta }_{\text{NLoS}}\right)·P_{n,k}^{\text{LoS}}+L_{n,k}^{\text{FS}}+{\eta }_{\text{NLoS}}.\hfill \end{array}\end{array}$$The Signal-to-Noise Ratio (SNR) between the n-th TP and the k-th DN is denoted by:

8

$$\begin{array}{c}\hfill {\gamma }_{n,k}=\frac{P_{\mu }}{{\sigma }_{0}B}·{10}^{-\frac{L_{n,k}}{10}},\end{array}$$where $P_{\mu }$ is the transmit power, ${\sigma }_0$ represents the noise power spectral density, and B denotes the bandwidth allocated to each DN.

Thus, the transmission rate between the UAV at the n-th TP and the k-th DN is as follows:

9

$$\begin{array}{c}\hfill R_{n,k}=B{log}_2\left(1,+,{\gamma }_{n,k}\right).\end{array}$$Since the energy consumed by task-related flight far exceeds that of communication [56], the contribution of communication energy on the total system’s energy consumption is negligible compared to propulsion energy. Therefore, we focus exclusively on the UAV’s propulsion energy.

Each DN can establish an effective connection with the nearest TP to conserve energy. Let Q denote the maximum number of data transmission links that a UAV can simultaneously establish with DNs at a TP. To represent connectivity between DNs and UAVs, we set $I_{n,k}=1$ if data transmission occurs between the n-th TP and the k-th DN; otherwise, $I_{n,k}=0$.

The UAV’s hover time at the n-th TP is expressed as:

10

$$\begin{array}{c}\hfill T_n=\underset{k\in \mathcal{A}(n)}{max}\left\{I_{n,k},·,\frac{S_k}{R_{n,k}}\right\},\end{array}$$where $\mathcal{A}(n)$ signifies the set of DNs connected to the n-th TP, and $S_k$ denotes the data offload size for the k-th DN.

Thus, the energy consumption during hovering at the n-th TP is given by:

11

$$\begin{array}{c}\hfill E_n=P_{\text{hover}}{T}_n,\end{array}$$where $P_{\text{hover}}$ denotes the hover power.

Dynamic and energy consumption model of quad-rotor UAV

Each UAV takes off from a ground-based starting point, flies to TPs to complete its assigned tasks, and then returns to the original starting point. The mission is divided into four main stages: ascent, horizontal flight, hovering, and descent.

We first focus on the horizontal flight and hovering phases, considering the quad-rotor UAV as a rigid body. We assume that the UAV’s horizontal flight speed is $v_h$ and the angular velocity of the motor is ${\omega }_h$.

Based on [57], the current $I_h$ and voltage $U_h$ of each motor at any given moment can be expressed as follows:

12

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I_h& =\frac{T_c}{C_t}{\omega }_h^2+I_0,\hfill \end{array}\end{array}$$

13

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U_h& =\frac{T_c{R}_0}{C_t}{\omega }_h^2+C_e{C}_p{\omega }_h+I_0{R}_0,\hfill \end{array}\end{array}$$where $U_0$, $I_0$, and $R_0$ represent the no-load voltage, no-load current, and motor resistance, respectively. $T_c$ denotes the torque coefficient, $C_t$ specifies the torque constant, $C_e$ represents the back-electromotive force constant, and $C_p$ is the unit conversion factor.

Therefore, the power $P_h$ of each motor can be computed as follows:

14

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill P_h& =U_h{I}_h\hfill \\ \hfill & =K_4{\omega }_h^4+K_3{\omega }_h^3+K_2{\omega }_h^2+K_1{\omega }_h+K_0,\hfill \end{array}\end{array}$$where $K_0=I_0^2{R}_0$, $K_1=C_p{C}_e{I}_0$, $K_2=\frac{2T_c{I}_0{R}_0}{C_t}$, $K_3=\frac{C_p{C}_e{T}_c}{C_t}$, and $K_4=\frac{T_c^2{R}_0}{C_t}$.

For the electrically propelled quad-rotor UAV, the power consumption during horizontal flight is determined by the combined contribution of all four motors. Let ${\alpha }_1$ denote the number of motors in the UAV. During constant-speed flight, the aggregate thrust $T_h$ generated by the motors and the corresponding aerodynamic drag $R_h$ are given as follows [58]:

15

$$\begin{array}{cc}& T_h={\alpha }_1{C}_f{\omega }_h^2,\hfill \end{array}$$

16

$$\begin{array}{cc}& R_h=C_r{v}_h^2,\hfill \end{array}$$where $C_f$ and $C_r$ are two constants representing the thrust coefficient and the fuselage drag coefficient, respectively.

In horizontal flight, the thrust’s horizontal component should overcome drag, while its vertical component should counterbalance the UAV’s gravitational force, as shown in Fig. 2. These components can be calculated as:

17

$$\begin{array}{cc}\hfill T_{h}cos\beta & =G,\hfill \end{array}$$

18

$$\begin{array}{cc}\hfill T_{h}sin\beta & =R_h,\hfill \end{array}$$where $\beta$ is the angle of attack, and G denotes the UAV’s weight.

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Fig. 2

Schematic of UAV forward flight

According to Eqs. (15)–(18), the motor’s angular velocity ${\omega }_h$ can be expressed as:

19

$$\begin{array}{c}\hfill {\omega }_h={({\alpha }_1{C}_f)}^{-\frac{1}{2}}{\left(G^2,+,C_r^2,v_h^4\right)}^{\frac{1}{4}}.\end{array}$$Hence, during horizontal flight and hovering, the power consumption can be expressed as:

20

$$\begin{array}{cc}\hfill P=& {\alpha }_1{P}_h\hfill \\ \hfill =& {\alpha }_1^{-1}{C}_f^{-2}{K}_4\left(G^2,+,C_r^2,v_h^4\right)\hfill \\ \hfill & +{\alpha }_1^{-1/2}{C}_f^{-3/2}{K}_3{\left(G^2,+,C_r^2,v_h^4\right)}^{\frac{3}{4}}\hfill \\ \hfill & +C_f^{-1}{K}_2{\left(G^2,+,C_r^2,v_h^4\right)}^{\frac{1}{2}}\hfill \\ \hfill & +{\alpha }_1^{1/2}{C}_f^{-1/2}{K}_1{\left(G^2,+,C_r^2,v_h^4\right)}^{\frac{1}{4}}+{\alpha }_1{K}_0.\hfill \end{array}$$Thus, by neglecting the additional time and energy losses from the UAV’s acceleration and deceleration during flight, the hovering power $P_{\text{hover}}$ can be determined by setting $v=0$ in Eq. (20).

Let $P_a$ and $P_d$ represent the ascending power and descending power, respectively. The power consumption in the vertical direction can be expressed as follows [59]:

21

$$\begin{array}{cc}\hfill P_a& =\frac{G}{2}{v}_a+\frac{G}{2}{\left(v_a^2,+,\frac{2G}{\rho S_r}\right)}^{\frac{1}{2}},\text{ascending},\hfill \end{array}$$

22

$$\begin{array}{cc}\hfill P_d& =\frac{G}{2}{v}_d-\frac{G}{2}{\left(v_d^2,-,\frac{2G}{\rho S_r}\right)}^{\frac{1}{2}},\text{descending},\hfill \end{array}$$where $v_a$ and $v_d$ denote the ascending and descending velocity, respectively. $\rho$ is the air density, and $S_r$ is the rotor disc area.

Therefore, the overall energy consumption for the mobility of the m-th UAV (excluding hovering) is given by:

23

$$\begin{array}{cc}\hfill E_m=& {\int }_0^{T_a}\left[P,\left(v_h,(t)\right),+,P_a,\left(v_a,(t)\right)\right]dt\hfill \\ \hfill & +\sum _{l=1}^{l_m-1}{\int }_0^{T_{h,l}}P\left(v_h,(t)\right)dt\hfill \\ \hfill & +{\int }_0^{T_d}\left[P,\left(v_h,(t)\right),+,P_d,\left(v_d,(t)\right)\right]dt,\hfill \end{array}$$where $T_a$ and $T_d$ denote the upward trajectory and downward trajectory times, respectively. $T_{h,l}$ represents the horizontal travel time from l-th TP to $l+1$-th TP in the path sequence ${\Lambda }_m$ of m-th UAV, $l\in {\mathcal{L}}_m$.

Problem formulation

Our objective is to minimize the energy consumption of the UAVs and address the path planning challenge for multiple UAVs under constrained resources, with the locations and quantities of TPs being unknown. We introduce a binary variable $J_{n,m}$ to represent the allocation status between UAVs and TPs. $J_{n,m}=1$ indicates that the n-th TP has been allocated to the m-th UAV for task offloading; otherwise, $J_{n,m}=0$ signifies that the n-th TP is not on the flight path of the m-th UAV.

Therefore, the problem can be formulated as follows:

24

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \underset{\left(x_n,,,y_n\right),N,{\Lambda }_m}{min}& \sum _{n=1}^N{E}_n+\sum _{m=1}^M{E}_m\hfill \\ \hfill \text{s.t.}& c_1:I_{n,k},J_{n,m}\in \{0,1\},\forall k\in \mathcal{K},n\in \mathcal{N},m\in \mathcal{M},\hfill \\ \hfill & c_2:\sum _{n=1}^N{I}_{n,k}=1,\forall k\in \mathcal{K},\hfill \\ \hfill & c_3:\sum _{k=1}^K{I}_{n,k}\le Q,\forall n\in \mathcal{N},\hfill \\ \hfill & c_4:\sum _{n=1}^N\sum _{k=1}^K{I}_{n,k}=K,\hfill \\ \hfill & c_5:\sum _{m=1}^M{J}_{n,m}=1,\forall n\in \mathcal{N},\hfill \\ \hfill & c_6:\sum _{n=1}^N\sum _{m=1}^M{J}_{n,m}=N,\hfill \\ \hfill & c_7:x_{min}\le x_m\le x_{max},\hfill \\ \hfill & c_8:y_{min}\le y_m\le y_{max},\hfill \\ \hfill & c_9:0\le v\le v_{max},\hfill \\ \hfill & c_{10}:X_m(0)=X_m^{\text{start}},X_m(T)=X_m^{\text{end}},\forall m\in \mathcal{M}.\hfill \end{array}\end{array}$$Here, $x_{min}$ and $x_{max}$ represent the lower and upper bounds of the UAVs’ positions along the X-axis, while $y_{min}$ and $y_{max}$ correspond to the lower and upper bounds of their positions along the Y-axis. Additionally, $v_{max}$ denotes the maximum allowable flying speed for the UAVs. Constraint $c_2$ ensures that each DN selects only one TP for data transmission. Constraint $c_3$ limits the UAV to providing computational services to a maximum of Q DNs at one TP. Constraint $c_4$ guarantees that all DNs are covered. Constraints $c_5$ and $c_6$ indicate that UAVs should visit all TPs. Constraint $c_{10}$ means that each UAV takes off from and lands at predetermined locations, where $X_m^{\text{start}}$ and $X_m^{\text{end}}$ represent the coordinates of the start and end points, respectively.

Proposed approach

In this section, we propose a variant of the DE algorithm to simultaneously optimize the location and number of TPs. The algorithmic framework is shown in Fig. 3, which takes the distribution of ground IoT devices as input, determines the high-altitude TP deployment for UAVs, and performs multi-UAV path planning to generate the final trajectories. Each deployment scheme is treated as an individual, with dynamic adjustments to the population size. The algorithm introduces three historical feedback-based mutation strategies to enhance solution quality and diversity, along with a new semi-adaptive control parameter strategy. Additionally, an end-to-end dynamic UAV allocation and integrated path optimization method is introduced for 3D UAV trajectory design.

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Fig. 3

A framework for collaborative spatial coordination in UAV deployment and trajectory planning

To tackle the comprehensive deployment problem, we introduce a semi-adaptive phased control parameter scheme designed to balance exploration and exploitation. This approach aims to mitigate the risk of misguided search directions arising from extreme parameter values generated by purely adaptive policies, thereby reducing the wastage of computational resources. Furthermore, in contrast to multi-operator DE variants that allocate resources through fixed assignments or alternating operator usage, we propose a parallel multi-operator mutation strategy that adaptively allocates resources based on solution quality and population diversity. Compared to a single fixed assignment, this approach can reduce the risk of premature loss of operator diversity. It is particularly well-suited for the multi-factor challenge of coordinating multiple UAVs, where factors such as the number of TPs, designated locations, operational sequence, and flight paths must be jointly addressed.

Population size reduction initialization

In UAV deployment optimization, each individual represents a complete deployment, where the number and positions of TPs vary simultaneously. We introduce a linear reduction mechanism to update the population size [48], preserving early-stage diversity and improving search efficiency as iterations progress. The population size of each generation G is expressed as:

25

$$\begin{array}{c}\hfill P{S}_{G+1}=\text{round}\left[\frac{P{S}_{\text{min}}-P{S}_{\text{ini}}}{FE{s}_{\text{Max}}},\times ,F,E,s,+,P,S_{\text{ini}}\right],\end{array}$$where $P{S}_{G+1}$, $P{S}_{\text{min}}$ and $P{S}_{\text{ini}}$ denote the population size at next generation $G+1$, the minimum population size, and the initial population size, respectively. Additionally, FEs and $FE{s}_{\text{Max}}$ denote the current and maximum number of fitness evaluations.

The detailed process of population initialization is provided in Algorithm 1. We set ${\mathcal{N}}_{max}$ as the maximum number of TPs in the initial population. Each complete deployment is represented as a set of coordinates, $\left\{X_1,,,X_2,,,\dots ,,,X_n\right\}$, where $n\le {\mathcal{N}}_{max}$. We randomly generate an initial population of $P{S}_{\text{ini}}$ individuals. Each individual has an encoding size of $3{\mathcal{N}}_{max}$, representing the position coordinates of ${\mathcal{N}}_{max}$ TPs. Then, each individual is checked for feasibility based on constraints. If any individual fails to meet the constraints, it is randomly regenerated until it becomes feasible.

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Algorithm 1

Setting Up the Initial Population $\mathcal{P}$

Collaborative adaptation evolution strategy

The mutation operation can be seen as a random linear combination of individuals from the parent population, while the crossover operation involves random exchanges across each dimension. The traditional DE algorithm typically uses a fixed scaling factor F and a constant crossover probability CR [60]. However, in complex optimization problems, relying on a single parameter generation can waste fitness evaluations and lead to premature convergence to local optima [61]. Therefore, we propose a semi-adaptive control parameter strategy based on historical feedback, combined with three distinct mutation strategies, to achieve elite-driven optimization through an archival mechanism.

Adaptive multi-operator mutation optimization strategy

We employ three mutation operators to generate vectors for each solution in the population [62]. In the initial phase, each operator evolves an equal number of individuals. As iterations progress, the number of individuals evolved by each mutation strategy is dynamically adjusted based on solution quality and diversity.

The subpopulation ${\mathcal{P}}_1$ use DE/current-to-$\rho$best with archive/1 as the mutation strategy. The archive stores replaced individuals, which can be reintroduced in subsequent mutations to maintain population diversity. The strategy is show in Eq. (26).

26

$$\begin{array}{c}\hfill {\mathbf{v}}_{i,z}={\mathbf{x}}_{i,z}+F_{i,z}·\left({\mathbf{x}}_{best,z}^{p_1},-,{\mathbf{x}}_{i,z}\right)+F_{i,z}·\left({\mathbf{x}}_{r_1,z},-,{\mathbf{x}}_{r_2,z}\right),\\ \hfill \end{array}$$where ${\mathbf{x}}_{i,z}$ represents the i-th individual in generation z, and ${\mathbf{v}}_{i,z}$ is the corresponding mutant vector. ${\mathbf{x}}_{r_1,z}$ is randomly selected from the population, ${\mathbf{x}}_{r_2,z}$ is randomly chosen from both the population and archive, and ${\mathbf{x}}_{best,z}^{p_1}$ is drawn from the top $p_1\%$ of optimal target vectors in the whole population, with $r_1\ne r_2\ne i$. Moreover, $F_{i,z}$ denotes the scale factor, which dynamically adapts and updates during the process.

The subpopulation ${\mathcal{P}}_2$ adopts a mutation strategy without archive, as shown in Eq. (27).

27

$$\begin{array}{c}\hfill {\mathbf{v}}_{i,z}={\mathbf{x}}_{i,z}+F_{i,z}·\left({\mathbf{x}}_{best,z}^{p_1},-,{\mathbf{x}}_{i,z}\right)+F_{i,z}·\left({\mathbf{x}}_{r_1,z},-,{\mathbf{x}}_{r_3,z}\right),\\ \hfill \end{array}$$where ${\mathbf{x}}_{r_3,z}$ is randomly selected from the whole population, $r_1\ne r_2\ne r_3\ne i$.

The subpopulation ${\mathcal{P}}_3$ is updated using a weighted-rand-to-$\rho$best mutation operator, defined as:

28

$$\begin{array}{c}\hfill {\mathbf{v}}_{i,z}=F_{i,z}·{\mathbf{x}}_{r_1,z}+\left({\mathbf{x}}_{best,z}^{p_2},-,{\mathbf{x}}_{r_3,z}\right),\end{array}$$where ${\mathbf{x}}_{best,z}^{p_2}$ is drawn from the top $p_2\%$ individuals.

In addition, to reduce the evaluation of duplicate and redundant solutions while preserving FEs, a semi-adaptive parameter mechanism is proposed. This mechanism retains successful parameters while dynamically adjusting them based on the evaluation process. The semi-adaptive adjustment process for the scaling factor $F_{i,z}$ is shown in Eq. (29).

29

$$\begin{array}{c}\hfill F_{i,z}=\left\{\begin{array}{cc}{r}_{F,1}+{\sigma }_F·rand{n}_i,\hfill & \text{if}FEs<{\mathrm{{\rm Y}}}_{F,1},\hfill \\ min\left(r_{F,2},,,r,a,n,d,c_i,\left({\mu }_{F,r_i},,,{\sigma }_F\right)\right),\hfill & \text{if}{\mathrm{{\rm Y}}}_{F,1}<FEs<{\mathrm{{\rm Y}}}_{F,2},\hfill \\ rand{c}_i\left({\mu }_{F,r_i},,,{\sigma }_F\right),\hfill & \text{otherwise},\hfill \end{array}\right)\\ \hfill \end{array}$$where ${\mu }_{F,r_i}$ represents the archive of successful historical control parameter, and $r_i$ is a random integer between 1 and the number of parameters in the control parameter pool. ${\mathit{randn}}_i$ and ${\mathit{randc}}_i$ follow Gaussian and Cauchy distribution, respectively, with ${\sigma }_F$ denoting the standard deviation. $r_{F,1}$ and $r_{F,2}$ are parameter constants affecting the updating process. ${{\rm Y}}_{F,1}$, ${{\rm Y}}_{F,2}$ denote the threshold values for each stage in the adaptive strategy.

Hybrid crossover strategy

Binomial and exponential crossovers are the two most commonly used methods and are typically implemented separately [63]. Here, we randomly apply both methods to develop a new hybrid crossover strategy. In generation z, the crossover operation is performed between individual ${\mathbf{x}}_{i,z}$ and the mutant individuals ${\mathbf{v}}_{i,z}$, denoted as follows:

30

$$\begin{array}{c}\hfill {\mathbf{u}}_{i,z}=\left\{\begin{array}{c}\text{if}rand\left[0,,,1\right]<p_u\hfill \\ \left\{\begin{array}{c}{\mathbf{v}}_{i,z},\text{if}rand\left[0,,,1\right]\le C{R}_{i,z}\cup z=z_{\mathit{rand}},\hfill \\ {\mathbf{x}}_{i,z},\text{otherwise},\hfill \end{array}\right)\hfill \\ \text{otherwise}\hfill \\ \left\{\begin{array}{c}{\mathbf{v}}_{i,z},\text{if}z={⟨n⟩}_D,{⟨n+1⟩}_D,...,{⟨n+L-1⟩}_D,\hfill \\ {\mathbf{x}}_{i,z},\text{otherwise},\hfill \end{array}\right)\hfill \end{array}\right)\\ \hfill \end{array}$$where $p_u$ is the selection probability parameter, D denotes the number of decision variables, and $z_{\mathit{rand}}$ is a randomly selected integer from the interval [1, D]. During exponential crossover, the integer n is randomly chosen from the range [1, D] to determine the starting position, and another integer L is selected from the same range to indicate the number of positions in the target individual to be replaced by the mutant individual. The notation ${⟨⟩}_D$ denotes a modulo operation that wraps around to the beginning when the index exceeds D, ensuring the crossover remains within the valid range.

The detailed process of crossover factor $C{R}_{i,z}$ is shown in Eq. (31).

31

$$\begin{array}{c}\hfill C{R}_{i,z}=\left\{\begin{array}{cc}max\left(r_{C,1},,,r,a,n,d,n_i,\left({\mu }_{CR,r_i},,,{\sigma }_{\mathit{CR}}\right)\right),\hfill & \text{if}FEs<{{\rm Y}}_{C,1},\hfill \\ max\left(r_{C,2},,,r,a,n,d,n_i,\left({\mu }_{CR,r_i},,,{\sigma }_{\mathit{CR}}\right)\right),\hfill & \text{if}{{\rm Y}}_{C,1}<FEs<{{\rm Y}}_{C,2},\hfill \\ rand{n}_i\left({\mu }_{CR,r_i},,,{\sigma }_{\mathit{CR}}\right),\hfill & \text{otherwise},\hfill \end{array}\right)\end{array}$$ where ${\mu }_{CR,r_i}$ represents the adaptive successful parameter of CR, and ${\sigma }_{\mathit{CR}}$ denotes the standard deviation. ${{\rm Y}}_{C,1}$ and ${{\rm Y}}_{C,2}$ denote the threshold values for the number of evaluations. $r_{C,1}$ and $r_{C,2}$ are the empirical parameters at different evolutionary stages, respectively.

Historical memory update based on fitness difference weighted strategy

In each iteration, if an offspring individual successfully replaces its parent in the selection phase, the control parameters F and CR used by the offspring are recorded as successful parameter values. To dynamically adjust F and CR, we introduce a historical memory mechanism based on these success parameters. These historical memories are denoted by the vectors ${\mu }_F=\left[{\mu }_{F,1},,,{\mu }_{F,2},\cdots ,{\mu }_{F,Len}\right]$ and ${\mu }_{\mathit{CR}}=\left[{\mu }_{CR,1},,,{\mu }_{CR,2},\cdots ,{\mu }_{CR,Len}\right]$, where Len denotes the length of the historical memory. When a mutated individual outperforms the initial one in fitness, the control parameters $F_i$ and $C{R}_i$ of the successful individual are stored in the corresponding auxiliary vectors $S_F$ and $S_{\mathit{CR}}$.

We first calculate the weights ${\omega }_c$ based on fitness differences, as given by Eq. (32).

32

$$\begin{array}{c}\hfill {\omega }_c=\frac{|f\left({\mathbf{x}}_{i,z}\right)-f\left({\mathbf{u}}_{i,z}\right)|}{{\sum }_{c=1}^{|S_F|}|f\left({\mathbf{x}}_{i,z}\right)-f\left({\mathbf{u}}_{i,z}\right)|},\end{array}$$where f denotes the objective function.

At the end of current generation z, we update the l-th elements of the recording memory, where $l\in [1,Len]$. If there are no successful individuals, the values from the previous generation are retained; otherwise, they are updated as follows:

33

$$\begin{array}{c}\hfill {\mu }_{F,l}=\frac{{\sum }_{c=1}^{|S_F|}{w}_c·S_F^2(c)}{{\sum }_{c=1}^{|S_F|}{w}_c·S_F(c)},\end{array}$$

34

$$\begin{array}{c}\hfill {\mu }_{CR,l}=\frac{{\sum }_{c=1}^{|S_{\mathit{CR}}|}{w}_c·S_{\mathit{CR}}^2(c)}{{\sum }_{c=1}^{|S_{\mathit{CR}}|}{w}_c·S_{\mathit{CR}}(c)}.\end{array}$$Moreover, the selection of strategy is dynamically adjusted throughout the optimization process [64]. At the end of each generation, we identify the optimal solution and refine the strategy selection by incorporating fitness differences and diversity.

A high-quality solution indicates proximity to the optimal value of the objective function. At the end of each generation, we assess the solution quality for each operator by identifying the optimal individual from that generation, which is expressed by:

35

$$\begin{array}{c}\hfill {\varpi }_s=\frac{f_{s,z}^{best}}{{\sum }_{s=1}^{n_s}{f}_{s,z}^{best}},\forall s=1,2,\dots ,n_s,\end{array}$$where ${\varpi }_s$ denotes the quality of operator s, $f_{s,z}^{best}$ is the optimal fitness solution achieved by operator s in generation z, and $n_s$ is the total number of operators.

In addition, the solution diversity metric ${\phi }_s$ is utilized to quantify the dispersion degree of the solutions, calculated by Eq. (36) and Eq. (37).

36

$$\begin{array}{cc}& {div}_s=\frac{{\sum }_{i=1}^{P{S}_s}dis\left({\mathbf{x}}_{i,z}^s,-,{\mathbf{x}}_{i,z}^{\text{best}}\right)}{P{S}_s},\forall s=1,2,\dots ,n_s,\hfill \end{array}$$

37

$$\begin{array}{cc}& {\phi }_s=\frac{{div}_s}{{\sum }_{s=1}^{n_s}{div}_s},\forall s=1,2,\dots ,n_s,\hfill \end{array}$$where $P{S}_s$ denotes the number of individuals evolved by operator s, and $dis\left({\mathbf{x}}_{i,z}^s,-,{\mathbf{x}}_{i,z}^{\text{best}}\right)$ denotes the Euclidean distance between the i-th individual and the best individual obtained by operator s.

Therefore, we integrate solution quality and diversity to assess each operator’s improvement performance, enabling adjustment of the number of individuals evolved by each operator. The number of solutions $P{S}_s$ evolved by each operator s is calculated as follows:

38

$$\begin{array}{cc}\hfill P{S}_s& =max\left({\lambda }_{\text{low}},,,min,\left({\lambda }_{\text{high}},,,\frac{\left(1,-,{\varpi }_s\right)+{\phi }_s}{{\sum }_{s=1}^{n_s}\left[\left(1,-,{\varpi }_s\right),+,{\phi }_s\right]}\right)\right)\hfill \\ \hfill & \times PS,\forall s=1,2,\dots ,n_s,\hfill \end{array}$$where ${\lambda }_{\text{low}}$ and ${\lambda }_{\text{high}}$ represent the lower and upper boundary values, respectively. Meanwhile, ${\sum }_{s=1}^{n_s}P{S}_s=PS$ ensuring that the total number of individuals evolved by all mutation strategies equals the population size.

Algorithm 2 shows the update method process. After initializing the population, we first divide it into $n_s$ subpopulations and initialize the control historical memories. Afterward, the algorithm proceeds with the iterative process for UAV deployment. we apply different mutation and crossover operations to individuals in each subpopulation. Then, we evaluate the performance of individuals based on fitness differences, retain the top-performing solutions and update control parameter history pools $⟨u_F,u_{\mathit{CR}}⟩$. To promote elite-driven evolution and prevent stagnation, the algorithm adjusts the number of evolved individuals for each operator by assessing solution quality and diversity. Furthermore, based on the evaluation process, the population size as well as parameters F and CR are dynamically adjusted. The entire process repeats until the maximum evaluation count is reached. Finally, the best individual in the new population $\mathcal{P}$ is accepted as the final deployment.

[See PDF for image]

Algorithm 2

Population Update $\mathcal{P}$

Trajectory planning optimization

In this phase, we tackle a mixed-integer non-convex optimization problem that simultaneously manages UAV scheduling and UAV-BS allocation. To efficiently plan UAV flight paths from at least one to a maximum of M bases, we employ a collective optimization heuristic algorithm. This algorithm integrates base selection, UAV allocation, and flight path planning into a unified framework. By automating the selection of bases and UAVs, it significantly reduces unnecessary energy consumption and flight time that would result from additional manual choices.

In the context of UAV deployment, the selection of the number and location of deployment locations, flight order, and trajectory are typically treated as independent optimization problems, often requiring experimenters to determine their respective optimal solutions separately [65]. Nevertheless, this characteristic independence often leads to suboptimal final trajectory generation, which constitutes an intrinsic decision-making deficiency within UAV planning. To address this limitation, we propose an end-to-end decision-making framework. Our algorithm enables the joint optimization of the number of TPs, the locations of TPs, multi-UAV flight sequence, and trajectories based on the distribution of DNs. The entire process eliminates the need for additional settings in intermediate steps, which theoretically ensures the optimality of the final decision outcomes.

As shown in Algorithm 3, we first determine the complete set of TPs’ locations and number. Before planning the UAV trajectories, we initialize the locations of all UAV-BSs. Then, for each UAV, an initial path is randomly generated to ensure that all TPs are covered. At each round of path planning, the algorithm reorganizes the path sequences, transfers TPs between UAVs, dynamically adjusts the current number of UAVs lenM, and reassign TPs accordingly under given constraints. The iterative calculations continue until the maximum number of iterations Maxiter is reached. In each round, the algorithm accepts new solutions that minimize the objective, gradually approaching a global optimum. Finally, the UAV trajectories are determined. Furthermore, the completion time of the entire MEC task depends on the return of the last UAV to its base. The algorithm can adjust the targeted optimization paths as needed, ensuring the efficient completion of the entire task.

[See PDF for image]

Algorithm 3

End-to-End Execution for Task Offloading and Trajectory Planning

Experimental studies

Parameter settings

We consider DNs to be randomly distributed within a $1000\text{m}\times 1000\text{m}$ area, with UAVs flying at a fixed altitude of $100\text{m}$ in the same horizontal plane. The UAV-BSs are located at the four corners of the square ground area. Each BS can launch up to one UAV, with at least one UAV active for the entire system. The number of UAVs, M, is an integer within the range $\in \left[1,,,4\right]$, and the number of motors is ${\alpha }_1=4$. The maximum vertical and horizontal speed is $v_{\mathit{max}}=10$$\text{m}/\text{s}$, and the maximum acceleration magnitude is $a_{\mathit{max}}=3$$\text{m}/{\mathrm{s}}^2$. The data sizes $S_k$, $k\in \mathcal{K}$, are randomly distributed in the range $[1,{10}^3]$ MB. The maximum number of data transmission links for a TP is $Q=5$. The communication parameters, UAV flight-related simulations, and other simulation settings are summarized in Table 1.

Table 1. Parameter settings

Notation	Definition	Value
H	UAV flight height	100 m
G	UAV weight	29.4 N
$f_c$	Carrier frequency	2 GHz
B	Channel bandwidth	1 MHz
${\eta }_{\text{LoS}}$	Additional pass loss for LoS links	1.6 dB
${\eta }_{\text{NLoS}}$	Additional pass loss for NLoS links	23 dB
${\sigma }_0$	Noise power spectral density	${10}^{-17}$ W/Hz
$p_{\mu }$	Transmit power	1 W
${\xi }_a$	The parameter of A2G path loss model	12.08
${\xi }_b$	The parameter of A2G path loss model	0.11
$T_c$	Torque coefficient	$8.891\times {10}^{-7}$$\text{N}·\text{m}/{(\text{rad}/\text{s})}^2$
$C_t$	Torque constant	$2.483\times {10}^{-2}$
$C_e$	Back-electromotive force constant	$2.6\times {10}^{-3}$
$C_f$	Thrust coefficient	$4.848\times {10}^{-5}$$\text{N}/{(\text{rad}/\text{s})}^2$
$C_r$	Fuselage drag coefficient	0.11 $\text{N}/{(\text{m}/\text{s})}^2$
$C_p$	Unit conversion factor	9.5493
$v_c$	The velocity of light	$3\times {10}^8$ m/s
$\rho$	Air density	1.205 $kg/m^3$
$S_r$	Rotor disc area	0.196 $m^2$
$I_0$	No-load current	0.3 A
$U_0$	No-load voltage	10 V
$R_0$	No-load resistance	0.4 $\mathrm{\Omega }$

Algorithms for comparison

To validate the performance of CSADE, we compare it with the DE algorithm [66] and the SHADE algorithm [67]. The DE algorithm, as a classical heuristic optimization method, has fixed control parameters throughout the evolutionary process. This means that the search efficiency often depends on the setting of the control parameters, making it easy to fall into a local optimal solution. Researchers are increasingly focusing on parameter adaptive methods. The SHADE algorithm is an improved version of the DE algorithm. It adopts an adaptive parameter mechanism to generate control parameters.

In the CSADE algorithm, the initial values of ${\mu }_F$ and ${\mu }_{\mathit{CR}}$ are set to 0.2. The scale factor $F$ is adjusted in stages using Cauchy distribution with a scale parameter ${\sigma }_F=0.1$. The parameters $r_{F,1}$ and $r_{F,2}$ are set to 0.5 and 0.7, respectively. The phase thresholds are defined as ${\mathrm{{\rm Y}}}_{F,1}=0.3FE{s}_{\text{Max}}$ and ${{\rm Y}}_{F,2}=0.6FE{s}_{\text{Max}}$. Additionally, the standard deviation ${\sigma }_{\mathit{CR}}$ is 0.1, with $r_{C,1}$ and $r_{C,2}$ set to 0.7 and 0.6, respectively. Its phase thresholds are set to ${\mathrm{{\rm Y}}}_{C,1}=0.3FE{s}_{\text{Max}}$ and ${{\rm Y}}_{C,2}=0.5FE{s}_{\text{Max}}$. We set the initial population size to $P{S}_{ini}=100$, with a minimum size of $P{S}_{min}=4$. The boundaries ${\lambda }_{\text{low}}$ and ${\lambda }_{\text{high}}$ are set to 0.1 and 0.9, respectively.

The parameters of the compared algorithms are from the corresponding references. In DE, the mutation probability and crossover probability are set to 0.5 and 0.9, respectively. In SHADE, the historical memory control parameters are initialized to 0.5. In order to obtain fair and consistent test results, the maximum number of fitness evaluations $FE{s}_{\text{Max}}$ is set to 150,000, with a uniform population size $\mathit{PS}$ of 100. Each of the three algorithms is independently executed 10 times on four different instances, with DN counts of $K=\{50,100,150,250\}$.

Performance evaluation under various instances

To illustrate the effectiveness of the CSADE algorithm, we give the comparison experiments with two other algorithms. The statistical results for the average deployment energy over 10 independent trials are reported in Table 2. In Table 2, the performance indicators include the best solution (Best), representing the minimum energy consumption of the system, as well as the mean solution (Mean) and the standard deviation (Std). As shown in the table, our proposed algorithm outperforms the other algorithms across all instances. Furthermore, as the number of DNs increases, the hover energy consumption also rises, yet CSADE consistently demonstrates superior performance. Specifically, when $K=100$, CSADE is better than DE and SHADE by 28.88% and 15.86%, respectively. When $K=250$, CSADE achieves improvements of 24.46% and 14.46% over DE and SHADE, respectively. This indicates that even as the spatial search dimension expands, the algorithm still obtains good results due to its effective adaptability and improvement strategies.

Table 2. Comparison of mean energy consumption results of different algorithms

K	Measure	DE	SHADE	CSADE
50	Best	1.1903E+06	9.4560E+05	8.5310E+05
	Mean	1.2223E+06	9.7794E+05	8.8057E+05
	Std	1.8587E+04	1.9545E+04	1.3097E+04
100	Best	2.1939E+06	1.8704E+06	1.5513E+06
	Mean	2.2581E+06	1.9081E+06	1.6057E+06
	Std	4.2973E+04	2.6989E+04	2.1742E+04
150	Best	2.9580E+06	2.5780E+06	2.1971E+06
	Mean	3.1295E+06	2.6671E+06	2.2583E+06
	Std	6.1594E+04	4.9115E+04	2.9268E+04
250	Best	4.7317E+06	4.2235E+06	3.6030E+06
	Mean	4.8793E+06	4.3072E+06	3.6846E+06
	Std	8.3148E+04	5.5545E+04	3.9039E+04

[See PDF for image]

Fig. 4

Box plot from ten repeated experiments in different data sets. a K = 50, b K = 100, c K = 150, d K = 250

To provide an intuitive comparison of the impact of different methods, we use box plots to represent the statistical distribution of the data. Figure 4 presents the statistical results for each algorithm, run 10 times on each dataset. Clearly, we can observe that CSADE achieves better mean and median values. The long line segments at the top and bottom of the blue box represent the maximum and minimum values of the data, respectively. In each instance, the worst value achieved by the proposed method is superior to the best value obtained by the comparison method. In particular, the interquartile range (IQR, 25%–75%) of the experimental results narrows significantly, as indicated by the length of the box. The overall data distribution is relatively concentrated, with the whiskers close to the median, indicating a small overall variability. Additionally, the scattered data points show no obvious outliers vertically, demonstrating a consistent overall performance in datasets. When $K=50$, the IQR of CSADE is $1.4473\times {10}^4$, reflecting reductions of 28.82% and 42.23% compared to DE and SHADE, respectively. When $K=150$, the IQR of CSADE is $2.0281\times {10}^4$, which is 53.39% and 63.63% lower than that of DE and SHADE, respectively. Thus, as the number of DNs increases, CSADE exhibits higher stability over multiple independent runs and a more centralized distribution of outcomes.

To evaluate the scalability and applicability of the proposed method in large-scale scenarios, we designed four sets of extended experiments. In each scenario, 400 DNs are randomly distributed within areas of increasing size. To accommodate the expanded coverage, the number of UAVs is incrementally increased. The four experimental settings are as follows: (A) 4 UAVs in a $1\text{km}\times 1\text{km}$ area; (B) 6 UAVs in a $2\text{km}\times 2\text{km}$ area; (C) 8 UAVs in a $3\text{km}\times 3\text{km}$ area; and (D) 10 UAVs in a $4\text{km}\times 4\text{km}$ area. For each region, UAV base stations are symmetrically and evenly deployed along the upper and lower boundaries. As shown in Fig. 5, a comparative analysis of different algorithms indicates that the method proposed in this study can effectively identify more optimal UAV deployment strategies and maintain excellent energy efficiency across diverse geographical environments and under continuously expanding system scales. Specifically, compared to traditional methods, our proposed method enhances energy efficiency by 27.7%, 51.8%, and 71.9%, respectively. As the coverage area expands and the number of UAVs increases, the performance advantages of our scheme become increasingly pronounced. Notably, with the expansion of the network scale, the path length UAVs are required to traverse for task execution also increases, leading to a significant rise in total system energy consumption, which further highlights the competitiveness of our approach.

Comparison of the algorithm convergence

Figure 6 shows the changes in the system’s average deployment energy consumption. It can be seen that under the same number of fitness evaluations, the CSADE algorithm all shows the advantages of stable and superior performance across different instance. In the initial phase, all algorithms rapidly reduce energy consumption, with SHADE showing a faster decrease compared to the others. However, taking $K=50$ as an example, CSADE’s surpasses SHADE in energy efficiency after $FEs=2700$. It maintains this advantage in subsequent evaluations, eventually converging to $8.9695\times {10}^5$ at $FEs=1.1578\times {10}^5$. As the number of evaluations increases, both DE and SHADE approach a stable state of convergence, whereas CSADE continues to decrease. Moreover, during evaluations on different data instances, CSADE maintains a relatively lower energy level, with convergence results outperforming those of other algorithms. This reflects and verifies that the improvement strategies effectively enhance its ability to escape local optima.

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Fig. 5

Comparison of energy consumption in networks across four geographic scales. A 4 UAVs for 1 km²; B 6 UAVs for 4 km²; C 8 UAVs for 9 km²; D 10 UAVs for 16 km²

[See PDF for image]

Fig. 6

Average energy consumption of UAV deployments in different data sets. a K = 50, b K = 100, c K = 150, d K = 250

Figure 7 illustrates the influence of DNs number on the energy consumption performance of the flight path. It can be clearly seen that the curves converge rapidly when $K=50$, $K=100$ and $K=150$, reaching a steady state with fewer evaluations. As the number of DNs increases, the convergence of the system slows down and the energy consumption of the UAV’s movement increases. This is primarily due to communication service limitations on the number of nodes a single TP can serve simultaneously. As a result, the UAV needs to fly more TPs, leading to a further increase in the UAV’s mobility cost. After convergence, the flight energy is reduced by 62.22%, 62.22%, 67.97%, and 66.69%, respectively, compared to the initial system flight energy consumption across four different instances.

[See PDF for image]

Fig. 7

Mobile energy consumption for network offloading tasks in UAVs

Verification of flight trajectory

Figure 8 depicts the TP deployments and the 3D trajectory of UAVs for four different DN distributions. As illustrated in Fig. 8, different colored lines indicate the flight trajectories of different UAVs. Morever, the blue dots distributed in the plane with $H=0$ represent DNs, and the gray dots distributed in the plane with $H=100$ represent TPs. All four UAVs take off. For $K=50,100,150,\text{and}250$, the corresponding numbers of TPs are 13, 21, 33, and 54, respectively. Our approach effectively ensures the deployment of TPs at minimal cost while adhering to service constraints. As the number of TPs increases, path complexity also rises, demonstrating the proposed method’s adaptability to complex 3D environments. Furthermore, it can be observed that each UAV takes off from its BS, sequentially visits its corresponding set of TPs to complete the service, and then returns to the origin BS. The UAV path ensures that each TP is effectively served and significantly reduces unnecessary round trips, providing an efficient end-to-end solution. This is a joint optimization of both local (individual UAV energy consumption) and global (total energy consumption and mission efficiency) benefits in order to avoid task delay caused by excessive pursuit of low energy consumption.

Conclusion

In this paper, we investigate a multi-UAV-assisted MEC system aimed at minimizing system overhead by jointly optimizing the number of UAV TP locations, 3D trajectories, and base station scheduling. We integrate the kinematic and physical properties of UAVs to develop a new energy model for path planning. In addition, we propose a dynamic population DE algorithm variant that integrates multiple improvement strategies to simultaneously determine both the number and locations of UAV TPs. Furthermore, we propose an end-to-end method for UAV dynamic assignment and flight path optimization to ensure efficient trajectory planning. Experimental results confirm the convergence of the proposed method, and show that it significantly reduces system energy consumption compared to the benchmark algorithm. In future work, we will consider dynamic ground node data acquisition and task partial offloading scenarios to improve network performance. Further studies include leveraging DRL to design hybrid cooperative UAV flight trajectories and scheduling strategies in complex environments.

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Fig. 8

UAV trajectories with different DNs distributions. a K = 50, b K = 100, c K = 150, d K = 250

Author contributions

Conceptualization: Fengling Huang, Quanbao Wang; Methodology: Fengling Huang, Xuqi Su; Writing - original draft preparation: Fengling Huang, Xuqi Su; Writing review and editing: Quanbao Wang, Fusen Guo.

Funding

This work was supported by the Fundamental Research Funds for the Central Universities.

Data availability

The codes and datasets generated or analyzed during the current study are available from the corresponding author upon reasonable request.

Declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Ethical Approval

Not applicable. The work presented in this research did not involve human participants or animal subjects, and therefore, did not require ethical approval concerning human or animal research.

Consent to participate

Not applicable.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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