Energy Efficiency and Total Mission Completion Time Tradeoff in Multiple UAVs-Mounted IRS-Assisted Data Collection System

1  Introduction

Sensors can be deployed in farmlands to collect environmental data such as soil moisture, air temperature, and wind speed, laying the foundation for Internet of Things (IoT) in precision agriculture [1]. However, limited energy storage restricts their operational lifespan. If sensor-collected data cannot be transmitted to the fusion center (FC) promptly, the finite memory capacity of sensors may lead to data overwriting. Furthermore, obstacles between sensors and the FC can result in incomplete data transmission due to the lack of a direct line-of-sight (LoS) path.

To address these issues, unmanned aerial vehicles (UAVs) have been leveraged as mobile base stations (BSs), relays, or data collectors [2]. Their high mobility and flexible deployment enable them to establish effective communication links, effectively mitigating non-line-of-sight (NLoS) problems. Nevertheless, UAVs are constrained by battery power, limiting their flight range and operational duration. This makes optimizing UAV trajectories crucial for energy-efficient data collection, especially when sensor nodes (SNs) are widely dispersed [3]. For tasks beyond the capability of a single UAV, collaborative multi-UAV systems are deployed [4], often by partitioning SNs among UAVs for full coverage [5]. Research in multi-UAV systems has focused on various objectives, including minimizing energy consumption [6], reducing the maximum task completion time [7], lowering costs [8], and enhancing data collection rates [9]. Further studies explore tradeoffs between performance metrics, such as balancing computation capacity and energy consumption [10] or optimizing energy usage vs. task completion time [11].

As another way to improve channel quality, intelligent reflecting surface (IRS) manipulates incident signal propagation [12], enabling communication between SNs and FC without deploying costly mobile base stations (BSs) or relays, thereby enhancing energy and spectral efficiency (EE and SE) for reliable system performance. Conventional IRS is installed on building surfaces [13], offering ease of installation but limited coverage and mobility. To address this, researchers propose integrating IRS with UAVs. UAV-mounted IRS (U-IRS) [14] delivers superior flexibility, link quality, and coverage for SNs. In such U-IRS-assisted IoT, SNs transmit signals to the FC via U-IRS, effectively bypassing NLoS channel limitations.

1.1 Related Work

As a transformative technology, U-IRS significantly enhances the performance of IoT. Existing studies broadly fall into single-U-IRS and multi-U-IRS-assisted IoT, as summarized below.

1.1.1 Single-U-IRS-Assisted IoT

Researchers primarily focus on enhancing capacity/achievable rate, reducing energy consumption, and improving EE. Notably, Ref. [14] pioneered the concept of U-IRS and demonstrated its advantage over terrestrial IRS in terms of average data rate and LoS probability, laying the foundation for subsequent studies. Subsequent studies typically involve the joint optimization of parameters such as SN transmit power, UAV trajectory, IRS phase shifts, and the number of reflecting elements (NoRE) to maximize the sum rate [15–17] or the minimum data rate [18]. Other works have extended these optimizations to maximize secrecy rate [19–21] or to design energy-aware trajectories [22–24]. Metrics like EE, defined as the total information bits per propulsion energy, have been quantified to explore rate-energy tradeoff [25], with further investigations into solar-powered U-IRS for EE maximization [26] and applications in maritime communications [27]. A max-min optimization framework for covert throughput and propulsion efficiency was proposed in [28].

However, a common shortcoming of these single-U-IRS studies is their inherent limitation in coverage and task capacity. The performance of a single U-IRS is bounded by the UAV’s limited flight endurance and range. Consequently, they are unsuitable for large-scale IoT deployments where SNs are widely dispersed, creating a clear motivation for employing multiple collaborative U-IRSs.

1.1.2 Multi-UAV IRS-Assisted IoT

For complex tasks exceeding single-UAV capabilities, multi-U-IRS collaboration is essential. However, research in this area remains sparse. Existing works have jointly optimized UAV placement, BS beamforming, and IRS passive beamforming to maximize the weighted sum rate [29], minimized UAV energy consumption in mobile-edge computing networks [30], derived outage probability for non-orthogonal multiple access (NOMA) systems [31], and minimized the maximum task offloading time by optimizing UAV positions [32].

1.1.3 Challenges in Existing Studies

Based on the above review, we identify the following specific gaps in the current literature on multi-U-IRS-assisted IoT:

i) Impact of NoRE: While parameters like trajectory and phase shifts are extensively optimized, with less consideration given to the impact of NoRE in the IRS. Although studies like [22,23] that analyze NoRE assume identical channel characteristics for all reflecting elements, resulting in low accuracy in the optimal NoRE.

ii) Optimization of total task completion time: In practical scenarios like smart agriculture, where frequent data collection is needed, minimizing the total mission completion time is vital for long-term cost efficiency. The number of UAVs, the energy provided to UAVs, and the flight trajectories of UAVs all affect the total mission completion time, which is related to costs. While prior multi-UAV research has focused on minimizing the maximum completion time (critical for time-sensitive applications like disaster relief or emergency communications), e.g., [7]. This metric does not guarantee a minimal total completion time. For instance, optimizing two UAVs’ completion times from (7, 2 s) to (6, 4 s) reduces the maximum time but increases the total time from 9 to 10 s. The optimization of total completion time in multi-U-IRS systems is rarely addressed.

iii) Tradeoff between EE and total time: EE and total task completion time are interdependent metrics crucial for system performance and cost. However, the tradeoff between these two key performance indicators remains significantly under-explored in the context of multi-U-IRS-assisted IoT.

1.2 Motivation and Contributions

The aforementioned gaps motivate our work. Specifically, we investigate the fundamental tradeoff between EE and the total task completion time in a multi-U-IRS-assisted IoT data collection system. The main contributions of this paper are fourfold:

•   We establish a data collection model for multi-U-IRS-assisted IoT. The expressions for the achievable rate, energy consumption, and system EE are formulated. On this basis, the total task completion time is mathematically characterized. Then, the impacts of UAV-SN association, NoRE, and UAVs trajectories on EE and total mission completion time are analyzed.

•   We propose a novel balanced objective function formulated as a weighted sum of the total task completion time and the reciprocal of EE. This formulation allows us to concurrently maximize EE and minimize the total time under practical constraints.

•   To solve the resulting non-convex and complex optimization problem, we develop an efficient alternating optimization algorithm. The algorithm decomposes the problem into three subproblems: (i) the UAV-SN association, reformulated as a multi-traveling salesman problem (mTSP) and solved via a genetic algorithm; (ii) the NoRE allocation, solved optimally using a conditional judgment-binary search (CJ-BS) algorithm by leveraging the proved unimodality of the objective function; and (iii) the UAVs trajectories, non-convex constraints are relaxed via slack variables and SCA is utilized to iteratively solve the convexified subproblems.

•   Simulation results illustrate the tradeoff between EE and total mission completion time, validating the effectiveness of the proposed scheme. The results demonstrate that the proposed scheme outperforms the static collection benchmark (the UAV hovers above SNs) [8] in terms of mission completion time, highlighting its practical value given the limited endurance and cost of UAVs.

1.3 Structure and Notations

The remainder of this paper is organized as follows: Section 2 presents the system model and problem formulation for multi-U-IRS-assisted data collection in IoT. Section 3 details the iterative algorithm for the joint optimization of UAV-SN association, the optimal NoRE, and UAVs trajectories. Section 4 presents the numerical results to verify the efficient performance of the proposed algorithm. Finally, Section 5 concludes this paper.

Notations: Scalars are denoted by italic letters, vectors by bold lowercase letters, and matrices by bold uppercase letters. (⋅)H and E(⋅) denote the conjugate transpose of a matrix or vector and the expectation operator, respectively. ‖⋅‖ denotes the Euclidean norm, and diag(⋅) represents a diagonal matrix. arg⁡(x) denotes a vector with each element being the phase of the corresponding element in x. RM×1 represents the space of M-dimensional real-valued vector. |G| denotes the cardinality of set G.

2  Sytem Model and Problem Formulation

2.1 System Model

In this paper, the system model for multi-U-IRS-assisted IoT data collection is illustrated in Fig. 1. The FC is located at coordinates D={0,0,0}, which is far from the sensors and obstructed by barriers, resulting in no direct line-of-sight (LoS) path between them. We deploy M (M>1) rotary-wing UAVs, each UAV equipped with IRS to assist relay sensor signals to the FC, where the set of UAVs is denoted as {um,1≤m≤M}. Each UAV flies at a constant altitude H, complying with safety regulations and avoiding obstacles. Each U-IRS comprises N reflecting elements, and the NoRE allocated to the SN k is denoted as Nk. A total of k sensors (K > M) are deployed on the ground, with the set of sensors denoted as {sk,1≤k≤K}. These sensors are stationary with known locations, and their horizontal coordinates are defined as {S}k∈R2×1. Each sensor is exclusively served by one U-IRS. Due to the number of sensors exceeding the number of UAVs (K > M), the SNs are partitioned into M disjoint groups. Let {Gm,1≤m≤M} represent the SN group associated with the UAV Um, and |Gm| denote the cardinality of the group, Gm⊆{sk,1≤k≤K}. We allocate non-overlapping frequency channels |Gm| to each UAV Um for collecting data from sensors in Gm. Sensors associated with the same UAV upload data using time-division multiple access (TDMA), eliminating intra-UAV interference. It is reasonable to assume that all SNs aim to upload their task input data as soon as possible. Therefore, the transmission power of each sensor is set to a constant value [32].

Figure 1: The multi-U-IRS-assisted IoT data collection model

This paper adopts the fly-hover-communicate protocol (FHCP) [33]: where UAVs utilize onboard IRS to assist in SN data collection only during hovering phases, while non-active SNs can enter a sleep mode [34], thereby extending the lifetime of IoT. Each U-IRS serves one sensor group, and each SN corresponds to a unique hovering point. After completing data collection at one hovering point, the UAV proceeds to the next one, e.g., the UAV Um exclusively collects data from SNs in Gm, with |Gm| hover points involved in this process. Consequently, the trajectory of the UAV Um starts from the start point, traverses all the hovering point |Gm|, finally flies to the end point. The 2D coordinates of the UAVs’ start and end points are predefined as qI,qF∈R2×1, respectively. Therefore, determining the UAVs’ trajectories requires solving for the optimal hovering point positions.

We define λm,k∈{0,1} as the SN-UAV association variable [35]. If the SN sk is associated to the UAV Um, λm,k=1; otherwise, λm,k=0. So Gm={sk|λm,k=1,∀k,1≤m≤M}, ensure exclusive association. Since a SN is only served by one U-IRS, we can get ∑m=1Mλm,k=1. πm=(πm,1,πm,2,⋯,πm,|Gm|) represents the visiting order of the SNs, the visit sequence of SNs in Gm can be expressed as (sπm,1,sπm,2,⋯,sπm,|Gm|). Define qπm,l∈R2×1,1≤l≤|Gm| as the 2-D coordinate of the hover point corresponding to the SN sπm,l associated with the UAV Um. The start and end point of the UAV Um are defined as qπm,0=qI and qπm,|Gm|+1=qF.

For the SN sπm,l, the received signal at the FC is

yπm,l=gNm,l,FCHΘgl,Nm,lPsx+ωπm,l(1)

where Ps is the transmit power of the SN sπm,l. Without loss of generality, hypothetically E(x)=0, E(x2)=1. ωπm,l∼CN(0,σ2) is the Gaussian noise with zero mean and variance σ2. Nm,l is the NoRE used by the SN sπm,l, 1≤Nm,l≤N. Let N^m,l denotes the index of a specific reflecting element in use, with N^m,l∈{1,2,⋯,Nm,l}. gNm,l,FCH=[g1,FC,g2,FC,⋯,gNm,l,FC] represents the channel power gain from the reflecting element to the FC, gNm,l,FCH∈C1×N. gl,Nm,l=[gl,1,gl,2,⋯,gl,Nm,l]H represents the channel power gain from the SN sπm,l to the reflecting element, gl,Nm,l∈CN×1. The channel power gain can be expressed as [36]: gN^m,l,FC=ρ0dN^m,l,FC−αςl, gl,N^m,l=ρ0dl,N^m,l−αςl, where ρ0 is the average channel power gain at a reference distance of 1 meter, dl,N^m,l represents the distance between SN sπm,l and the reflecting element N^m,l in the IRS, dl,N^m,l=H2+‖qπm,l−{S}πm,l‖2. dNm,l,FC=H2+‖qπm,l‖2. dN^m,l,FC represents the distance from the reflecting element N^m,l in the IRS to the FC (approximated as UAV-to-FC distance due to small IRS size), dN^m,l,FC=H2+‖qπm,l‖2. α represents the path loss exponent. ςk describes the medium and small-scale Rayleigh fading component of the channel, which is an exponential variable that satisfies the independent and same distribution, with E(|ςk|2)=1. Define θN^m,l=ejθN^m,l, Θ=diag[θ1,θ2,⋯,θN^m,l,⋯,θNm,l]. θN^m,l∈[0,2π) is the phase shift of the reflecting element N^m,l, which can be expressed as [37]: θN^m,l=−arg⁡(gN^m,l,FCgl,N^m,l).

The instantaneous achievable rate corresponding to the SN sπm,l can be expressed as a function of the NoRE Nm,l and the position of the hover point qπm,l.

Rπm,l{Nm,l,qπm,l}=Blog2(1+Ps(∑N^m,l=1Nm,l|gN^m,l,FCgl,N^m,l|)2σ2)(2)

where B is the bandwidth allocated to the UAV Um. The frequency-division multiplexing access mechanism is adopted to avoid inter-UAV interference, which assumes that different UAVs operate on distinct frequencies during data collection and transmission.

The SE is defined as

SEπm,l{Nm,l,qπm,l}=Rπm,l{Nm,l,qπm,l}B=log2(1+Ps(∑N^m,l=1Nm,l|gN^m,l,FCgl,N^m,l|)2σ2)(3)

2.2 EE Model

The total energy consumption of the system during UAVs-assisted service comprises the following components: the energy consumption for traveling of the UAVs, the total UAVs hovering energy consumption, the energy consumption of reflecting element in IRS for phase shifting, the energy consumption of SNs (including signal transmission and intrinsic circuit power), the intrinsic energy consumption of FC. The total energy consumption can be formulated as

Em=EUAV,f+EUAV,h+EIRS+EGm+EFC=Epmr∑l=1|Gm|+1‖qπm,l−qπm,l−1‖+∑l=1|Gm|τπm,l(PUAV(0)+Nm,lPIRS+ηPs+PC,πm,l+PFC)=Epmr∑l=1|Gm|+1‖qπm,l−qπm,l−1‖+∑l=1|Gm|τπm,l(Nm,lPIRS+Pelse,πm,l)(4)

The calculation methods for Epmr and PUAV(V) refer to [33]. Epmr is the maximum-range (MR) propulsion energy consumption of the UAV per unit traveling distance in Joule/meter, representing the minimum UAV energy consumption per unit traveling distance. PUAV(V) is the power consumption of the rotor-wing UAV, when V=0, and PUAV(0) denotes the UAV’s hovering power consumption when the UAV hovers. Pelse,πm,l=PUAV(0)+ηPs+PC,πm,l+PFC, τπm,l is the hovering time of the UAV Um corresponding to the SN sπm,l. PIRS is the power consumption when the reflecting elements of IRS provide the phase shift. η denotes the power amplifier efficiency, PC,πm,l is the intrinsic circuit power consumption of the SN sπm,l, and PFC represents the intrinsic power consumption of the FC.

The total energy consumption and EE of the system can be expressed as (5) and (6), respectively, where Qπm,l is data amount that need to be collected from the SN sπm,l.

Etotal=∑m=1MEm=Epmr∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖+∑m=1M∑l=1|Gm|τπm,l(Nm,lPIRS+Pelse,πm,l)(5)

EEtotal=∑m=1M∑l=1|Gm|Qπm,lEpmr∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖+∑m=1M∑l=1|Gm|τπm,l(Nm,lPIRS+Pelse,πm,l)(6)

2.3 Mission Completion Time

In multi-U-IRS-assisted IoT data collection, the total mission completion time comprises two components: (1) The flight time of UAVs: The time required for UAVs to fly from the starting point, traverse all hovering points in sequence, and finally reach the endpoint. (2) The hovering time of UAVs: The time spent at each hovering point, during which the IRS in UAVs assists in transmitting SN signals to the FC. After completing data transmission from the associated SNs, UAVs terminates hovering and proceeds to the next hovering point immediately. Set Tf to the UAVs total flight time, it is associated with the total flight distance of the UAVs. So the total mission completion time can be expressed as

Tf=∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖Vmr(7)

where Vmr is the MR speed, which is the minimum UAV flight speed with minimal energy consumption under the FHCP [33].

Let Th be the total hovering time of the UAVs, which is related to the UAVs hovering time at each hovering point and can be expressed as

Th=∑m=1M∑l=1|Gm|τπm,l(8)

Thus, the total task completion time can be expressed as [35]

Ttotal=Tf+Th=∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖Vmr+∑m=1M∑l=1|Gm|τπm,l(9)

As shown in (9), both Ttotal and EE are associated with the flight distance and hovering time of UAVs, indicating that these two objectives are highly coupled.

2.4 Problem Formulation

Unlike single-U-IRS-assisted IoT data collection, in multi-U-IRS-assisted IoT data collection, we must consider both maximizing EE and minimizing the total mission completion time to fully demonstrate the advantages of multi-UAV systems. Eqs. (6) and (9) reveal that EE and mission completion time are closely related to three optimization parameters: the SN-UAV association {λm,k}, NoRE {Nm,l}, and the hovering point positions {qπm,l}. In this paper, we aim to achieve a balanced optimization of EE and total mission completion time by jointly optimizing these three parameters. To facilitate analysis, we define as (10).

χ=1EEtotal=Epmr∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖+∑m=1M∑l=1|Gm|τπm,l(Nm,lPIRS+Pelse,πm,l)∑m=1M∑l=1|Gm|Qπm,l(10)

Thus the problem of EE maximization is transformed into the minimization problem of χ.

According to the above analysis, the problem can be formulated as a multi-objective optimization problem (P1).

(p1):min{λm,k},{Nk},{qπm,l}⁡χ(11a)

min{λm,k},{Nk},{qπm,l}⁡Ttotal(11b)

s.t.∑m=1Mλm,k=1,1≤k≤K,∀k(11c)

Gm={sk|λm,k=1,∀k,1≤m≤M}(11d)

λm,k∈{0,1},∀m,k(11e)

1≤Nm,l≤N,∀m,l(11f)

qπm,0=qI,qπm,|Gm|+1=qF(11g)

τπm,l(ηPs+PC,πm,l)≤Eπm,l,max,∀m,l(11h)

Epmr∑l=1|Gm|+1‖qπm,l−qπm,l−1‖+∑l=1|Gm|τπm,l(PUAV(0)+Nm,lPIRS)≤Em,max,∀m,l(11i)

τπm,lBlog2(1+Ps(∑N^m,l=1Nm,l|gN^m,l,FCgl,N^m,l|)2σ2)≥Qπm,l,∀m,l(11j)

Eq. (11c)–(11e) indicates the association between the UAV and SN. Eq. (11f) represents the constraint on the NoRE. It is the upper bound of the NoRE, which means that the number of allocated elements can not exceed the most NoRE that the UAV carried. Eq. (11g) defines the UAV trajectory’s start and end point constraints. Eq. (11h) indicates the energy budget constraint of the SN, where Eπm,l,max denotes the maximum energy budget of a SN sπm,l. Eq. (11i) represents the UAV energy constraint. Due to the limited endurance time of UAVs, the mission must be completed within the UAV’s operational duration. Eq. (11j) represents the throughput constraint, ensuring complete data collection from all SNs.

Lemma 1: At the optimal solution to problem (P1), we have

τπm,lBlog2(1+Ps(∑N^m,l=1Nm,l|gN^m,l,FCgl,N^m,l|)2σ2)=Qπm,l,∀m,l(12)

Proof: Lemma 1 can be shown by contradiction. If there is an optimal solution τπm,l∗ making τπm,l∗Blog2(1+Ps(∑N^m,l=1Nm,l|gN^m,l,FCgl,N^m,l|)2σ2)>Qπm,l,∀m,l. We can find τπm,l′<τπm,l∗ to satisfy these constraints. In other words, τπm,l∗ can be decreased with other variables fixed, then χ and Ttotal decrease, and all other constraints are still satisfied, contradicting Pareto optimality, the lemma is proved.□

As revealed by Lemma 1, for (P1), the optimal hovering time in (P1) exactly matches the time required to complete data transmission from the SNs. Upon completing data collection, the UAV ceases hovering and proceeds to the next hovering point along a straight trajectory. We assume that all SNs have identical data upload requirements and the number of information bits that need to be collected from every SN is Q~, so we derive ∑m=1M∑l=1|Gm|Qπm,l=K⋅Q~. Following lemma 1, the hovering time of the UAV Um at the hover point associated with the SN sπm,l can be expressed as

τπm,l{Nm,l,qπm,l}=Q~Rπm,l{Nm,l,qπm,l}(13)

As shown in the expression for EE in (6), the EE depends on the UAV-SN association {λm,k}, NoRE {Nm,l}, and hovering point position {qπm,l}. Similarly, the total mission completion time is also influenced by these three parameters. Taking the NoRE as an example: (2) indicates that the achievable rate of the associated SN increases with NoRE, thereby reducing the UAV hovering time required for transmit the target data amount Q~ and shortening the total mission completion time. However, increasing the NoRE also raises energy consumption, which negatively impacts the system EE. This implies the NoRE minimizing mission completion time does not necessarily correspond to the maximum EE. Similar tradeoff analyses apply to the other two parameters. Therefore, a fundamental tradeoff between EE and mission completion time can be explored. To characterize the tradeoff, we define a weighted sum as

Λω=ωwfψcfχ+(1−ωwf)Ttotal(14)

where 0≤ωwf≤1 is the weighting factor, ψcf is the compensation factor for χ. Since the χ and Ttotal are at different levels, ψcf is therefore introduced to scale these two terms to a comparable numerical range, this enables the two variables to be of the same order of magnitude, ensuring that the weight 0≤ωwf≤1 can effectively and intuitively adjust the tradeoff between energy efficiency and total mission completion time. Finally, (P1) is reformulated as

(P2):min{λm,k},{Nk},{qπm,l}⁡Λω(15a)

s.t.  (11c)−(11i)(15b)

0≤ωwf≤1(15c)

According to (P2) and (12), ωwf and (1−ωwf) determine the priority of enlarging the system’s EE and decreasing the total mission completion time, respectively. Note that the case of ωwf=1 is equivalent to the χ minimization problem, which equates to the EE maximization problem. ωwf=0 is equivalent to the total mission completion time minimization problem. Obviously, we can always obtain the unique optimal tradeoff between the EE and the total mission completion time for any ωwf.

3  Algorithm Design

(P2) is a constrained combinatorial optimization problem, where the objective function Λω depends on UAV-SN association {λm,k}, NoRE {Nm,l}, and hovering point positions {qπm,l}. The coupling of these three variables in constraints (11g)–(11i), along with the non-convexity of (11i) and λm,k is a binary function in (11d), renders (P2) intractable for direct optimization. We decompose (P2) into three sub-problems to optimize the UAV-SN association optimization, NoRE, and the hovering point position separately. For each sub-problem, we assume the other variables are fixed and optimize only the targeted variable. The alternating optimization of these three variables is then employed to solve (P2).

To facilitate subsequent analysis, we expand and reorganize the mathematical expression of Λω, and combine like terms. The reformulated expression of Λω is given as

Λω=ωwfψcfχ+(1−ωwf)Ttotal=ωwfψcfEpmr∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖+∑k=1Kτk(NkPIRS+Pelse,k)K⋅Q~+(1−ωwf)(∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖Vmr+∑m=1M∑l=1|Gm|τπm,l)=(ωwfψcfEpmrK⋅Q~+1−ωwfVmr)∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖+∑m=1M∑l=1|Gm|τπm,l(ωwfψcf(Nm,lPIRS+Pelse,πm,l)K⋅Q~+(1−ωwf))=(ωwfψcfEpmrK⋅Q~+1−ωwfVmr)∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖⏟part 1+∑m=1M∑l=1|Gm|Q~Blog2(1+Ps(∑Nm,l=1N|gNm,l,FCgl,Nm,l|)2σ2)(ωwfψcf(Nm,lPIRS+Pelse,πm,l)K⋅Q~+(1−ωwf))⏟part 2(16)

Eq. (16) intuitively illustrates the correlation among UAV-SN association {λm,k}, NoRE {Nm,l}, hovering point position {qπm,l}, and the objective function Λω, thereby facilitating subsequent analysis.

The alternating optimization process proceeds as follows.

3.1 Algorithm Design

When the NoRE {Nm,l} and the hovering point {qπm,l} are given, the achievable rate of the associated SN and the corresponding UAV hovering time become fixed. Consequently, part 2 in (16) becomes a constant. For the UAV-SN association {λm,k} requiring optimization, we only need to analyze part 1 in (16). Under these conditions, (P2) can be reformulated as

(P3):min{λm,k}⁡(ωwfψcfEpmrK⋅Q~+1−ωwfVmr)∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖(17a)

s.t.  (11c)−(11e),(11g),(15c)(17b)

The objective function of (P3) can be expanded as

(ωwfψcfEpmrK⋅Q~+1−ωwfVmr)∑m=1M∑l=1|Gm|+1‖qπm,l−qπm,l−1‖=(ωwfψcfEpmrK⋅Q~+1−ωwfVmr)(∑l=1|G1|+1‖qπ1,l−qπ1,l−1‖+∑l=1|G2|+1‖qπ2,l−qπ2,l−1‖+⋯+∑l=1|GM|+1‖qπM,l−qπM,l−1‖)(18)

As shown in (18), the minimization of the objective function in optimization problem (P3) corresponds to minimizing the flight distances of UAVs serving different SN groups, which constitutes a multiple Traveling Salesman Problem (m-TSP) with predetermined initial and final locations. Specifically, this involves multi-salesmen starting from predefined origins, each visiting a distinct subset of cities (where each city is visited exactly once by one salesman), and all salesmen ultimately arriving at predetermined destinations. The goal is to find the solution that minimizes the total travel distance. Since the m-TSP is NP-hard, optimization problem (P3) also falls into the NP-hard category. Consequently, genetic algorithm can obtain an efficient approximate solution for (P3).

3.2 Optimizing NoRE

When given the UAV-SN association {λm,k} and hover point position {qπm,l}, the UAV trajectory and the SN groups associated with each UAV are determined, and the visit sequence of SNs within each assigned group becomes deterministic. Therefore, part 1 in (16) becomes deterministic. For the NoRE {Nm,l} requiring optimization, we only need to focus on minimizing part 2 in (16). Under this condition, (P2) can be reformulated as (P4).

(P4):min{Nm,l}∑m=1M∑l=1|Gm|Q~Blog2⁡(1+Ps(∑n=1Nm,l|gn,FCgl,n|)2σ2)×(ωwfψcf(Nm,lPIRS+Pelse,πm,l)K⋅Q~+(1−ωwf))(19a)

s.t.  (11f),(11h)−(11i),(15c)(19b)

By expanding the objective function in (P4), we can get (20), which is shown at the bottom of page 9. Where γm,l=ωwfψcfPIRSK⋅Q~, βNm,l≥βNm,l+1, δ=Psσ2, βNm,l=|gNm,l,FCgl,Nm,l|, and in descending order, that is βNm,l≥βNm,l+1.

∑m=1M∑l=1|Gm|Q~Blog2(1+Ps(∑Nm,l=1N|gNm,l,FCgl,Nm,l|)2σ2)(ωwfψcf(Nm,lPIRS+Pelse,πm,l)K⋅Q~+(1−ωwf))=∑m=1M∑l=1|Gm|Q~Blog2(1+Ps(∑Nm,l=1N|gNm,l,FCgl,Nm,l|)2σ2)(Nm,lωwfψcfPIRSK⋅Q~+(ωwfψcfPelse,πm,lK⋅Q~+(1−ωwf)))=∑m=1M∑l=1|Gm|Q~Blog2(1+δ(∑Nm,l=1NβNm,l)2)(Nm,lγm,l+ζm,l)=Q~Blog2(1+δ(∑N1,1=1NβN1,1)2)(N1,1γ1,1+ζ1,1)⏟part (1,1)+Q~Blog2(1+δ(∑N1,2=1NβN1,2)2)(N1,2γ1,2+ζ1,2)⏟part (1,2)+⋯+Q~Blog2(1+δ(∑NM,|Gm|=1NβNM,|Gm|)2)(NM,|Gm|γM,|Gm|+ζM,|Gm|)⏟part (M,|Gm|)(20)

First, we select part (m, l) in (20) for analysis, subsequently extending the methodology to the aggregate sum of all parts. According to our prior work [38], when δ(∑Nm,l=1NβNm,l)2≥1, Rπm,l(Nm,l+1)−Rπm,l(Nm,l)≥Rπm,l(Nm,l+2)−Rπm,l(Nm,l+1). Proposition 1: F(Nm,l)=Q~Blog2(1+δ(∑Nm,l=1NβNm,l)2)(Nm,lγm,l+ζm,l) is a unimodal function.

Proof: See Appendix A.□

According to proposition 1, it can be seen that F(Nm,l) is a unimodal function, and the CS-BJ algorithm in our existing research [38] is applied to solve the optimal NoRE N~m,l to obtain the minimum value of F(Nm,l). Similarly, in (18), we can use the CS-BJ algorithm to solve the minimum value of each part, thus obtaining the minimum value of the objective function in (P4). The CS-BJ algorithm for (P4) is summarized in Algorithm 1.

F(Nm,l) can be written as

F(Nm,l)=Q~Blog2(1+δ(∑Nm,l=1NβNm,l)2)(Nm,lγm,l+ζm,l)=Q~γm,lB⋅Nm,l+ζm,lγm,llog2(1+δ(∑Nm,l=1NβNm,l)2)(21)

From (21), it is evident that as a unimodal function, the peak of the unimodal function F(Nm,l) corresponds to a specific NoRE N~m,l and ζm,lγm,l. Different values of ωwf yield distinct ζm,lγm,l, and then obtain distinct optimal NoRE N~m,l, which also affect the value of Λω. Thereby ωwf is critical to achieve the balance between EE and the total mission completion time. The impact of ωwf is analyzed in the simulations.

3.3 Optimizing UAV Trajectory

Given UAV-SN association {λm,k} and NoRE {Nm,l}, the SN group associated with each UAV is also determined, and the visit sequence of SNs within each SN group and NoRE used by each SN is determined. According to (16), the hover point {qπm,l} is related to both part 1 and part 2. Compared to (P2), the constraints involved in {λm,k} and {Nm,l} need not be considered. So (P2) can be rewritten as

(P5):min{qπm,l}⁡Λω(22a)