AoI-Minimal Task Assignment and Trajectory Optimization in Multi-UAV-Assisted Wireless Powered IoT Networks

Abstract

This paper investigates the energy transfer and data collection problem of multiple unmanned aerial vehicle (UAV)-assisted wireless-powered Internet of Things (IoT) networks. To ensure information freshness for IoT devices and reduce UAVs’ energy consumption, we minimize the average Age of Information (AoI) of IoT devices by jointly optimizing the energy harvesting (EH) and data collection time for IoT devices, the selection of data collection points (DCPs), DCP-IoT associations, and task assignment, flight speed, and trajectories of UAVs, subject to the limited endurance of UAVs. As this problem is nonconvex, we propose a novel DCP association and trajectory-planning scheme that seeks age-optimal solutions through an iterative three-step process. First, we calculate the EH and data collection time for IoT devices using Karush–Kuhn–Tucker (KKT) conditions. Then, we introduce an optimal hovering time allocation-based affinity propagation (OHTAP) clustering algorithm to determine optimal DCP locations and establish DCP-IoT associations. Finally, we develop two algorithms to optimize UAVs’ trajectories: an improved partheno-genetic algorithm with enhancement mechanisms (EIPGA) and a hybrid algorithm that combines improved MinMax k-means clustering with EIPGA. Numerical results confirm that our scheme consistently outperforms benchmark schemes in AoI performance and solution stability across diverse scenarios.

1. Introduction

Unmanned aerial vehicles (UAVs), also commonly known as drones, have attracted significant attention in recent years for various applications, such as disaster rescue, aerial inspection, and Internet of Things (IoT). In wireless IoT networks, UAVs are frequently deployed as mobile base stations, wireless relays, or data collectors to improve communication reliability and ensure timely coverage [1,2]. UAV-assisted IoT networks offer superior environmental adaptability and communication quality compared to terrestrial networks, making them a promising solution to support the rapidly growing and highly dynamic wireless data traffic in future 6G networks.

Recently, UAV-assisted IoT networks have been widely adopted in various time-sensitive scenarios [3,4,5], in which data generated by IoT devices often have strict freshness requirements to avoid errors or catastrophic consequences caused by outdated information. The freshness of the received information is thus a critical concern in time-sensitive systems. To quantify the freshness of information, a new performance metric named Age of Information (AoI) was introduced in [6]. AoI represents the time elapsed since the latest update was generated [7], which captures the freshness of information from the destination’s perspective.

Due to the increasing popularity of UAVs in IoT systems and the significance of AoI in time-sensitive systems, AoI-oriented UAV-assisted IoT network designs have been extensively studied. For example, in [8], the authors aimed to minimize both the maximum AoI (Max-AoI) and average AoI (Avg-AoI) of all sensor nodes (SNs) by optimizing the UAV’s trajectory under the ideal fly–hover–collect strategy. In [9], the authors divided all SNs into non-overlapping clusters and assigned each cluster a data collection point (DCP), then optimized the UAV’s trajectory and DCP-IoT association to minimize the Max-AoI and Avg-AoI. In [10], the authors derived the peak AoI (PAoI) and Avg-AoI for UAV-assisted data collection and dissemination on a graph, then minimized both by optimizing the UAV’s random flight trajectory. In [11], the authors proposed a UAV-aided ground nodes (GNs) localization and communication framework that jointly optimizes localization accuracy, beamwidth of GNs, bandwidth, and the UAV’s trajectory to minimize the Avg-AoI. Inspired by advances in artificial intelligence, the authors of [12] utilized an end-to-end framework that integrates a clustering module and an Enhanced Pointer Neural Network (EPNet) to minimize the Max-AoI and Avg-AoI. In [13], the authors jointly optimized the selection of hovering positions and their visiting sequence using a transformer and the weighted ${\mathrm{A}}^{*}$ algorithm to minimize the total AoI of the collected data. Collectively, these studies demonstrate that AoI-oriented network designs ensure information freshness in time-sensitive systems, in contrast to traditional energy efficiency-oriented and throughput-oriented network designs [14,15,16,17,18].

However, the above research predominantly addresses single-UAV trajectory optimization to improve AoI performance. Given the size, weight, and power (SWAP) limitations of UAVs [2], a single UAV is insufficient to perform data collection missions for large-scale IoT networks within stringent time requirements. This limitation has spurred interest in AoI-oriented multi-UAV collaborative data collection for IoT networks. In [19], the authors proposed an energy-constrained multi-UAV cooperative data collection framework for time-sensitive dense IoT networks to minimize the Max-AoI. In [20], the authors aimed to minimize the PAoI and Avg-AoI of all SNs by optimizing the number of DCPs, the SN-DCP and DCP-UAV associations, and the trajectories of UAVs, subject to the endurance of the UAV. The heuristic algorithms were used to find the optimal trajectory in [19,20], whereas in [21,22], deep reinforcement learning-based (DRL) algorithms were introduced to solve the trajectory-planning problem. In [21], the authors proposed a QMIX-based algorithm to jointly optimize UAV trajectories and UAV-SN scheduling to minimize the expected total AoI over time. In [22], the authors proposed a Deep Q-network (DQN)-based algorithm to jointly minimize the weighted Avg-AoI and transmission power of the IoT devices.

Despite these advancements, prior studies often neglect energy constraints or assume ample energy availability in IoT devices. In fact, most IoT devices are powered by batteries with limited capacities, which requires periodic recharging or replacement. Manual intervention for battery maintenance is highly inefficient or even impractical, especially in remote areas such as deserts or dense forests. Motivated by the success of integrating radio frequency (RF)-based Wireless Power Transfer (WPT) into UAV-assisted IoT networks [23,24,25,26], the research focus has progressively shifted to AoI-oriented UAV-assisted wireless-powered IoT network designs. For example, in [27], the authors adopted dynamic programming (DP) and ant colony (AC) algorithms to jointly optimize UAVs’ trajectory, energy harvesting (EH) time, and data collection time with the objective of minimizing the Avg-AoI of all SNs. In [28], the authors formulated the Avg-AoI minimization problem as a Markov problem with large state spaces and proposed a DQN-based algorithm to jointly optimize the UAV’s trajectory, information transmission, and EH scheduling of the SNs. In [29], the authors derived a lower bound on covertness constraint based on the Kullback–Leibler divergence and jointly optimized the WPT duration, the data transmission duration, and the WPT transmission power to minimize the average covert AoI. For multi-UAV-assisted wireless-powered IoT networks, the authors of [30] proposed a DRL-based algorithm under time-varying channel conditions to minimize the sum of AoI of all the IoT devices. In [31], the authors aimed to minimize the Avg-AoI by jointly optimizing the selection of hovering positions, hovering time, and flight paths of UAVs, and relay pairing relationships among UAVs. In [32], the authors jointly optimized the number and selection of hovering positions, the UAV-IoT association, the flight speed and trajectories of UAVs, the computing resources, and the EH and data transmission time of IoT devices to minimize the Avg-AoI.

AoI-oriented multi-UAV-assisted energy transfer and data collection have emerged as a promising solution, providing sustainable energy support for IoT devices, addressing the energy storage limitations of each UAV, and ensuring data freshness [31,32]. However, the large-scale deployment of IoT devices and the cooperative operation of multiple UAVs present several challenges in multi-UAV-assisted wireless-powered IoT networks: (i) how to optimize the trajectories of multi-UAVs to cooperatively cover all IoT devices with the fewest DCPs, reduce the UAVs’ energy consumption, and minimize the Avg-AoI; (ii) the determination and assignment of DCP tasks to each UAV are complicated by factors such as the energy capacity of each UAV, the distribution of IoT devices, and the number of UAVs and IoT devices; (iii) in order to obtain more accurate measurements of Avg-AoI and energy consumption, it is necessary to consider models that closely align with real-world systems; (iv) how to design an algorithm that balances accuracy and efficiency for the Avg-AoI minimization problem, particularly in large-scale scenarios.

To address the above challenges, in this paper, we delve into the problem of AoI-oriented multi-UAV cooperative energy transfer and data collection, in which multiple energy-limited rotary-wing UAVs are dispatched to collect update packets from all IoT devices. Different from [27,30], we account for the number and distribution of IoT devices, as well as the number and energy capacities of UAVs. Our research aims to minimize the Avg-AoI of all IoT devices and reduce UAV energy consumption. The main contributions are summarized as follows.

• We formulate the multi-UAV-assisted energy transfer and data collection problem to minimize the Avg-AoI in wireless-powered IoT networks by jointly optimizing the IoT devices’ uploading sequence, the locations of DCPs, the DCP-IoT associations, task assignment, the EH time, the data collection time, flight speed, and UAVs’ trajectories.
• To tackle this NP-hard optimization problem, we decouple the original problem into three subproblems through theoretical derivation: hovering time allocation, DCP-IoT association, and multi-UAV cooperative task assignment and trajectory optimization.
• The first subproblem, proven to be convex, is solved using the Karush–Kuhn–Tucker (KKT) conditions. The second subproblem, fundamentally a clustering problem, is addressed with an optimal hovering time allocation-based affinity propagation (OHTAP) clustering algorithm. The final subproblem is modeled as a stage-weighted multiple-traveling-salesmen problem (MTSP) and solved using an improved partheno-genetic algorithm with enhancement mechanisms (EIPGA) or a hybrid algorithm that combines improved MinMax k-means (IMinMax k-means) clustering with EIPGA.
• Our proposed scheme significantly outperforms benchmark schemes in terms of Avg-AoI optimization and solution stability, as evidenced by extensive experiments. Furthermore, we analyze the impacts of various factors on Avg-AoI, including the number of IoT devices and UAVs, the selection of EH models, and the energy storage, flight speed, and acceleration of UAVs.

The remainder of this paper is organized as follows. Section 2 describes the system model for multi-UAV-assisted wireless-powered IoT networks. Section 3 formulates the problem of Avg-AoI-minimal energy transfer and data collection. Section 4 details the implementation steps of our iterative three-step scheme. Section 5 presents simulation experiments to validate the effectiveness of the proposed scheme. Finally, we briefly conclude the paper in Section 6.

2. System Model

2.1. Network Description

We consider a multi-UAV-assisted wireless-powered IoT network that consists of one data center (DC) $a_0$, M rotary-wing UAVs equipped with half-duplex hybrid access points (HAPs), denoted by $\mathcal{U}=\left\{u_1,\dots ,u_M\right\}$, and K ground IoT devices, denoted by $\mathcal{A}=\left\{a_1,\dots ,a_K\right\}$. The IoT devices are assumed to be remotely distributed over a wide area to perform environmental sensing tasks. Without loss of generality, we adopt a three-dimensional (3D) Cartesian coordinate system, where the horizontal locations of the DC $a_0$ and each IoT device $a_k\in \mathcal{A}$ are fixed and represented by the vector ${\mathbf{w}}_k=\left[x_k,y_k\right]\in {\mathbb{R}}^2,k\in \left\{0,1,\cdots ,K\right\}$. Throughout the mission, all UAVs execute the fly–hover–collect strategy [27] and the stop–wait strategy [7] to ensure stable transmissions and collision avoidance. The acceleration of each UAV for switching between hovering and flying states is assumed to be constant, denoted by $acc$. Additionally, the set of data collection points (DCPs) at which the UAVs hover to transfer energy and collect data is denoted by $\mathcal{C}=\left\{c_1,c_2,\cdots ,c_L\right\}$. The location of each DCP $c_l$ is given by ${\mathbf{w}}_l^c=\left[x_l^c,y_l^c,H\right]\in {\mathbb{R}}^3$, where H is a constant that specifies the vertical flight altitude of UAVs relative to the DC and IoT devices. Overall, Figure 1 depicts the process of multi-UAV-assisted energy transfer and data collection.

Through the DCPs, the connections between UAVs and IoT devices can be indirectly established. For clarity, we use binary indicators ${\xi }_{m,l}$ and ${\rho }_{l,k}$ to represent association between UAV $u_m$ and DCP $c_l$, and between DCP $c_l$ and IoT device $a_k$, respectively. If UAV $u_m$ hovers at DCP $c_l$ to service IoT devices, then ${\xi }_{m,l}=1$; otherwise, ${\xi }_{m,l}=0$. Similarly, if DCP $c_l$ is associated with IoT device $a_k$, then ${\rho }_{l,k}=1$; otherwise, ${\rho }_{l,k}=0$. Accordingly, the set of DCPs visited by UAV $u_m$ and the set of IoT devices associated with DCP $c_l$ are denoted by ${\mathcal{C}}_m=\left\{c_l\mid {\xi }_{m,l}=1,c_l\in \mathcal{C}\right\}$ and ${\mathcal{A}}_l=\left\{a_k\mid {\rho }_{l,k}=1,a_k\in \mathcal{A}\right\}$, respectively. It is assumed that each DCP is visited by only one UAV and each IoT device is associated with only one DCP, resulting in the following constraints:

$$\begin{array}{c}\hfill \sum _{m=1}^M{\xi }_{m,l}=1,{\xi }_{m,l}\in \{0,1\},l=1,\dots ,L,\end{array}$$

(1)

$$\begin{array}{c}\hfill \sum _{l=1}^L{\rho }_{l,k}=1,{\rho }_{l,k}\in \{0,1\},k=1,\dots ,K.\end{array}$$

(2)

Additionally, the total number of IoT devices visited by the UAVs through all the DCPs must satisfy

$$\begin{array}{c}\hfill \sum _{m=1}^M\sum _{l=1}^L\sum _{k=1}^K{\xi }_{m,l}·{\rho }_{l,k}=K.\end{array}$$

(3)

When the UAVs hover at the DCPs, they can establish communication links with associated IoT devices using various multiple access schemes. In this paper, we assume the Time Division Multiple Access (TDMA) scheme, whereby IoT devices associated with each DCP sequentially upload data to the UAVs. If $c_l$ is the j-th DCP in the trajectory of UAV $u_m$, then $c_l$, ${\mathbf{w}}_l^c$, and ${\mathcal{A}}_l$ are relabeled as $q_{m,\left(j\right)}$, ${\mathbf{w}}_{m,\left(j\right)}^c$, and ${\mathcal{S}}_{m,\left(j\right)}$, respectively. Similarly, if $a_k$ is the i-th IoT device that uploads data to UAV $u_m$ hovering at DCP $q_{m,\left(j\right)}$, then $a_k$, ${\mathbf{w}}_k$, $t_{m,l,k}^{\mathrm{EH}}$, and $t_{m,l,k}^{\mathrm{CD}}$ are relabeled as $s_{m,\left(j\right)}^{\left(i\right)}$, ${\mathbf{w}}_{m,\left(j\right)}^{\left(i\right)}$, $t_{m,\left(j\right)}^{\mathrm{EH}\left(i\right)}$, and $t_{m,\left(j\right)}^{\mathrm{CD}\left(i\right)}$, respectively. Thus, given ${\mathcal{C}}_m$ and $a_0$, the trajectory of UAV $u_m$ can be represented by a permutation ${\mathit{q}}_m=\left[a_0,q_{m,\left(1\right)},q_{m,\left(2\right)},\cdots ,q_{m,\left(\left|{\mathcal{C}}_m\right|\right)},a_0\right]$, where $\left|{\mathcal{C}}_m\right|$ is the cardinality of the set ${\mathcal{C}}_m$.

2.2. Channel Modeling

The communication links between UAVs and IoT devices mainly depend on the propagation environment, which can be either line-of-sight (LoS) or non-line-of-sight (NLoS). According to [33], the path loss for both LoS and NLoS links between IoT device $a_k$ and UAV $u_m$ hovering at DCP $c_l$ can be expressed in decibels (dB) as follows:

$$\begin{array}{c}\hfill L_{m,l,k}^{LoS}=10\alpha ·log\left({\displaystyle \frac{4\pi f_c{d}_{m,l,k}}{c}}\right)+{\eta }_{LoS},\end{array}$$

(4)

$$\begin{array}{c}\hfill L_{m,l,k}^{NLoS}=10\alpha ·log\left({\displaystyle \frac{4\pi f_c{d}_{m,l,k}}{c}}\right)+{\eta }_{NLoS},\end{array}$$

(5)

where $d_{m,l,k}$ denotes the distance between IoT device $a_k$ and UAV $u_m$ hovering at DCP $c_l$, $\alpha$ is the path loss exponent, $f_c$ is the carrier frequency, c is the speed of light, and ${\eta }_{LoS}$ and ${\eta }_{NLoS}$ are the excessive path losses for LoS and NLoS links, respectively. The probability of an LoS link between IoT device $a_k$ and UAV $u_m$ hovering at DCP $c_l$ is given by [34]

$$\begin{array}{c}\hfill P_{m,l,k}^{LoS}={\displaystyle \frac{1}{1+aexp\left[-b\left({\theta }_{m,l,k}-a\right)\right]}},\end{array}$$

(6)

where a and b are constants determined by environmental conditions and the carrier frequency, respectively. ${\theta }_{m,l,k}=\left({\displaystyle \frac{180}{\pi }}\right)arcsin\left({\displaystyle \frac{H}{d_{m,l,k}}}\right)$ is the elevation angle between IoT device $a_k$ and UAV $u_m$ hovering at DCP $c_l$. Given $P_{m,l,k}^{LoS}$, the corresponding probability of an NLoS link is $P_{m,l,k}^{NLoS}=1-P_{m,l,k}^{LoS}$. Based on Equations (4)–(6), the average path loss for the UAV-IoT communication can be expressed as

$$\begin{array}{c}\hfill L_{m,l,k}=P_{m,l,k}^{LoS}·L_{m,l,k}^{LoS}+P_{m,l,k}^{NLoS}·L_{m,l,k}^{NLoS}.\end{array}$$

(7)

Accordingly, the average channel gain is $G_{m,l,k}={\displaystyle \frac{1}{L_{m,l,k}}}$. When UAV $u_m$ returns to DC $a_0$, we assume that it offloads the data at a constant power $P_{\mathrm{OFF}}$ immediately. The data offloading rate $R_{\mathrm{OFF}}$ and data offloading time $t_m^{\mathrm{OFF}}$ can be calculated by

$$\begin{array}{c}\hfill R_{\mathrm{OFF}}=B·{log}_2\left(1+{\displaystyle \frac{G_0·P_{\mathrm{OFF}}}{{\sigma }_w^2}}\right),\end{array}$$

(8)

$$\begin{array}{c}\hfill t_m^{\mathrm{OFF}}={\displaystyle \frac{{\sum }_{k=1}^{K_m}{D}_k}{R_{\mathrm{OFF}}}},\end{array}$$

(9)

where B and ${\sigma }_w^2$ are the channel bandwidth and the noise power at the UAV receiver, respectively. $G_0$ is the average channel gain when UAVs hover over $a_0$. $K_m={\sum }_{l=1}^L{\sum }_{k=1}^K{\xi }_{m,l}·{\rho }_{l,k}$ represents the total number of IoT devices serviced by UAV $u_m$.

2.3. Energy Harvesting Model and Data Collection Model

The IoT devices utilize the harvested energy to transmit data to the UAVs. To account for various non-linear elements in practical RF-based EH circuits, such as diodes, diode-connected transistors, and circuit saturation characteristics, we introduce a non-linear EH model [35,36] based on actual EH circuit measurement data. The power harvested at IoT device $a_k$ can be described as

$$\begin{array}{c}\hfill P_{m,l,k}^{\mathrm{EH}}=P_{max}·{\displaystyle \frac{1-e^{-C·P_{\mathrm{e}}·G_{m,l,k}}}{1+e^{-C·\left(P_{\mathrm{e}}·G_{m,l,k}-D\right)}}},\end{array}$$

(10)

where $P_{max}$ represents the maximum output direct current power when the EH circuit reaches saturation, and $P_{\mathrm{e}}$ denotes the fixed transmission power of the UAVs. The constants C and D are related to specific circuit characteristics, with values dependent on resistance, capacitance, and circuit sensitivity. Thus, the energy harvested at IoT device $a_k$ during the EH time interval $t_{m,l,k}^{\mathrm{EH}}$ is given by

$$\begin{array}{c}\hfill E_{m,l,k}^{\mathrm{EH}}=P_{m,l,k}^{\mathrm{EH}}·t_{m,l,k}^{\mathrm{EH}}.\end{array}$$

(11)

In the data collection stage, we consider only the dominant transmission power of IoT devices and neglect their circuit power, as described in [14]. Additionally, it is assumed that all UAVs operate on orthogonal frequency channels, allowing multiple UAVs to transfer energy or collect data simultaneously without interference. The data uploading rate between IoT device $a_k$ and UAV $u_m$ hovering at DCP $c_l$ during the data collection period $t_{m,l,k}^{\mathrm{CD}}$ can be calculated by

$$\begin{array}{c}\hfill R_{m,l,k}=B·{log}_2\left(1+{\displaystyle \frac{G_{m,l,k}·E_{m,l,k}^{\mathrm{EH}}}{t_{m,l,k}^{\mathrm{CD}}·{\sigma }_w^2}}\right),\end{array}$$

(12)

where $E_{m,l,k}^{\mathrm{EH}}$ is allocated uniformly over $t_{m,l,k}^{\mathrm{CD}}$. Let $D_k$ represent the updated perceptual data volume at IoT device $a_k$ within the $t_{m,l,k}^{\mathrm{CD}}$. To ensure the data uploading requirement of each IoT device $a_k$, we must satisfy

$$\begin{array}{c}\hfill t_{m,l,k}^{\mathrm{CD}}·{log}_2\left(1+{\displaystyle \frac{G_{m,l,k}·P_{m,l,k}^{\mathrm{EH}}·t_{m,l,k}^{\mathrm{EH}}}{t_{m,l,k}^{\mathrm{CD}}·{\sigma }_w^2}}\right)\ge {\overline{D}}_k,\end{array}$$

(13)

where ${\overline{D}}_k={\displaystyle \frac{D_k}{B}},k=1,2,\cdots ,K$.

2.4. Energy Consumption Model of UAVs

In general, the total energy consumption of a UAV mainly consists of communication energy consumption and propulsion energy consumption. The communication energy consumption of a UAV mainly occurs during the process of communication circuitry operations, signal processing, signal radiation/reception, etc. [7]. Assume that the communication-related power of UAVs remains constant $P_{\mathrm{c}}$, and the communication energy consumption of UAV $u_m$ can be calculated by

$$\begin{array}{c}\hfill E_m^{\mathrm{C}}=P_{\mathrm{c}}·\left[\sum _{l=1}^{\left|{\mathcal{C}}_m\right|}\sum _{k=1}^{\left|{\mathcal{A}}_l\right|}\left(t_{m,l,k}^{\mathrm{CD}}+t_{m,l,k}^{\mathrm{EH}}\right)+t_m^{\mathrm{OFF}}\right].\end{array}$$

(14)

Throughout the mission, we disregard energy consumption during the take-off and landing stages. As derived in [37], we adopt a power consumption model (in watts) to represent the UAV’s propulsion power consumption, which is approximated by

$$\begin{array}{cc}\hfill P\left(V\right)& =P_0\left(1+{\displaystyle \frac{3V^2}{U_{\mathrm{tip}}^2}}\right)+{\displaystyle \frac{1}{2}}{d}_0{\rho }_{0}sA{V}^3+P_i{\left(\sqrt{1+{\displaystyle \frac{V^4}{4v_0^4}}}-{\displaystyle \frac{V^2}{2v_0^2}}\right)}^{1/2},\hfill \end{array}$$

(15)

where $P_0$ and $P_i$ are two constants representing the profile power and induced power of UAV in hovering state, $U_{\mathrm{tip}}$ is the tip speed of the rotor blade, $v_0$ denotes the mean rotor-induced speed in hovering, and $d_0$, ${\rho }_0$, s, and A are the fuselage drag ratio, the air density, rotor solidity, and rotor disc area, respectively. More detailed information about the parameters and the power consumption model can be found in [37]. Then, the total propulsion energy consumption of UAV $u_m$ can be calculated by

$$\begin{array}{cc}\hfill E_m^{\mathrm{P}}& =\sum _{l=0}^{\left|{\mathcal{C}}_m\right|}\left[2·{\int }_0^{t_{m,l}^{acc}}P(V\left(t\right)dt+P\left(V\right)·t_{m,l}^{\mathrm{V}}\right]+P\left(0\right)·\left[\sum _{l=1}^{\left|{\mathcal{C}}_m\right|}\sum _{k=1}^{\left|{\mathcal{A}}_l\right|}\left(t_{m,l,k}^{\mathrm{CD}}+t_{m,l,k}^{\mathrm{EH}}\right)+t_m^{\mathrm{OFF}}\right],\hfill \end{array}$$

(16)

where $t_{m,l}^{acc}$ and $t_{m,l}^{\mathrm{V}}$ denote the acceleration time of UAV $u_m$ and the time that UAV $u_m$ flies at the speed V, respectively. $V\left(t\right)=V_0+acc·t$ represents linear acceleration or deceleration, and $V_0$ is the initial speed. Since acceleration and deceleration in this paper are symmetric, the energy consumed during deceleration is equal to that during acceleration. On the above basis, the total energy consumption of UAV $u_m$ is $E_m^{\mathrm{sum}}=E_m^{\mathrm{C}}+E_m^{\mathrm{P}}$.

3. Problem Formulation

3.1. The Avg-AoI of IoT Devices

The freshness of information is crucial for time-sensitive IoT applications. To evaluate information freshness, we utilize the AoI metric, which is defined as the time interval between the UAV’s initiation of data collection from the device and the completion of data offloading at the DC. In this paper, we adopt the generate-at-will model [9], wherein each IoT device generates environmental sensing data at arbitrary times, packages the data into a packet with a timestamp, and forwards it to the UAV. The sensing and sampling times of each IoT device are very brief and can be negligible. Therefore, the AoI of the data collected by UAV $u_m$ from IoT device $s_{m,\left(j\right)}^{\left(i\right)}$ at time $T_m$ is given by

$$\begin{array}{c}\hfill A_{m,\left(j\right)}^{\left(i\right)}\left(T_m\right)={\left(T_m-T_{m,\left(j\right)}^{\mathrm{CD}\left(i\right)}\right)}^{+},\end{array}$$

(17)

where ${\left(x\right)}^{+}=max\{0,x\}$, $T_{m,\left(j\right)}^{\mathrm{CD}\left(i\right)}$ and $T_m$ denote the timestamps at which UAV $u_m$ begins collecting data from IoT device $s_{m,\left(j\right)}^{\left(i\right)}$, and completes the data offloading task, respectively.

For clarity, the time sequence of UAV $u_m$ is illustrated in Figure 2. It can be seen that $A_{m,\left(j\right)}^{\left(i\right)}\left(T_m\right)$ mainly consists of the EH time, the data collection time, the flight time, and the data offloading time. In particular, $A_{m,\left(j\right)}^{\left(i\right)}\left(T_m\right)$ can be labeled as $A_{m,\left(j\right)}^{\left(i\right)}\left(\mathit{\rho },\mathit{\xi },{\mathit{t}}_m^{\mathrm{EH}},{\mathit{t}}_m^{\mathrm{CD}},{\mathit{q}}_m\right)$, which is calculated as

$$\begin{array}{c}\hfill \begin{array}{c}{A}_{m,\left(j\right)}^{\left(i\right)}\left(\mathit{\rho },\mathit{\xi },{\mathit{t}}_m^{\mathrm{EH}},{\mathit{t}}_m^{\mathrm{CD}},{\mathit{q}}_m\right)={\displaystyle \sum _{n=i}^{\left|{\mathcal{S}}_{m,\left(j\right)}\right|}}\left(t_{m,\left(j\right)}^{\mathrm{CD}\left(n\right)}\right)+{\displaystyle \sum _{p=j+1}^{\left|{\mathcal{C}}_m\right|}}\left(t_{m,\left(p\right)}^{\mathrm{H}}\right)+{\displaystyle \sum _{p=j}^{\left|{\mathcal{C}}_m\right|}}\left(t_{m,\left(p\right)}^{\mathrm{F}}\right)+t_m^{\mathrm{OFF}},\hfill \end{array}\end{array}$$

(18)

where $t_{m,\left(j\right)}^{\mathrm{H}}=t_{m,\left(j\right)}^{\mathrm{EH}}+t_{m,\left(j\right)}^{\mathrm{CD}}$ represents the time that UAV $u_m$ hovers at DCP $c_l$ to service IoT devices, and $t_{m,\left(j\right)}^{\mathrm{F}}$ is the flight time of UAV $u_m$ from DCP $q_{m,\left(j\right)}$ to $q_{m,(j+1)}$. Hence, the Avg-AoI of all data collected from K IoT devices can be calculated as

$$\begin{array}{c}\hfill \begin{array}{c}{A}^{\mathrm{avg}}\left(\mathit{\rho },\mathit{\xi },{\mathbf{T}}^{\mathrm{EH}},{\mathbf{T}}^{\mathrm{CD}},\mathbf{Q}\right)\hfill \\ ={\displaystyle \frac{1}{K}}{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{j=1}^{\left|{\mathcal{C}}_m\right|}}{\displaystyle \sum _{i=1}^{\left|{\mathcal{S}}_{m,\left(j\right)}\right|}}{A}_{m,\left(j\right)}^{\left(i\right)}\left(\mathit{\rho },\mathit{\xi },{\mathit{t}}_m^{\mathrm{EH}},{\mathit{t}}_m^{\mathrm{CD}},{\mathit{q}}_m\right)\hfill \\ ={\displaystyle \frac{1}{K}}{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{j=1}^{\left|{\mathcal{C}}_m\right|}}\left\{{\displaystyle \sum _{i=1}^{\left|{\mathcal{S}}_{m,\left(j\right)}\right|}}i·t_{m,\left(j\right)}^{\mathrm{CD}\left(i\right)}+\left(t_{m,\left(j\right)}^{\mathrm{H}}+t_{m,\left(j\right)}^{\mathrm{F}}\right)·{\displaystyle \sum _{n=1}^{j-1}}\left|{\mathcal{S}}_{m,\left(n\right)}\right|+\left(t_{m,\left(j\right)}^{\mathrm{F}}+t_m^{\mathrm{OFF}}\right)·\left|{\mathcal{S}}_{m,\left(j\right)}\right|\right\}\hfill \end{array}\end{array}$$

(19)

We can see that once the sets of ${\mathcal{C}}_m$ and ${\mathcal{S}}_{m,\left(j\right)}$ are determined, the last term in the right-hand side (RHS) of Equation (19) is only related to the flight trajectory ${\mathit{q}}_m$. The second term depends on the flight trajectory ${\mathit{q}}_m$ and the hovering time ${\mathit{t}}_m^{\mathrm{H}}$. To calculate the first term, in addition to determining the data collection time ${\mathit{t}}_m^{\mathrm{CD}}$, the data uploading sequence of IoT devices in each DCP $q_{m,\left(j\right)}$ should also be known. Referring to the Rearrangement Inequality [38], the optimal value of the first term can be obtained by sorting $\left\{t_{m,\left(j\right)}^{\mathrm{CD}\left(1\right)},t_{m,\left(j\right)}^{\mathrm{CD}\left(2\right)},\dots ,t_{m,\left(j\right)}^{\mathrm{CD}\left(\left|{\mathcal{S}}_{m,\left(j\right)}\right|\right)}\right\}$ in descending order for each $j\in \left\{1,2,\dots ,\left|{\mathcal{C}}_m\right|\right\}$. Let ${\mathrm{\Gamma }}_{m,\left(j\right)}=\left\{{\pi }_{m,\left(j\right)}^1,{\pi }_{m,\left(j\right)}^2,\dots ,{\pi }_{m,\left(j\right)}^{\left|{\mathcal{S}}_{m,\left(j\right)}\right|}\right\}$ represent a permutation of IoT device labels in ${\mathcal{S}}_{m,\left(j\right)}$. The optimal value of the first term is then expressed as

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{K}}{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{j=1}^{\left|{\mathcal{C}}_m\right|}}{\displaystyle \sum _{i=1}^{\left|S_{m,\left(j\right)}\right|}}\left(i·t_{m,\left(j\right)}^{\mathrm{CD}\left({\pi }_{m,\left(j\right)}^i\right)}\right),\end{array}$$

(20)

3.2. Avg-AoI-Optimal Multi-UAV-Assisted Energy Transfer and Data Collection Problem

Our objective is to minimize the Avg-AoI of all IoT devices by jointly optimizing the required EH and data collection time for each IoT device, the locations of DCPs, the DCP-IoT and UAV-DCP associations, the flight speed, and the UAVs’ trajectories, subject to the limited endurance of UAVs. The optimization problem is mathematically formulated as follows:

$$\begin{array}{cc}\hfill \mathbf{P}\mathbf{1}:& \underset{\mathit{\rho },\mathit{\xi },{\mathbf{T}}^{\mathrm{EH}},{\mathbf{T}}^{\mathrm{CD}},\mathbf{Q}}{\min }{A}^{avg}\left(\mathit{\rho },\mathit{\xi },{\mathbf{T}}^{\mathrm{EH}},{\mathbf{T}}^{\mathrm{CD}},\mathbf{Q}\right)\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \sum _{m=1}^M{\xi }_{m,l}=1,{\xi }_{m,l}\in \{0,1\},\forall l=1,\dots ,L,\hfill \end{array}$$

(21a)

$$\begin{array}{cc}\hfill & \sum _{l=1}^L{\rho }_{l,k}=1,{\rho }_{l,k}\in \{0,1\},\forall k=1,\dots ,K,\hfill \end{array}$$

(21b)

$$\begin{array}{cc}\hfill & \sum _{m=1}^M\sum _{l=1}^L\sum _{k=1}^K{\xi }_{m,l}·{\rho }_{l,k}=K,\hfill \end{array}$$

(21c)

$$\begin{array}{cc}\hfill & t_{m,l,k}^{\mathrm{CD}}\ge 0,t_{m,l,k}^{\mathrm{EH}}\ge 0,\hfill \end{array}$$

(21d)

$$\begin{array}{cc}\hfill & t_{m,l,k}^{\mathrm{CD}}·{log}_2\left(1+{\displaystyle \frac{G_{m,l,k}·P_{m,l,k}^{\mathrm{EH}}·t_{m,l,k}^{\mathrm{EH}}}{t_{m,l,k}^{\mathrm{CD}}·{\sigma }_w^2}}\right)\ge {\overline{D}}_k,\hfill \end{array}$$

(21e)

$$\begin{array}{cc}\hfill & V=\underset{V\ge 0}{argmin}{\displaystyle \frac{P\left(V\right)}{V}},\hfill \end{array}$$

(21f)

$$\begin{array}{cc}\hfill & E_m^{\mathrm{sum}}\le E_{\max },\forall m=1,\dots ,M.\hfill \end{array}$$

(21g)

Here, constraints (21a)–(21c) denote the UAV-DCP association constraints, DCP-IoT association constraints, and the limitations on the number of IoT devices that each UAV can service, respectively; constraint (21d) ensures that the required EH time and data collection time for each IoT device are non-negative; constraint (21e) establishes the energy and data causality requirements for IoT devices, guaranteeing successful data transmission; constraint (21f) specifies the optimal UAV speed that maximizes the total traveling distance for a given onboard energy, known as the maximum-range (MR) speed; constraint (21g) imposes energy storage limitations for each UAV, where $E_{\max }$ is the maximum available energy of each UAV.

The problem $\mathbf{P}\mathbf{1}$ is difficult to solve directly due to several reasons. First, the optimization variables $\left\{\mathit{\rho },\mathit{\xi }\right\}$ for DCP-IoT and UAV-DCP associations are binary and thus, (21a)–(21c) involve integer constraints. Second, the task assignment and the visiting order of DCPs for each UAV are closely intertwined with the EH time and the data collection time for each IoT device. Third, the Avg-AoI of IoT devices is an implicit function of the variables $\left\{\mathit{\rho },\mathit{\xi },\mathbf{Q}\right\}$. Finally, the dimensions of the variables $\left\{\mathit{\xi },{\mathit{q}}_m\right\}$ vary with the number of UAVs M, which depends on the number and spatial distribution of IoT devices as well as the energy capacity of each UAV.

4. Methodology

4.1. Solution Architecture

As analyzed above, the Avg-AoI of all IoT devices is represented as the weighted sum of the EH time, the data collection time, the flight time, and the data offloading time. Additionally, the EH time and the data collection time for each IoT device is independent of the UAVs’ trajectories throughout the energy transfer and data collection mission. Therefore, we propose a novel DCP association and trajectory-planning scheme that utilizes an iterative three-step process to address the Avg-AoI-optimal multi-UAV-assisted energy transfer and data collection problem $\mathbf{P}\mathbf{1}$. As illustrated in Figure 3, the scheme consists of two main modules: (i) DCP-IoT association and optimization, intended to minimize the EH and data collection time of IoT devices, and (ii) task assignment and trajectory optimization, designed to reduce the UAVs’ flight time. The DCP-IoT association and optimization module is further divided into two submodules: hovering time allocation and DCP selection.

In a nutshell, the coordinates of all IoT devices $\left\{{\mathbf{w}}_1,{\mathbf{w}}_2,\cdots ,{\mathbf{w}}_K\right\}$ are first sent to the hovering time allocation module, which determines the optimal EH and data collection time allocation for one UAV hovering at DCP $c_l$ to service IoT device $a_k$. Then, the DCP selection module determines the set of DCPs $\left\{c_1,c_2,\cdots ,c_L\right\}$ and the binary DCP-IoT association matrix $\rho ={\left[{\rho }_{l,k}\right]}_{L\times K}$. Meanwhile, the optimal EH and data collection time for IoT devices $\left\{\mathbf{T}\mathrm{EH}\ast ,\mathbf{T}\mathrm{CD}\ast \right\}$ and the data uploading order ${\mathrm{\Gamma }}_{m,\left(j\right)}$ can be obtained. After that, the task assignment and trajectory optimization module finds the set of DCPs ${\mathcal{C}}_m$ visited by UAV $u_m$ and the energy-constrained optimal trajectories ${\mathit{q}}_m$ to achieve the optimal flight time and data offloading time. Lastly, the minimum Avg-AoI of all IoT devices is calculated based on the data collection time, the EH time, the flight time, and the data offloading time.

4.2. DCP-IoT Association and Optimization

For the DCP-IoT association and optimization problem, the goal is to minimize the total hovering time of UAVs and the number of DCPs by jointly optimizing the DCP locations, the DCP-IoT association, and the minimum EH and data collection time for each IoT device.

$$\begin{array}{cc}\hfill \mathbf{P}\mathbf{2}:& \underset{\left(\mathit{\rho },{\mathbf{T}}^{\mathrm{EH}},{\mathbf{T}}^{\mathrm{CD}}\right)}{\min }\sum _{l=1}^K\sum _{k=1}^K\left(t_{l,k}^{\mathrm{EH}}+t_{l,k}^{\mathrm{CD}}\right)·{\rho }_{l,k}+\tau ·\sum _{l=1}^K{\rho }_{l,l}\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \sum _{l\in {\mathcal{N}}^{+}\left(k\right)}{\rho }_{l,k}=1,\forall k,\hfill \end{array}$$

(22a)

$$\begin{array}{cc}\hfill & {\rho }_{l,l}=\underset{k\in {\mathcal{N}}^{+}\left(l\right)}{\max }{\rho }_{l,k},\forall l,\hfill \end{array}$$

(22b)

$$\begin{array}{cc}\hfill & {\rho }_{l,k}\in \{0,1\},\forall l,k.\hfill \end{array}$$

(22c)

where $\mathcal{N}\left(k\right)=\left\{l\mid ∥{\mathbf{w}}_l-{\mathbf{w}}_k∥\le r,l\ne k,a_l\in \mathcal{A}\right\}$ denotes the index set of the neighboring nodes of IoT device $a_k$ within a coverage radius r, and ${\mathcal{N}}^{+}\left(k\right)=\left\{k\right\}\cup \mathcal{N}\left(k\right)$. The parameter $\tau$ represents the ability of an IoT device to act as a cluster centroid, which is a non-negative weight factor. When $\tau =0$, each IoT device serves as its own DCP. Noticeably, an appropriate $\tau$ is helpful to balance the hovering time and flight time of the UAVs, thereby further reducing the Avg-AoI.

Clearly, the optimization variables $\mathit{\rho }$ for the DCP-IoT association are binary; thus, integer constraints are introduced in constraints (22a)–(22c). Furthermore, even with a fixed binary DCP-IoT association matrix $\mathit{\rho }$, the variables $\left\{t_{l,k}^{\mathrm{EH}},t_{l,k}^{\mathrm{CD}}\right\}$ in the objective function depend on the solutions for optimal hovering time allocation, which must satisfy the energy and data causality constraints. As a result, problem $\mathbf{P}\mathbf{2}$ is hard to solve directly. To improve the tractability of problem $\mathbf{P}\mathbf{2}$, we decompose it into two subproblems: the hovering time allocation subproblem $\mathbf{P}\mathbf{3}$ and the DCP selection subproblem $\mathbf{P}\mathbf{4}$. For problem $\mathbf{P}\mathbf{3}$, the goal is to achieve the optimal EH time and the data collection time allocation $\left\{t_{l,k}^{\mathrm{EH}},t_{l,k}^{\mathrm{CD}}\right\},l,k\in \left\{1,2,\cdots ,K\right\}$ for each pair of DCP $c_l$ and IoT device $a_k$, while ensuring that each IoT device $a_k$ can upload the required amount of data using the harvested energy.

$$\begin{array}{cc}\hfill \mathbf{P}\mathbf{3}:& \underset{\left(t_{l,k}^{\mathrm{EH}},t_{l,k}^{\mathrm{CD}}\right)}{\min }\sum _{l=1}^K\sum _{k=1}^K\left(t_{l,k}^{\mathrm{EH}}+t_{l,k}^{\mathrm{CD}}\right)\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& t_{l,k}^{\mathrm{EH}}\ge 0,t_{l,k}^{\mathrm{CD}}\ge 0,\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\hfill & t_{l,k}^{\mathrm{CD}}·{log}_2\left(1+{\displaystyle \frac{G_{l,k}·P_{l,k}^{\mathrm{EH}}·t_{l,k}^{\mathrm{EH}}}{t_{l,k}^{\mathrm{CD}}·{\sigma }_w^2}}\right)\ge {\overline{D}}_k.\hfill \end{array}$$

(23b)

With any given optimal hovering time allocation solution $\left\{t_{l,k}^{\mathrm{EH}\ast },t_{l,k}^{\mathrm{CD}\ast }\right\}$, the goal of problem $\mathbf{P}\mathbf{4}$ is to minimize both the total hovering time of UAVs and the number of DCPs by determining the optimal DCP positions and establishing DCP-IoT associations.

$$\begin{array}{cccc}\hfill & \mathbf{P}\mathbf{4}:\hfill & \hfill & \underset{\mathit{\rho }}{\min }\sum _{l=1}^K\sum _{k=1}^K\left(t_{l,k}^{\mathrm{EH}\ast }+t_{l,k}^{\mathrm{CD}\ast }\right)·{\rho }_{l,k}+\tau ·\sum _{l=1}^K{\rho }_{l,l}\hfill \\ \hfill & \mathrm{s}.\mathrm{t}.\hfill & \hfill & \left(22\mathrm{a}\right)-\left(22\mathrm{c}\right).\hfill \end{array}$$

(24)

Based on the obtained DCP set and DCP-IoT association, the optimal EH and data collection time $\left\{{\mathbf{T}}^{\mathrm{EH}\ast },{\mathbf{T}}^{\mathrm{CD}\ast }\right\}$ for each IoT device can be derived.

4.2.1. Optimization of Hovering Time Allocation

When one UAV hovers at DCP $c_l$ to service IoT device $a_k$, the minimum EH time and data collection time depend solely on the data amount generated by IoT device $a_k$. Additionally, constraints (23a) and (23b) are independent for different IoT devices. Consequently, problem $\mathbf{P}\mathbf{3}$ can be viewed as $K^2$ independent minimization problems, each aiming to minimize the sum of EH time and data collection time for an individual IoT device.

Lemma 1.

Problem $\mathit{P}\mathit{3}$ is a convex optimization problem.

Proof. 

First, the objective function of $\mathbf{P}\mathbf{3}$ is an affine function with respect to $\left\{t_{l,k}^{\mathrm{EH}},t_{l,k}^{\mathrm{CD}}\right\}$. Second, constraint (23a) is evidently convex. Finally, for constraint (23b), given $c_l$, $a_k$ and $D_k$, the terms $G_{l,k}$, $P_{l,k}^{\mathrm{EH}}$ and ${\overline{D}}_k$ can be viewed as constants. By calculating the Hessian matrix [39] of the function $f(x,y)=-xln(1+{\displaystyle \frac{by}{x}})$, it can be known that $f(x,y)$ is a convex function. Consequently, constraint (23b) is also convex. Based on the definition of the convex optimization problem [39], Lemma 1 is proved.    □

Suppose that the optimal solution to $\mathbf{P}\mathbf{3}$ is $\left\{t_{l,k}^{\mathrm{EH}\ast },t_{l,k}^{\mathrm{CD}\ast }\right\}$. Through contradiction, we can easily derive that in the optimal solution to $\mathbf{P}\mathbf{3}$, the constraints (23b) for all IoT devices should be active. In other words, the optimal EH time and data collection time must satisfy

$$\begin{array}{c}\hfill t_{l,k}^{\mathrm{CD}\ast }·{log}_2\left(1+{\displaystyle \frac{G_{l,k}·P_{l,k}^{\mathrm{EH}}·t_{l,k}^{\mathrm{EH}\ast }}{t_{l,k}^{\mathrm{CD}\ast }·{\sigma }_w^2}}\right)={\overline{D}}_k,\end{array}$$

(25)

Otherwise, we can always adjust $\left\{t_{l,k}^{\mathrm{EH}},t_{l,k}^{\mathrm{CD}}\right\}$ to satisfy the equality without decreasing the objective value.

According to Lemma 1 and Equation (25), we adopt the Lagrange multiplier method [39] to solve problem $\mathbf{P}\mathbf{3}$. The Lagrangian function of $\mathbf{P}\mathbf{3}$ is

$$\begin{array}{c}\hfill \begin{array}{c}L\left({\mathbf{T}}^{\mathrm{EH}},{\mathbf{T}}^{\mathrm{CD}},\mathit{\mu }\right)={\displaystyle \sum _{l=1}^K}{\displaystyle \sum _{k=1}^K}\left(t_{l,k}^{\mathrm{EH}}+t_{l,k}^{\mathrm{CD}}\right)+{\displaystyle \sum _{l=1}^K}{\displaystyle \sum _{k=1}^K}{\mu }_{l,k}\left[{\overline{D}}_k-t_{l,k}^{\mathrm{CD}}{log}_2\left(1+{\displaystyle \frac{G_{l,k}·P_{l,k}^{\mathrm{EH}}·t_{l,k}^{\mathrm{EH}}}{t_{l,k}^{\mathrm{CD}}·{\sigma }_w^2}}\right)\right]\hfill \end{array}\end{array}$$

(26)

where $\mu =\left\{{\mu }_{l,k}\right\}$ is the non-negative Lagrangian dual variable associated with the constraint (23b).

Applying KKT conditions [39] and Equation (25), we can obtain the system of equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\partial L}{\partial t_{l,k}^{\mathrm{CD}}}}=1-{\mu }_{l,k}^{\ast }·\left({\displaystyle \frac{{\overline{D}}_k}{t_{l,k}^{\mathrm{CD}\ast }}}-{\displaystyle \frac{2^{{\overline{D}}_k/t_{l,k}^{\mathrm{CD}\ast }}-1}{ln2·2^{{\overline{D}}_k/t_{l,k}^{\mathrm{CD}\ast }}}}\right)=0,\hfill \\ {\displaystyle \frac{\partial L}{\partial t_{l,k}^{\mathrm{EH}}}}=1-{\displaystyle \frac{{\mu }_{l,k}^{\ast }·G_{l,k}·P_{l,k}^{\mathrm{EH}}}{{\sigma }_w^2·ln2·2^{{\overline{D}}_k/t_{l,k}^{\mathrm{CD}\ast }}}}=0,\hfill \\ t_{l,k}^{\mathrm{CD}\ast }·{log}_2\left(1+{\displaystyle \frac{G_{l,k}·P_{l,k}^{\mathrm{EH}}·t_{l,k}^{\mathrm{EH}\ast }}{t_{l,k}^{\mathrm{CD}\ast }·{\sigma }_w^2}}\right)={\overline{D}}_k,\hfill \end{array}\right.\end{array}$$

(27)

By solving the system of Equation (27), the optimal hovering time allocation $\left\{t_{l,k}^{\mathrm{EH}\ast },t_{l,k}^{\mathrm{CD}\ast }\right\}$ for one UAV hovering at DCP $c_l$ to service IoT device $a_k$ can be obtained by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{t}_{l,k}^{\mathrm{CD}\ast }={\displaystyle \frac{ln2·{\overline{D}}_k}{\mathcal{W}\left(\frac{{\gamma }_{l,k}-1}{e}\right)+1}},\hfill \\ t_{l,k}^{\mathrm{EH}\ast }={\displaystyle \frac{\left(2^{{\overline{D}}_k/t_{l,k}^{\mathrm{CD}\ast }}-1\right)·t_{l,k}^{\mathrm{CD}\ast }}{{\gamma }_{l,k}}},\hfill \end{array}\right.\end{array}$$

(28)

where $\mathcal{W}\left(·\right)$ is the Lambert function, ${\gamma }_{l,k}={\displaystyle \frac{\left(G_{l,k}·P_{l,k}^{\mathrm{EH}}\right)}{{\sigma }_w^2}}$.

4.2.2. Optimization of DCP Selection

Problem $\mathbf{P}\mathbf{4}$ is essentially a clustering process that aims to classify all IoT devices into L exclusive clusters and determine the clustering centroid. To reduce the search space for the optimal DCP locations, we propose an optimal hovering time allocation-based affinity propagation (OHTAP) clustering algorithm to determine the optimal DCP locations and establish appropriate DCP-IoT associations.

The OHTAP algorithm mainly relies on a “message passing” mechanism. It utilizes the similarity between pairs of IoT devices to partition K IoT devices into L exclusive clusters and subsequently designates some IoT devices as DCPs at which UAVs hover. Formally, let $\mathbf{S}={\left(s_{l,k}\right)}_{K\times K}$ represent the similarity matrix. The similarity $s_{l,k}$ between IoT devices $a_l$ and $a_k$ is defined as the hovering time that one UAV hovers over candidate IoT device $a_l$ to service IoT device $a_k$, which can be expressed as

$$\begin{array}{c}\hfill s_{l,k}=\left\{\begin{array}{cc}-\left(t_{l,k}^{\mathrm{EH}}+t_{l,k}^{\mathrm{CD}}\right)-\tau ,\hfill & l=k,\hfill \\ -\left(t_{l,k}^{\mathrm{EH}}+t_{l,k}^{\mathrm{CD}}\right),\hfill & l\ne k.\hfill \end{array}\right.\end{array}$$

(29)

In the OHTAP clustering process, two types of bi-directional messages, namely responsibility and availability, are passed and iterated between adjacent IoT devices to produce clustering results. Let $\mathbf{R}={\left(r_{l,k}\right)}_{K\times K}$ and $\mathbf{E}={\left(e_{l,k}\right)}_{K\times K}$ represent the responsibility matrix and availability matrix, respectively. The iterative messages $r_{l,k}$ and $e_{l,k}$ are updated according to the following rules until convergence:

$$\begin{array}{c}\hfill r_{l,k}=s_{l,k}-\underset{j\in {\mathcal{N}}^{+}\left(k\right)\setminus \left\{l\right\}}{max}\left\{s_{j,k}+e_{j,k}\right\},\end{array}$$

(30)

$$\begin{array}{c}\hfill e_{l,k}=\left\{\begin{array}{cc}{\displaystyle \sum _{i\in \mathcal{N}\left(l\right)}}max\left\{r_{l,i},0\right\},\hfill & l=k,\hfill \\ min\left\{0,r_{l,l}+{\displaystyle \sum _{i\in \mathcal{N}\left(l\right)\setminus \left\{k\right\}}}max\left\{r_{l,i},0\right\}\right\},\hfill & l\ne k.\hfill \end{array}\right.\end{array}$$

(31)

Here, $r_{l,k}$ denotes the suitability of IoT device $a_l$ to serve as a cluster centroid for IoT device $a_k$, whereas $e_{l,k}$ denotes the suitability of IoT device $a_k$ to select $a_l$ as its cluster centroid. A damping factor $\lambda \in \left[0,1\right]$ is introduced at each iteration to mitigate numerical oscillations during the message update process.

When convergence is reached, the DCP-IoT association matrix $\mathit{\rho }$ can be obtained by

$$\begin{array}{c}\hfill {\rho }_{l,k}=\left\{\begin{array}{cc}1,\hfill & l=\underset{j\in \mathcal{L}}{\mathrm{argmin}}\left(s_{j,k}\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(32)

where $\mathcal{L}=\left\{l\mid r_{l,l}+e_{l,l}>0\right\}$ denotes the indices of the initial cluster centroids. Subsequently, two DCP candidate sets $\left\{c_l^m\right\}$ and $\left\{c_l^{1c}\right\}$ along with their respective positions $\left\{{\mathbf{w}}_l^m\right\}$ and $\left\{{\mathbf{w}}_l^{1c}\right\}$ can be determined by solving the mean problem and the 1-center problem separately for each cluster of IoT devices:

$$\left\{\begin{array}{c}{\mathbf{w}}_l^m={\displaystyle \frac{{\displaystyle \sum _{k=1}^K}{\rho }_{l,k}{\mathbf{w}}_k}{{\displaystyle \sum _{k=1}^K}{\rho }_{l,k}}},\hfill \\ {\mathbf{w}}_l^{1c}=\underset{{\mathbf{w}}_l^{1c}}{\mathrm{argmin}}\underset{a_l|{\rho }_{l,k}=1}{\max }∥{\mathbf{w}}_k-{\mathbf{w}}_l^{1c}∥.\hfill \end{array}\right.$$

(33)

Based on the DCP candidate sets, the optimal set of DCPs $\left\{c_l\right\}$ and the corresponding positions $\left\{{\mathbf{w}}_l^c\right\}$ can be obtained by calculating and comparing the hovering time. The procedure is detailed in Algorithm 1.

Algorithm 1  The OHTAP clustering algorithm with parameter $\tau$

Input: The system parameters $(K,M,V,H,P_{\mathrm{c}},P_e,\mathrm{etc}.)$ for all IoT devices and UAVs, the optimal hovering time allocation $\left\{t_{l,k}^{\mathrm{EH}\ast },t_{l,k}^{\mathrm{CD}\ast }\right\}$ that one UAV hovers at DCP $c_l$ to service IoT device $a_k$, and a sufficiently large constant $ite{r}_{\max }$.   1: Calculate optimal hovering time allocation $\left\{t_{l,k}^{\mathrm{EH}\ast },t_{l,k}^{\mathrm{CD}\ast }\right\}$ that one UAV hovers at DCP $c_l$ to service IoT device $a_k$ according to Equation (28);   2: Initialize the parameter: $iter=0$, and $r_{l,k}=e_{l,k}=0$;   3: while $iter\le ite{r}_{\max }$ do   4:    Update messages $\left\{r_{l,k},e_{l,k}\right\}$ by Equations (30) and (31);   5:    if converge then   6:      break   7:    end if   8:    $iter\leftarrow iter+1$;   9: end while  10: Obtain the DCP-IoT association matrix $\mathit{\rho }=\left\{{\rho }_{l,k}\right\}$ by Equation (32);  11: Find two DCP candidate sets by Equation (33), and then calculate and compare the hovering time to determine the set of DCPs $\left\{c_l\right\}$ and the corresponding positions $\left\{{\mathbf{w}}_l^c\right\}$; Output: The set of DCP $\mathcal{C}$, the location of each DCP $\left\{{\mathbf{w}}_l^c\right\}$, and the DCP-IoT matrix $\mathit{\rho }$.

4.3. Task Assignment and UAV Trajectory Optimization

For the task assignment and trajectory-planning problem, the goal is to find the optimal UAV-DCP association and the age-optimal trajectories of energy-constrained UAVs, using the given optimal solution $\left\{\mathit{\rho }\ast ,{\mathbf{T}}^{\mathrm{EH}\ast },{\mathbf{T}}^{\mathrm{CD}\ast }\right\}$ to problem $\mathbf{P}\mathbf{2}$

$$\begin{array}{cc}\hfill & \mathbf{P}\mathbf{5}:\underset{\left(\mathit{\rho }\ast ,\mathit{\xi },{\mathbf{T}}^{\mathrm{EH}\ast },{\mathbf{T}}^{\mathrm{CD}\ast },\mathbf{Q}\right)}{\min }{A}^{avg}\left(\mathit{\rho }\ast ,\mathit{\xi },{\mathbf{T}}^{\mathrm{EH}\ast },{\mathbf{T}}^{\mathrm{CD}\ast },\mathbf{Q}\right)\hfill \\ \hfill & \mathrm{s}.\mathrm{t}.\left(21\mathrm{a}\right),\left(21\mathrm{c}\right),\left(21\mathrm{f}\right),\left(21\mathrm{g}\right).\hfill \end{array}$$

(34)

Problem $\mathbf{P}\mathbf{5}$ can essentially be viewed as a stage-weighted MTSP process, which is known to be NP-hard. As discussed in [8,9], the optimal Avg-AoI trajectory of the UAV is a stage-weighted shortest Hamiltonian path when no energy constraint is imposed. This trajectory can be found using a genetic algorithm (GA) [40] or other appropriate methods.

Let $q_m^o$ and $q_m^{\mathrm{TSP}}$ represent the age-optimal trajectory of UAV $u_m$ without energy constraints and its corresponding traveling salesman problem (TSP) trajectory, respectively. It is well established that the TSP trajectory of UAV $u_m$ is the shortest and consumes the least energy at a given MR speed. Given that $E_m^{\mathrm{TSP}}\le E_{\max }$, we analyze the energy-constrained age-optimal trajectory under the following three cases. Case 1: When ${max}_m{E}_m^{\mathrm{sum}}\le E_{\max }$, the age-optimal solution is $q_m^{\ast }=q_m^o$. Case 2: If the UAV’s energy capacity satisfies ${max}_m{E}_m^{\mathrm{TSP}}\le E_{\max }\le {max}_m{E}_m^{\mathrm{sum}}$, an age-optimal trajectory for each UAV can be found using intelligent search methods, such as GA. Case 3: If the UAV’s energy capacity satisfies $E_{\max }<{max}_m{E}_m^{\mathrm{TSP}}$, no feasible trajectories exist for all UAVs under the given energy constraints. In this scenario, a UAV-expansion strategy is executed, i.e., one more UAV is added, and then the processes of task assignment and trajectory planning are repeated.

The key to solving problem $\mathbf{P}\mathbf{5}$ lies in determining $q_m^o$. To achieve this, we develop two algorithms. One is an improved partheno-genetic algorithm with enhancement mechanisms (EIPGA), which incorporates a “mutation-before-selection” mechanism and a hybrid selection operator to enhance global optimization capabilities. The other is a hybrid algorithm that combines IMinMax k-means clustering with EIPGA, where the former achieves an equilibrium-based task assignment, and the latter optimizes the trajectory for each UAV.

4.3.1. EIPGA-Based Algorithm

The improved partheno-genetic algorithm (IPGA) [41] replaces the crossover operation in traditional GAs with multiple mutation operations, making it well suited for solving the TSP and other combinatorial optimization problems [41,42]. Although the IPGA offers advantages such as simple structure, high efficiency, and fast convergence, it often neglects valuable information from slightly inferior individuals, leading to limited global search capability. To overcome this limitation, we propose an improved partheno-genetic algorithm with enhancement mechanisms (EIPGA), which incorporates the concepts of “mutation-before-selection” and a hybrid selection operator. For clarity, the proposed EIPGA-based algorithm is summarized in Algorithm 2, with key details outlined below:

Algorithm 2  EIPGA-based task assignment and trajectory optimization algorithm

Input: The system parameters $(K,M,V,H,P_{\mathrm{c}},P_e,\mathrm{etc}.)$ for all IoT devices and UAVs, the DCP set $\mathcal{C}=\left\{c_1,c_2,\cdots ,c_L\right\}$, the DCP-IoT association matrix $\mathit{\rho }$, the optimal hovering time allocation $\left\{t_{l,k}^{\mathrm{EH}\ast },t_{l,k}^{\mathrm{CD}\ast }\right\}$, and the EIPGA related parameters $\left\{N_p,N_g,N_{new}\right\}$. 1: Generate an initial population of $N_p$ chromosomes, and set $n\leftarrow 1$; 2: while $n\le N_g$ do 3:    Generate $N_{new}$ offspring chromosomes of $N_p$ chromosomes according to mutation operators and breakpoint update operator; 4:    Calculate the Avg-AoI $A_{new}^{\mathrm{avg}}$ of $N_{new}$ chromosomes according to Equation (19); 5:    Use a hybrid selection operator on $N_{new}$ chromosomes to obtain a new population of $N_p$ chromosomes and calculate relevant $A_p^{\mathrm{avg}}$; 6:    Add n by one: $n\leftarrow n+1$; 7: end while 8: Find the optimal task assignment and UAV trajectories $\left\{{\mathit{\xi }}^o,{\mathbf{Q}}^o\right\}=\arg {min}_{\mathit{\xi },\mathbf{Q}}{A}_p^{\mathrm{avg}}$. Output: The UAV-DCP association matrix ${\mathit{\xi }}^o$, the optimal flight trajectories of UAVs ${\mathbf{Q}}^o$.

• Chromosome structure: The Path-breakpoint sequence encoding method [41] is adopted to represent the chromosome structure of the solution. It ensures that except for the DC, each DCP is visited by one UAV only once. As shown in Figure 4, the first part of the chromosome encodes a permutation of integers from 1 to L, indicating the L DCPs visited by all UAVs. The second part of the chromosome delineates the breakpoints that divide the DCP permutation into M segments. Each segment corresponds to the path assigned to a particular UAV.
• Mutation operators: The mutation operator randomly selects individuals and modifies specific entries within their chromosomes. As illustrated in Figure 5, the algorithm implements four types of mutation operations: Flip, Swap, Left Slide (LSlide), and Right Slide (RSlide). A breakpoint update operator is also included to adjust the partition points within the chromosome. This operator distinguishes the task set and trajectory for each UAV, facilitating equitable distribution of workloads among UAVs.
• Hybrid selection operator: The selection operation identifies individuals with higher fitness based on the principle of “survival of the fittest”. To preserve genetic diversity and prevent high-fitness individuals from dominating the population, a hybrid selection operator that combines elite selection, roulette selection, and original population generation is introduced. During the i-th population update, the new population is generated by sequentially selecting $0.7N_p$, $0.2N_p$ and $0.1N_p$ individuals using elite selection, roulette selection, and original population generation operators, respectively.

4.3.2. Hybrid Algorithm

In this part, we propose a hybrid algorithm that solves the task assignment and trajectory-planning problem by combining IMinMax k-means clustering and EIPGA, as summarized in Algorithm 3. The IMinMax k-means clustering [43] minimizes the maximum intra-cluster variance, rather than the sum of intra-cluster variances in traditional k-means clustering [44]. This approach ensures an equilibrium-based task assignment among the UAVs, thereby reducing the Avg-AoI. Below are the detailed steps of the IMinMax k-means clustering algorithm.

Step 1: Initialize the locations of cluster centroids ${\mathbf{x}}_m^{\left(0\right)}$ for each $m\in \left\{1,2,\cdots ,M\right\}$ using random equilibrium space partition with $k_0$, and set $t=0,p_{\mathrm{init}}=0$, and $w_m^{\left(0\right)}=1/M$.

Step 2: Construct a weighted formulation ${\mathcal{E}}_w$ of the sum of the intra-cluster variances to mimic the behavior of the maximum variance criterion

$${\mathcal{E}}_w=\sum _{m=1}^M{w}_m^p\sum _{l=1}^L{\xi }_{m,l}{∥c_l-{\mathbf{x}}_m∥}^2,\mathrm{subject}\mathrm{to}\sum _{m=1}^M{w}_m=1,w_m\ge 0,$$

(35)

where $w_m$ is a weight factor that controls cluster variances, and $p\in \left[0,1\right)$ regulates the sensitivity of weight updates to relative differences in cluster variances. The higher the values of p, the larger the relative differences in variances among clusters, and vice versa.

Algorithm 3  Hybrid algorithm for UAV task assignment and trajectory optimization

Input: The system parameters $(K,M,V,H,P_{\mathrm{c}},P_e,\mathrm{etc}.)$ for all IoT devices and UAVs, the DCP set $\mathcal{C}=\left\{c_1,c_2,\cdots ,c_L\right\}$, the DCP-IoT association matrix $\mathit{\rho }$, the optimal hovering time allocation $\left\{t_{l,k}^{\mathrm{EH}\ast },t_{l,k}^{\mathrm{CD}\ast }\right\}$, the EIPGA related parameters $\left\{N_p,N_g^{\prime },N_{new}^{\prime }\right\}$, and the clustering parameters $\left(k_0,p_{\max },p_{\mathrm{step}},\beta ,\epsilon ,t_{\max }\right)$.   1: Initialize the locations of cluster centroids ${\mathbf{x}}_m^{\left(0\right)}$ for every $m\in \left\{1,2,\cdots ,M\right\}$ based on random equilibrium space partition with $k_0$, and set $w_m^{\left(0\right)}=1/M,t=0,p_{\mathrm{init}}=0$;   2: Construct a weighted formulation of the sum of the intra-cluster variances according to Equation (35);   3: while $\left|{\mathcal{E}}_w^t-{\mathcal{E}}_w^{t-1}\right|>\epsilon$ or $t<t_{\max }$ do   4:    for all  ${\xi }_{m,l}$, $m=1$ to M, $l=1$ to L do   5:      Update the cluster assignments $\left\{{\xi }_{m,l}^t\right\}$ using Equation (36);   6:    end for   7:    if empty or singleton clusters have emerged then   8:      Reduce p by $p_{\mathrm{step}}$: $p\leftarrow p-p_{\mathrm{step}}$;   9:      if $p<p_{\mathrm{init}}$ then  10:        return NULL;  11:      end if  12:      ${\xi }_{m,l}^t={\left[{\xi }^{\left(p\right)}\right]}_{m,l},w_m^{t-1}={\left[{\mathbf{w}}^{\left(p\right)}\right]}_m$;  13:    end if  14:    for all ${\mathbf{x}}_m$, $m=1$ to M do  15:      Update the cluster centroids $\left\{{\mathbf{x}}_m^t\right\}$ according to Equation (37);  16:    end for  17:    if $p<p_{\max }$ then  18:      ${\xi }^{\left(p\right)}=\left\{{\xi }_{m,l}^t\right\},{\mathbf{w}}^{\left(p\right)}=\left\{w_m^{t-1}\right\},p\leftarrow p+p_{\mathrm{step}}$;  19:    end if  20:    for all $w_m$, $m=1$ to M do  21:      Update the weights $\left\{w_m^t\right\}$ based on Equations (37) and (38);  22:    end for  23:    Add t by one: $t\leftarrow t+1$;  24: end while  25: Obtain the UAV-DCP association matrix ${\mathit{\xi }}^o$, and find trajectories of UAVs using EIPGA. Output: The UAV-DCP association matrix ${\mathit{\xi }}^o$, the optimal flight trajectories of UAVs ${\mathbf{Q}}^o$.

Step 3: Update the cluster assignments ${\xi }_{m,l}^t$ for each DCP $c_l$ to the nearest cluster centroid ${\mathbf{x}}_m^{t-1}$. The cluster assignments ${\xi }_{m,l}^t$ at t-th iteration are given by

$${\xi }_{m,l}^t=\left\{\begin{array}{c}1,m=\underset{1\le m^{\prime }\le M}{argmin}{\left(w_{m^{\prime }}^{t-1}\right)}^p{∥c_l-{\mathbf{x}}_{m^{\prime }}^{t-1}∥}^2,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

(36)

Step 4: Verify the cluster results. If an empty or singleton cluster appears, reduce p by $p_{\mathrm{step}}$ and revert to the previous assignments and weights. The cluster centroid ${\mathbf{x}}_m^t$ at the t-th iteration is updated as

$${\mathbf{x}}_m^t={\displaystyle \frac{{\displaystyle \sum _{l=1}^L}{\xi }_{m,l}^t{c}_l}{{\displaystyle \sum _{l=1}^L}{\xi }_{m,l}^t}}.$$

(37)

Step 5: Update the weights. If $p<p_{\max }$, save the cluster assignments and the weights $\left\{{\xi }_{m,l}^t,w_m^{t-1}\right\}$. Also, update $p=p+p_{\mathrm{step}}$. Afterward, incorporate the weight constraints ${\sum }_{m=1}^M{w}_m=1$ and $w_m\ge 0$ into ${\mathcal{E}}_w$ using the Lagrange multiplier and set the derivatives with respect to $w_m$ to zero. Consequently, the closed-form solution of $w_m$ is determined by

$$w_m={\displaystyle \frac{{\mathcal{V}}_m^{\frac{1}{1-p}}}{{\displaystyle \sum _{m^{\prime }=1}^M}{\mathcal{V}}_{m^{\prime }}^{\frac{1}{1-p}}}},$$

(38)

where ${\mathcal{V}}_m={\sum }_{l=1}^L{\xi }_{m,l}{∥c_l-{\mathbf{x}}_m∥}^2$ is the cluster variance. As $1/(1-p)>0$ for $0\le p<1$, larger cluster variances ${\mathcal{V}}_m$ lead to higher weights $w_m$. To improve the stability of the algorithm, a memory factor $\beta \in [0,1]$ is introduced into the weights, i.e., $w_m^t=\beta w_m^{t-1}+(1-\beta )w_m$.

Step 6: Check the convergence. If $\left|{\mathcal{E}}_w^t-{\mathcal{E}}_w^{t-1}\right|\le \epsilon$ or $t\ge t_{\max }$, terminate the algorithm. Otherwise, update $t=t+1$ and return to Step 3. Here, $t_{\max }$ is the predetermined maximum number of iterations, and $\epsilon$ is a tiny convergence threshold.

After completing the task assignment, a relative equilibrium in task assignment is achieved. The EIPGA-based algorithm is then used to find the age-optimal trajectory for each UAV.

4.4. Impact of the AoI Performance

In our proposed scheme, the OHTAP clustering algorithm is designed to determine the locations of DCPs and establish the DCP-IoT association. The optimization objective of this algorithm is related to the EH time and data collection time of IoT devices. The EH time and data collection time are parts of the IoT devices’ Avg-AoI. Consequently, the Avg-AoI is influenced by the results of the OHTAP clustering algorithm. Based on the results of the OHTAP clustering algorithm, we perform EIPGA-based task assignment and use the trajectory-planning algorithm to establish the UAV-DCP association and find age-optimal trajectories through all the DCPs. The EIPGA algorithm incorporates a “mutation-before-selection” mechanism and a hybrid selection operator to enhance global optimization capabilities. Through iteration and recursion, it determines the DCP sequence and age-optimal trajectory for each UAV. However, as the number of IoT devices K increases, the computational efficiency and precision of the algorithm may decrease. To address this issue, we develop a hybrid-based task assignment and trajectory-planning algorithm that combines the task assignment capability of IMinMax k-means clustering with the trajectory optimization capability of EIPGA. The clustering component ensures equilibrium-based task assignment among UAVs, while the EIPGA component optimizes trajectories, resulting in a reduced Avg-AoI and improved computational efficiency.

4.5. Complexity Analysis

To sum up, the complete solution scheme for problem $\mathbf{P}\mathbf{1}$ is outlined in Algorithm 4. It consists of three main steps: (1) determine the optimal hovering time allocation using KKT conditions; (2) run Algorithm 1 with parameter $\tau$ to determine DCP sets and establish the DCP-IoT association; (3) apply Algorithm 2 or Algorithm 3 to find the age-optimal trajectories through all the DCPs. The last two steps are performed alternately until the convergence condition is met.

We analyze the complexity of Algorithm 1. According to [45], the messages $r_{l,k}$ and $e_{l,k}$$\left(\forall l,k\right)$ are calculated in $\mathcal{O}\left(K^3\right)$ time per iteration, where K is the number of IoT devices. The algorithm terminates after at most $ite{r}_{\max }$ iterations, resulting in a total time complexity of $\mathcal{O}\left(K^{3}ite{r}_{\max }\right)$. The space complexity is $\mathcal{O}\left(K^2\right)$, since the memory used to store the messages $r_{l,k}$ and $e_{l,k}$ is reused.

Algorithm 4  The age-optimal multi-UAV-assisted energy transfer and data collection algorithm

Input: The system parameters $(K,M,V,H,P_{\mathrm{c}},P_e,\mathrm{etc}.)$ for all IoT devices and UAVs, a small constant $\mathsf{\Delta }\tau$, and a sufficiently large constant ${\tau }_{\max }$.   1: Initialize the parameter: $\tau =0$;   2: while $\tau \le {\tau }_{\max }$ do   3:    Run Algorithm 1 with parameter $\tau$ to obtain the DCP-IoT association matrix $\mathit{\rho }=\left\{{\rho }_{l,k}\right\}$, the set of DCPs $\mathcal{C}=\left\{c_1,c_2,\cdots ,c_L\right\}$, and the location of each DCP $\left\{{\mathbf{w}}_l^c\right\}$;   4:    Calculate the optimal EH and data collection time for IoT devices $\left\{{\mathbf{T}}^{\mathrm{EH}\ast },{\mathbf{T}}^{\mathrm{CD}\ast }\right\}$ by Equation (28), and determine data uploading order ${\mathsf{\Gamma }}_{m,\left(j\right)}$ based on Equation (20);   5:    Generate initial waypoints set of UAVs ${\mathbf{Q}}_0$;   6:    Run Algorithm 2 or Algorithm 3 to obtain UAV-DCP association matrix and UAVs’ trajectories $\left\{{\mathit{\xi }}^o,{\mathbf{Q}}^o\right\}$, and then determine the energy consumption $\left\{E_m^{\mathrm{sum}},E_m^{\mathrm{TSP}}\right\}$;   7:    if ${max}_m{E}_m^{\mathrm{sum}}\le E_{\max }$ then   8:      The Aol-optimal solution ${\mathit{q}}_m^{*}={\mathit{q}}_m^o$;   9:    else if ${max}_m{E}_m^{\mathrm{TSP}}\le E_{\max }$  10:      Find a trajectory for each UAV by EIPGA;  11:    else if ${max}_m{E}_m^{\mathrm{TSP}}>E_{\max }$  12:      No feasible solution exists and the processes of task assignment and trajectory planning are repeated with $M\leftarrow M+1$;  13:    end if  14:    Calculate the Avg-AoI $A^{\mathrm{avg}}$ using Equation (19).  15:    Update the parameter $\tau \leftarrow \tau +\mathsf{\Delta }\tau$;  16: end while  17: Find the minimum Avg-AoI with optimal parameter $\tau \ast$; Output: The set of DCP positions $\mathcal{C}$, the binary DCP-IoT and UAV-DCP association matrix $\left\{\mathit{\rho }\ast ,\mathit{\xi }\ast \right\}$, the hovering time allocation $\left\{{\mathbf{T}}^{\mathrm{EH}\ast },{\mathbf{T}}^{\mathrm{CD}\ast }\right\}$, the flight trajectories of UAVs $\mathbf{Q}\ast$, and the minimum Avg-AoI.

We also examine the time and space complexity of Algorithm 2, which finds near age-optimal trajectories for Avg-AoI through recursion and iteration. The time complexity is approximately $O\left(N_g{N}_{new}(L+M+log{N}_{new})\right)$, and the space complexity is $O\left(N_{new}(L+M)\right)$, where $N_g$ and $N_{new}$ are the maximum number of generations and the number of offspring produced per iteration, respectively.

For Algorithm 3, the complexity analysis follows. The IMinMax k-means clustering algorithm has time and space complexities of approximately $O\left(ML{t}_{\max }\right)$ and $O\left(ML\right)$, respectively. When combined with the complexity analysis of Algorithm 2, the total time and space complexities of Algorithm 3 are $O(ML{t}_{\max }+N_g^{\prime }{N}_{new}^{\prime }(L+Mlog{N}_{new}^{\prime }))$ and $O\left(\max \left\{ML,N_{new}^{\prime }L,N_{new}^{\prime }M\right\}\right)$, where $N_g^{\prime }\left(N_g^{\prime }\ll N_g\right)$ and $N_{new}^{\prime }\left(N_{new}^{\prime }\ll N_{new}\right)$ are the maximal generation and the number of offspring produced per iteration, respectively.

Therefore, the complexity of Algorithm 4 is obtained. The time and space complexities of the OHTAP-based EIPGA scheme are $O\left({\displaystyle \frac{{\tau }_{\max }}{\mathsf{\Delta }\tau }}\left[K^{3}ite{r}_{\max }+N_g{N}_{new}(L+M+log{N}_{new})\right]\right)$ and $\mathcal{O}\left(K^2\right)$, respectively. Similarly, the time and space complexities of the OHTAP-based hybrid scheme are $O\left({\displaystyle \frac{{\tau }_{\max }}{\mathsf{\Delta }\tau }}\left[K^{3}ite{r}_{\max }+ML{t}_{\max }+N_g^{\prime }{N}_{new}^{\prime }(L+Mlog{N}_{new}^{\prime })\right]\right)$ and $\mathcal{O}\left(K^2\right)$, respectively.

5. Experiments

In this section, we conduct a series of experiments to evaluate the performance of the proposed DCP association and trajectory-planning scheme.

5.1. Simulation Settings

We consider a multi-UAV-assisted wireless-powered IoT network that includes one DC $a_0$ and K IoT devices. The DC is located at the coordinate origin $(0,0)$, while the IoT devices are randomly distributed within a 2D area of 1000 m × 1000 m. As per [37], the key parameters of the UAV’s power consumption are configured as $d_0=0.6$, ${\rho }_0=1.225 \mathrm{kg}/{\mathrm{m}}^3$, $A=0.503 {\mathrm{m}}^2$, $v_0=4.03$, $U_{\mathrm{tip}}=120$, $P_0=79.86 \mathrm{W}$, $P_i=88.36 \mathrm{W}$, and $s=0.05$. The main parameters for the IMinMax k-means are set to $p_{\mathrm{init}}=0$, $p_{\max }=0.5$, $p_{\mathrm{step}}=0.01$, $\beta =0.3$, $\epsilon ={10}^{-6}$, and $t_{\max }=500$ [43]. Unless stated otherwise, additional parameters are listed in

Table 1. Simulation settings.

Notation	Description	Value
B	System bandwidth	$10 \mathrm{MHz}$
${\sigma }_w^2$	Noise Power	$-110 \mathrm{dBm}$
H	Constant Flight altitude	$30 \mathrm{m}$
$P_{max}$	Maximum Output DC Power	$24 \mathrm{mW}$
$D_k$	Size of Data Packet	$1 \mathrm{Mbits}$
$P_{\mathrm{c}}$	Communication-Related Power	$5 \mathrm{W}$
${\eta }_{\mathrm{LoS}}$	Excessive Path Loss of LoS Link	$1 \mathrm{dB}$
${\eta }_{\mathrm{NLoS}}$	Excessive Path Loss of NLoS Link	$20 \mathrm{dB}$
$f_c$	Carrier Frequency	$2 \mathrm{GHz}$
$P_e$	Transmission Power	$1 \mathrm{W}$
$P_{\mathrm{OFF}}$	Offloading Power	$1 \mathrm{W}$
$\alpha$	Path Loss Exponent	2
$(a,b)$	Environment Constants	9.61, 0.16
$(C,D)$	EH Circuit Constants	150, 0.014

, where propagation environment parameters a and b and excessive path losses ${\eta }_{\mathrm{LoS}}$, ${\eta }_{\mathrm{NLoS}}$ are derived from the Urban environment defined in [34]. All experiments are implemented in MATLAB 2022b, and simulation results are averaged over 50 experimental realizations.

5.2. Baseline Description

To facilitate a rational comparison of the performance of the proposed schemes, we implement four benchmark schemes: the OA-based scheme, the AP-based scheme, the energy-based scheme, and the SCADC-based scheme, all under identical conditions. Each benchmark is briefly described below.

• OA-based scheme [8]: A one-to-one association method where each DCP is associated with a single IoT device. The UAV-DCP task assignment and each UAV’s trajectory are optimized using a hybrid-based algorithm.
• AP-based scheme [9]: An improved version of the OA-based scheme, where all IoT devices are clustered into M clusters using the AP algorithm, and then the UAV-DCP task assignment and each UAV’s trajectory are optimized using a GA-based algorithm.
• Energy-based scheme [12]: An energy-optimal scheme that employs Google OR-Tools to solve the TSP trajectory for each UAV, based on the results of IMinMax k-means.
• SCADC-based scheme [20]: An improved version of the AP-based scheme, where the DCP-IoT association, UAV-DCP association, and each UAV’s trajectory are determined using the SCADC algorithm, kernel k-means algorithm, and an improved ant colony optimization (ACO) algorithm, respectively.

Given the NP-hard nature of the AoI-oriented multi-UAV-assisted energy transfer and data collection problem, the above four benchmarks solve the age-optimal task assignment and trajectory planning in two stages. The first stage determines the DCP positions of all UAVs using clustering algorithms and establishes DCP-IoT associations. The second stage finds the age-optimal flight trajectory for each UAV.

5.3. Comparison of the Proposed Schemes with Benchmarks

We compare the performance of the proposed OHTAP-based scheme with benchmark schemes for AoI-oriented task assignment and trajectory optimization, considering IoT device sizes ranging from 50 to 2000. The Avg-AoIs, variances, and runtimes of each scheme are presented in Figure 6. The Avg-AoIs achieved by all solution schemes increase, because when the number of IoT devices grows, the system becomes denser and the number of IoT devices associated with DCPs visited by each UAV will also increase, which affects not only the EH and data collection time, but also the amount of data to be forwarded, resulting in longer hovering and flight times for the UAVs. Unlike the OA-based scheme, which treats each IoT device as a DCP, other schemes that perform optimal DCP selection can better balance the hovering time and flight time of UAVs across various scenarios. As expected, the OA-based scheme is markedly inferior to the others, with the performance gap widening as the number of IoT devices increases. Among the solution schemes, our OHTAP-based EIPGA scheme delivers the best performance in small-scale networks with 50 IoT devices, but its search efficiency decreases as the network size grows. In contrast, our OHTAP-based hybrid scheme consistently achieves the best Avg-AoI and the smallest variances across IoT networks of varying sizes, demonstrating superior performance in both Avg-AoI optimization and stability. While the runtime of all schemes increases with the number of IoT devices, the OHTAP-based hybrid scheme maintains high computational efficiency, with its runtime remaining under 30 s even for networks with up to 2000 devices, ensuring its feasibility for real-time, large-scale deployment.

Subsequently, we maintain 200 IoT devices randomly distributed within the wireless network and evaluate the proposed OHTAP-based scheme by increasing the number of UAVs and the energy capacity of each UAV separately. Figure 7a shows the impact of the number of UAVs on the Avg-AoI for different solution schemes. For a network with up to 200 IoT devices, at least two UAVs, each with an energy capacity of 120 KJ, are required. As the number of UAVs increases, the Avg-AoI for all schemes exhibits a decreasing trend. The reason is that an increase in the number of UAVs will reduce the tasks assigned to each UAV, thereby shortening their total flight time, hovering time, and data offloading time. It is observed again that our OHTAP-based hybrid scheme significantly outperforms the other five schemes. Figure 7b shows the relationship between UAVs’ energy capacity and the number of UAVs necessary for different solution schemes. With the improvement in energy capacity of each UAV, the number of UAVs necessary for energy transfer and data collection decreases across all schemes. Conversely, when the energy capacity is restricted, additional UAVs are required to ensure efficient energy transfer and data collection, thereby minimizing the Avg-AoI. In detail, the energy-based TSP trajectory scheme is the least energy-intensive among all schemes but does not attain the minimum Avg-AoI. The proposed OHTAP-based hybrid scheme delivers the best Avg-AoI under the same number of UAVs and onboard energy storage while consuming less energy, underscoring its effectiveness for age-optimal task assignment and trajectory planning.

5.4. Comparison of Different Clustering Algorithms

Table 2. The minimum Avg-AoI with different clustering algorithms.

The Number of IoT Devices		200 ($\mathit{\tau }=3$)			500 ($\mathit{\tau }=3$)			1000 ($\mathit{\tau }=5$)			1500 ($\mathit{\tau }=5$)			2000 ($\mathit{\tau }=4$)
Methods		The Minimum Average AoI
		mean	best	var	mean	best	var	mean	best	var	mean	best	var	mean	best	var
k-means	k-means++	112.27	89.05	266.29	145.07	131.50	70.95	189.84	178.00	51.25	220.88	209.78	39.22	249.70	238.73	24.39
	IMinMax k-means	99.59	89.55	130.97	142.52	129.47	260.75	187.29	174.83	73.23	217.99	207.97	35.49	243.91	236.05	7.97
k-means++	k-means++	91.05	86.47	6.58	134.12	126.32	15.67	178.52	169.12	12.99	212.57	204.75	18.63	242.24	231.67	22.86
	IMinMax k-means	89.45	85.80	5.07	130.69	124.87	6.57	173.83	169.31	8.96	207.40	202.19	7.56	236.06	231.13	5.59
k-medoids	k-means++	99.89	96.05	6.16	132.99	126.68	11.45	176.06	171.59	6.17	209.59	204.57	13.38	236.63	228.70	9.52
	IMinMax k-means	97.40	95.13	1.30	130.36	127.46	1.74	172.27	169.74	2.17	204.89	201.81	2.52	231.52	229.11	1.95
OHTAP	k-means	89.03	84.93	7.01	127.59	123.17	6.44	164.03	158.37	7.28	199.93	192.85	11.46	230.89	223.89	9.67
	k-means++	89.21	85.78	3.65	127.13	122.28	6.36	164.08	159.43	6.67	199.07	193.37	10.29	231.07	222.88	9.29
	Global k-means	88.23	88.04	0.07	127.74	127.33	0.07	166.25	165.06	0.28	195.77	194.96	0.28	232.98	231.62	0.45
	IMinMax$\mathit{k}$-means	85.82	84.85	0.32	122.49	121.14	0.75	159.14	157.81	0.41	194.47	191.49	1.62	225.15	222.78	1.10

presents a comparison of different clustering algorithms used in our proposed scheme, which was validated as a superior solution in the previous subsection. Specifically, we evaluate the performance of OHTAP clustering, k-means clustering, k-means++ clustering, and k-medoids clustering for DCP-IoT association and optimization, as well as k-means clustering, k-means++ clustering, global k-means clustering, and IMinMax k-means clustering for multi-UAV task assignment, within a range of IoT device quantities $K\in \left\{200,500,1000,1500,2000\right\}$. For each network scale, we use the optimal clustering parameter $\tau$ across all clustering algorithms. The results show that our OHTAP-based IMinMax k-means algorithm outperforms all other algorithms by a large margin, indicating superior accuracy and stability. In particular, the OHTAP-based DCP-IoT association and optimization algorithm performs better than the other three clustering algorithms. Furthermore, the IMinMax k-means clustering algorithm excels in task assignment due to its ability to realize equilibrium-based task assignment among UAVs, which improves the efficiency of data collection and processing across the network.

Recall that the number of DCPs identified by the OHTAP clustering algorithm is determined by the preference parameter $\tau$. To examine the impact of the preference parameter $\tau$ on age-optimal task assignment and trajectory planning, we fix $K=1500$ and simulate different DCP-IoT association algorithms and UAV-DCP task assignment algorithms over the range $\tau \in \left\{0,1,\cdots ,20\right\}$. As shown in Figure 8a, the OHTAP-based IMinMax k-means clustering achieves the best Avg-AoI performance at $\tau =5$. Furthermore, all algorithms exhibit a consistent trend, with the Avg-AoI decreasing initially and then increasing as $\tau$ grows. This trend is further illustrated in Figure 8b, which shows that both the total EH time and data collection time rise with $\tau$, whereas the total flight time falls. The underlying cause is a reduction in the number of DCPs identified by the OHTAP clustering algorithm as $\tau$ increases. Therefore, the growing distance between some IoT devices and their associated DCPs extends the time required for EH and data collection, while the age-optimal trajectories consisting of the DC and fewer DCPs are shorter, reducing the total flight time. These findings underscore the importance of identifying the optimal $\tau *$ for age-optimal multi-UAV-assisted network planning.

5.5. Comparison of Different EH Models

Figure 9 shows the minimum Avg-AoI and harvested power for varying numbers of IoT devices (K) under different EH models. These models include a linear model with $\eta =0.9$, a piecewise linear model with $\eta =0.6$, and a non-linear model characterized by parameters $P_{max}=24 \mathrm{mW}$, $C=150$ and $D=0.014$. The parameters of the non-linear EH model used in the simulations are derived from practical EH circuit measurements [35,36]. It is worth noting that the maximum conversion efficiency of EH circuits is typically below $90\%$ [46], constrained by factors such as impedance matching, device parasitics, and harmonic generation. In Figure 9a, the linear EH model unexpectedly outperforms the non-linear model in terms of Avg-AoI across all tested K values, despite the latter more accurately capturing the non-linear features of real-world systems. Additionally, the linear model with higher $\eta$ yields the lowest Avg-AoI among all the EH models. To clarify this counterintuitive result, Figure 9b compares the output powers of the non-linear EH model and the linear EH model with different conversion efficiencies $\eta$. At low input RF power levels (approximately 0–0.5 mW), the output powers of all EH models increase nearly linearly, whereas the linear EH model with the higher $\eta$ results in a larger output power bias compared to the non-linear EH model. That is to say, the higher the conversion efficiency of the linear EH model, the greater the harvested power, leading to an excessively optimistic Avg-AoI expectation (i.e., a smaller Avg-AoI). Therefore, the non-linear EH model, based on real EH circuit measurements, provides more accurate Avg-AoI measurements and optimization results for UAV-assisted wireless-powered IoT networks.

5.6. Impacts of the UAVs’ Flight Speed and Acceleration

Once the DCP-IoT association and UAV-DCP task assignment are determined, the Avg-AoI of IoT devices and the energy consumption of UAVs depend solely on the UAVs’ flight time, which is influenced by their flight speed and acceleration. To examine the impact of flight speed when the proposed OHTAP-based hybrid scheme is used, we report the variations in the minimum Avg-AoI of IoT devices and the energy consumption of UAVs with respect to flight speed in Figure 10. The Avg-AoI of IoT devices decreases rapidly at first and then stabilizes as flight speed rises, as shown in Figure 10a. This trend can be attributed to the flight dynamics of the UAV, where acceleration prevents the UAVs from reaching their target speeds when flying between closely spaced DCPs. In other words, beyond a certain flight speed, further increases in speed no longer reduce the total flight time, resulting in a stabilized Avg-AoI. Meanwhile, the energy consumption of UAVs initially decreases significantly as flight speed rises, but starts to increase gradually after the speed exceeds the MR speed $V_{mr}$, as shown in Figure 10b. The reason lies in the fact that when flight speed rises, the flight time shortens and then levels off, whereas the propulsion power consumption decreases but then increases. Moreover, it is evident that the MR speed, as implemented in our OHTAP-based hybrid scheme, achieves near-minimum Avg-AoIs and minimizes energy consumption, demonstrating that the MR speed strikes a favorable balance between these two key performance metrics for varying numbers of IoT devices.

Figure 11 shows the minimum Avg-AoI and energy consumption achieved by our OHTAP-based hybrid scheme under four acceleration settings. When the acceleration parameter is set to $acc=0$, UAVs are assumed to fly at a constant speed $V_{mr}$, disregarding the effects of acceleration and deceleration. This simplification excludes the durations of acceleration and deceleration, leading to an underestimation of UAVs’ flight times along age-optimal trajectories. Consequently, the calculated Avg-AoI and energy consumption are artificially lower than their actual values. As expected, the results demonstrate optimal performance in terms of both Avg-AoI and energy consumption for $acc=0$. Furthermore, as the number of IoT devices increases, the gap between the minimum Avg-AoI and energy consumption calculated under the uniform flight and uniformly accelerated flight models widens. Therefore, the uniformly accelerated flight model better captures UAV motion dynamics and provides more reliable performance evaluations. It is also observed that the higher acceleration in the uniformly accelerated flight model reduces both the minimum Avg-AoI and energy consumption. In the end, the optimization of flight speed and the consideration of acceleration are crucial for the analysis of both Avg-AoI and energy consumption in age-optimal multi-UAV-assisted IoT networks, particularly in large-scale dense IoT networks.

6. Conclusions

This paper studied AoI-oriented task assignment and trajectory planning for multi-UAV-assisted wireless-powered IoT networks. Taking into account the strict energy and data causality constraints of IoT devices, as well as the limited endurance capability of UAVs, we formulated an optimization problem to minimize the Avg-AoI of all IoT devices by co-optimizing the EH and data collection time for IoT devices, the selection of DCPs, DCP-IoT association, and task assignment, flight speed, and trajectories of UAVs. Due to the nonconvex nature of this optimization problem, we decoupled it into three subproblems through theoretical derivation, and then proposed a novel DCP association and trajectory-planning scheme based on an iterative three-step process using KKT conditions, the OHTAP clustering algorithm, and the EIPGA/hybrid-based algorithm. Simulation results demonstrate that the proposed scheme consistently outperforms benchmark schemes in Avg-AoI performance and solution stability across various scenarios. In particular, our scheme is scalable up to 2000 IoT devices while maintaining significant advantages in accuracy and efficiency. Furthermore, by incorporating acceleration and speed optimization into age-optimal trajectory planning, UAVs achieve more accurate measurements of Avg-AoI and energy consumption, striking a balance between the two metrics. Additionally, we analyzed several factors that impact the Avg-AoI, including the number of IoT devices and UAVs, the selection of EH models, and the energy storage of UAVs. Proper adjustment of these parameters can strike a balance between the Avg-AoI of IoT devices and the energy consumption of UAVs, thereby further improving the overall performance of the system. In future research, we intend to extend our work to scenarios with dynamically mobile IoT devices and optimize UAV flight altitudes. We also want to integrate more advanced EH models, such as adaptive EH models, and explore emerging methodologies, such as DRL-based approaches.