Virtual Force-Based Swarm Trajectory Design for Unmanned Aerial Vehicle-Assisted Data Collection Internet of Things Networks

Abstract

In this paper, the problem of trajectory design for unmanned aerial vehicle (UAV) swarms in data collection Internet of Things (IoT) networks is studied. In the considered model, the UAV swarm is deployed to patrol a designated area and collect status information from sensors monitoring physical processes. The sense-collect-interchange-explore (SCIE) protocol is proposed to regulate UAV actions, ensuring synchronization and adaptability in a distributed manner. To maintain real-time monitoring while reducing data transmission, we introduce status freshness, which is an extension of age of information (AoI) and allows negative values to reflect the swarm’s predictive capabilities. The trajectory design problem is then formulated as an optimization problem to minimize average status freshness. A virtual force-based algorithm is developed to solve this problem, where UAVs are influenced by attractive forces from sensors and repulsive forces from neighbors. These forces guide UAVs toward sensors requiring data transmission while reducing communication overlap. The proposed distributed algorithm allows each UAV to independently design its trajectory, reducing redundancy and enhancing scalability. Simulation results show that the proposed method can significantly reduce average status freshness under the same energy efficiency conditions compared to artificial potential field algorithm. The proposed method also achieves significantly reduction in terms of communication overhead, compared to fully connected strategies, ensuring scalability in large-scale UAV deployments.

1. Introduction

The Internet of Things (IoT) is crucial in shaping future network infrastructures, facilitating a wide range of applications through extensive IoT device connectivity [1]. These applications span from smart cities and industrial manufacturing to telemedicine and environmental monitoring. As artificial intelligence and smart manufacturing technologies advance, they empowered IoT networks and devices with enhanced intelligence and autonomy [2]. Meanwhile, the emergence of the sixth-generation (6G) networks underscore the connectivity across diverse environments [3]. However, the traditional IoT relying on ground base stations faces new challenges in achieving broad network coverage and efficient deployment. This evolution fosters the integration of traditional ground networks and airborne networks [4], positioning unmanned aerial vehicles (UAVs) as key enablers for realizing ubiquitous three-dimensional (3D) communications.

Benefiting from UAVs’ flexible mobility and high probability of line-of-sight (LoS) connections, UAV-enabled IoT networks enhance data transmission, seamless network coverage, and establishing temporary communication. In recent years, numerous studies have focused on enhancing UAV communication performance, such as reconfigurable intelligent surfaces (RIS) [5] and advanced beamforming techniques [6], further improving their efficiency and reliability. As aerial base stations and mobile relays, UAVs effectively connect to IoT devices in remote or inaccessible regions, facilitating the collection and analysis of real-time data, thereby improving the functionality and efficiency of IoT networks. Despite these advancements, the physical constraints of a single UAV, such as its limited energy storage and computing power, curtail its flight time and payload capacity. Thereby, the capabilities of a single UAV may not reach the demands of real-time data transmission in prolonged IoT networks.

To address these challenges, the concept of UAV swarms has emerged as a promising solution to large-scale and complex tasks [7]. Different from naively increasing the number of UAVs, utilizing them in a swarm offers significant advantages. A UAV swarm can achieve more complex coordinate with distributed decision-making strategy. For the expansions of future IoT networks into complex and unreachable geographical environments, this collective approach enables a cost-effective, highly flexible deployment solution. Furthermore, UAV swarms can achieve adaptive topology structures, mitigating the impact of the number changes of UAVs while maintaining network performance. This adaptability enhances the IoT network’s robustness and reliability in real-time data collection and processing. Nevertheless, the deployment of UAV swarms introduces its own challenges, including inter-UAV communication, swarm management, and collective decision-making processes. Addressing these issues is essential to fully unlock the capabilities and advantages of UAV swarms, enhancing the efficiency and functionality of IoT networks.

1.1. Related Works

The existing literature [8,9,10,11] has studied a number of problems related to enhancing the functionality and efficiency of IoT networks. The authors in [8] deployed aerial base stations on UAVs to construct heterogeneous space-air-ground networks, addressing the explosion connection challenges posed by massive IoT devices. A cluster-based energy-efficient resource allocation mechanism was proposed in this architecture to minimize energy consumption, prompting green communication. In [9], the integration of mobile edge computing into IoT networks was studied to enable IoT devices to offload computation-intensive and delay-sensitive tasks to the network edge, ensuring high quality of services. The study also introduced a non-orthogonal multiple access technique to enhance massive connectivity and optimize energy efficiency. The work in [10] studied the problem of dynamic real-time resource allocation in IoT networks by training a nested neural network to optimize resource allocation in multi-device collaboration tasks, achieving high performance decision-making. In [11], the IoT network was modeled as a directed graph, and a graph neural network-based framework was proposed to address the high complexity of device-to-device resource allocation, demonstrating generalizability and accelerating the optimization of wireless resources. However, most of the existing works [8,9,10,11] focused on improving communication performance in IoT networks through network architecture and advanced optimization algorithms, often overlooking the analysis of data characteristics. Analyzing data characteristics can significantly reduce the amount of data transmission required in the IoT network by leveraging the computational capabilities of devices to estimate missing data. Furthermore, a detailed analysis of actual data transmission intervals is crucial for evaluating the accuracy of data estimation by devices. This approach enables the dynamic adjustment of transmission frequency, facilitating real-time monitoring of data changes within the network.

A number of existing works [12,13,14,15,16] have studied the applications of multi-UAV assisted communication systems. The authors of [12] introduced a blockchain-enabled multi-UAV system to ensure the security and energy efficient of IoT networks. In this system, each UAV follows a fixed trajectory and must mine charging coins through positive behaviors, such as forwarding data and maintaining model records, to stay active in the communication network. In [13], the authors proposed an iterative single-head attention mechanism for multi-UAV path planning. An additional communication helper was used to collect state embeddings from UAVs and distribute an attention score vector to each UAV, aiding in effective coordination. The work in [14] investigated network utility maximization in a dual-functional radar-communication multi-UAV network, where a central station determined the resource allocation strategy and provided the instructions to each UAV. The authors of [15] addressed the deployment and association problem of multi-UAV assisted cellular networks using a centralized multi-agent Q-learning algorithm. The algorithm was trained offline and applied online once the training process was complete. In [16], the authors proposed a centralized algorithm to achieve on-demand coverage with minimum number of UAVs, and a distributed algorithm based on artificial potential field (APF) theory to allow each UAV autonomously controls its position for moving UEs. However, most of the existing works [12,13,14,15] rely on centralized or discrete multi-UAV network designs, where either one or multiple central nodes control the entire UAV network or each UAV operates independently without impacting others. These designs often lack the flexible and autonomy needed for collaboration among multiple UAVs, as they are constrained by relatively fixed UAV network topologies. As a result, they fail to fully exploit the capabilities of UAVs functioning as a swarm. By exploring swarm behaviors inspired by natural systems, the scalability, adaptability, and efficiency of multi-UAV system can be significantly enhanced, particularly in dynamic and resource-constrained environments.

1.2. Contributions

The main contribution of this paper is to provide a decentralized trajectory design method for UAVs in a swarm to collect status information from sensors and minimize the average status freshness. Our key contributions are summarized as follows:

• IoT network design: We propose a data collection IoT network where a UAV swarm is deployed to monitor physical processes in the designated area. The UAVs patrol in the area, collecting status information from sensors associated with the physical processes. The behavior of the UAV swarm is regulated through the sense-collect-interchange-explorer (SCIE) protocol, ensuring synchronization and adaptability of UAV actions within the swarm.
• Status freshness extension: To enable real-time monitoring of physical processes while reducing data transmission, we introduce status freshness as an extension of the traditional age of information (AoI). Status freshness corresponds the actual collect interval and estimation error and allows for negative values to reflect the predictive capability of the UAV swarm, effectively quantifying how long sensors can avoid transmitting without compromising monitoring quality. Based on the concept of status freshness, the trajectory design problem is formulated as an optimization problem to minimize the average status freshness.
• Virtual force-based algorithm: To address this problem, we develop a virtual force-based algorithm. Each UAV is influenced by attractive forces from sensors and repulsive forces from neighbor UAVs. On the one hand, attractive forces guide the UAVs closer to sensors requiring data transmission. On the other hand, the repulsive forces help minimize the overlap of communication areas among UAVs, enhancing coverage efficiency. The resultant virtual force determines the trajectory of each UAV at the current time slot. The proposed algorithm allows each UAV in the swarm to independently design its trajectory based on its local information, reducing redundant information exchange between UAVs. This decentralized approach provides a low-complexity solution, making it practical for real-world deployments.

Simulation results show that the proposed algorithm effectively reduces the average status freshness while minimizing redundant transmissions. Compared to the traditional artificial potential field (APF) algorithm [16], the proposed algorithm can achieve lower average status freshness with close energy efficiency. Compared to fully connected strategies, the swarm-based interchange strategy achieves significant reduction in data transmission overhead. Furthermore, the dynamic adaptability of the proposed algorithm ensures efficient swarm deployment and optimized resource utilization in large-scale scenarios.

Table 1. Differences between the proposed solution and existing works.

Reference	Objective	UAV	Trajectory Design	Adaptability	Scalability
[8]	Energy efficiency	Multiple	No	Moderate	Moderate
[9]	Energy efficiency	N/A	N/A	Low	Low
[10]	Dynamic resource allocation	N/A	N/A	Low	Low
[11]	Link scheduling, channel and power allocation	N/A	N/A	Low	Moderate
[12]	Security, energy efficiency	Multiple	No	Moderate	Moderate
[13]	Faster travel, collision avoidance	Multiple	Centralized design	Low	Low
[14]	Data rate, fairness	Multiple	No	Low	Low
[15]	User association	Multiple	No	Low	Low
[16]	On-demand coverage	Multiple	Distributed design	Moderate	High
Our work	Status freshness	Swarm	Distributed design	High	High

UAV: Unmanned aerial vehicle. N/A: Not applicable.

summarizes the differences of the proposed algorithm compared to previous reported studies.

1.3. Organization

The rest of this paper is organized as follows. The system model and problem formulation are described in Section 2. The virtual force-based algorithm is introduced in Section 3. In Section 4, simulation results are presented and analyzed. Finally, conclusions are drawn in Section 5.

2. System Model and Problem Formulation

We consider a remote area with complex terrain, which is equipped with a set of M IoT sensors, denoted as $\mathcal{M}$. The sensors have sufficient storage, computing, and transmission capabilities. Each sensor is dedicated to continuously monitoring the real-time status of a specific physical process. The physical processes monitored by these sensors are related to the changes in environmental and geographical conditions, such as temperature, humidity, and groundwater levels. Real-time data transmission ensures ongoing surveillance of the monitored physical processes. Thereby, the IoT network is capable of prompt decisions and actions, even geographical disaster predictions, based on the most current status information. However, due to the remote location and complex terrain of the considered area, it is quite a dangerous, laborious, and costly task to manually deploy traditional ground base stations as data collection centers. To address these obstacles, as illustrated in Figure 1, a swarm of U UAVs has been deployed to patrol the area, denoted as $\mathcal{U}$. Each single UAV acts as a distributed data collector, tasked with collecting real-time status information from the sensors. This enables the entire UAV swarm serves as the data center and decision maker in the IoT network. For ease of reading, we summarize the main notations in this paper in

Table 2. List of primary notations.

Type	Notations	Description
Index	m	Index of IoT sensors
	$i,j$	Index of UAVs
	t	Index of Time slots
Quantity	M	Number of IoT sensors
	U	Number of UAVs in the swarm
	T	Number of time slots
Set	$\mathcal{M}$	Set of IoT Sensors
	$\mathcal{U}$	Set of UAVs in the swarm
	${\mathcal{M}}_{it}^{\mathrm{C}}$	Set of sensors in communication area
	${\mathcal{M}}_{it}^{\mathrm{S}}$	Set of sensors in sensing area
	${\mathcal{U}}_{it}^{\mathrm{N}}$	Set of neighbor UAVs
Position	${\mathbf{w}}_m$	Coordinates of a sensor
	${\mathbf{u}}_{it}$	Coordinates of a UAV
	$\mathbf{U}$	Trajectory of UAV swarm
Indicator	$a_{imt}$	Transmission indicator, element of $\mathbf{A}$
	$v_{ijt}$	Neighbor UAV indicator, element of ${\mathcal{V}}_{it}$
Status Freshness	${\lambda }_{mt},{\lambda }_{mt}^{\prime }$	Status freshness of physical process, non-negative value
	${\Lambda }_{imt},{\Lambda }_{imt}^{\prime }$	Status freshness at a UAV, non-negative value
	${\tau }_{mt}$	Minimum sampling interval
Virtual Force	${\mathit{F}}_{imt}^{\mathrm{A}},{\mathit{F}}_{it}^{\mathrm{A}}$	Attractive force, total attractive force
	${\mathit{F}}_{ijt}^{\mathrm{R}},{\mathit{F}}_{it}^{\mathrm{R}}$	Repulsive force, total repulsive force
	${\mathit{F}}_{it}$	Total virtual force
	$K^{\mathrm{A}},K^{\mathrm{R}}$	Coefficients of attractive force, repulsive force
Time Duration	$T_{imt}^{\mathrm{T}}$	Transmission time for uploading status freshness
	$T^{\mathrm{S}},T^{\mathrm{C}},T^{\mathrm{I}},T^{\mathrm{E}}$	Sense, collect, interchange, and explore stage time
	$T^{\mathrm{H}},T^{\mathrm{F}}$	Hovering, flying time
Radius	$R^{\mathrm{C}}$	Radius of UAV communication area
	$R^{\mathrm{S}}$	Radius of UAV sensing area
	$R^{\mathrm{I}}$	Radius of UAV inter-swarm communication Ideal minimum distance between UAVs
	$R^{\mathrm{B}}$	UAV’s minimum distance from edge
Distance	$d_{ijt}^{\mathrm{S}}$	Distance between a pair of UAVs in the swarm
	$d_{imt},d_{imt}^{\mathrm{P}}$	Distance, horizontal distance between a UAV and a sensor
Vector	${\mathit{d}}_{ijt}^{\mathrm{S}}$	Distance vector from UAV i to UAV j
	${\mathit{d}}_{imt},{\mathit{d}}_{imt}^{\mathrm{P}}$	Distance vector, horizontal distance vector from UAV i to sensor m
	${\mathit{d}}_{it}$	Displacement vector of a UAV
	${\mathit{V}}_{it}$	Velocity vector of a UAV
Others	H	UAV altitude
	$V_{it}$	Speed scalar of a UAV
	$P_{it}^{\mathrm{F}},P_i^{\mathrm{H}}$	Propulsion, hovering power
	$E_{it},E^{\mathrm{B}}$	Propulsion energy consumption, battery capacity
	$c_{imt},P_m^{\mathrm{T}}$	Transmission rate, transmission power of sensor m
	$D^{\mathrm{O}}$	Minimum unit of communication overhead

.

Actually, in the proposed model, the real-time monitoring of physical processes can be collaboratively optimized at both the sensor and the UAV swarm levels. On the one hand, the sensors analyze the status dynamics of the monitored physical processes to obtain the frequency and intensity of data changes. Thereby, the sensors can adaptively adjust the timing and amount of data transmission and save communication resources of the UAV swarm. On the other hand, the UAV swarm serves a dual role in the IoT network. Each UAV is associated with sensors in its communication area to satisfy the real-time data transmission requirements. Additionally, UAVs are able to detect the transmission requirements of sensors in their sensing area. In response to these transmission requests, the UAV swarm dynamically adjusts its formation and topology to ensure efficient data collection and surveillance of physical processes.

Next, we first model the monitored physical processes as nonlinear time-varying dynamics, and derive the freshness of the status information. Subsequently, we introduce the UAV energy consumption model and the network topology of the UAV swarm, followed by the communication models of sensor-to-UAV. Finally, the problem of minimizing the average status freshness is formulated.

2.1. Nonlinear Time-Varying Dynamics of Physical Process

In the studied model, sensors are aligned with the continuous physical processes they are tasked to monitor. Through the observation of real-time data of these processes, the sensors collect status information at specified intervals and request to send data packets to the associated UAVs. Due to the limited wireless resources and locations of the UAVs in swarm, sensors can not transmit the collected status information at every time slot. Thereby, the UAV swarm can only achieve real-time surveillance by estimating the missing status based on the limited collected information. However, the error between the status estimated by the UAV swarm and the status monitored by the sensors prevents the IoT network from accurately capturing the changes in the physical processes, thereby affecting the decision-making of the network.

Aiming at reducing the estimation errors, the sensors model the monitored physical processes by analyzing historical real status. Only when the estimation error exceeds a threshold, the sensors require transmitting collected status to the associated UAV for continuous monitoring. As a result, the transmission requirements can be also reduced. Each sensor m models the monitored physical process as a nonlinear time-varying dynamic [17], which can be formulated as follows:

$${\mathit{s}}_{m\left(t+1\right)}={\mathit{A}}_m{\mathit{s}}_{mt}+f_m\left({\mathit{s}}_{mt}\right)+{ϵ}_{mt},$$

(1)

where ${\mathit{s}}_{mt}\in {\mathbb{R}}^{N_m}$ denotes the status vector corresponding to the physical process at time slot t, monitored by sensor m. Here, $N_m$ signifies the dimensionality of the data related to the physical process, indicating the number of distinct data elements involved. ${\mathit{A}}_m$ represents the linear transformation coefficient matrix, $f\left(·\right)$ indicates a nonlinear mapping function satisfying $f\left(\mathbf{0}\right)=\mathbf{0}$, and ${ϵ}_{mt}$ denotes the stochastic error term, assumed to be independent of the physical process. The formulated model (1) provides a comprehensive framework for capturing the complex behaviors of physical processes by incorporating both linear dynamics and nonlinear interactions. For example, in the context of modeling the air quality index using (1), the linear component is utilized to represent wind force [18], while the nonlinear component accounts for the effects of precipitation [19].

Limited communication resources prevent sensors from transmitting every collected status information. Thereby, when there is a failure in uploading status information, it becomes necessary to estimate the current status based on historical observations. Using (1), the estimated status of physical process can be derived by [20]

$${\widehat{\mathit{s}}}_{mt}={\mathit{A}}_m^{{\delta }_{mt}}{\mathit{s}}_{m\left(t-{\delta }_{mt}\right)}+\sum _{k=1}^{{\delta }_{mt}}{\mathit{A}}_m^{k-1}{f}_m\left({\mathit{s}}_{m\left(t-k\right)}\right)+\sum _{k=1}^{{\delta }_{mt}}{ϵ}_{m\left(t-k\right)},$$

(2)

where ${\delta }_{mt}$ indicates the interval from the latest sample. We assume that the expectation of the error term ${ϵ}_{mt}$ in (1) is zero. Consequently, the expectation of the cumulative error term ${\sum }_{k=1}^{{\delta }_{mt}}{ϵ}_{m\left(t-k\right)}$ is also considered to be zero. Given the estimated status vector at time slot t, the estimation error is further defined as ${\Delta }_{mt}\triangleq {\widehat{\mathit{s}}}_{mt}-{\mathit{s}}_{mt}$. Note that estimating the status information offers a means to continue monitoring the variation in physical processes within an acceptable error margin. Nevertheless, sustained upload failures can result in the accumulation of estimation errors, eventually leading to significant inaccuracies that compromise the monitor on the physical processes. Therefore, it is essential to find out the duration of upload failures that can be tolerated before losing track of the physical processes. To this end, we need to calculate the maximal variation frequency of the physical processes. The derivation of the estimation error with respect to time, which is given by

$${\displaystyle \frac{\mathrm{d}{\Delta }_{mt}}{\mathrm{d}t}}=\underset{\mathrm{First}-\mathrm{order}}{\underbrace{\phantom{\left({\mathit{A}}_{mt}+{\mathit{J}}_{f_m}\left({\mathit{s}}_{mt}\right)\right){\Delta }_{mt}}\left({\mathit{A}}_{mt}+{\mathit{J}}_{f_m}\left({\mathit{s}}_{mt}\right)\right){\Delta }_{mt}}}+\underset{\mathrm{High}-\mathrm{order}}{\underbrace{\phantom{\left({\mathit{A}}_{mt}+{\mathit{J}}_{f_m}\left({\mathit{s}}_{mt}\right)\right){\Delta }_{mt}}o\left(\parallel {\Delta }_{mt}\parallel \right)}},$$

(3)

where the first term represents the first-order approximation with ${\mathit{J}}_{f_m}\left({\mathit{s}}_{mt}\right)$ being the Jacobian matrix of function $f_m$, and the second term indicates a high-order approximation that is negligible [21]. Subsequently, we diagonalize the coefficient matrix associated with the first-order approximation in (3), which is given by

$${\mathit{A}}_{mt}+{\mathit{J}}_{f_m}\left({\mathit{s}}_{mt}\right)=\mathit{P}\mathrm{diag}\left({\mu }_{1t},\dots ,{\mu }_{N_{m}t}\right){\mathit{P}}^{-1},$$

(4)

where $\mathrm{diag}\left(·\right)$ represents the construction of a diagonal matrix, ${\mu }_{n_{m}t}$ for $n_m=1,\dots ,N_m$, represents the eigenvalues, and $\mathit{P}$ indicates the invertible matrix formed by the eigenvectors corresponding to ${\mathit{A}}_{mt}+{\mathit{J}}_{f_m}\left({\mathit{s}}_{mt}\right)$. Based on the result of (4), the maximal variation frequency of the physical process being monitored by sensor m in time-domain is calculated as [22]

$${\Omega }_{mt}=\sqrt{{\displaystyle \frac{\parallel {\Delta }_{mt}{\parallel }_2^2}{{\xi }_m^2}}-\underset{n_m}{min}{\mu }_{n_{m}t}^2},$$

(5)

where $n_m=1,\dots ,N_m$, and ${\xi }_m$ indicates the minimal frequency that sensor m can resolve. Based on the Nyquist sampling theory, the maximal sampling interval of the physical process monitored by sensor m is given by

$${\tau }_{mt}={\displaystyle \frac{\pi }{{\Omega }_{mt}}}.$$

(6)

From (5) and (6), it is evident that the maximal sampling interval ${\tau }_{mt}$ is negatively correlated with the estimation error ${\Delta }_{mt}$. This is due to the fact that, a larger estimation error indicates that the physical process undergoes rapid and significant changes, necessitating more frequent data uploads to ensure the variations in the physical process are captured accurately. Conversely, a smaller estimation error suggests that the physical process changes more slowly and with less magnitude, allowing the variation trends of the physical process to be understood even without frequent updates of status information.

2.2. Status Freshness of the Monitored Physical Processes

Given the previous model of physical processes, sensors can help evaluate the estimation error of the swarm, based on the last collected time slot and the changes in the monitored physical processes. Next, we first extend the concept of AoI and further define the metric of status freshness to assess the UAV swarm in monitoring physical processes.

When the UAV swarm is first deployed, $t=0$, the physical process monitored by each sensor is unknown to the UAVs. The status freshness of physical process monitored by sensor m can be initially set as a sufficiently large value as follows:

$${\lambda }_{m0}={\lambda }_{max}.$$

(7)

Unlike traditional AoI [23], which simply resets to zero after an information update, the status freshness considered in this paper is updated based on the value determined by (6) whenever sensor m uploads the monitored status to its associated UAV i. Conversely, if sensor m does not upload its status information, the status freshness ${\lambda }_{mt}$ increases over time, similar to traditional AoI. Therefore, the status freshness is given by

$${\lambda }_{mt}=\left\{\begin{array}{ccc}\hfill & \lceil -{\tau }_{mt}\rceil ,\hfill & \hfill \exists a_{imt}=1,\\ \hfill & {\lambda }_{m\left(t-1\right)}+1,\hfill & \hfill \forall a_{imt}=0,\end{array}\right.$$

(8)

where the operator $\lceil ·\rceil$ is used to round up the integer. $a_{imt}$ is the transmission indicator, with $a_{imt}=1$ representing that sensor m upload the monitored status information to UAV i at time slot t, and otherwise, $a_{imt}=0$; the status freshness increases over time. From (8), we can see that the status freshness ${\lambda }_{mt}$ is updated to a negative value $\lceil -{\tau }_{mt}\rceil$ when sensor m uploads data at time slot t. The negative value of the status freshness indicates that even with up to ${\lambda }_{mt}$ time slots where the status information is not uploaded by sensor m, the UAV swarm can still predict the dynamics in the monitored physical process using the previous collected data.

As shown in Figure 2, compared to traditional AoI, the status freshness indicates the duration that a sensor can pause status updates after a single update, while the UAV swarm can still monitor the dynamics of the physical process through prediction.

2.3. UAV Energy Consumption Model

The energy consumption of UAV consists of two main components: communication related energy and the propulsion energy [24]. As propulsion energy is generally much larger than communication related energy, this paper focuses solely on the former [25]. According to the derivation in [24], the propulsion power of UAV i at time slot t is given by

$$P_{it}^{\mathrm{F}}\left(V_{it}\right)=\underset{\mathrm{blade}\mathrm{profile}}{\underbrace{\phantom{P_{\mathrm{b}}\left(1+{\displaystyle \frac{3V_{it}^2}{V_{\mathrm{tip}}^2}}\right)}{P}_{\mathrm{blade}}\left(1+{\displaystyle \frac{3V_{it}^2}{V_{\mathrm{tip}}^2}}\right)}}+\underset{\mathrm{induced}}{\underbrace{\phantom{P_{\mathrm{b}}\left(1+{\displaystyle \frac{3V_{it}^2}{V_{\mathrm{tip}}^2}}\right)}{P}_{\mathrm{induced}}{\left(\sqrt{1+{\displaystyle \frac{V_{it}^4}{4V_{\mathrm{induced}}^4}}}-{\displaystyle \frac{V_{it}^2}{2V_{\mathrm{induced}}^2}}\right)}^{1/2}}}+\underset{\mathrm{parasite}}{\underbrace{\phantom{P_{\mathrm{b}}\left(1+{\displaystyle \frac{3V_{it}^2}{V_{\mathrm{tip}}^2}}\right)}{\displaystyle \frac{1}{2}}{K}_{\mathrm{parasite}}{V}_{it}^3}},$$

(9)

where $P_{\mathrm{blade}}$ and $P_{\mathrm{induced}}$ are constants representing the blade profile power and induced power during hovering, respectively. $V_{it}$ is the flight speed of UAV i at time slot t, $V_{\mathrm{tip}}$ denotes the rotor blade’s tip speed, $V_{\mathrm{induced}}$ represents the mean rotor induced speed during hovering, and $K_{\mathrm{parasite}}$ is the parasite coefficient influenced by the fuselage drag ratio, rotor solidity, air density, and rotor disc area. From (9), the propulsion power consumption of a UAV including three components:

a.

Blade profile power: consumed due to profile drag caused by the blade’s airfoil shape, typically varying with the rotor’s rotational speed.

b.

Induced power: consumed due to induced drag, which mainly dominates during hovering and low-speed flight.

c.

Parasite power: consumed due to parasite drag, which increases significantly with flight speed.

When UAV i hovers to collect status information and share its own knowledge, substituting $V_{it}=0$ into (9) simplifies the propulsion power. The hovering power is given by

$$P_i^{\mathrm{H}}=P_{it}^{\mathrm{F}}\left(0\right)=P_{\mathrm{blade}}+P_{\mathrm{induced}}.$$

(10)

The total propulsion energy consumption of UAV i at time slot t is given by

$$E_{it}=P_{it}^{\mathrm{F}}{T}^{\mathrm{F}}+P_i^{\mathrm{H}}{T}^{\mathrm{H}},$$

(11)

where $T^{\mathrm{F}}$ and $T^{\mathrm{H}}$ represent the flying time and hovering time of the UAV, respectively. Each UAV carries a battery with limited energy $E^{\mathrm{B}}$. If the battery is depleted, the UAV must exit the designated area.

2.4. Network Topology of the UAV Swarm

We assume that each UAV i is positioned at the coordinate ${\mathbf{u}}_{it}=\left(x_{it}^{\mathrm{U}},y_{it}^{\mathrm{U}}\right)$ during time slot t. The initial deployment position of UAV i is denoted as ${\mathbf{u}}_{i0}=\left(x_{i0}^{\mathrm{U}},y_{i0}^{\mathrm{U}}\right)$. The altitude of the entire UAV swarm is set at a constant height H. As shown in Figure 1, the communication area of each UAV is a circular region centered at the UAV’s ground projection, with a radius of $R^{\mathrm{C}}$. The sensors in the communication area of a UAV can transmit the monitored status to the UAV. Similarly, the sensing area is a circular region with a radius of $R^{\mathrm{S}}$. The UAV senses the transmission requirements of the sensors in the sensing area, as well as the status freshness of the monitored physical processes. From Figure 1, it is clearly evident that the set of sensors within a UAV’s communication area constitutes a subset of those within its sensing area.

In the studied system, the network topology of the UAV swarm is time-varying and established solely based on the distances between the UAVs, represented as ${\mathcal{V}}_t={\left[v_{ijt}\right]}_{U\times U}$. The distance between UAV i and UAV j is denoted as $d_{ijt}^{\mathrm{S}}=\parallel {\mathbf{u}}_{it}-{\mathbf{u}}_{jt}\parallel$, where $\parallel ·\parallel$ represents the Euclidean norm. Any pair of UAVs with a distance less than the inter-swarm communication distance $R^{\mathrm{I}}$ is considered neighboring and can communicate, denoted as $v_{ijt}=1$. Otherwise, $v_{ijt}=0$ indicates that UAV i and UAV j are disconnected at time slot t. We denote the set of neighbor UAVs of UAV i during time slot t as ${\mathcal{U}}_{it}^{\mathrm{N}}=\left\{j|v_{ijt}=1\right\}$. The number of neighbor UAVs of UAV i at time slot t is calculated as $|{\mathcal{U}}_{it}^{\mathrm{N}}|$, where the operator $|·|$ calculates the number of elements in the set. Leveraging this dynamic topology, the network adapts seamlessly to changes in the swarm composition. For instance, UAVs with depleted batteries can exit the swarm without disrupting the overall network connectivity, as the topology dynamically adjusts to reflect these changes. Additionally, new UAVs can be supplemented into the swarm and integrated into the network based on their distances to existing UAVs. This adaptability ensures the sustained operation of the UAV swarm, enhances its resilience, and guarantees long-term performance despite the energy constraints and mechanical failures of individual UAVs.

In each time slot, UAV i updates the knowledge of sensors in its sensing area. This knowledge includes the UAV’s understanding of the current status information and the transmission requirements of the associated sensors, which also reflects the status freshness of the monitored physical processes. Acting as part of the UAV swarm, UAV i shares its updated knowledge with the neighbor UAVs to ensure collaboration within the swarm. The information shared between a UAV and its neighbor UAVs constitutes the communication overhead in the IoT network. The knowledge related to an individual sensor is denoted as $D^{\mathrm{O}}$. During a time slot, if a UAV updates the knowledge of $M^{\prime }$ sensors in its sensing area, the total information it shares with neighbor UAVs amounts to $M^{\prime }\times D^{\mathrm{O}}$.

It is worth noting that due to the time-varying network topology of the UAV swarm and the absence of a fixed central UAV to synchronize information across all the UAVs, the status freshness known by each individual UAV may differ. UAV i’s knowledge of status freshness aligns with the freshness monitored by the sensors only when it senses the transmission requirements. Additionally, UAV i can further update its knowledge of status freshness based on the information shared by the neighbor UAVs. Consequently, we represent the status freshness of the physical process monitored by sensors m at UAV i during time slot t as

$${\Lambda }_{imt}=\left\{\begin{array}{ccc}\hfill & {\lambda }_{mt},\hfill & \hfill \parallel {\mathbf{u}}_{it}-{\mathbf{w}}_m\parallel <R^{\mathrm{S}},\\ \hfill & min\left\{{\Lambda }_{im\left(t-1\right)}+1,{\Lambda }_{jmt}|j\in {\mathcal{U}}_{it}^{\mathrm{N}}\right\},\hfill & \hfill \parallel {\mathbf{u}}_{it}-{\mathbf{w}}_m\parallel \ge R^{\mathrm{S}},\end{array}\right.$$

(12)

where ${\mathbf{w}}_m=\left(x_m^{\mathrm{S}},y_m^{\mathrm{S}}\right)$ denotes the location of sensor m. From (12) we can see that, if sensor m is in the sensing area of UAV i during time slot t, UAV i can align its status freshness with sensor m by sensing the transmission requirements. Otherwise, if sensor m is outside the sensing area of UAV i, UAV i compares its knowledge with the information shared by neighbor UAVs, and updates its status freshness to the minimum value observed. Notably, if UAV i has no neighbors, its knowledge of status freshness increases over time.

From the perspective of the UAV swarm, status freshness also reflects the urgency for a sensor to transmit the monitored status information. Higher freshness indicates a greater deviation between the UAV’s prediction of the monitored physical process and the actual real-time status. Consequently, the status freshness known to each UAV influences its choice of hovering position for the next time slot, causing the network topology of the UAV swarm to change accordingly.

2.5. Sensor-to-UAV Communication Model

The sensors within the communication area of UAV i during time slot t are associated with UAV i and can transmit their monitored status information to UAV i. To model the sensor-to-UAV communication links, we use the probabilistic line-of-sight (LoS) channel model [26]. The LoS probability of data transmission from sensor m to UAV i is given by

$${\gamma }_{imt}^{\mathrm{LoS}}={\displaystyle \frac{1}{1+Xexp\left(-Y\left[{\varphi }_{imt}-X\right]\right)}},$$

(13)

where X and Y are environment-specific constants. The elevation angle ${\varphi }_{imt}$, from sensor m to UAV i in degrees, is calculated as ${\varphi }_{imt}={\displaystyle \frac{180}{\pi }}{sin}^{-1}\left({\displaystyle \frac{H}{d_{imt}}}\right)$. The distance $d_{imt}$, between UAV i and sensor m at time slot t, is defined as $d_{imt}\triangleq \sqrt{\parallel {\mathbf{u}}_{it}-{\mathbf{w}}_m{\parallel }^2+H^2}$.

Channel gains for LoS and non-LoS (NLoS) conditions when transmitting data from sensor m to UAV i are

$$g_{imt}^{\mathrm{LoS}}=d_{imt}^{-\alpha },$$

(14)

$$g_{imt}^{\mathrm{NLoS}}=\eta d_{imt}^{-\alpha },$$

(15)

where $\alpha$ represents the path loss exponent, and $\eta$ signifies the additional attenuation factor caused by NLoS conditions.

The overall channel gain between sensor m and UAV i during time slot t is approximated as [26]

$${\overline{g}}_{imt}={\gamma }_{imt}^{\mathrm{LoS}}{g}_{imt}^{\mathrm{LoS}}+{\gamma }_{imt}^{\mathrm{NLoS}}{g}_{imt}^{\mathrm{NLoS}},$$

(16)

where ${\gamma }_{im}^{\mathrm{NLoS}}=1-{\gamma }_{im}^{\mathrm{LoS}}$ indicating the NLoS probability.

We assume that the UAVs in the swarm simultaneously collect data from their associated sensors, each operating on distinct frequencies. According to the Shannon capacity theorem, the data transmission rate from sensor m to UAV i is given by

$$c_{imt}=B_i{log}_2\left(1+{\displaystyle \frac{P_m^{\mathrm{T}}{\overline{g}}_{imt}}{{\sigma }^2}}\right),$$

(17)

where $B_i$ is the bandwidth allocated to UAV i, $P_m^{\mathrm{T}}$ is the transmission power of sensor m, and ${\sigma }^2$ is the noise power at the receiver. Consequently, the transmission time from sensor m to UAV i is determined by

$$T_{imt}^{\mathrm{T}}={\displaystyle \frac{D_m^{\mathrm{S}}}{c_{imt}}}.$$

(18)

In each time slot, UAV i can choose several sensors in its communication area and sequentially collect data from the chosen sensors. A maximum transmission time for a time slot is denoted by $T^{\mathrm{C}}$ to limit the number of sensors that uploading status information to a single UAV.

2.6. Problem Formulation

In the studied model, we aim to deploy the UAV swarm to monitor the dynamic changes in physical processes within the designated area. Due to the limited wireless resources, sensors can not upload all their monitored status to the UAV swarm, necessitating status prediction to support the surveillance. From the above derivations, we also conclude that maintaining the freshness of status information can effectively reduce the estimation errors. Therefore, the optimization problem is defined as minimizing the average status freshness as follows:

$$\begin{array}{cc}\hfill \underset{T\to +\infty }{lim}\underset{\mathrm{A},\mathrm{U}}{min}& {\displaystyle \frac{1}{T}}\sum _{t=1}^T\sum _{m=1}^M{\lambda }_{mt}^{\prime },\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \sum _m{a}_{imt}{T}_{imt}^{\mathrm{T}}\le T^{\mathrm{C}},i\in \mathcal{U},t=1,2,\dots ,T,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & \parallel {\mathbf{u}}_{it}-{\mathbf{w}}_m\parallel \le R^{\mathrm{C}},\forall a_{imt}=1,\hfill \end{array}$$

(21)

where $\mathbf{A}={\left[a_{imt}\right]}_{U\times M\times T}$ records the indicators of sensor m uploading status information to UAV i in time slot t, $\mathbf{U}={\left[{\mathrm{u}}_{\mathrm{it}}\right]}_{U\times T}$ is the trajectory of the UAV swarm. In (19), we ignore the negative values of the status freshness, which indicate periods where updates can be paused. Instead, we only consider the positive value ${\lambda }_{mt}^{\prime }=max\left\{{\lambda }_{mt},0\right\}$ to represent the duration for which the sensor requires transmission but has not yet been responded. Constraint (20) ensures that the total upload time in a time slot can not exceed the threshold $T^{\mathrm{C}}$. Constraint (21) ensures that the status transmission is only feasible when the sensor m is within the UAV i’s communication area.

3. Virtual Force-Based Algorithm for Swarm Trajectory Design

To solve the status freshness minimization problem in (19), we propose an algorithm based on virtual forces. This algorithm is inspired by the artificial potential field theory [27], and aims to achieve effective data collection and dynamic task response in complex environments. Compared to existing trajectory design methods based on reinforcement learning [28] and convex optimization [29], the virtual force-based algorithm offers lower computational complexity and better real-time adaptability, enabling rapid deployment and dynamic adjustments of the UAVs. Moreover, the proposed algorithm supports distributed decision-making, where each UAV independently determines its trajectory based on local information and interactions with neighbor UAVs, thereby enabling the swarm to collaborate effectively. It also provides the scalability for the swarm, allowing it to adapt to changes in the number of UAVs.

Next, we begin by designing the SCIE protocol to regulate the actions of UAVs in the swarm. We then introduce the components of the virtual force and explain how the virtual force-based algorithm facilitates the swarm trajectory design. Finally, we analyze the time complexity of the proposed algorithm.

3.1. UAV Swarm SCIE Protocol

Inspired by the observe–orient–decision–action (OODA) loop [30], we first propose a SCIE protocol that supports the UAV data collections from the sensors and information sharing in the swarm network. The OODA loop conceptualizes and optimizes the actions of fighter pilots in combat. Aligned with the four stages in OODA loop, as illustrated in Figure 3, the behaviors of each UAV in each time slot t can be decomposed as follows:

a.

Sense (observe): At the beginning of each time slot t, UAV i arrives at a new position, defining its current sensing area and communication area. Each UAV i hovers for a duration of $T^{\mathrm{S}}$, during which it senses the transmission requirements and the status freshness from the sensors within its sensing area, as well as the positions of neighbor UAVs. Simultaneously, the UAV schedules the transmission requirements, determining whether sensors within its communication area are allowed to transmit and the order of transmission.

b.

Collect (orient): Based on the scheduling from the previous stage, the sensors granted transmission permission can upload their monitored status information. Within the same time slot t, transmission from different sensors are managed using a time-division approach. To ensure synchronized actions across the UAV swarm, the total transmission time for each UAV i must not exceed $T^{\mathrm{C}}$, as specified by (20). If the remaining transmission time is insufficient to complete another sensor’s transmission, the remaining time is left as stub.

c.

Interchange (decision): After receiving the monitored status information, each UAV i updates its own knowledge and transmits data packages with neighbor UAVs to share information. The shared data includes both the updated status information and the status freshness in the current time slot. Based on its own knowledge and shared information from neighbor UAVs, each UAV i can then determine the hovering position for next time slot, aiming to minimizing the status freshness. This entire process, including the knowledge sharing and decision-making, lasts for a duration of $T^{\mathrm{I}}$.

d.

Explore (action): All U UAVs take a duration of $T^{\mathrm{E}}$ to fly to their new hovering positions, preparing for the next time slot.

These four stages, sense, collect, interchange, and explore, form a continuous loop that constitutes the SCIE protocol, designed to enable all individual UAVs to function as a unified swarm. In the decision-making process for data collection and trajectory design, each individual UAV relies not only on the information gathered from its own communication and sensing areas, but also incorporates the shared information from neighbor UAVs, contributing to the overall optimization of the swarm operation.

Note that, in the first three stages—sense, collect, and interchange—the UAVs hover stationary, leading to the relationship $T^{\mathrm{H}}=T^{\mathrm{S}}+T^{\mathrm{C}}+T^{\mathrm{I}}$. The UAVs only move to their next hovering position during the final explore stage, resulting in $T^{\mathrm{F}}=T^{\mathrm{E}}$. The time settings for each stage of the SCIE protocol need to be adapted to the specific conditions of different scenarios. On the one hand, the total duration of a single time slot must be aligned with the frequency at which sensors monitor status information about physical processes, ensuring synchronization between the UAV swarm and the sensors. On the other hand, the duration proportion of each stage within a time slot should be pre-designed based on specific conditions, such as the UAV-to-sensor ratio and the ratio of UAV speed to the size of the designated area. For example, when the number of UAVs is relatively low compared to the number of sensors, the proportion of the collect stage should be increased, allowing each UAV to collect status information from more sensors within a single time slot. Similarly, when the UAV flight speed is relatively low compared to the range of the designated area, the proportion of the explore stage should be appropriately increased to provide UAVs with more flight time to reach their hovering positions for the next time slot.

3.2. Components of the Virtual Force-Based Algorithm

In the interchange stage of the SCIE protocol, the UAV swarm must determine the hovering positions for the next time slot. In the studied model, the network topology of the UAV swarm is time-varying, with no fixed central UAV serving as a global controller. Consequently, each UAV independently identifies its next hovering position based on its updated knowledge and shared information from neighbor UAVs.

On the one hand, to reduce the average status freshness of the monitored physical processes, each UAV aims to position itself near areas with greater transmission demands and higher status freshness, thereby completing sensor transmission requirements and reducing the status freshness. On the other hand, to minimize overlap with the communication areas of neighbor UAVs, each UAV maintains a certain distance from its neighbors. Therefore, we introduce virtual forces, based on APF theory [16], to quantify the attraction of transmission requirements and the repulsion of the neighbor UAVs.

In the studied model, UAV i is subject to two types of virtual forces: (i) the attractive force ${\mathit{F}}_{imt}^{\mathrm{A}}$, by which sensor m pulls UAV i to a closer position for uploading monitored status information, as shown in Figure 4a; and (ii) the repulsive force ${\mathit{F}}_{ijt}^{\mathrm{R}}$, by which neighbor UAV j pushes UAV i out of their overlapping communication area, as shown in Figure 4b. In time slot t, the total virtual force that is subject to UAV i is calculated by

$${\mathit{F}}_{it}={\mathit{F}}_{it}^{\mathrm{A}}+{\mathit{F}}_{it}^{\mathrm{R}},$$

(22)

where the total virtual force ${\mathit{F}}_{it}$ is the vector sum of two types of virtual forces, ${\mathit{F}}_{it}^{\mathrm{A}}$ and ${\mathit{F}}_{it}^{\mathrm{R}}$ represent the total attractive force and the total repulsive force exerted on UAV i at time slot t, respectively. Different from the relationship between force and motion in classical physics, here we simply relate the virtual force exerted on the UAV to its velocity, ${\mathit{V}}_{it}={\mathit{F}}_{it}$. The displacement is then calculated by ${\mathit{d}}_{it}={\mathit{F}}_{it}{T}^{\mathrm{F}}$. The hovering position at next time slot $t+1$ can be calculated as ${\mathbf{u}}_{i\left(t+1\right)}={\mathbf{u}}_{it}+{\mathit{d}}_{it}$. Next, we define each type of virtual force in details.

3.2.1. The Attractive Force

To upload the real-time status information to the UAV swarm, sensors send transmission requests to nearby UAVs. This prompts the UAV to move closer and the sensors in UAV’s communication area to transmit status information. The attractive force ${\mathit{F}}_{imt}^{\mathrm{A}}$ exerted on UAV i by sensor m at time slot t is defined as

$${\mathit{F}}_{imt}^{\mathrm{A}}=\left\{\begin{array}{ccc}\hfill & K^{\mathrm{A}}{\Lambda }_{imt}^{\prime }{\mathit{d}}_{imt}^{\mathrm{P}},\hfill & \hfill d_{imt}^{\mathrm{P}}\le R^{\mathrm{C}},\\ \hfill & K^{\mathrm{A}}{\Lambda }_{imt}^{\prime }{\left({\displaystyle \frac{R^{\mathrm{C}}}{d_{imt}^{\mathrm{P}}}}\right)}^{\beta }{\mathit{d}}_{imt}^{\mathrm{P}},\hfill & \hfill R^{\mathrm{C}}<d_{imt}^{\mathrm{P}}\le R^{\mathrm{S}},\\ \hfill & \mathbf{0},\hfill & \hfill d_{imt}^{\mathrm{P}}\ge R^{\mathrm{S}},\end{array}\right.$$

(23)

where $K^{\mathrm{A}}$ is the coefficient of the attractive force related to status freshness of UAV i, the value of $K^{\mathrm{A}}$ is derived in Appendix A. $\beta$ is the force loss exponent, and ${\Lambda }_{imt}^{\prime }=max\left\{{\Lambda }_{imt},0\right\}$ ensures the weight of status freshness a non-negative value. ${\mathit{d}}_{imt}^{\mathrm{P}}={\mathbf{u}}_{it}-{\mathbf{w}}_m$ represents the vector from UAV i’s projection position on the ground to the location of sensor m at time slot t, which reflects the direction of the attractive force. $d_{imt}^{\mathrm{P}}=\parallel {\mathit{d}}_{imt}^{\mathrm{P}}\parallel$ is the horizontal distance from the projection of UAV i to sensor j. From (23), we can see that the attractive force exerted by sensors located within the communication area of UAV i increases linearly with the horizontal distance. For sensors that locate within the sensing area but outside the communication area, the attractive force is inversely proportional to the horizontal distance. We can also see that, for the sensors located within the sensing area of UAV i, the attractive force is linearly proportional to the status freshness ${\Lambda }_{imt}$ known to UAV i. In particular, if a sensor is just located on the edge of the communication area and the sensing area of a UAV, the attractive forces calculated under both definition are equal. Moreover, sensors located outside the sensing area exert no attractive force on UAV i. This design allows each individual UAV to prioritize sensors that are closer to them during data collection, thereby reducing the overall flight distance and energy consumption. Since propulsion energy is the primary contributor to UAV energy consumption, reducing flight distances directly enhances energy efficiency. This design allows UAVs to conserve energy and extend their operating time, thereby improving the overall system’s performance on minimizing the average status freshness. At the same time, it encourages UAVs to focus on collecting status information from physical processes with higher status freshness, effectively reducing the system average status freshness.

As illustrated in Figure 4a, the direction of the total attractive force follows the parallelogram rule, but its magnitude is normalized based on the number of sensors exerting the attractive force, which is given by

$${\mathit{F}}_{it}^{\mathrm{A}}={\displaystyle \frac{1}{|{\mathcal{M}}_{it}^{\mathrm{S}}|}}\sum _{m\in {\mathcal{M}}_{it}^{\mathrm{S}}}{\mathit{F}}_{imt}^{\mathrm{A}},$$

(24)

where ${\mathcal{M}}_{it}^{\mathrm{S}}$ is the set of sensors located within UAV i’s sensing area at time slot t. The normalization of total attractive force ensures that a UAV does not experience unbounded force from any direction, preventing it from moving excessively far and straying from the areas where data collection is necessary.

3.2.2. The Repulsive Force

To ensure efficient data collection within the designated area and minimize the system average status freshness, the UAV swarm aims to achieve dynamic seamless coverage. During deployment, the UAVs strategically disperse to cover the area with minimal overlap while maintaining continuous coverage. As shown in Figure 5, the cellular structure effectively addresses this issue. The ideal minimum distance $R^{\mathrm{I}}$ between UAVs is determined by the radius $R^{\mathrm{C}}$ of their communication area, with $R^{\mathrm{I}}=\sqrt{3}{R}^{\mathrm{C}}$. The repulsive force $F_{ijt}^{\mathrm{R}}$ exerted on UAV i by UAV j at time slot t is defined as

$${\mathit{F}}_{ijt}^{\mathrm{R}}=\left\{\begin{array}{ccc}\hfill & -K^{\mathrm{R}}\left(R^{\mathrm{I}}-d_{ijt}^{\mathrm{S}}\right){\displaystyle \frac{{\mathit{d}}_{ijt}^{\mathrm{S}}}{d_{ijt}^{\mathrm{S}}}},\hfill & \hfill d_{ijt}^{\mathrm{S}}<R^{\mathrm{I}},\\ \hfill & 0,\hfill & \hfill d_{ijt}^{\mathrm{S}}\ge R^{\mathrm{I}},\end{array}\right.$$

(25)

where $K^{\mathrm{R}}$ is the coefficient of the repulsive force, derived in Appendix B. The ${\mathit{d}}_{ijt}^{\mathrm{S}}={\mathbf{u}}_{it}-{\mathbf{u}}_{jt}$ is the vector from UAV i to UAV j, $d_{ijt}^{\mathrm{S}}=\parallel {\mathit{d}}_{ijt}^{\mathrm{S}}\parallel$ is the distance between them. For UAVs in the swarm, the repulsive force is mutual, such that ${\mathit{F}}_{ijt}^{\mathrm{R}}=-{\mathit{F}}_{jit}^{\mathrm{R}}$.

From (25) we can see that, if the distance between a pair of UAVs, $d_{ijt}^{\mathrm{S}}$, is less than the ideal minimum distance $R^{\mathrm{I}}$, a repulsive force pushes them away. Conversely, if the distance exceeds $R^{\mathrm{I}}$, the repulsive force becomes zero. Similar to the total attractive force, the total repulsive force of UAV i is also normalized as

$${\mathit{F}}_{it}^{\mathrm{R}}={\displaystyle \frac{1}{|{\mathcal{U}}_{it}^{\mathrm{N}}|}}\sum _{j\in {\mathcal{U}}_{it}^{\mathrm{N}}}{\mathit{F}}_{ijt}^{\mathrm{R}}.$$

(26)

3.3. Implementation of the Trajectory Design

Given the SCIE protocol of the UAV swarm’s behaviors and the virtual force exerted on UAVs, we further explain how to use the proposed algorithm to solve the problem in (19). The concept of virtual forces has typically been used to assist the motion control of agents to accomplish a fixed task objective in previous studies, such as communication coverage in response to static communication demands [16], and obstacle avoidance with a clear destination [31]. In the proposed IoT network, the transmission requirements are time-varying and depends on the dynamics of the monitored physical processes. Additionally, the trajectory of each UAV in the swarm is self-determined, rather than being controlled by a single centralized UAV. Thereby, the trajectory design of the UAV swarm must not only account for the distances between UAVs and sensors, but also dynamically respond to the status freshness of the monitored physical processes, as well as the positions of adjacent UAVs. Based on the staged behaviors of SCIE protocol and components of virtual forces, the virtual force-based trajectory design algorithm is summarized in Figure 6. Next, we introduce the algorithm implementation stage-by-stage.

For each time slot t, the UAVs first determine their communication area and sensing area based on the current position ${\mathbf{u}}_{it}$. The sensors within these areas are denoted as ${\mathcal{M}}_{it}^{\mathrm{C}}$ and ${\mathcal{M}}_{it}^{\mathrm{S}}$, respectively. Each UAV then senses and updates the status freshness ${\Lambda }_{imt}$ for the physical processes monitored by sensors in its sensing area. Subsequently, the UAV sorts the sensors in its communication area in descending order of the updated status freshness to prioritize the transmission schedule.

Based on the sorting result, the UAV prioritizes collecting of status information with higher status freshness. Each UAV can allocate at most $T^{\mathrm{C}}$ for data collection, with each sensor uploading its status sequentially, requiring a transmission time of $T_{imt}^{\mathrm{T}}$. The UAV continues collecting status until the remaining time is insufficient to complete the transmission from the next sensor. Sensors that have successfully uploaded their status information simultaneously update their status freshness according to the calculated maximum sampling interval ${\tau }_{mt}$ and synchronize this update with the associated UAV.

After completing data collection, the UAV first identifies the positions of its neighbor UAVs and interchange known knowledge with them. Each UAV then updates own knowledge combining the shared information from neighbor UAVs and calculate the attractive force ${\mathit{F}}_{imt}^{\mathrm{A}}$ exerted by sensors in the sensing area and repulsive force ${\mathit{F}}_{ijt}^{\mathrm{R}}$ exerted by neighbor UAVs. The total virtual force ${\mathit{F}}_{it}$ then determines the movement vector ${\mathit{d}}_{it}$.

If a UAV has no neighbor UAVs and no sensors within its sensing area, i.e., ${\mathit{F}}_{it}={\mathit{d}}_{it}=\mathbf{0}$, it indicates that the UAV is disconnected from the entire network. Therefore, we assign a random movement vector to these UAVs, allowing them to explore other areas and rejoin the swarm, thereby continuing their data collection tasks collaboratively. Before the UAV actually moves, it must verify that the target position lies within the designated area. If the position exceeds this boundary, the movement vector needs to be scaled down so that the UAV maintains lie in the area during next time slot $t+1$ with a minimum distance of $R^{\mathrm{B}}={\displaystyle \frac{\sqrt{2}}{2}}{R}^{\mathrm{C}}$ from the area’s edge, as illustrated in Figure 5. After that, each UAV flies to the next hovering position ${\mathbf{u}}_{i\left(t+1\right)}$ and begins the tasks for next time slot $t+1$.

3.4. Time Complexity of the Virtual Force-Based Algorithm

The virtual force-based algorithm enables each UAV to independently design its trajectory based on local information, eliminating the need for a centralized control node to manage all UAVs. This distributed approach significantly reduces computational bottlenecks, particularly in large-scale UAV swarms. To better understand the computational efficiency, we provide a detailed analysis of the time complexity by examining its core operations.

For each UAV, the time complexity is determined by the following primary operations:

a.

Identifying Sensor and neighbor UAVs ($\mathcal{O}\left(M+U\right)$): Each UAV identifies nearby sensors within its sensing area and neighbor UAVs within the inter-swarm communication distance. This involves evaluating the distance between the UAV and each sensor or UAV, leading to a complexity of $\mathcal{O}\left(M\right)$ for M sensors, and $\mathcal{O}\left(U\right)$ for U UAVs.

b.

Updating status freshness ($\mathcal{O}\left(M\right)$): The status freshness for each sensor within the UAV’s sensing area is updated based on the latest collected status information. This involves iterating over the M sensors.

c.

Sorting sensors by status freshness $\mathcal{O}\left(M{log}_{2}M\right)$: To prioritize data collection, sensors are sorted based on their status freshness. A sorting algorithm, such as quicksort or heapsort, introduces a complexity of $\mathcal{O}\left(M{log}_{2}M\right)$ [32].

d.

Calculating repulsive and attractive forces ($\mathcal{O}\left(M+U\right)$): For attractive forces, the UAV computes interactions with sensors within its sensing area, with a complexity of $\mathcal{O}\left(M\right)$. For repulsive forces, the UAV considers the effect of neighbor UAVs within the inter-swarm communication distance, leading to an additional complexity of $\mathcal{O}\left(U\right)$.

Consider that $U<M$ in most scenarios, the total complexity for a single UAV is dominated by $\mathcal{O}\left(M{log}_{2}M\right)$, with the sensor-related operations being the primary contributors. Note that, in the calculation of complexity, we use the total number of sensors M. For a single UAV, however, only the sensors within its sensing range are relevant, and their number $|{\mathcal{M}}_{it}^{\mathrm{S}}|$ is significantly smaller than M. As a result, the actual computational complexity for each UAV is lower than the theoretical maximum complexity $\mathcal{O}\left(M{log}_{2}M\right)$.

In contrast, a centralized decision-making approach must compute the attractive and repulsive forces, update status freshness, and sort sensors for all UAVs globally. This requires calculating interactions between every pair of UAVs and sensors, leading to a significantly higher complexity $\mathcal{O}\left(U^2+M{log}_{2}M\right)$.

Leveraging the distributed advantages of the swarm-based approach, the proposed trajectory design algorithm reduces the decision dissemination process required in centralized methods, thereby avoiding additional latency that could impact the system’s real-time performance and overall efficiency in minimizing the average status freshness. Furthermore, since each UAV can independently compute its own trajectory based on local information with a low complexity of $O\left(M{log}_{2}M\right)$, the proposed method avoids the computational bottleneck caused by the quadratic term of U in centralized approaches. This makes the proposed method more scalable and better suited to accommodate an increasing number of UAVs, especially in large-scale UAV networks.

4. Simulation Results

In our simulations, we consider a square area with a side length of 600 m, in which the UAV swarm is deployed to collect status of physical processes from sensors. The UAVs keep an altitude $H=100$ m and a maximal flight speed $v=30$ m/s during movement. For implementing and verifying the proposed swarm-based algorithm, we use the Matlab tools for simulation. The status information of the physical processes is extracted from the real data of PM2.5 density [33]. Unless state otherwise, the number of UAVs is 10 and the number of sensors is 30. The primary parameters we used during the simulations are listed in

Table 3. System parameters.

Parameters	Description	Value
H	UAV altitude	100 m
$V_{it}$	UAV maximal flight speed	30 m/s
$E^{\mathrm{B}}$	UAV battery capacity	${10}^6$ J
$\alpha$	Path loss exponent	2
$\eta$	NLoS attenuation factor	0.3
$X,Y$	Environment constants	11.95, 0.136
${\sigma }^2$	Noise power	−84 dBm
$B_i$	Bandwidth	1 MHz
$P_m^{\mathrm{T}}$	Transmission power	0.2 W
$\beta$	Force loss exponent	1
$R^{\mathrm{C}}$	Radius of UAV communication area	80 m
$R^{\mathrm{S}}$	Radius of UAV sensing area	150 m
$T^{\mathrm{S}}$	Sense time	5 s
$T^{\mathrm{C}}$	Collect time	20 s
$T^{\mathrm{I}}$	Interchange time	5 s
$T^{\mathrm{E}}$	Explorer time	5 s

NLoS: non line-of-sight

. We compare the proposed method with traditional APF algorithm [16]. Next, we first validate the feasibility of the proposed swarm-based algorithm, and then analyze its efficiency in reducing the average status freshness under varying numbers of UAVs and sensors.

Figure 7 illustrates the trajectory design of the UAV swarm in response to virtual forces over different steps. The orange circles indicate the locations of sensors, where larger circle indicating the sensor with higher status freshness. The UAV swarm, represented by blue square markers, is initially placed near the center of the designated area at $t=0$, as shown in Figure 7a. The green arrows show the direction of virtual forces acting on the UAVs, with longer arrow indicating larger virtual force. At $t=2$, as shown in Figure 7b, we can see that the UAVs in the swarm are primarily influenced by the repulsive forces from neighbor UAV. This causes the UAVs farther from the center of the designated area to spread out, while those closer to the center remain in place due to the relative balance of repulsive exerted on them from all directions. As time progresses, at $t=3,5,\mathrm{and}8$, as shown in Figure 7c–e, respectively, we can see a noticeable reduction in the number of circles, as well as a decrease in their sizes, which indicates that the UAV swarm, using the proposed virtual force-based method, is effectively collecting data and reducing the status freshness of the monitored physical processes. At $t=13$, as shown in Figure 7f, we can see that while the majority of circles have either disappeared or significantly decreased in size, while most UAVs are now evenly distributed across the designated area. The virtual forces on the UAVs achieve relative balance, demonstrating the long-term effectiveness and stability of the proposed algorithm. This result highlights the ability of the UAV swarm to adapt and respond efficiently over time without reliance on a centralized controller.

Figure 8 illustrates the variation in average status freshness and average virtual force over time slots. The blue solid line represents the average status freshness, while the red dashed line indicates the average virtual force, with their corresponding values shown on the left and right vertical axes, respectively. In the initial phase of the deployment, time slots 0-5, both the average status freshness and virtual force are relatively high. This is due to two main factors. On the one hand, the UAVs are positioned close to each other, resulting in strong repulsive forces that enable the UAV swarm to quickly disperse across the designated area, thereby achieving efficient coverage. On the other hand, the status information monitored by the sensors has not yet been collected by the UAVs, leading to extremely poor status freshness and causing the UAVs to experience strong attractive forces from the sensors. In the rapid adjustment phase, time slots 5–10, both metrics experience a significant decline. The drop of average status freshness indicates that the UAV swarm quickly completes most of the status information collection tasks. Simultaneously, the virtual force reduction reflects that the UAVs have reached more stable positions and reduced excessive movement. During the steady phase, time slots 10–50, the average status freshness stabilized at a low value with minor fluctuations. This stems from the dynamic changes in the monitored physical processes, resulting in new transmission requirements. Similarly, the average virtual force oscillates within a small range. This indicates that the UAV swarm has reached a relatively stable state while still being capable of promptly responding to dynamic transmission requirements within the area.

Figure 9 illustrates the comparison of real-time status of the monitored physical processes and the estimated status. The blue solid line represents the real data of PM 2.5 density, while the green dashed line corresponds to estimated status where the traditional AoI is used as the weight of attractive forces, and the red dashed line shows estimated status using the proposed status freshness as the weight. Overall, both approaches show a close alignment with the real data for most time slots. However, in regions where the PM 2.5 density changes rapidly, as highlighted in the zoomed-in section of Figure 9, the proposed method can more accurately capture these fluctuations, while the AoI-based method shows some deviations. This suggests that status freshness reflects better changes in the monitored physical processes, thereby enhancing the prediction accuracy in dynamic environments. Using status freshness as the attractive force enables UAVs to better assess the urgency of sensor status updates. As a result, it allows for more efficient allocation of wireless resources of UAV swarm.

Figure 10 demonstrates how varying the number of sensors affects the average status freshness when the UAV swarm size is fixed at $U=10$. A box plot is used to summarize the distribution of the average status freshness over time. In the box plot, the central line represents the median value, while the edges of the box indicate the first and third quartiles. The whiskers extend to the minimum and maximum values within 1.5 times the interquartile range, and the asterisk denotes the mean value. From Figure 10 we can see that as the number of sensors increases, both the median and mean status freshness values show an upward trend. This indicates that the UAV swarm faces increasing difficulty in maintaining low status freshness with higher sensor density. This is due to the face that higher sensor density leads to increased status transmission requirements. Consequently, each individual sensor must wait for more time slots to update its status information to the UAV swarm.

In Figure 11, we show how varying the number of UAVs affects the average status freshness when the number of sensors is fixed at $M=30$. From Figure 11 we can observe that as the number of UAVs increases, both the median and mean values of the average status freshness show a downward trend. This indicates that adding more UAVs enhances the swarm’s ability to collect status information and maintain low status freshness. The improvement is due to the increased coverage and data collection capacity provided by a larger number of UAVs, which allows sensors to update their status information more frequently. Consequently, each sensor experiences shorter waiting time slots before its status information is collected by the UAV swarm, leading to a reduction in overall status freshness. As a result, the UAV swarm can better capture the dynamic changes in the monitored physical processes.

In Figure 12, we compare the average status freshness under the trajectory designed by the proposed algorithm with that of the APF algorithm. To reduce the influence of randomness, we statistically analyze the average status freshness over the first 50 time slots across 500 independent runs. From Figure 12, we can see that, for both algorithms, the average status freshness decreases as the number of UAVs increases, and slightly increases as the number of sensors grows. This steams from increasing UAVs reduces the workload per UAV, allowing each UAV to collect status information from sensors more frequently, thereby reducing the status freshness. On the other hand, increasing the number of sensors raises the overall transmission requirements, leading to longer intervals between updates for each sensor, which results in higher status freshness. However, under the current parameter settings, more than 10 UAVs essentially saturate the designated area, meaning that a further increase in UAVs has a minimal impact on improving coverage efficiency. As a result, even with a significant increase in the number of sensors, the status freshness only exhibits a slight growth, as the UAV swarm has already reached its maximum coverage potential. From Figure 12 we can also see that, under different UAV and sensor configurations, the proposed algorithm consistently achieves lower status freshness compared to the APF algorithm. This is because the proposed virtual force-based algorithm incorporates status freshness as a weight when calculating attractive forces, prioritizing the urgency of transmission requests, rather than solely focusing on the distance to sensors as in the traditional APF. Additionally, when calculating repulsive forces, the proposed algorithm ensures that UAVs can quickly spread out, thereby efficiently expanding the communication coverage of the UAV swarm and reducing communication area overlap between UAVs. These features enable the UAV swarm to respond promptly to communication requirements within the area, thereby enhancing the performance in reducing status freshness.

Table 4. Energy consumption (in Joules) per time slot as the number of UAVs varies.

Number of UAVs	APF	Proposed
5	585.2	587.3
10	587.2	588.3
15	588.2	589.3
20	588.7	589.4

presents the energy consumption (in Joules) per time slot as the number of UAVs varies. As shown in Table 4, it can be observed the energy consumption shows a slight increase as the number of UAVs grows, and both algorithms show a similar growth trend. This is because, at the specified flight speed, the difference between hovering power and flying power is negligible. As a result even though the UAVs may travel different distances under the two algorithms, the energy consumption remains nearly identical as long as the operation time is the same. Combining with the results from the previous figure, it is evident that the proposed algorithm can achieve a lower status freshness under the same energy consumption, leading to a better energy efficiency. Under the current parameter settings, an individual UAV can operate for more than 160 time slots. This helps reduce the occurrence of UAV failures due to battery depletion, thereby decreasing the frequency of UAV replacements, contributing to a more stable and resilient data collection system with enhanced adaptability and long-term operational reliability.

Figure 13 shows the effect of varying the number of UAVs on the communication overhead during the first 50 time slots under different interchange strategies, with the number of sensors fixed at $M=15\mathrm{or}30$. In the interchange stage of the SCIE protocol, each UAV shares its known information with neighbor UAVs. In our simulations, we compare this swarm-based strategy with a fully connected (FC) strategy, where updated information is sent to all UAVs in the swarm. From Figure 13 we can see that, a higher number of sensors results in an increased communication overhead for both strategies. This is because a greater number of sensors leads to more status updates that need to be shared among UAVs. Additionally, we can also see that, using the swarm-based strategy keeps the communication overhead relatively stable as the number of UAVs increases. In contrast, the FC strategy results in a significant increase as the number of UAVs grows. This is due to the fact that, under the swarm-based strategy, the amount of communication overhead each UAV consumes is only related to the number of its neighbor UAVs instead of the scale of the UAV swarm. In contrast, under the FC strategy, UAVs need to share information with all other UAVs in the swarm, which is directly affected by the number of UAVs in the swarm. While this approach enables each UAV to obtain global status freshness of monitored physical processes, it is often inefficient in practice. Sensors located far from a UAV typically do not exert significant attractive forces, indicating the UAV is unlikely to prioritize moving to distant areas to collect status. As a result, much of the communication overhead in the FC strategy is redundant. Swarm-based strategy allows for efficient transmission between UAVs with saved wireless resources.

5. Conclusions

This paper presents a decentralized trajectory design approach for UAV swarms in data collection IoT networks. By introducing the SCIE protocol, UAV actions are regulated to ensure synchronization and adaptability within the swarm. The concept of status freshness extends traditional AoI and corresponds the actual collect interval and estimation error of the UAV swarm, enabling efficient scheduling and real-time monitoring while reducing data transmission. The proposed virtual force-based algorithm dynamically adjusts UAV trajectories by combining attractive forces from sensors and repulsive forces from neighbor UAVs, achieving efficient coverage and minimized communication overlap. Simulation results demonstrate the effectiveness of the proposed method in reducing average status freshness and communication overhead, while ensuring scalability and adaptability for large-scale UAV deployments.

Despite its contributions, this study has certain limitations. For instance, the proposes method does not explicitly consider environmental effects, such as weather conditions or obstacles, which could affect UAV trajectories and communications. Furthermore, the scalability to extremely large networks with hundreds of UAVs and sensors needs further validation. Future work will address these limitations through more comprehensive simulations and real-world field tests.

In summary, this work provides a practical and efficient solution for real-time monitoring in UAV-assisted IoT networks, offering key contributions to decentralized UAV swarm control, the novel application of status freshness, and the development of scalable algorithms suitable for large-scale networks. These contributions pave the way for further research and practical applications in UAV-assisted IoT systems.