Decentralized control design for UAV swarms communication

Abstract

Unmanned Aerial Vehicles (UAVs) are aircraft systems that operate remotely or autonomously using on-board computers without the need for a human pilot. This paper investigates the split control design of UAV swarms utilizing the master–slave paradigm, which improves UAV operations in complicated situations, including search and rescue missions, border monitoring, and disaster response. The proposed system ensures strong communication and steady flight formations by combining genetic algorithms with A* algorithms for effective trajectory planning. We specifically designed a new ZigBee-based communications protocol to address the unique challenges associated with UAV communications within FANETs (flying ad hoc networks). We cover the decentralized architecture of the FANET framework. Finally, we test the efficiency of our protocol by integrating a Raspberry Pi 3 Model B board with the XBEE PRO S3B 915 MHz module into a DJI Phantom 3 Standard UAV. Findings showed that the UAV and ground station successfully transmitted images across different test conditions while maintaining consistent performance levels.

Article Highlights

The work uses stable formations and decentralized control to improve UAV swarm efficiency in difficult missions.

In UAV networks, a ZigBee-based communication protocol guarantees reliable data sharing under a variety of circumstances.

Optimized route planning with hybrid algorithms enhances obstacle navigation, energy efficiency, and UAV flight stability.

Introduction

UAV utilization has increased dramatically in a variety of areas, including the military, emergency response, and agriculture, because of technological advancements. This article investigates autonomous swarm formation based on the master–slave idea, which allows for different UAV formations such as matrix, linear, and circular. We employ trajectory algorithms to rectify faults in real-time movement, which improves UAV operations in complicated scenarios such as search and rescue missions, border monitoring, and disaster response[1].

We’ll talk about how to build basic components, hovering stability dimensions, movement descriptions, and quad copters. The proposed strategy depends on decentralized control algorithms, which are necessary for reliable communication and stable flying formations in UAV swarms [2].

Further, we create a unique ZigBee-based communication protocol for flying ad hoc networks (FANETs). We tested this protocol on a Raspberry Pi 3 Model B board, an XBEE PRO S3B 915 MHz module, and a DJI Phantom 3 Standard UAV. Additionally, the integration process involved ensuring that the system could handle the data processing demands of real-time image transmission and maintain stable communication links during flight. The findings showed effective picture transmission and consistent performance across a variety of circumstances.

We investigated how to keep swarm networks connected using the master–slave technique, and we developed a model for wireless route loss based on Friis and ground-ray reflections. Using these route loss models, we created the link budget, simulated changes in communication link performance at different distances, and provided reliable communication within the UAV swarm [3].

The goal of this study is to increase UAV autonomy and efficiency by developing improved control algorithms and robust communication protocols, opening the door for new applications in a variety of disciplines. We proposed a system that utilizes Flying Ad-hoc Networks (FANETs) and ZigBee technology to establish robust communication links and message-based communication strategies between UAVs and the ground control centre (GCC) for efficient data transmission and enhanced daily inspections. We developed a wireless path loss model, drawing from Friis and ground-ray reflection models and simulations, to guarantee dependable communication among the UAV swarm [4]. The study demonstrates significant coverage expansion and cost reduction. This paper is structured as follows: Section I describes the introduction to multi-UAV swarm communication. Section II discusses related works; Section III presents the current state of FANET in terms of architectures, algorithm overview, and applications; Section IV elaborates on the proposed swarm communication UAV strategy; Section V outlines the mission planner; Section VI analyses the obtained results; and Section VII concludes and suggests future direction.

Related work for swarm communication:

Recent research has looked at several communication technologies and tactics for improving UAV networks, with ZigBee appearing as a potential choice for UAV telemetry, swarm coordination, and mobile edge computing (MEC) systems. Unmanned aerial vehicles (UAVs) provide line-of-sight communication, flexibility, and 3D positioning, improving connectivity for IoT device [18]. Studies have focused on optimizing UAV-enabled relaying for MEC systems, minimizing mission completion time, and ensuring data security. UAV communication networks have explored ZigBee, a low-power consumption, cost-effective protocol, particularly in applications requiring dependable telemetry, data relay, or low-cost fly-by-sensor control.

For instance, Zhou et al. [5] effectively facilitated location estimation using a Kalman filter by incorporating ZigBee into a UAV telemetry system. Ueyama et al. [8] developed a UAV system using ZigBee with the aim of enhancing the resilience of wireless sensor networks (WSNs) during a disaster. The UAVs served as data couriers or routers to ensure connectivity during the disaster. Mykytyn et al. [10] devised a ZigBee-based telemetry system to transmit data from UAVs to ground control and the system to ensure its reliability in transmitting data packets of limited duration.

Other researchers have utilized ZigBee to emphasize the communication and coordination of UAV swarms. In order to optimize swarm performance and increase UAV-to-UAV communication during search and rescue operations, Bousbaa et al. [12] implemented geographic routing. Using DJI Phantom UAVs, Pereira et al. [5] evaluated a ZigBee-based multi-UAV communication protocol, which demonstrated dependable data exchange between UAVs and ground control in a variety of flight scenarios. These results emphasize ZigBee’s function as a resilient and adaptable protocol for applications that require minimal or nonexistent infrastructure, such as remote monitoring and disaster recovery.

This Table 1 analyses recent studies on UAV and ZigBee technology, emphasizing their objectives, technologies employed. The promise of ZigBee to provide telemetry, coordination, and job offloading in MEC systems has been validated by these research, which have recently offered important insights into UAV communication technology. They emphasize ZigBee’s adaptability in environments that lack infrastructure, such as remote IoT device management and disaster recovery [18]. The ZigBee protocol, with its low data rate, can become congested as the number of UAVs grows, potentially leading to delays and data loss. To mitigate these challenges, the system can incorporate strategies such as limiting message frequency, employing data compression, or using hierarchical clustering to manage larger swarms more effectively. This comprehensive framework establishes a solid foundation for the utilization of UAVs in future 6G networks to attain URLLC for MEC applications [18, 19].

Table 1. Summarizes the recent studies for swarm UAV communication

Author and Ref	Purpose	Technology	Key findings
Zhou et al. [5]	Telemetry information exchange	ZigBee	Reliable UAV location estimation with Kalman filter
Sineglazov and Daskal [6]	UAV navigation	IEEE 802.15.4	Feasibility of wireless network for location estimation
Bacco et al. [7]	Smart farming performance evaluation	IEEE 802.15.4	Aerial mobility restricted transmission range to one third of expected value
Ueyama et al. [8]	Disaster resilience in wireless sensor networks	ZigBee	UAVs used as routers/data mules, effective in disaster scenarios
Mushtaq et al. [9]	Fly-by sensors (FBS) control	ZigBee	Cost-effective, low power, reliable and secure operation
Mykytyn et al. [10]	Telemetry system development	ZigBee	Successful transmission of messages with a maximum length of 120 characters
Ullah et al. [11]	Hybrid network communication	IEEE 802.15.4 + WiFi	Network performance impacted by UAV movement and antenna positioning
Bousbaa et al. [12]	UAV swarm coordination for search missions	Geographic Routing	Improved UAV-to-UAV communication performance through geographic routing
Wheeb et al. [13]	Ad-hoc network for UAV swarm control	Ad-hoc network	Focus on connectivity and maximizing coverage area
Arul et al. [14]	ZigBee adapters for micro drones	ZigBee	Limited to lab environment, compatible with complex mobility patterns
Su et al. [15]	Cellular networks for UAV swarm connectivity	Cellular networks	Analysed network delay impacts on stability, validated through simulations
Ren et al. [16]	Controller based on virtual leader structure	Virtual leader	Achieved pre-defined formation in time-varying topology, using Lyapunov-based algorithms
El-Emarry et al. [17]	Improves the network connectivity with minimum energy consumption in user equipment(UE)	6G Mobile network	Improves the connectivity, To minimize the UE energy consumption during task offloading process using UAV

FANET

A FANET is a communication network that employs an ad-hoc method to support applications requiring UAVs to collaborate on a mission [21]. Ensuring the network’s resilience and efficient handling of communication interruptions is crucial. The collaborative aspect of these applications necessitates addressing specific issues to ensure accurate information exchange during the mission. Because these networks are decentralized and there is no central coordinating entity, communication coordination functions play an important role in this scenario.

A. Architecture

FANET designs usually have two basic communication modes: UAV-UAV (U2U) and UAV-infrastructure (U2I). These designs may be classified as centralized UAV swarm communication strategy and decentralized UAV communication systems [ 21& 22]. We can classify the architecture of centralized UAV networks into three types: star networks, multi-star networks, and direct networks. The ground station channels all messages between UAVs through its star-connected non-terminal links. The ground station coordinates with all the UAVs to establish direct links. This is a basic and easy-to-implement structure that is well-suited for small-scale use, but it is very latent and includes only one Ground control point. This is shown in a Fig. 1a centralized UAV network directly connected to the ground control point. The multi-star network is a more advanced version of said networks, in which groups of UAVs connect to a master UAV that communicates with the group’s ground control station. In Fig. 1b represents the decentralized UAV communication network which improves system redundancy and flexibility, it complicates hierarchy even more. The directly connected architecture on the other hand uses just one GCC for connectivity that results into delays because there is no direct UAV-UAV connection.

Fig. 1 [Images not available. See PDF.]

a Centralized UAV swarm communication networks, b Decentralized UAV swarm communication networks

Swarm Interaction among drones is modelled based on separation, alignment, and cohesion principles. To maintain the safe distance between the UAV consider the Eq. 1 for separation.

1

$$F_s=k_s\sum _{j\in N_i}\frac{(x_i-x_j)}{{\Vert x_i-x_j\Vert }^2}$$where $k_s$ is a separation constant $x_i$ and $x_j$ are position of the drone I and j and $N_i$ represents the neighbouring UAV. Alignment of each UAV’s velocity with its neighbour

2

$$F_a=k_a\sum _{j\in N_i}(v_i-v_j)$$where $k_a$ is an alignment constant and $v_{i}and{v}_j$ are velocity vector of UAV.

3

$$F_c=k_c(x_c-x_i)$$where $k_c$ is a cohesion constant and $x_c$ is a centroid position of the swarm These equations ensure stable interaction and coordination among the UAV during adaptive environment changes. Our proposed FANET architecture employs a master–slave approach to enhance coordination and communication efficiency. The master UAV oversees the swarm, regulating data transmission and guiding the formation. The centralized control improves coordination; however, it creates a single point of failure should the master UAV experience a malfunction. We propose a resilience strategy for implementations in which, upon the failure of the master UAV, another UAV will autonomously assume the role of master. For cluster formation, we first determine the optimal number of clusters, taking into account the Euclidean distance between each UAV in relation to the centroid number. We then calculate the centroid by averaging each cluster, and repeat this process until the centroid remains unchanged.

4

$$N_c=\frac{1}{\Vert S_i\Vert }\sum _{U_i\in S_i}{U}_i$$

The set of all UAVs is assigned to a cluster. A k-means clustering algorithm will optimize the selection of this backup UAV based on criteria such as energy levels, distance to the ground control station, and coverage capability [23]. This decentralized fallback mechanism makes sure that the system keeps running by letting a different UAV take over. This enhances the system’s reliability while preserving the benefits of the master–slave coordination structure.

B. Algorithm overview

Mission planner software, as suggested in the protocol, will generate a swarm and, from there, determine possible flight paths for the master UAV, whereby there is stability in keeping a formation when it routes through mission points [20]. The formation of flying follows a master–slave model, so once all drones reach target locations, this singular drone synchronizes them into a team by making them follow its course. The master mission provides modified waypoint coordinates to the slaves for use during take-off. In between going to one waypoint and another, UAVs all over wait until they get their next destination point from the main drone. This entire flock flies at the same speed; hence, remaining in a stable flight formation is easy. Coordinating signals from the master UAV direct its slaves, who respond with acknowledgements before setting waypoints where data can be collected about different objects in their environment. Different types of swarming, like linear, matrix, or circle, are available through tools within the mission planner that have unique features and varying trade-offs, such as communication reliability versus coverage area. Linear maintains equidistant spacing between UAVs, while matrix has them arranged in square grids, while circular places slaves around masters who stay at their centre. More waypoints slow down aircraft motion, resulting in synchronization issues caused by the increased complexity of missions.

The algorithm flowchart begins by initializing the UAV mission and reading the GPS coordinates of the waypoints from the mission planning software. The system configures the communication ports to ensure the proper transmission of data between the UAVs. The system then determines if the UAVs are armed and ready; if not, it resumes the start-up procedure. Once equipped, the system selects the appropriate flight mode for the environment and launches the UAVs from their base. The UAVs fly past each waypoint in order, maintaining their flight formation. When the UAVs reach the final waypoint, they return to the launch point (RTL) and complete the mission by turning down the equipment. The algorithm assures that UAVs can fly independently, preserving formation and completing mission goals with minimum human interaction. The flow chart of UAV mission algorithm is shown in Fig. 2. For the stability and control of UAV is important to movement within swarm. We employ the Proportional-Integral-Derivative (PID) control algorithm to adjust rotor speeds, achieving stable hover and flight. Altitude is control by the equation$$F_z={m.g+k}_p\left(z_d,-,z\right)+k_d\frac{d\left(z_d,-,z\right)}{\mathit{dt}}$$$$\tau =k_p\left({\theta }_d,-,\theta \right)+k_i\int \left({\theta }_d,-,\theta \right)dt+k_d\frac{d\left({\theta }_d,-,\theta \right)}{\mathit{dt}}$$where $k_p,k_{i}and{k}_d$ are the PID constant; z is the altitude and $\theta$ represents the orientation of UAV. It navigates predefined waypoints using GPS and an A* pathfinding algorithm, adjusting dynamically in response to obstacles.

Fig. 2 [Images not available. See PDF.]

Flowchart for the proposed UAV mission algorithm

Proposed communication strategy

To ensure the safe and efficient execution of UAV operations, a comprehensive communication plan is necessary. To efficiently manage UAV operations, this method calls for the use of multiple ground control systems (GCS) and communication protocols. Key components include MA Proxy, a lightweight and extendable ground control system for UAVs that supports the MAV Link protocol; Mission Planner, a Windows-compatible ground control station for configuring and dynamically controlling UAVs; APM Planner 2.0, an open-source ground station framework for MAV Link-based autopilots that runs on multiple operating systems; and Q Ground Control, a GCS that automatically detects as well as attaches to vehicles, providing. Figure 3 shows that view of mission planner from Google satellite and Fig. 4 show the creation of waypoint navigation.

Fig. 3 [Images not available. See PDF.]

Google satellite view of mission planner

Fig. 4 [Images not available. See PDF.]

Creation waypoint navigation

The trajectory planning algorithm is a combination of genetic and A* algorithms, including route planning and formation control modules. The genetic algorithm optimizes UAV flight trajectories by emulating natural selection process. It determines the most efficient paths using criteria like distance, energy consumption, and obstacle avoidance. Through repeated selection, crossover, and mutation processes, the algorithm converges on optimum pathways that balance these characteristics, allowing for successful long-term swarm planning. Real-time pathfinding uses the A* algorithm, allowing UAVs to dynamically adjust their trajectories in response to environmental changes and obstacle presence. The A* algorithm calculates the shortest feasible path using heuristic cost, and entails determining the geometric route in the operational space and planning the trajectory in the joint space, taking into account kinematic and dynamic constraints. These algorithms enhance the UAV swarm’s efficiency by providing both optimized paths for long-distance travel and responsive adjustments for real-time navigation. This hybrid approach allows the system to adapt effectively to varying mission conditions. The formation control module creates unique pathways for every UAV to maintain formation during flight and avoid obstacles. The trajectory-planning algorithm and the desired formation shape provide the reference path for this task, its different path trajectory planning is represented in Fig. 5. Setting the home position at the UAV’s arming location or using a rally point as an alternative return point, entering waypoints and commands through a point-and-click interface on Google Maps, and managing altitude relative to the launch position, with Google Earth topology data adjusting the desired altitude to reflect ground height, are all part of the mission planning process. After writing the mission to the autopilot and storing it in EEPROM, you can check it by reading it back. Figure 6 shows that Mission Planner for UAV swarm communication.

Fig. 5 [Images not available. See PDF.]

Different path trajectory planning a Triangle path, b Linear path, c Circle path

Fig. 6 [Images not available. See PDF.]

Copter mission planner for UAV swarm communication

Mission Planner uses the Flight Data panel to test the firmware and execute simulations to confirm algorithm effectiveness, making testing and validation crucial. Safety and performance considerations include ensuring that trajectory planning does not generate incompatible forces and torques; avoiding non-smooth trajectories that can cause vibrations and damage actuators; and executing low-jerk trajectories for greater accuracy and speed. This complete communication approach incorporates ground control systems and protocols to aid in UAV trajectory planning and mission execution, ensuring safe and efficient operations via rigorous testing and validation procedures.

Simulation setup and test methodology

The UAV trajectory planning and control system’s test methodology takes a multi-phase approach to verify both security and effectiveness before real flights begin. We first perform simulations using Python and Simulink to evaluate and authenticate the effectiveness of the suggested algorithms. For instance, we divide the firefighting mission into two phases: in phase one, the UAVs establish the necessary formation geometry (like a triangle or parallelogram), and in phase two, they adhere to the ground station’s prescribed trajectory through the mission planner. The UAV’s control system tackles the trajectory planning issue by creating reference inputs that ensure the intended motion occurs within its kinematic and dynamic restrictions. We then carry out the proposed trajectory in the joint space, considering kinematic and dynamic constraints to minimize mechanical resonance modes and reduce vibration. We also use MAV Proxy and other ground control systems during testing to verify the proper configuration and operation of the UAVs. This involves establishing waypoints, designating home locations, and ensuring proper return-to-launch (RTL) procedures. Mission Planner, the ground control software, allows for mission planning, waypoint input and real-time monitoring, ensuring that the UAVs carry out their intended tasks correctly. This rigorous testing technique assures that the UAVs function safely, efficiently, and effectively, with all possible hazards addressed prior to real deployment. Figure 7 illustrate the understanding altitude in UAV.

Fig. 7 [Images not available. See PDF.]

Understanding altitude in Ardupilot

We built a simulation setup using Python and Simulink to evaluate the UAV trajectory planning and communication techniques. The simulation environment replicates real-world settings, ensuring the safety and usefulness of the suggested algorithms prior to actual flights. We divide the UAVs’ mission into two phases: formation control and trajectory following.

In Phase 1, UAVs combine to produce the required formation geometry (such as a triangle or parallelogram). In Phase 2, the base station uses a mission planner to set a trajectory for the UAVs to follow. The trajectory planning challenge entails creating reference inputs for the UAV control system and ensuring that the intended motion is achieved while conforming to the UAV’s kinematic and dynamic restrictions. Table 2 shows the components of Simulation setup.

Table 2. Components and simulation setup specification values

Components	Specification
Motor	960 kv
Propeller	9 inch
Battery	5200 mAh (4S)
Battery voltage	16.8 V
Battery energy	314,496 J
Mass of UAV	1.66 kg
Thrust for take-off	2.071 kg
Pitch angle	5°
Vehicle velocity	3 m/s
Battery efficiency	70%

This Fig. 8 shows a new way to formulate and solve the three-dimensional route planning problem using particle swarm optimization. The unmanned aerial vehicles’ primary design point is reflective of their original trajectory. Particle swarm optimization generates alternative paths. Three specified objectives guide the optimization of B-spline curves. The Table 3 represents the path, distance and time taken for the number of waypoints.

Fig. 8 [Images not available. See PDF.]

Path planning without drone optimization

Table 3. Path planning for no of waypoints

No of waypoints	Distance(m)	Time (s)	Path
0	239	150	0–1-2–3-4–5-6–7-8–0
1	227	146	0–4-6–7-3–2-1–5-8–0
2	225	145	0–1-2–4-6–7-3–5-8–0
3	210	140	0–2-3–7-6–4-5–1-8–0
4	209	139	0–1-4–5-2–3-6–7-8–0
5	207	139	0–1-2–7-3–6-4–5-8–0
6	196	135	0–1-2–3-7–6-4–5-8–0

Maximum altitude: 30 m; Operation area: 200 sq.; Distance travelled: 239 m

Results and discussion

The statistics show the energy consumption and performance parameters of a UAV under various situations and improvements.

During the UAV-to-ground station communication testing, we found various restrictions and difficulties. Notably, the UAVs’ short flight length and autonomy limited the frequency of testing and travel distances. Furthermore, we detected differences in communication performance across diverse environmental situations. For example, interference from barriers and changes in the UAV’s height caused discrepancies in data transmission stability and success rates. These criteria highlight areas for improvement in the procedure, especially in adapting to environmental fluctuations. Future system iterations will look at solutions like signal enhancement and autonomous altitude changes to help reduce these issues and increase overall communication dependability. The Fig. 9 shows the energy consumption for flying and hovering modes as a function of the number of ants, with flying energy gradually decreasing from roughly 15,000 J to 13,000 J and hovering energy remaining constant at around 7,500 J. This shows that increasing the number of ants decreases the amount of energy needed for flight, potentially owing to improved flight pathways or aerodynamic efficiency. Table 4 shows the packet delivery ratio and energy consumption of different UAV nodes.

Fig. 9 [Images not available. See PDF.]

Theoretical energy consumed during various light modes

Table 4. Different nodes of PDR and energy consumption (UAV)

Parameters	Node 1	Node 2	Node 3	Node 4	Average
Packet delivery ratio	95	91	92	96	93.5
Energy consumption (in J)	0.1	0.16	0.2	0.21	0.1675

The Figs. 10 and 11 shows the current and voltage with time. Figure 10 represents the current consumption before the path optimization, Fig. 11 shows the current consumption after the path optimization The green line represents voltage and the red line represents current, which fluctuates but averages about 8A. After the path optimization the current and voltage fluctuation is reduced with in time seconds these observations show a constant voltage supply with slight fluctuations in current use, emphasizing the UAV’s consistent power needs throughout operation.

Fig. 10 [Images not available. See PDF.]

Current consumption for before optimization path

Fig. 11 [Images not available. See PDF.]

Current consumption for optimized path

The Fig. 12 shows overall energy use before and after optimization across time. Both lines begin at 0 kJ and climb linearly, however the optimized instance (orange line) increases at a slower pace, reaching around 21 kJ after 200 s against 23 kJ in the non-optimized example (blue line). Table 5 illustrates that the optimization successfully decreases total energy usage, hence increasing the UAV’s energy efficiency.

Fig. 12 [Images not available. See PDF.]

Before optimization vs after optimization—energy consumed by UAV

Table 5. Comparison of parameters before and after optimization

Parameter	Before optimization	After optimization
Current (A)	8	8(average, fewer spikes)
Voltage (V)	16.5	16.5(with reduced fluctuation)
Energy consumption (kJ)	23	21
Efficiency improvement	–	− 8.7% reduction in energy

Conclusion and future work

In a FANET framework, this research successfully established a master–slave control design ZigBee-based communication protocol for multi-UAV systems. The system exhibited robust picture transmission between the UAV and the ground control centre (GCC) under diverse operating conditions, as shown by a sequence of tests. In the first scenario, which included a stationary UAV, picture decoding achieved a perfect success rate of 100%. The success percentages declined to 90% and 80% in scenarios two and three, respectively, as the tests included dynamic mobility and increased complexity. The ultimate scenario, which emphasized the execution of multi-UAV delivery, resulted in a success rate of 75%. Overall, the communication protocol demonstrated stability, as evidenced by successful picture transmission in 42 out of 50 sessions, resulting in an 84% success rate. Furthermore, there were no significant differences in transmission duration across various situations. However, several limitations were noted, including the restricted flight duration and autonomy of the UAV, which limited the frequency of tests and travel distances.

Future research will go beyond communication protocols to enhance UAV autonomy in complex situations. This research will incorporate sophisticated features such as obstacle avoidance, target tracking, and dynamic mission adaptation, while also evaluating network behaviour such as latency and throughput. UAVs can carry out more autonomous activities, especially in difficult areas, by combining machine learning with computer vision for course planning and obstacle identification. Additionally, we will investigate hybrid communication protocols that combine ZigBee with technologies like LoRa (long range) to facilitate data-intensive jobs and improve scalability and adaptability in multi-UAV swarm operations.

Author contributions

1 and 2,3 author contribute in conceptualization, writing, simulation result analysis, author contribute simulation result and result analysis, supervision, author 4 and 5 contribute in supervision, revision, I would like to submit our manuscript titled “Decentralized Control Design for UAV Swarms Communication.” for publication in your esteemed journal. The paper provides control design for UAV swarms Communication with FANET and improves the effective data transmission, coverage. The analysis includes simulation results to validate the effectiveness of these techniques in enhancing system performance. We applied to this journal through the organizing committee of Energy Summit 2024. We declare that this manuscript is original, has not been published before, and is not currently being considered for publication elsewhere. As the corresponding author I confirm that the manuscript has been read and approved for submission by all the named authors. We know of no conflicts of interest associated with this publication and there has been no significant financial support for the work that could have influence its outcomes.

Funding

No Funding.

Data availability

This study did not involve the use of any external datasets. All findings are derived from theoretical computations and simulations, which are fully detailed in the manuscript.

Declarations

Ethics approval and consent to participate

Not Applicable.

Competing interests

The authors declare no competing interests.

Publisher's Note

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

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