Bridging ACO-Based Drone Logistics and Computing Continuum for Enhanced Smart City Applications

Abstract

In the context of evolving Smart Cities, the integration of drone technology and distributed computing paradigms presents significant potential for enhancing urban infrastructure and services. This paper proposes a comprehensive approach to optimizing urban delivery logistics through a cloud-based model that employs Ant Colony Optimization (ACO) for planning and Model Predictive Control (MPC) for trajectory tracking within a broader Computing Continuum framework. The proposed system addresses the Capacitated Vehicle Routing Problem (CVRP) by considering both drone capacity constraints and autonomy, using the ACO-based algorithm to efficiently assign delivery destinations while minimizing travel distances. Collision-free paths are computed by using a Visibility Graph (VG) based approach, and MPC controllers enable drones to adapt to dynamic obstacles in real time. Additionally, this work explores how clusters of drones can be deployed as edge devices within the Computing Continuum, seamlessly integrating with IoT sensors and fog computing infrastructure to support various urban applications, such as traffic management, crowd monitoring, and infrastructure inspections. This dual-architecture approach, combining the optimization capabilities of ACO with the flexible, distributed nature of the Computing Continuum, allows for scalable and efficient urban drone deployment. Simulation results validate the effectiveness of the proposed model in enhancing delivery efficiency and collision avoidance while demonstrating the potential of integrating drone technology into Smart City environments for improved data collection and real-time response.

1. Introduction

1.1. Background

In recent years, Smart Cities have increasingly relied on advanced technologies such as the Internet of Things (IoT) and Unmanned Aerial Vehicles (UAVs) to optimize infrastructure, improve the quality of life, and enhance sustainability. Among these technologies, drones have emerged as pivotal elements, serving a multitude of urban applications, from traffic management and environmental monitoring to emergency response and infrastructure inspections. Leveraging their bird’s eye view and rapid deployment capabilities, drones have become integral to the “Computing Continuum”, a paradigm that seamlessly integrates cloud, fog, and edge computing to provide real-time data processing and decision-making across distributed systems [1].

As Smart Cities evolve, there is a growing need to integrate drones into a more extensive and cohesive infrastructure that spans across various layers of computation, from cloud-based services to localized edge and fog computing. This integration is part of a broader paradigm known as the Computing Continuum [2], which enables seamless interaction between different computing resources, ensuring that data collected by drones and IoT sensors is processed efficiently and in real time, regardless of where it originates. Within this context, drones are no longer merely isolated devices but act as active participants in a distributed network capable of adapting to the dynamic and complex nature of urban environments. This capability allows for enhanced data collection, processing, and decision-making, establishing drones as indispensable assets for Smart City applications [3].

Despite the growing interest in deploying drone fleets for urban tasks, a significant challenge lies in optimizing their operations, especially for applications such as urban delivery logistics, where efficient route planning and real-time adaptability are crucial. Urban delivery presents unique challenges due to the complexity of city layouts, obstacles, varying traffic conditions, and the need to balance drone energy autonomy and payload constraints. To address these challenges, advanced algorithms and control strategies must be employed to ensure that drones can navigate efficiently, avoid collisions, and adapt to dynamic changes in their environment [4,5].

One of the fundamental aspects of UAV path planning is determining feasible and optimal flight paths, especially in the context of multiple UAVs working collaboratively [6,7]. UAVs in a formation should be considered as a collective system rather than isolated entities. This approach has proven beneficial in tasks such as surveillance, search and rescue, and package delivery, where coordinated movements enhance efficiency and coverage [8].

Several classical algorithms have been adapted and extended for UAV path planning. Graph-based algorithms, such as Dijkstra and A*, offer a structured method for finding the shortest path in a given environment, resulting in effective scenarios where UAVs must navigate through predefined waypoints while avoiding obstacles [9,10,11]. However, these methods become computationally expensive in dynamic or large-scale environments.

To address the limitations of traditional graph-based methods, alternative approaches have emerged. Rapidly exploring Random Trees (RRT) and its variations, such as RRT*, have gained popularity for their ability to generate paths efficiently in high-dimensional spaces [12,13]. However, they can lead to suboptimal solutions, becoming ineffective in scenarios requiring high precision.

Another interesting approach to UAV path planning is based on artificial potential fields, which treat UAVs and obstacles as points with attractive and repulsive forces, respectively. This method has proven effective for real-time guidance, allowing UAVs to respond dynamically to changes in the environment [14,15,16]. However, one of the main challenges with potential fields is the presence of local minima, which can trap UAVs in suboptimal positions and necessitate additional strategies for escape.

Natural optimization techniques, inspired by biological systems, have also been extensively applied to UAV path planning. Algorithms such as Ant Colony Optimization (ACO) and Particle Swarm Optimization (PSO) leverage collective behavior observed in nature to explore the search space and identify optimal paths [17,18,19,20]. These methods are particularly advantageous in complex scenarios, at the cost of a higher computational burden. In particular, over the years, several authors have demonstrated how ACO algorithms represent a valid support for the solution of the CVRP problem. The authors in [21] developed an ACO algorithm for the CVRP, showing how the ant algorithms are competitive with other metaheuristics for solving CVRP. In [22], the authors solved two most important problems in distribution logistics, such as loading of the freight into the vehicles and the successive routing of the vehicles along the road network, developing an ACO algorithm for the resolution of the latter problem. Given the NP-hard nature of the Vehicle Routing Problem with Drone (VRPD), the authors in [23] demonstrate the effectiveness of the proposed SCO algorithm to solve this kind of problem. Furthermore, to solve the problem of instant delivery for multi-UAV systems, the authors in [24] proposed two advanced ACO algorithms, whose effectiveness was proven through experimental simulations.

Optimal control and Model Predictive Control (MPC) have also emerged as powerful techniques for UAV path planning and trajectory tracking, particularly in dynamic and uncertain environments [25,26,27,28,29]. This approach is highly effective for scenarios where UAVs need to adapt to dynamic obstacles, such as other moving vehicles or pedestrians in urban settings.

When considering multi-UAV systems, cooperative path planning (CPP) becomes a critical aspect. In contrast to single-UAV path planning, CPP requires the consideration not only of individual UAV objectives but also the overall mission goals, ensuring that the entire fleet works in harmony to achieve a shared objective [30]. This requires advanced algorithms capable of managing multiple UAVs simultaneously, balancing factors such as collision avoidance, task allocation, and energy efficiency.

For instance, in cooperative target assignment and path planning, the integration of decision theory can be effective to balance individual UAV preferences with overall mission requirements. In [31], the authors proposed a combination of decision theory and Voronoi-based path planning to ensure that UAVs can navigate in formation while avoiding obstacles and optimizing their flight paths. Another interesting approach is introduced in [32], where the authors employ reinforcement learning to handle dynamic environments, allowing UAVs to adapt their routes in response to changes in real time.

For large-scale missions, clustering-based approaches can ensure that UAVs distribute workload efficiently, as proposed in [33], where the multi-UAV dynamic mission assignment is based on an improved k-means clustering algorithm to manage reconnaissance missions by grouping targets and optimizing reconnaissance sequences using the dynamic Dijkstra algorithm.

A connectivity-based prioritization method that integrates dynamic positioning and path planning [34] can enable multi-UAV systems to maintain connectivity between targets and relay UAVs, enhancing operational efficiency in large-scale disaster scenarios.

Reinforcement learning (RL) techniques have also been extensively applied to improve path planning and target assignment in multi-UAV systems. In [35], a multi-agent RL-based dynamic path planner is present, able to establish relay chains between targets and a ground base station in real time. In [36], authors demonstrated the effectiveness of a multi-agent path planning model using Deep Reinforcement Learning, which offered adaptability to different team sizes and communication constraints. The work in [37] presents a priority-aware task assignment and path planning (AMTP) algorithm based on actor–critic multi-agent reinforcement learning for heterogeneous UAV operations to manage task completion rates, energy efficiency, and load balancing. Another idea is to employ a distributed control approach to overcome communication difficulties and centralization issues in multi-UAV formations, integrating an enhanced distributed Ant Colony Optimization (ACO) algorithm with Q-Learning to handle multi-task goal assignments and allowing UAVs to adaptively respond to changing environments [38]. Another example of the use of ACO-based algorithms is in [39], combining a bidirectional ACO and discrete Honey Badger Algorithm (BACOHBA) for task assignment with a Honey Badger–Fruit Fly Algorithm (HBAFOA) for path planning.

However, the potential of multi-agent systems in the context of Smart Cities needs the integration within a robust computational infrastructure that spans cloud, fog, and edge computing [40,41,42]. This interconnected ecosystem enhances data processing and decision-making, as well as ensuring seamless coordination among UAVs, enabling real-time adaptability in dynamic urban environments [43]. Recent advancements in IoT Edge/Fog computing have played a pivotal role in this evolution [44], providing the necessary computational support to optimize UAV operations and mission execution within a broader Smart City computing continuum [45,46].

A significant advancement in IoT Edge/Fog deployments is their adaptability, which is highlighted in [47], where the authors present an enhancement of OpenStack to enable Function-as-a-Service (FaaS) on IoTs. Their approach notably decreased response times and saved bandwidth by situating computational resources nearer to data sources, improving IoT service performance, especially in urban applications such as traffic management and environmental monitoring. This adaptability perfectly matches multi-UAV systems, allowing drones to process and transmit data more efficiently, thus supporting the broader Smart City Computing Continuum.

In [48], the authors investigated infrastructure management in Italian Smart Cities, focusing on Cyber-Physical Systems (CPSs) within cooperative environments to enhance data collection and analysis efficiency. Their approach aligns with optimizing drone operations in Smart Cities, where real-time data analytics enable responsive decision-making.

Pujol et al. [49] introduced a methodology employing the Markov Blanket and Free Energy Principle to manage Compute Continuum systems. Their work diverges from traditional client–server architectures, advocating adaptive management strategies, particularly in applications like Edge Intelligence (EI). This adaptability is crucial for multi-UAV systems that must navigate and respond to dynamic environments, further emphasizing the need for intelligent path planning and data processing capabilities.

Cloud integration is essential to advance UAV autonomy and operational efficiency. The authors in [50,51] combined UAV autopilot systems with cloud computing, which markedly improved mission planning and real-time data analysis, while Luo et al. [52] developed a UAV-cloud system for disaster detection that integrated video acquisition, data scheduling, and measurement of network state. Both studies highlight the potential of cloud computing to extend UAV capabilities, enabling more effective responses in complex scenarios.

Mahmoud et al. [53] expanded this concept by presenting a UAV-cloud platform that manages distributed UAV applications using cloud services and APIs, addressing the challenges of processing power and energy resources. This work showcases the feasibility of leveraging cloud integration for efficient UAV resource management, which is critical for large-scale operations involving multiple UAVs.

The integration of cloud-based solutions has also addressed challenges associated with real-time monitoring and flight path adjustments [54], introducing a cloud-based UAV Flight Tracker to manage UAV density in urban areas with collision avoidance and dynamic path adjustments.

The combination of UAVs with edge and cloud systems has also been explored using advanced machine learning techniques. Cheng et al. [55] proposed a Federated Deep Reinforcement Learning (FDRL) framework to optimize task offloading and energy allocation in UAV operations within a mobile edge–cloud continuum. Their framework improved resource management in wireless networks, demonstrating the potential for reinforcement learning to improve UAV task efficiency and execution.

In [56], the authors examined the use of a cloud-to-edge-to-IoT continuum for search and rescue (SAR) operations, emphasizing the importance of seamless integration across different computing layers for efficient data management and decision-making in disaster scenarios, which is particularly relevant to multi-UAV systems operating in real time in complex environments.

Lastly, Zeng et al. [57] introduced the Computing in the Network (COIN) concept, which integrates in-network computation and processing within the edge–cloud continuum, supporting applications like self-driving vehicles and VR/AR. This concept aligns with the requirements of multi-UAV systems, where in-network processing can significantly reduce latency and improve operational efficiency.

Collectively, these studies underscore the importance of integrating UAVs with CC frameworks, enabling enhanced real-time data processing, communication, and mission planning by exploiting the whole infrastructure [58]. This trend aligns closely with the advancements in cooperative path planning and target assignment discussed earlier, where multi-agent systems, reinforcement learning, and optimization algorithms play critical roles in achieving efficient multi-UAV coordination [59,60].

By incorporating edge, fog, and cloud facilities, multi-UAV systems can dynamically adapt to changing environments, optimize task allocation, and improve collision avoidance, thereby enhancing their overall effectiveness in Smart City applications and beyond.

1.2. Problem Statement

This paper addresses the challenge of efficiently managing multi-UAV operations in Smart Cities. It leverages the Compute Continuum (CC) to enable scalable, real-time decision-making and resource allocation to manage the guidance of a UAV formation from the beginning throughout the whole mission. The Compute Continuum, integrating cloud, fog, and edge computing, offers a promising solution by distributing processing power, enabling real-time data analysis, and providing adaptive control mechanisms.

This paper integrates a novel architecture that exploits CC with an approach to identify, control, and rearrange the optimal mission path planning for drone logistics. Consider a formation of m UAVs with an assigned depot, which corresponds to the departure point. Each UAV has its own payload capacity for transporting goods to a destination point. Several targets are considered in a logistic mission, which must be reached by UAVs to deploy some goods. Each target has its requested capacity. The objective of the mission is to satisfy the requested capacities of every target by deploying the formation of UAVs. To achieve these goals, our solution employs state-of-the-art methodologies based on Visibility Graphs, Ant Colony Optimization, and Model Predictive Control to address key challenges in path planning, dynamic resource allocation, scalable fleet management, and collaborative decision-making. These methodologies are integrated into the CC architecture to enable an efficient and scalable deployment of UAVs in Smart City environments.

This paper investigates the following aspects of CC management for multi-UAV systems:

• Path Planning: Design path planning algorithms to find the optimal path in terms of fuel consumption to guide UAVs to targets.
• Dynamic Resource Allocation: Develop algorithms that enable the cloud to dynamically allocate computing resources, storage capacity, and communication bandwidth to UAVs based on their current needs and mission requirements.
• Scalable Fleet Management: Design a scalable cloud architecture that can efficiently manage hundreds or even thousands of UAVs, ensuring consistent performance and reliability as the fleet size grows.
• Collaborative Decision-Making: The exploitation of CC facilities enables collaborative decision-making among multiple UAVs, enabling them to coordinate their actions effectively and achieve common goals.
• Data Analytics and Insights: Leverage cloud-based data analytics tools to extract valuable insights from the vast amount of data collected by UAVs. This can include identifying patterns, predicting future trends, and optimizing operational strategies.
• Data Integration and Interoperability: Mutual data exchange among UAVs, applications, and Smart Cities is a key element involving multiple stakeholders who share technologies and data.

2. Mission Planning: Path Optimization and Target Assignment

The proposed methodology integrates a Visibility Graph-based approach to precompute feasible trajectories between mission targets, as well as the Ant Colony Optimization (ACO) algorithm to efficiently assign tasks and routes to UAVs, considering their capacities and mission constraints.

2.1. Problem Definition and Inputs

The proposed methodology assumes the following:

• A single starting depot for all UAVs, denoted by $B_0$.
• A known number, m, of UAVs, each with a specific payload capacity Q.
• A set of $N_t$ mission targets, denoted by $B_j$, with $j\in 1,2,\dots ,N_t$, each with a specific demand for goods $d_j$.
• A static environment where obstacles are known a priori.
• The CC infrastructure as defined in previous works [50,51] and briefly outlined in Section 2.2.

The goal is to find an optimal set of routes for UAVs, $\mathcal{R}=\{{\mathtt{R}}_1,{\mathtt{R}}_2,\dots ,{\mathtt{R}}_m\}$, that satisfies all target demands $d_j$, with $j\in 1,2,\dots ,N_t$, minimizes total travel distance, and respects UAV capacity constraints.

The proposed workflow (Figure 1) begins with the reception of a set of delivery requests, processed by a scheduling system, responsible to plan missions. The scheduling system executes the optimization procedure in Section 2.4 and Section 2.5. For each mission, the scheduling system considers a set of constraints that influence the assignment of drones to specific tasks. Once the mission is planned, the drone management system receives the instructions and checks the availability of drones. If there are enough drones ready, the system proceeds; otherwise, it waits for more to become available. Drones are classified as either ready or undergoing charging, and a list of available units is returned for mission execution. A communication channel is established to ensure drone connectivity and integrate them with CC facilities, including environmental data and city control utilities. Finally, drones execute their missions, and upon completion, the system logs the success and updates drone availability.

2.2. Command and Control Infrastructure Overview

The CC infrastructure is based on a modular cloud/fog computing architecture designed to enable reliable coordination and monitoring of UAV operations.

At a high level, the system is composed of three main layers:

• Cloud: It is responsible for high-level mission planning, resource allocation, and overall network orchestration. It hosts the service logic, network control agents, and communication middleware.
• Fog Node: Located closer to the UAVs, this layer provides real-time mission visualization and local decision support, integrating optimization procedures (e.g., through a Ground Control Station such as QGroundControl).
• Edge/IoT Node: Onboard nodes to manage sensor data, basic autonomy functions, and communication with upper layers.

This architecture allows UAVs to operate in a distributed and coordinated manner, ensuring low-latency control, scalability, and resilience. A simplified diagram of the infrastructure is shown in Figure 2.

2.3. Preparatory Phase for Drones Coordination Process

The drones’ coordination, as stated in [50,51], involves the entire city infrastructure of CC in different phases and with different roles. This way, it is clear that the coordination of activities, by considering the heterogeneous nature of the urban environment, assumes a hierarchical structure in which the “Cloud Application Orchestrator” and “Cloud Fog Coordinator” (running, respectively, in the cloud and the fog nodes; see [51]) play a central role in managing the drone fleet by optimizing the deployment, operation, and coordination of the drones within a Smart City environment. In more detail, its role is crucial to support the VG algorithm in its path optimization tasks.

Once the cluster is created, the ”Cloud Fog Coordinator” feeds the VG with information about known obstacles in the flight area at the chosen flight altitude before assigning missions to the drones. To achieve this goal, the ”Cloud Fog Coordinator” extracts data related to building occupancy at the flight altitude from the area map portion; this is identified as a geo-shape, commonly a rectangle surrounding the starting and arriving points. This task is performed by querying the Smart City Cloud services, which are exposing no-SQL Big Data facilities (i.e., MongoDB, which exposes native functions to support geo-shape management). In this way, the Smart City stores and provides the city building occupancy views representing the real status at different quotas; moreover, data are updated by the “Cloud Fog Coordinators” routines that collect data coming back from flying drones regarding obstacles identified. Data stored in a city’s Big Data facilities are collected in objects following the structure reported in the Listings 1 and 2.

Details about the algorithm to identify connected points in the area are in Appendix A.

Listing 1. Prototype of the points collection.

{  "_id":ObjectId,     // MongoDB auto-generated unique ID  "uuid":String,       // Unique identifier for the point  "lat":Number,         // Latitude of the point  "long":Number,       // Longitude of the point  "height":Number,   // Height of the point  "id_building":String }

Listing 2. Prototype of the buildings collection.

{  "_id":ObjectId,     // MongoDB auto-generated unique ID  "uuid":String,       // Unique identifier for the building  "heights":[{"height":Number,"points":[String]},   ...// Array of UUIDs referencing points at the same quota  ]// Array of height levels with points }

2.4. Trajectory Precomputation Using VG

The first phase involves the precomputation of feasible trajectories between all relevant positions in the environment, using a path optimization algorithm based on the VG technique, which ensures that trajectories are both collision-free and optimized for travel distance.

Path planning involves the solution of a minimization problem, whose objective function is the length of the trajectory, being the optimality linked to fuel/energy consumption. To take into account the regulations governing drone operations, it is assumed that drones fly at a predetermined altitude. This assumption allows for reducing the path planning problem to a bidimensional problem.

Consider the path planning problem between the starting point A and the target point B. Let $A=(x_A;y_A)$ and $B=(x_B;y_B)$ be the starting and target points, respectively, and ${\psi }_A$ and ${\psi }_B$ are starting and final directions.

The considered constraints are related to the presence of obstacles or no-fly zones in the operational environment.

For this purpose, in the presence of obstacles, the path planning problem can be converted into a minimum cost path search over the Visibility Graph (VG) built on the application scenario, considering the following assumptions:

• Obstacles are approximated with polygons.
• The minimum turn radius $R_{min}$ is shorter than any obstacle edge and the distance between the points A/B and any obstacle vertex.
• The starting and target heading ${\psi }_A$ and ${\psi }_B$ are locally obtained using a smoothing procedure with a negligible increase in the total path length.

Let us consider N polygonal obstacles in the environment. Each obstacle ${\mathfrak{P}}_p$ is described by an ordered sequence of $n_p$ vertices, denoted by the set ${\mathcal{I}}_p=\left\{V_1^p,V_2^p,\dots ,V_{n_p}^P\right\}$. The standard Visibility Graph $\mathbb{G}=\left(\mathcal{V},\mathcal{A}\right)$ is an undirected graph, whose node set $\mathcal{V}$ is composed of all the vertices of the obstacles, i.e., $\mathcal{V}={\bigcup }_{p=1}^N{\mathcal{I}}_p$; the arc set $\mathcal{A}$ contains the segment $\overline{V_i{V}_j}$ between the visible nodes $V_i$ and $V_j$, i.e., $\overline{V_i{V}_j}$ does not intersect any obstacle ${\mathfrak{P}}_p$, with $p\in \{1,2,\dots ,N\}$. The reduced Visibility Graph (RVG) $\overline{\mathbb{G}}=\left(\overline{\mathcal{V}},\overline{\mathcal{A}}\right)$ benefits from a reduction in the number of nodes $\overline{\mathcal{V}}\subseteq \mathcal{V}$ and feasible edges $\overline{\mathcal{A}}\subseteq \mathcal{A}$, without compromising the quality of the solution [61,62]. The node set is composed of the starting and target points A, B and the convex hull of any obstacle ${\mathfrak{P}}_p$, whereas the arc set $\overline{\mathcal{A}}$ is made of the edges of the convex hull of the obstacles and the common tangents to each couple of polygons/obstacles.

Given a point P and an obstacle ${\tilde{\mathfrak{P}}}_p$, two tangent segments can be drawn starting from P and touching a vertex of ${\tilde{\mathfrak{P}}}_p$, denoted by $\overline{V_{t_1}^{p}P}$, $\overline{V_{t_2}^{p}P}$.

Between two convex obstacles ${\tilde{\mathfrak{P}}}_p$ and ${\tilde{\mathfrak{P}}}_q$, four tangent segments can be defined, denoted by $\overline{V_{t_1^{+}}^p{V}_{t_1^{+}}^q}$, $\overline{V_{t_2^{-}}^p{V}_{t_2^{-}}^q}$, $\overline{V_{t_3^{+}}^p{V}_{t_3^{-}}^q}$, and $\overline{V_{t_4^{-}}^p{V}_{t_4^{-}}^q}$. Each tangent passes through a vertex ${\tilde{\mathfrak{P}}}_p$ and a vertex of $\tilde{\mathfrak{P}}q$. The outer tangents $\overline{V_{t_1^{+}}^p{V}_{t_1^{+}}^q}$ and $\overline{V_{t_2^{-}}^p{V}_{t_2^{-}}^q}$ do not intersect the area between the two polygons. Consequently, the vertices touched by these tangents are on the same side of the line connecting the centroids of the two polygons. The inner tangents $\overline{V_{t_3^{+}}^p{V}_{t_3^{-}}^q}$ and $\overline{V_{t_4^{-}}^p{V}_{t_4^{-}}^q}$ pass between the two polygons, touching a vertex of ${\tilde{\mathfrak{P}}}_p$ on the opposite side of the vertex touched on ${\tilde{\mathfrak{P}}}_q$.

The shortest path on RVG represents the optimal sequence of waypoints $\left(A,V_1,\dots ,V_{N_s},B\right)$ defining the flight path. A detailed description of the algorithm is provided in Appendix B.

To take into account the constraints on the minimum turn radius $R_{min}$ and on the initial and final directions, ${\psi }_A$ and ${\psi }_B$, we can build four Dubins paths [62,63]. The optimal path is the shortest one.

2.5. Ant Colony Optimization for CVRP

The Capacitated Vehicle Routing Problem (CVRP) can be defined on the graph $\mathbb{H}=(\mathcal{N},\mathcal{P})$, where $\mathcal{N}=\{B_0,B_1,\dots ,B_{N_t}\}$ is the set of nodes, representing the depot and mission targets, and $\mathcal{P}\subseteq \mathcal{N}\times \mathcal{N}$ is the set of arcs, that represent the precomputed feasible trajectories with their associated costs. In this context, $B_0$ represents the depot position, while the other nodes $\{B_1,\dots ,B_{N_t}\}$ correspond to mission targets. Each edge $(B_i,B_j)\in \mathcal{P}$ has an associated cost $c_{ij}$, representing the effort required to traverse the edge, such as distance or fuel consumption.

Additionally, each target node $B_i$ has a demand $d_i$ (with $d_0=0$ at the depot), and each UAV has a capacity Q that limits the total demand it can serve in a single route. The number of available UAVs is m. Since the number of available UAVs is m, for each vehicle $s\in \{1,2,\dots ,m\}$, the ordered sequence of visited targets must be computed ${\mathtt{R}}_s=(B_0,B_{s_1},B_{s_2},\dots ,B_{s_p},B_0)$, with $B_{s_i}\in \mathcal{N}$.

The objective of the CVRP is to determine a set of routes $\mathcal{R}=\{{\mathtt{R}}_1,{\mathtt{R}}_2,\dots ,{\mathtt{R}}_m\}$ such that the total cost L of all routes is minimized:

$$minL=\sum _{s=1}^m\sum _{(B_i,B_j)\in {\mathtt{R}}_s}{c}_{ij}.$$

(1)

The problem must satisfy the following constraints:

• The total demand served in each route ${\mathtt{R}}_s$ must not exceed the UAV capacity Q:

  $$\sum _{B_j\in {\mathtt{R}}_s}{d}_j\le Q\forall s$$

  (2)
• Each target must be visited exactly once.

To solve the CVRP, we employ the Ant Colony Optimization (ACO) algorithm, a bio-inspired metaheuristic that mimics the behavior of ants in finding optimal paths through pheromone deposition and probabilistic exploration.

The ACO algorithm operates on the graph $\mathbb{H}$, where each arc $(B_i,B_j)$ is associated with a pheromone value ${\tau }_{ij}$ that reflects the desirability of selecting that edge, and a heuristic value ${\eta }_{ij}=1/c_{ij}$, which prioritizes shorter or less costly edges. Each ant constructs a feasible solution (a set of routes) by probabilistically choosing edges based on the pheromone and heuristic values.

The algorithm initializes the pheromone levels uniformly as ${\tau }_{ij}^0={\tau }_0$, where ${\tau }_0$ is a constant. At each iteration, ants build solutions starting at the depot $B_0$. The transition probability for an ant at node $B_i$ to move to node $B_j$ is given by

$${\varrho }_{ij}={\displaystyle \frac{{\left[{\tau }_{ij}\right]}^{\alpha }{\left[{\eta }_{ij}\right]}^{\beta }}{{\sum }_{k\in {\tilde{\mathcal{N}}}_i}{\left[{\tau }_{ik}\right]}^{\alpha }{\left[{\eta }_{ik}\right]}^{\beta }}},$$

(3)

where $\alpha$ and $\beta$ are parameters that control the relative importance of the pheromone and heuristic values, and ${\tilde{\mathcal{N}}}_i$ is the set of nodes that the ant can visit next without violating capacity constraints.

Once all ants complete their routes, the algorithm evaluates the solutions and updates the pheromone levels. The pheromone update rule is

$${\tau }_{ij}=(1-\rho ){\tau }_{ij}+\mathrm{\Delta }{\tau }_{ij},$$

(4)

where $\rho$ is the pheromone evaporation rate, and $\mathrm{\Delta }{\tau }_{ij}$ is the contribution of all ants to edge $(B_i,B_j)$:

$$\mathrm{\Delta }{\tau }_{ij}=\sum _{l=1}^{N_{\mathrm{ants}}}\mathrm{\Delta }{\tau }_{ij}^l,\mathrm{\Delta }{\tau }_{ij}^l=\left\{\begin{array}{cc}{\displaystyle \frac{1}{C^l}},\hfill & \mathrm{if}\mathrm{edge}(B_i,B_j)\mathrm{is}\mathrm{in}\mathrm{the}l\mathrm{-th}\mathrm{ant}\mathrm{solution},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(5)

Here, $C^l$ is the total cost of the solution constructed by the l-th ant.

The algorithm iteratively refines the solutions through pheromone updates and probabilistic exploration until a termination criterion is met, such as a maximum number of iterations or convergence to a stable solution (see Algorithm 1).

Algorithm 1: Ant Colony Optimization for CVRP.

2.6. Model Predictive Control for UAV Trajectory Tracking

Once the mission planning is completed, the computed reference trajectories are passed to the onboard control systems of the UAVs. The onboard controllers must ensure that each UAV accurately follows its assigned trajectory while respecting the dynamic and operational constraints of the vehicle. This task is particularly challenging in scenarios involving dynamic obstacles or multiple UAVs, where collision avoidance must be guaranteed in real time. Model Predictive Control (MPC) is employed as the control strategy due to its ability to handle multi-variable systems with constraints in a predictive and optimal manner.

Model Predictive Control (MPC) is a control strategy widely used for trajectory tracking in UAVs. The MPC framework predicts the future states of the system over a finite prediction horizon $[k,k+n_p]$, where k is the current time step, and determines the optimal sequence of control inputs by solving a constrained optimization problem at each step. The first control action of the optimal sequence is applied to the system, and the process is repeated in a receding horizon manner.

At each time step k, the MPC solves the following optimization problem to control the horizontal motion:

$${\mathbf{U}}_i^{\ast }\left(k\right)=arg\underset{{\mathbf{U}}_i\left(k\right)}{min}J({\mathit{\xi }}_i\left(k\right|k),{\mathbf{U}}_i\left(k\right),k),$$

(6)

subject to

$$\begin{array}{cc}\hfill {\mathit{\xi }}_i(k+p+1|k)=\mathbf{f}({\mathit{\xi }}_i(k+p|k),{\mathbf{u}}_i(k+p|k),k),& p\in [0,n_p],\end{array}$$

(7)

$$\begin{array}{c}\hfill {\mathit{\xi }}_i\left(k\right|k)={\mathit{\xi }}_i\left(k\right),\end{array}$$

(8)

$$\begin{array}{cc}\hfill \mathit{g}({\mathit{\xi }}_i(k+p|k),{\mathbf{u}}_i(k+p|k),k)\le \mathbf{0},& p\in [0,n_p],\end{array}$$

(9)

where ${\mathbf{U}}_i\left(k\right)=[{\mathbf{u}}_i\left(k\right|k),{\mathbf{u}}_i(k+1|k),\dots ,{\mathbf{u}}_i(k+n_c|k)]$ is the sequence of control inputs over the control horizon $n_c$ (with $n_c\le n_p$). The state vector ${\mathit{\xi }}_i(k+p|k)$ is predicted using the system dynamics $\mathbf{f}$, and $J(·)$ is the cost function to be minimized. The constraints $\mathit{g}(·)$ include state and input limits as well as collision avoidance constraints.

To predict the UAV trajectory, a discrete-time linear state-space model is used:

$${\mathit{\xi }}_i(k+1)=\mathbf{\Psi }{\mathit{\xi }}_i\left(k\right)+\mathbf{W}{\mathbf{u}}_i\left(k\right),$$

(10)

where the following are used:

• ${\mathit{\xi }}_i\left(k\right)={[x_i\left(k\right),y_i\left(k\right),v_{i,x}\left(k\right),v_{i,y}\left(k\right)]}^T$ is the state vector, including position ${\mathbf{P}}_i\left(k\right)={[x_i\left(k\right),y_i\left(k\right)]}^T$ and velocity ${\mathbf{V}}_i\left(k\right)={[v_{i,x}\left(k\right),v_{i,y}\left(k\right)]}^T$,
• ${\mathbf{u}}_i\left(k\right)={[a_{i,x}\left(k\right),a_{i,y}\left(k\right)]}^T$ is the control input vector representing the accelerations,
• $\mathbf{\Psi }$ and $\mathbf{W}$ are the system matrices:

  $$\mathbf{\Psi }=\left[\begin{array}{cccc}1& 0& \mathrm{\Delta }t& 0\\ 0& 1& 0& \mathrm{\Delta }t\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],\mathbf{W}=\left[\begin{array}{cc}0& 0\\ 0& 0\\ \mathrm{\Delta }t& 0\\ 0& \mathrm{\Delta }t\end{array}\right].$$

The MPC minimizes a quadratic cost function defined as

$$J({\mathit{\xi }}_i\left(k\right|k),{\mathbf{U}}_i\left(k\right),k)=\sum _{p=0}^{n_p}{\mathbf{e}}_{{\mathit{\xi }}_i}{(k+p|k)}^T{\mathbf{Q}}_{\mathit{\xi }}{\mathbf{e}}_{{\mathit{\xi }}_i}(k+p|k)+\sum _{p=0}^{n_c}{\mathbf{u}}_i{(k+p|k)}^T{\mathbf{R}}_{\mathit{u}}{\mathbf{u}}_i(k+p|k)$$

(11)

where ${\mathbf{e}}_{{\mathit{\xi }}_i}(k+p|k)={\mathit{\xi }}_i(k+p|k)-{\mathit{\xi }}_i^{\mathrm{ref}}(k+p|k)$ is the tracking error between the predicted state and the reference trajectory. The matrices ${\mathbf{Q}}_{\mathit{\xi }}$ and ${\mathbf{R}}_{\mathit{u}}$ are positive definite weighting matrices for the state error and control effort, respectively.

The optimization problem is subject to both physical and safety constraints. These include limits on velocity, acceleration, and the requirement to avoid collisions with obstacles or other UAVs.

The UAV velocity and acceleration are bounded by physical constraints:

$$v_{i,min}\le \parallel {\mathbf{V}}_i(k+p|k)\parallel \le v_{i,max},\forall p\in [0,n_p],$$

(12)

$${\mathbf{u}}_{i,min}\le {\mathbf{u}}_i(k+p|k)\le {\mathbf{u}}_{i,max},\forall p\in [0,n_p]$$

(13)

To ensure collision avoidance, the UAV trajectory must maintain a safe distance $d_{\mathrm{safety}}$ from the other UAVs or moving obstacles, as modeled in [64].

Constraints defined in (12) describe a non-convex region. To handle this problem, the admissible regions are approximated using polytopes that can be defined through linear constraints. This approximation introduces binary variables to select the active region for the UAV trajectory, resulting in a mixed-integer formulation.

The inclusion of binary variables ${\delta }_{i,s}^v(k+p|k)$ to represent the activation of specific polytopic regions reformulates the MPC problem as a Mixed-Integer Quadratic Programming (MIQP) problem [64].

The MIQP formulation allows the MPC to handle complex, non-convex constraints while ensuring global feasibility. Efficient numerical solvers can be employed to compute the optimal control inputs in real time. According to the receding horizon approach, only the first control input ${\mathbf{u}}_i^{\ast }\left(k\right|k)$ is applied at each time step, and the process is repeated at the next step.

3. Results

In this section, the main objective is to evaluate the effectiveness of the proposed solution, integrating RVG, ACO, and MPC algorithms. An in-depth analysis of the preparatory phase for the drone coordination process performed by the CC that was discussed in previous works [50,51]. Although the considered scenario does not reflect the operational scale of high-volume systems, it serves to demonstrate the feasibility of the procedure and the scalability for localized delivery networks, where moderate numbers of drones and requests are still relevant.

Table 1. Main procedure parameters.

Parameter	Value
Number of Ants	200
Maximum number of epochs	300
Safety distance $d_{safety}$ (m)	1.00
Maximum speed norm $v_{i,max}$ (m/s)	15.00
Minimum speed norm $v_{i,min}$ (m/s)	0.00
Maximum acceleration ${\mathit{u}}_{i,max}$	$\left[5.00,5.00,2.00\right]$
Cruise speed $\parallel {\mathit{V}}_{\mathit{i}}\parallel$ (m/s)	5.00
Vertical speed (for take-off and landing) (m/s)	2.00
Cruise altitude (m)	20.00
Simulation sampling time (s)	0.10
MPC prediction horizon $n_p=n_c$	10

summarizes the main procedure parameters considered in the execution of test cases.

The scenario was tested in a simulated environment using MATLAB/Simulink R2024b suite to emulate the drones’ flight dynamics. Two test cases were considered, with different numbers of UAVs and customers to be served. In both cases, drones have a payload of 5 kg and operate in an urban area of $1200\times 1200$ m.

3.1. Test Case #1

The first test case with only six UAVs ($m=6$) and nine custumers ($N_t=9$) is considered to show the effectiveness of the proposed solution, supplying visual results also in terms of trajectories.

Table 2. Test case #1: Depot and customer data.

ID	x (m)	y (m)	Demand (kg)
$B_0$ (Depot)	500	750	0
$B_1$	50	264	4
$B_2$	14	400	3
$B_3$	53	115	2
$B_4$	682	50	1
$B_5$	700	488	2
$B_6$	940	684	3
$B_7$	800	894	4
$B_8$	440	1040	1
$B_9$	320	1039	2

reports the coordinates of the depot and targets for test case #1.

In this scenario, the RVG algorithm was executed to build the graph for the CVRP problem, as summarized in

Table 3. Test case #1: CVRP Graph $\mathbb{H}$ representing the length of RVG planned trajectories between area locations.

	${\mathit{B}}_0$	${\mathit{B}}_1$	${\mathit{B}}_2$	${\mathit{B}}_3$	${\mathit{B}}_4$	${\mathit{B}}_5$	${\mathit{B}}_6$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$
${\mathit{B}}_{\mathbf{0}}$	0.0	675.6	603.0	786.2	731.8	434.9	452.4	436.2	364.0	356.3
${\mathit{B}}_{\mathbf{1}}$	675.6	0.0	151.7	195.5	687.7	739.8	1051.5	1105.5	900.3	891.7
${\mathit{B}}_{\mathbf{2}}$	603.0	151.7	0.0	347.0	756.4	734.8	1005.8	1036.2	801.9	767.2
${\mathit{B}}_{\mathbf{3}}$	786.2	195.5	347.0	0.0	632.3	765.2	1073.9	1213.8	1039.9	1032.7
${\mathit{B}}_{\mathbf{4}}$	731.8	687.7	756.4	632.3	0.0	454.3	691.0	1008.7	1090.0	1075.2
${\mathit{B}}_{\mathbf{5}}$	434.9	739.8	734.8	765.2	454.3	0.0	313.6	583.3	718.4	780.7
${\mathit{B}}_{\mathbf{6}}$	452.4	1051.5	1005.8	1073.9	691.0	313.6	0.0	348.9	654.3	758.7
${\mathit{B}}_{\mathbf{7}}$	436.2	1105.5	1036.2	1213.8	1008.7	583.3	348.9	0.0	426.9	554.9
${\mathit{B}}_{\mathbf{8}}$	364.0	900.3	801.9	1039.9	1090.0	718.4	654.3	426.9	0.0	133.4
${\mathit{B}}_{\mathbf{9}}$	356.3	891.7	767.2	1032.7	1075.2	780.7	758.7	554.9	133.4	0.0

Minimum and maximum lengths are highlighted in bold.

.

The ACO algorithm was applied to determine the optimal UAV allocation strategy, as reported in

Table 4. Test case #1: Mission assignment and route length obtained with ACO.

UAV	Assigned Targets Number	Route Length (m)
UAV 1	${\mathtt{R}}_1=\{B_0,B_5,B_4,B_0\}$	1621
UAV 2	${\mathtt{R}}_2=\{B_0,B_8,B_9,B_0\}$	854
UAV 3	${\mathtt{R}}_3=\{B_0,B_6,B_0\}$	905
UAV 4	${\mathtt{R}}_4=\{B_0,B_7,B_0\}$	872
UAV 5	${\mathtt{R}}_5=\{B_0,B_1,B_0\}$	1351
UAV 6	${\mathtt{R}}_3=\{B_0,B_2,B_3,B_0\}$	1736

.

After RVG+ACO-based mission planning, the optimized paths were assigned to the drones and simulated, considering a cruise speed of $V_c=5$ m/s and an altitude of $h=20$ m.

Figure 3 shows a top-view of the scenario to depict the drones’ tracked trajectories to serve customers in an urban environment. The analysis highlights that every drone was able to complete its own assigned route with minimal deviation with respect to the planned path.

Table 5. Test case #1: Simulation results summary.

Parameter	Value
Maximum tracking error (m)	5.36
Average tracking error (m)	0.09
Maximum horizontal speed (m/s)	8.45
Average horizontal speed (m/s)	4.71
Minimum mutual distance (m)	1.02
Minimum planned mutual distance (m)	0.59
Minimum flight time (s)	221
Average flight time (s)	301
Maximum flight time (s)	413

resumes the results obtained through simulation. Little variation between planned and tracked paths can be highlighted, with an average tracking error of $0.09$ m and a maximum tracking error of $5.36$ m due to some intersections between planned routes, which asked drones to perform collision avoidance maneuvers. However, the minimum mutual distance is always over the assigned separation distance of one meter. The variability of flight times, between 221 and 413 s, exhibits the obvious different complexity of assigned routes, but they are always under the maximum flight time of a typical multi-rotor drone.

Figure 4 shows the minimum distance between vehicles during the flight (excluding the time in the depot) to verify the effectiveness of the proposed solution to complete each delivery without any collision. Being the departure in common between vehicles, the minimum distance constraint is disabled for vehicles in the depot. Under this assumption, the minimum distance constraint is always satisfied during the flight.

Table 6. Test case #1: Time to complete planning phases, measured on a MacBook Pro with Apple M2 chip, using a preliminary MATLAB implementation.

Planning Phase	Time to Complete (s)
RVG Graph $\overline{\mathbb{G}}$	261
CVRP Graph $\mathbb{H}$	0.15
ACO Solution	0.92

resumes the time to complete each planning phase: the building phase of RVG graph $\overline{\mathbb{G}}$, the construction of CVRP Graph $\mathbb{H}$, and the ACO solution. As depicted, with a low number of customers, the RVG graph construction phase is the most time-consuming, mainly dependent on the obstacle configuration. It is highly suitable for cloud-based parallelization, and it can be performed offline, since the resulting graph can be incrementally updated when new customers or obstacles are introduced.

Finally, Figure 5 shows the horizontal speed profiles $\parallel {\mathit{V}}_{\mathit{i}}\parallel$ of the i-th UAV, with $i=1,2\dots m$. It is worth noting that there are several acceleration and deceleration maneuvers: at the beginning and at the end of the route, at the arrival at the customers, and in the proximity of some collision avoidance situations.

3.2. Test Case #2

The second test case was considered to evaluate the scalability of the proposed procedure, taking into account $m=30$ UAVs and $N_t=99$ customers to be served. As in the previous test case, the drones have a payload of 5 kg and operate in an urban area of $1200\times 1200$ m.

The RVG algorithm was executed to build the graph for the CVRP problem.

Table 7. Test case #2: CVRP Graph summary.

Parameter	Value
Maximum Length (m)	1435
Minimum Length (m)	4.639
Maximum n. of edges	13
Minimum n. of edges	2
Average n. of edges	4.68

shows the main results for the CVRP Graph.

The ACO algorithm was applied to determine the optimal UAV allocation strategy, as reported in

Table 8. Test case #2: ACO results summary.

Parameter	Value
N. of needed UAVs	27
Overall Length (m)	36,019
Max route length (m)	2108
Min route length (m)	143
Average route length (m)	1334

.

As shown in

Table 9. Test case #2: Time to complete planning phases, measured on a MacBook Pro with Apple M2 chip, using a preliminary MATLAB implementation.

Planning Phase	Time to Plan (s)
RVG Graph $\overline{\mathbb{G}}$	549
CVRP Graph $\mathbb{H}$	35
ACO Solution	104

, the times required to complete each planning phase increase with the complexity of the scenario. It is worth noting that these results were obtained using a non-optimized MATLAB implementation running on a standard laptop (MacBook M2 Pro). However, all algorithms can be parallelized, and the use of a CC architecture to exploit distributed computing becomes essential for scalability.

After mission planning, the optimized paths were assigned to the drone and simulated, considering a cruise speed of $\parallel {\mathit{V}}_i\parallel =$ 5 m/s and an altitude of $h=20$ m.

Table 10. Test case #2: Simulation results summary.

Parameter	Value
Maximum tracking error (m)	21.37
Average tracking error (m)	0.11
Maximum horizontal speed (m/s)	11.08
Average horizontal speed (m/s)	4.52
Minimum mutual distance (m)	1.05
Minimum planned mutual distance (m)	0.01
Minimum flight time (s)	137
Average flight time (s)	367
Maximum flight time (s)	529

summarizes the results obtained in the simulation. Analogously to test case #1, little variation between planned and tracked paths can be highlighted, with an average tracking error of 0.11 m and a maximum tracking error of 21.37 m. Here, the maximum tracking error was larger than the previous and it cannot be considered small. However, the greater complexity of the scenario with many UAVs and the presence of some intersections between planned routes (highlighted by a minimum planned mutual distance of 0.01 m) forced drones to perform several collision avoidance maneuvers. However, the minimum mutual distance is always over the assigned separation distance of one meter. The variability of flight times, between 137 s and 529 s, exhibits the obvious different complexity of assigned routes, but they are always under the maximum flight time of a typical multi-rotor drone.

4. Conclusions

This paper presents a solution that exploits the Compute Continuum (CC) architecture for distributing computing modules to manage the formation of UAVs in a logistics scenario. The proposed approach is considered an efficient solution to path planning, dynamic resource allocation, scalable fleet management, and collaborative decision-making, integrating state-of-the-art methodologies in an advanced CC scheme. The numerical results, obtained through the simulation of a realistic scenario, have proved the effectiveness of the proposed solution. The test cases, involving two different compositions of UAVs, deployed in an urban scenario, highlighted the ability to optimize the distribution of the targets among the drones, minimizing the trajectories and guaranteeing the delivery of the requested goods, without collisions. Although the absence of a direct comparison with other techniques, the proposed methodology is considered among the most effective techniques for solving CVRP [21,22,23,24]. It is important to notice that the computing architecture is scalable and extensible, considering also different planning problems as the Multi-Depot Vehicle Routing Problem (MDVRP) [65], a relevant variant of VRP in large-scale scenarios. The modularity of the framework allows for the integration of several optimization algorithms, and future works can go into detail for the simultaneous integration of several algorithmic solutions to optimize the adaptability to various delivery problems.

The integration of the CC enables an advantage over traditional approaches, allowing real-time data analysis and improved computational resources management, which is fundamental to making the solution scalable. Furthermore, it enables, without any degradation in the performance of the UAV, the quasi-real-time update of the urban map realized as a parallel activity by the flight control modules on fog nodes.

Future works must address some key challenges still unexplored, such as secure communication channels and the use of different types of UAVs (ground-based, aerial fixed-wing, and rotary-wing), possibly through experimental test cases in emulated or real scenarios. Furthermore, preliminary experimental tests can be conducted considering hardware-in-the-loop simulation to effectively validate the implementation of the proposed system before deploying it to a Smart City context. Additionally, the integration of fog and edge nodes can be tested in controlled VICON-based environments to prove the effectiveness of MPC-based tracking and collision avoidance.