In recent years, the rapid advancement of technology has propelled the development of unmanned aerial vehicle (UAV) systems, elevating their research standards and expanding their application domains. Currently, UAVs are widely used in forest firefighting, earthquake search and rescue, military reconnaissance, and notably, in-flight refueling. However, due to their compact size, drones often have limited fuel capacity, which restricts their operational radius and endurance during missions. To address these limitations, autonomous aerial refueling (AAR) technology has emerged as a critical research focus, enabling extended flight times and ranges for drone formations. This article delves into the optimization of docking trajectories for multi-drone formation refueling, a pivotal phase in AAR that directly impacts mission success and safety. I will present a comprehensive approach that integrates high-fidelity computational fluid dynamics (CFD) modeling with an improved ant colony optimization (ACO) algorithm to ensure safe and efficient trajectory planning for drone formations during refueling operations.
The autonomous aerial refueling process for drone formations typically encompasses five stages: rendezvous, formation flying, docking, refueling, and separation. Among these, the docking phase is particularly crucial, as it involves the precise alignment of the receiver drone with the refueling drogue or boom, often in the presence of complex aerodynamic wake effects from the tanker aircraft. Ensuring safety during this phase is paramount, as the receiver drone must navigate hazardous zones characterized by vortex-induced roll moments that can compromise stability. Traditional path-planning methods often prioritize trajectory length or algorithmic efficiency but may overlook flight safety considerations specific to drone formation refueling. In this work, I focus on optimizing docking trajectories by modeling dangerous wake regions as obstacles and employing an enhanced ACO algorithm that balances path length and safety through adaptive weighting mechanisms.
The foundation of this research lies in accurately characterizing the flight environment for drone formation refueling. Using CFD simulations, I compute the roll moment coefficients experienced by a receiver drone behind a tanker aircraft across a discretized three-dimensional space. This space is divided into cubic cells, with each cell vertex assigned a coordinate and a corresponding roll moment coefficient. By comparing these coefficients to a safety threshold—set at 50% of the maximum roll damping moment coefficient that the receiver drone’s ailerons can provide—I identify hazardous points. The danger level at each point is quantified as the ratio of the roll moment coefficient to the threshold, resulting in a matrix of coordinates and risk values: $(x_d, y_d, z_d, R_d)$. This point-cloud representation serves as the obstacle map for trajectory planning, ensuring that the drone formation avoids areas where aerodynamic forces could lead to loss of control. The cell size is set to 1 meter to balance modeling accuracy and computational efficiency, capturing fine details of the wake without overwhelming the planning algorithm.
To formalize the trajectory optimization problem for drone formation refueling, I define two cost functions that reflect key objectives: minimizing trajectory length and maximizing flight safety. For a planned trajectory composed of $n$ segments, smoothed using B-spline curves, the trajectory length cost $f_L$ is the sum of the lengths $l_i$ of each segment:
$$f_L = \sum_{i=1}^{n} l_i$$
The flight safety cost $f_S$ accounts for proximity to hazardous points. Each hazard point is surrounded by a spherical collision-avoidance zone with a radius of $\frac{\sqrt{3}}{2}$ meters, which encloses all dangerous regions within the point source. If a trajectory point $(x_i, y_i, z_i)$ lies outside all collision zones, the safety cost is the sum of the danger levels $R_i$ at each point; otherwise, a large penalty constant is applied to deter entry into hazardous areas:
$$f_S = \begin{cases} \sum_{i=1}^{n} R_i, & \text{if } \forall \, \|(x_d, y_d, z_d) – (x_i, y_i, z_i)\| > \frac{\sqrt{3}}{2} \\ 1000, & \text{if } \exists \, \|(x_d, y_d, z_d) – (x_i, y_i, z_i)\| \leq \frac{\sqrt{3}}{2} \end{cases}$$
These cost functions are combined into a single objective function using adaptive weights $\alpha$ and $\beta$, which are dynamically adjusted based on the drone’s position relative to hazards and the goal. This adaptive approach allows the drone formation to prioritize safety when near dangers and shorten paths when in safe zones, enhancing overall performance for multi-drone operations.
Trajectory planning for drone formation refueling involves selecting an optimization algorithm capable of handling complex, high-dimensional search spaces. Traditional methods like A* search, rapid-exploration random trees (RRT), and artificial potential fields often suffer from slow convergence, local optima entrapment, or inefficiency in 3D environments. In contrast, swarm intelligence algorithms, such as ant colony optimization (ACO), offer robust global search capabilities and flexibility, making them suitable for multi-drone formation scenarios. The basic ACO algorithm mimics ant foraging behavior, where ants deposit pheromones on paths, and subsequent ants choose routes based on pheromone concentration and heuristic information like distance. The pheromone update rule is given by:
$$\tau_{ij}(t+1) = (1 – \rho) \tau_{ij}(t) + \Delta \tau_{ij}(t)$$
where $\rho$ is the evaporation rate, and $\Delta \tau_{ij}(t)$ is the pheromone increment from all ants on the path from node $i$ to $j$ in iteration $t$, typically set as the inverse of the fitness value. However, standard ACO converges slowly and may produce suboptimal initial solutions, which is problematic for real-time drone formation refueling. To address these issues, I propose an improved ACO (IACO) algorithm incorporating opposition-based learning and fuzzy-controlled adaptive weighting.
Opposition-based learning accelerates convergence by generating initial solutions that are closer to the optimum. In the context of drone formation trajectory planning, the initial ant population is randomly placed in the 3D environment, with positions $X = (X_1, X_2, \dots, X_m)$, where $X_i = (x_i, y_i, z_i)$. For each position, an opposite position $X’_i = (x’_i, y’_i, z’_i)$ is computed relative to a reference point, such as the tanker’s left wingtip trailing edge coordinate $(C_x, C_y, C_z)$. The opposite coordinates are derived as $x’_i = 2C_x + 50 – x_i$, $y’_i = y_i$, and $z’_i = 2C_z – z_i$, reflecting symmetry about the tanker’s plane. The fitness of both original and opposite positions is evaluated, and the superior individuals form the new initial population. This process enhances the quality of starting solutions, leading to faster pheromone accumulation and reduced computation time for drone formation planning.
Furthermore, I introduce a variable section interval strategy to improve obstacle avoidance without excessive computational load. Instead of fixed intervals between trajectory waypoints, I use smaller intervals (e.g., 5 meters) near hazardous zones and larger intervals (e.g., 10 meters) in safe regions. This adapts the planning resolution to the environment’s risk profile, ensuring precise navigation around dangers while maintaining efficiency. The adaptive weighting of cost functions is managed via a fuzzy logic controller with two inputs: the distance to the goal $d_g$ and the minimum distance to any hazard point $d_p$. The outputs are the weights $\alpha$ and $\beta$ for the trajectory length and safety costs, respectively. The input and output variables are fuzzified into linguistic sets such as “close,” “medium,” and “far,” with membership functions defined as follows:
| Variable | Linguistic Sets | Membership Function Type |
|---|---|---|
| $d_g$ | PS (close), PM (medium), PF (far) | Triangular |
| $d_p$ | PS (close), PM (medium), PF (far) | Triangular |
| $\alpha$ | PS (small), PM (medium), PF (large) | Triangular |
| $\beta$ | PS (small), PM (medium), PF (large) | Triangular |
The fuzzy rule base, comprising nine rules, dictates weight adjustments. For example, if $d_g$ is close and $d_p$ is close, $\alpha$ is set to medium and $\beta$ to medium, balancing both objectives. The combined fitness function becomes:
$$f = \alpha f_L + \beta f_S$$
where $f_L$ and $f_S$ are normalized to the range [0,1] to prevent dominance by either cost. Normalization is performed as:
$$f_{Li} = l_i / \sum_{i=1}^{n} l_i, \quad f_{Si} = \begin{cases} R_i / \sum_{i=1}^{n} R_i, & \text{if safe} \\ 1000 / n, & \text{if hazardous} \end{cases}$$
This fuzzy-adaptive mechanism ensures that the drone formation dynamically adjusts its trajectory priorities, enhancing safety during critical docking maneuvers while minimizing path length in open areas.
The improved ACO algorithm for drone formation refueling docking trajectory optimization proceeds through the following steps, illustrated in a flowchart format. First, the flight environment is modeled using CFD-derived hazard points. The ant population is initialized with opposition-based learning to generate high-quality starting solutions. During each iteration, ants select path nodes based on pheromone levels and heuristic information, with section intervals varied according to hazard proximity. After all ants complete their paths, the fitness is evaluated using the adaptive weighted cost function, and pheromones are updated accordingly. The process repeats until convergence or a maximum iteration count is reached, yielding an optimal trajectory that is subsequently smoothed with B-spline curves for flyability. This approach not only accelerates convergence but also improves solution quality for complex multi-drone formation scenarios.
To validate the performance of the IACO algorithm, I conduct benchmark tests using standard multi-objective optimization functions from the IEEE CEC 2009 suite, such as UF1, UF2, UF3, and UF7. These functions pose conflicting objectives similar to the trade-off between trajectory length and safety in drone formation planning. The results, averaged over 10 runs with 100 ants and 100 iterations, demonstrate that IACO achieves better optimal fitness values and faster convergence compared to standard ACO. For instance, on UF2, IACO reduces the average fitness from 0.403 to 0.01 and cuts convergence time to 23.08% of that required by ACO. This confirms the enhanced global search capability and efficiency of the proposed improvements, which are essential for real-time applications in drone formation refueling.
For practical evaluation, I simulate both single-drone and multi-drone formation refueling scenarios under various wind conditions. The tanker aircraft is modeled as a large transport configuration (CRM), and the receiver drones are based on the “Predator” UAV model. Flight conditions include a Mach number of 0.3, an angle of attack of 3 degrees, and ambient pressure and temperature of 70,121 Pa and 268.66 K, respectively. In the single-drone case, the receiver starts from an observation zone on the tanker’s left side, moves to a pre-docking position, and finally docks with the right-wing refueling drogue. For multi-drone formations, up to four receiver drones are arranged in a 60-degree formation with two-wing-span intervals on the tanker’s left, following NATO ATP-56(B) procedures. The drones execute docking sequentially, with collision avoidance ensured by non-intersecting trajectories or safe time separations exceeding 0.15 seconds (based on wing-span and speed ratios).
The simulation results for single-drone refueling without crosswinds show that IACO generates trajectories that closely follow safety-optimal paths near hazards and length-optimal paths in safe regions, effectively balancing both objectives. Quantitative metrics are summarized in the table below:
| Trajectory Method | Path Length Cost $f_L$ | Flight Safety Cost $f_S$ | Convergence Time Reduction |
|---|---|---|---|
| Standard ACO | 141.7624 | 21.4546 | Baseline |
| Improved ACO (IACO) | 143.6143 | 17.9958 | 12.5% faster |
| Length-Optimal (L) | 107.9865 | N/A | N/A |
| Safety-Optimal (S) | N/A | 12.7638 | N/A |
IACO increases path length by only 1.4% compared to standard ACO but improves safety by 16.12%, while also reducing convergence time. With crosswinds of 10 m/s, the hazard zones shift due to altered wake vortices, and IACO adapts by planning trajectories through safe gaps between tail vortices, further showcasing its robustness. In this case, IACO reduces path length by 15.9% and safety cost by 1.9% relative to standard ACO, with a 5.6% acceleration in convergence.
For multi-drone formation refueling, the results underscore the scalability of the IACO algorithm. Without crosswinds, the planned trajectories for two drones exhibit no collisions, with time separations at potential intersection points exceeding 0.35 seconds, well above the safety threshold. The combined costs for the formation are presented in the following table:
| Trajectory Method | Total Path Length Cost $f_L$ | Total Flight Safety Cost $f_S$ | Convergence Time Reduction |
|---|---|---|---|
| Standard ACO | 242.3359 | 36.7035 | Baseline |
| Improved ACO (IACO) | 237.7808 | 29.9589 | 18% faster |
IACO achieves a 1.9% reduction in path length and an 18.4% improvement in safety, along with an 18% faster convergence, demonstrating its efficacy for coordinated drone formation operations. Under crosswind conditions, IACO further optimizes the trajectories, reducing path length by 5.0% and safety cost by 11.4% compared to standard ACO, without introducing collision risks. These findings highlight the algorithm’s ability to handle dynamic environmental changes, which are common in real-world drone formation refueling missions.
The success of the IACO algorithm in drone formation trajectory planning can be attributed to several key factors. The opposition-based learning accelerates initial solution generation, reducing the number of iterations needed to find near-optimal paths. The adaptive weighting mechanism, driven by fuzzy logic, allows the drone formation to respond intelligently to varying risk levels, prioritizing safety when necessary without unnecessarily elongating trajectories. Additionally, the variable section interval strategy enhances obstacle avoidance precision while maintaining computational efficiency. These improvements collectively address the limitations of traditional ACO, making it suitable for complex, safety-critical applications like multi-drone formation refueling.
In terms of computational complexity, the IACO algorithm maintains a polynomial time complexity similar to standard ACO, but with reduced iteration counts due to faster convergence. The CFD-based environment modeling, while computationally intensive, is performed offline, allowing real-time trajectory planning during operation. For larger drone formations, the algorithm can be parallelized, as each ant’s path evaluation is independent, scaling well with increasing numbers of drones. This scalability is crucial for future applications involving swarms of drones in refueling scenarios, where coordination and safety are paramount.
Looking ahead, several extensions to this work could further enhance drone formation refueling capabilities. Integrating machine learning techniques, such as reinforcement learning, could enable adaptive tuning of algorithm parameters based on historical flight data. Additionally, incorporating real-time sensor feedback from the drone formation could allow dynamic updates to the hazard map, accounting for unforeseen atmospheric disturbances or tanker maneuvers. The IACO algorithm could also be extended to handle more complex constraints, such as fuel consumption minimization or communication latency in multi-drone networks. These advancements would contribute to fully autonomous, resilient refueling systems for drone formations operating in diverse environments.
In conclusion, this article presents a comprehensive framework for optimizing refueling docking trajectories in multi-drone formation scenarios. By leveraging high-fidelity CFD modeling to identify hazardous wake regions and developing an improved ant colony optimization algorithm with opposition-based learning and fuzzy-adaptive weighting, I have demonstrated significant enhancements in trajectory safety and efficiency. The simulation results confirm that the proposed IACO algorithm outperforms standard ACO in both single-drone and multi-drone formation settings, offering faster convergence, shorter paths, and higher safety margins. As drone technologies continue to evolve, such trajectory optimization methods will play a vital role in enabling safe and sustainable autonomous operations, from military refueling missions to commercial logistics. The integration of these algorithms into real-time control systems promises to unlock new possibilities for extended endurance and coordinated flight in drone formations, paving the way for broader adoption of aerial refueling in unmanned aviation.