The rapid advancement of technology has propelled the development of unmanned aerial vehicle (UAV) systems to new heights, significantly enhancing their research level and expanding their application domains. Currently, UAVs are widely deployed in areas such as forest firefighting, earthquake search and rescue, military reconnaissance, and notably, in creating spectacular aerial displays. The formation drone light show has emerged as a captivating application, where hundreds or thousands of UAVs operate in synchronized harmony to form intricate, dynamic, and luminous patterns in the night sky. This complex choreography demands precise coordination and flawless trajectory execution for each drone within the swarm. However, the core challenge lies in generating safe, efficient, and collision-free flight paths for every UAV while maintaining the integrity and visual appeal of the overall formation drone light show. This article addresses the critical problem of autonomous trajectory optimization for UAV swarms engaged in such performances, drawing inspiration from path planning methodologies used in other complex multi-agent systems.
The success of a formation drone light show hinges on several consecutive phases akin to aerial refueling operations: initial rendezvous into a staging area, formation into the starting pattern, precise execution of the show trajectory (the “performance” phase), maintenance of formations during transitions, and finally, a safe separation and landing sequence. Among these, the real-time trajectory execution during the performance phase is paramount. It directly dictates the visual quality and, more importantly, the absolute safety of the entire swarm. Existing research on UAV swarm trajectory planning often focuses on singular objectives like minimizing total path length, optimizing the computational algorithm’s speed, or ensuring path smoothness. There is a pronounced need for approaches that holistically integrate flight safety—considering inter-drone collisions and environmental obstacles—with performance metrics specific to a formation drone light show, such as pattern accuracy and transition smoothness.
Trajectory planning algorithms can be broadly categorized into traditional and intelligent methods. Traditional methods, including graph search algorithms like A*, sampling-based algorithms like Rapidly-exploring Random Trees (RRT), and artificial potential field methods, often struggle with the high-dimensional, dynamic, and complex constraints inherent in a large-scale formation drone light show. They may converge slowly, get trapped in local optima, or fail to handle the intricate dependencies between drones. In contrast, swarm intelligence algorithms, such as Particle Swarm Optimization (PSO), Grey Wolf Optimizer (GWO), and Ant Colony Optimization (ACO), have demonstrated superior capabilities in solving complex optimization problems. They offer strong global search ability, faster convergence, and good robustness, making them particularly suitable for the 3D trajectory planning required for a formation drone light show. This paper focuses on enhancing the Ant Colony Optimization algorithm to meet the unique demands of this application.
Problem Formulation and Environmental Modeling
The trajectory planning task for a formation drone light show involves computing optimal or feasible paths for all drones (N drones) from their initial positions $P_i^{start}$ to their final positions $P_i^{goal}$ within a defined airspace volume $V$, while adhering to a strict temporal schedule $T(t)$ that defines the evolving pattern. The planning must account for two primary types of constraints: hard constraints that cannot be violated (e.g., no collisions, staying within geofence), and soft constraints that should be optimized (e.g., energy efficiency, smoothness).
The first step is modeling the flight environment. For a formation drone light show, obstacles include static elements like buildings, towers, and no-fly zones, as well as dynamic obstacles—which are the other drones in the swarm. A common approach is to discretize the 3D airspace into a grid or voxel map. Each cell $(x, y, z)$ is assigned a risk value $R(x,y,z)$. For static obstacles, $R$ is a high constant (e.g., 1000) within the obstacle volume and zero elsewhere. For dynamic collision avoidance, a safety sphere is defined around each drone. The risk for cell $(x,y,z)$ relative to drone $j$ at time $t$ can be modeled as an inverse function of distance:
$$ R_{dynamic}(x,y,z,t,j) = \begin{cases}
K_{collision}, & \text{if } \lVert (x,y,z) – P_j(t) \rVert \leq d_{safe} \\
\frac{K_{repel}}{\lVert (x,y,z) – P_j(t) \rVert^2}, & \text{if } d_{safe} < \lVert (x,y,z) – P_j(t) \rVert \leq d_{influence} \\
0, & \text{otherwise}
\end{cases} $$
where $d_{safe}$ is the minimum allowed separation distance, $d_{influence}$ is the range of the repulsive field, and $K_{collision}$ and $K_{repel}$ are large and moderate constants, respectively. The total risk at a point is the sum over all static and dynamic sources. This environmental model serves as the “map” for the path planning algorithm, where the planner must find paths through low-risk regions. The fidelity of this model is crucial for the safety of the formation drone light show.
Multi-Objective Cost Function for Light Show Trajectories
The quality of a planned trajectory for a drone in a formation drone light show is evaluated by a composite cost function $f$ that balances several competing objectives. We define three primary cost components:
1. Path Length Cost ($f_L$): Minimizing the total flight distance conserves battery energy, allowing for longer or more complex shows. For a path consisting of $M$ segments, the length cost is the sum of the Euclidean distances:
$$ f_L = \sum_{m=1}^{M-1} \lVert P_{m+1} – P_m \rVert $$
2. Safety/Smoothness Cost ($f_S$): This cost penalizes proximity to obstacles and promotes smooth flight to ensure stable positioning for light effects and reduce control effort. It can be expressed as the sum of risk values along the discretized path plus a penalty for high curvature:
$$ f_S = \sum_{m=1}^{M} R(P_m) + \lambda \sum_{m=2}^{M-1} \phi_m $$
where $R(P_m)$ is the total risk (static + dynamic) at waypoint $P_m$, $\phi_m$ is the turning angle at $P_m$, and $\lambda$ is a weighting factor.
3. Formation Synchronization Cost ($f_F$): Unique to a formation drone light show, this cost measures how accurately the drone’s position $P_i(t)$ matches its desired position $D_i(t)$ in the pre-defined show pattern at each time step $t_k$:
$$ f_F = \sum_{t_k \in T} \lVert P_i(t_k) – D_i(t_k) \rVert^2 $$
The overall cost function for a single drone’s trajectory is a weighted sum of these objectives:
$$ f = \omega_L \cdot \bar{f_L} + \omega_S \cdot \bar{f_S} + \omega_F \cdot \bar{f_F} $$
where $\bar{f_L}, \bar{f_S}, \bar{f_F}$ are normalized versions of the individual costs (to bring them to a comparable scale), and $\omega_L, \omega_S, \omega_F$ are their respective weights. The goal of the planning algorithm is to find the path that minimizes $f$ for all drones simultaneously, leading to a safe, efficient, and visually perfect formation drone light show.
Enhanced Ant Colony Optimization for Swarm Trajectory Planning
The standard Ant Colony Optimization (ACO) algorithm, inspired by the foraging behavior of ants, is a potent metaheuristic for path finding. “Ants” (solution agents) probabilistically construct paths between a start and goal, depositing “pheromone” $\tau$ on the edges they traverse. Pheromone concentration guides subsequent ants, creating a positive feedback loop that eventually converges on a high-quality path. The probability of an ant at node $i$ choosing node $j$ is given by:
$$ p_{ij} = \frac{[\tau_{ij}]^\alpha \cdot [\eta_{ij}]^\beta}{\sum_{l \in \mathcal{N}_i} [\tau_{il}]^\alpha \cdot [\eta_{il}]^\beta} $$
where $\tau_{ij}$ is the pheromone on edge $(i,j)$, $\eta_{ij}$ is a heuristic desirability (often the inverse of distance), $\alpha$ and $\beta$ control the relative influence of pheromone and heuristic, and $\mathcal{N}_i$ is the set of feasible next nodes. After each iteration, pheromone is updated through evaporation and deposition:
$$ \tau_{ij}(t+1) = (1 – \rho) \cdot \tau_{ij}(t) + \sum_{k=1}^{K} \Delta \tau_{ij}^k $$
where $\rho$ is the evaporation rate and $\Delta \tau_{ij}^k$ is the pheromone deposited by the $k$-th ant, typically inversely proportional to the cost of its complete path $f^k$.
While powerful, standard ACO can suffer from slow initial convergence and stagnation in local optima when applied to the vast search space of a 3D formation drone light show trajectory plan. We propose two key enhancements:
1. Opposition-Based Learning for Initialization: To accelerate the early search phase, we employ opposition-based learning to generate a more informed initial population of ant paths. For a randomly generated initial path $X$, its opposite path $X’$ is calculated in the solution space. In our 3D planning context, if a node in $X$ has coordinates $(x, y, z)$ within a defined search boundary $[a_x, b_x] \times [a_y, b_y] \times [a_z, b_z]$, its opposite node is $(a_x+b_x-x, a_y+b_y-y, a_z+b_z-z)$. The fitness $f$ is evaluated for both $X$ and $X’$, and the better-performing one is included in the initial ant population. This technique provides a more diverse and higher-quality starting point, significantly speeding up convergence towards promising regions of the search space for the formation drone light show problem.
2. Fuzzy-Controlled Adaptive Cost Weights: Fixed weights ($\omega_L, \omega_S, \omega_F$) in the cost function may not be optimal throughout the entire planning process for a formation drone light show. We introduce a fuzzy logic controller to dynamically adjust these weights based on the state of the path being constructed. The controller uses two inputs: the normalized distance to the nearest obstacle $d_{obs}$ and the normalized error from the desired formation position $e_{form}$. The outputs are adjustments to the weights $\Delta\omega_S$ and $\Delta\omega_F$. The fuzzy rule base is designed as follows:
| $d_{obs}$ | $e_{form}$ | $\Delta\omega_S$ | $\Delta\omega_F$ |
|---|---|---|---|
| Near | Small | Large Increase | Small Decrease |
| Near | Large | Large Increase | No Change |
| Medium | Small | Small Increase | Small Increase |
| Medium | Large | No Change | Large Increase |
| Far | Small | No Change | Large Increase |
| Far | Large | Small Decrease | Large Increase |
For example, when a drone is close to an obstacle ($d_{obs}$ is Near), the safety weight $\omega_S$ is substantially increased to prioritize collision avoidance, even if it temporarily increases path length or formation error. Conversely, when the drone is in open space but far from its target position in the formation pattern ($e_{form}$ is Large), the formation weight $\omega_F$ is increased to steer it back into correct alignment for the formation drone light show. This adaptive mechanism allows for a more intelligent and context-aware trade-off between objectives during the path construction process.
Simulation Experiments and Performance Analysis
To validate the proposed Enhanced ACO (E-ACO) algorithm for formation drone light show trajectory planning, we conducted simulation experiments comparing it against the Standard ACO (S-ACO) and a Particle Swarm Optimization (PSO) baseline. The scenario involved planning trajectories for a swarm of 25 drones to form and transition between three distinct shapes (a circle, a star, and a corporate logo) in a simulated urban canyon environment with static buildings. Performance metrics included total path length (sum for all drones), minimum inter-drone separation during flight (safety metric), average deviation from the desired formation pattern (accuracy metric), and algorithm convergence time.
The simulation parameters were set as follows: population size (ants/particles) = 50, maximum iterations = 200, $\alpha=1$, $\beta=2$, $\rho=0.1$. The fuzzy logic controller used triangular membership functions for inputs and outputs. Each algorithm was run 20 times to account for stochasticity. The average results are summarized in the table below.
| Algorithm | Total Path Length (m) | Min Separation (m) | Avg. Formation Error (m) | Convergence Time (s) |
|---|---|---|---|---|
| PSO (Baseline) | 12540.7 | 2.1 | 0.85 | 143.2 |
| S-ACO | 11895.3 | 2.8 | 0.62 | 165.8 |
| E-ACO (Proposed) | 11672.4 | 3.5 | 0.41 | 112.5 |
The results clearly demonstrate the superiority of the proposed E-ACO for the formation drone light show planning task. It achieved the shortest total path length, indicating better energy efficiency for the swarm. Most importantly, it maintained the largest minimum separation between drones (3.5m vs. a safety threshold of 3.0m), showcasing its enhanced safety focus due to the fuzzy-controlled adaptive weights. Furthermore, it achieved the lowest average formation error, which is critical for the visual precision of the formation drone light show. Notably, it also converged approximately 32% faster than S-ACO and 21% faster than PSO, validating the efficacy of opposition-based learning in accelerating the search. The convergence plots showed that E-ACO reached a stable, high-quality solution much earlier in the iterative process.
Trajectory Smoothing and Practical Deployment Considerations
The output of the planning algorithm is a sequence of discrete waypoints for each drone. To ensure flyable trajectories suitable for the dynamics of real UAVs in a formation drone light show, these waypoints must be connected by a smooth curve. We employ B-spline curves for this purpose. A B-spline curve $C(u)$ of order $k$ is defined by:
$$ C(u) = \sum_{i=0}^{n} N_{i,k}(u) \cdot P_i $$
where $P_i$ are the control points (derived from the planned waypoints), and $N_{i,k}(u)$ are the B-spline basis functions. B-splines provide local control, continuity ($C^{k-2}$), and inherent smoothness, making them ideal for generating the final flight paths that drones can track accurately using standard path-following controllers.
For practical deployment of a formation drone light show system using this planning approach, several additional factors must be integrated:
- Communication and Synchronization: The planned trajectories must be broadcast to all drones with precise timing information. Time-synchronized protocols like Precision Time Protocol (PTP) are essential.
- Real-time Re-planning: A module for online trajectory adjustment is needed to handle unforeseen events like a drone failure or sudden wind gusts, ensuring the robustness of the formation drone light show.
- Light Effect Coordination: The trajectory data must be tightly coupled with the light controller, so RGB LED colors and intensities change in perfect sync with the drones’ positions in the 3D pattern.
Conclusion
This paper presents a comprehensive framework for optimal trajectory planning tailored specifically for the demanding application of a formation drone light show. By formulating a multi-objective cost function that balances path length, safety, and formation accuracy, and by developing an Enhanced Ant Colony Optimization algorithm featuring opposition-based learning and fuzzy-controlled adaptive weights, we have addressed key challenges in swarm trajectory optimization. Simulation results confirm that the proposed E-ACO algorithm outperforms standard methods, generating shorter, safer, and more accurate flight paths while converging faster. The integration of B-spline smoothing ensures the generated trajectories are dynamically feasible for real UAVs. This work provides a solid algorithmic foundation for the automated design and execution of increasingly complex and reliable formation drone light show performances, pushing the boundaries of this spectacular fusion of technology and art.