The spectacular, synchronized aerial ballets of formation drone light shows have captivated global audiences, transforming the night sky into a dynamic, luminous canvas. Behind the artistry lies a profound challenge of coordination and decision-making: how to optimally assign hundreds, sometimes thousands, of individual drones to specific spatial and temporal points in a complex four-dimensional trajectory, all while ensuring robustness against real-world uncertainties. Interestingly, this core problem of optimal assignment under imperfect information mirrors critical challenges in military operations, such as coordinating an unmanned aerial vehicle (UAV) formation for a synchronized strike. In such tactical scenarios, commanders must allocate assets to targets despite “grey” information—incomplete data, sensor limitations, and the unpredictable decisions of an adversary. This article transposes a sophisticated mathematical framework developed for combat modeling—the Grey Bi-Matrix Game—into the domain of large-scale automated choreography for formation drone light shows. We will explore how this framework provides a resilient, strategic approach to trajectory assignment, ensuring stunning visual coherence even when facing the inevitable uncertainties of wind, communication latency, and individual drone performance variance.
The choreography of a formation drone light show is fundamentally a large-scale, dynamic assignment problem. Each drone is an agent that must move from its current state (position, battery level) to a defined state in a future keyframe of the show. The collective goal is to achieve the desired visual pattern with perfect synchrony, minimal energy expenditure, and maximal safety margins. Traditional methods often rely on pre-computed, deterministic paths or centralized optimizers that assume a perfect world. However, in practice, a swarm operates in a “grey” environment. Information is incomplete: real-time battery drain estimates are approximate, localized wind gusts are unpredictable, and the precise communication delay to each drone may vary. Furthermore, the “preferences” or constraints of the system (e.g., prioritizing the integrity of a central letter in a logo over a peripheral swirl) introduce a strategic element akin to the “payoff” in a game. The concept of a formation drone light show thus evolves from a simple playback script to a strategic, adaptive system.
The Core Concept: Grey Systems and Strategic Games
The term “grey” in systems theory describes situations where information is partially known and partially unknown. This is distinct from “black” (completely unknown) or “white” (completely known). In a formation drone light show, the precise performance envelope of every drone at every moment is a grey parameter—we have a likely range, but not an exact value. Game theory, specifically bi-matrix games, models strategic interactions between two rational players, each with their own set of strategies and payoffs for every combination of choices.
In our adapted model for a formation drone light show, we reconceptualize the two players not as adversaries, but as two conflicting objectives within the choreography system:
- Player A (The Artistic Objective): This player’s goal is to maximize the visual fidelity and aesthetic payoff of the show. Its “strategies” are the possible assignments of drones to target positions in each keyframe.
- Player B (The Operational Objective): This player’s goal is to maximize safety, efficiency, and robustness. Its “strategies” represent the system’s prioritization schemes—for example, strategies that favor minimizing total distance traveled, balancing battery use, or avoiding predicted high-wind zones.
The interaction is a game because improving the artistic payoff (e.g., by demanding drones make sharp, energy-intensive maneuvers for a better visual) often conflicts with the operational payoff (which favors gentle, energy-conserving paths). The system must find an equilibrium assignment that strategically balances these competing demands under grey conditions.
Mathematical Framework: The Grey Bi-Matrix Game for Drone Choreography
Let’s formalize the model for a formation drone light show. Consider a transition between two keyframes in a show, involving \( m \) drones and \( n \) target positions in the next keyframe. Typically, \( m = n \) for a one-to-one assignment.
1. Strategy Sets:
Let \( I \) be the set of all possible assignment strategies for Player A (Artistic). Each pure strategy \( i \in I \) is a permutation assigning each drone to a unique target. With \( m \) drones, \( |I| = m! \).
Let \( J \) be the set of all operational prioritization strategies for Player B. Each strategy \( j \in J \) could be a weighting vector over cost factors (distance, energy, risk). The number of strategies can be defined based on discretized weight combinations.
2. Grey Payoff Matrices:
The payoffs are not fixed numbers but intervals, reflecting the grey nature of the system. We define two matrices:
- \( \otimes A = [\otimes a_{ij}] \): The grey artistic payoff when Artistic strategy \( i \) meets Operational strategy \( j \).
- \( \otimes B = [\otimes b_{ij}] \): The grey operational payoff for the same combination.
A grey number \( \otimes g \) is denoted as \( \otimes g \in [\underline{g}, \overline{g}] \), where \( \underline{g} \) is the lower bound and \( \overline{g} \) is the upper bound.
The payoff for a given strategy pair \( (i, j) \) is calculated based on drone and target properties. We define:
- \( V = (v_1, v_2, …, v_n) \): The “visual importance” weight of each target position. A position at the center of a logo has a higher \( v_l \).
- \( S = (s_1, s_2, …, s_m) \): The “state health” score of each drone, based on its remaining battery, estimated reliability, etc.
- \( H = [h_{kl}] \): The “aesthetic fidelity” score if drone \( k \) is assigned to target \( l \). This can be a function of the drone’s capability (e.g., color brightness) and the target’s requirement.
- \( C = [c_{kl}] \): The “operational cost” if drone \( k \) moves to target \( l \). This is a grey number \( \otimes c_{kl} \in [\underline{c}_{kl}, \overline{c}_{kl}] \) based on uncertain wind and performance.
Let \( \textbf{1}^i_{kl} \) be an indicator that is 1 if strategy \( i \) assigns drone \( k \) to target \( l \), and 0 otherwise. Let \( \textbf{1}^j_{kl} \) be an indicator that is 1 if operational strategy \( j \) applies a high cost-mitigation focus to the pair (k,l), and 0 otherwise.
The grey payoffs can be modeled as:
$$ \otimes a_{ij} = \sum_{k=1}^m \sum_{l=1}^n \textbf{1}^i_{kl} \cdot \left( \xi_A \cdot v_l \cdot h_{kl} – \omega_A \cdot \textbf{1}^j_{kl} \cdot \otimes c_{kl} \right) $$
$$ \otimes b_{ij} = \sum_{k=1}^m \sum_{l=1}^n \textbf{1}^j_{kl} \cdot \left( \xi_B \cdot s_k / \otimes c_{kl} – \omega_B \cdot \textbf{1}^i_{kl} \cdot v_l \right) $$
Here, \( \xi_A, \omega_A, \xi_B, \omega_B \) are tuning parameters that weight the importance of reward versus cost for each player, shaping the artistic-operational trade-off for the formation drone light show.
3. The Equilibrium Solution:
We seek a mixed-strategy Nash equilibrium—a pair of probability distributions \( x^* = (x_1^*, …, x_{|I|}^*) \) over artistic strategies and \( y^* = (y_1^*, …, y_{|J|}^*) \) over operational strategies—such that neither objective can unilaterally increase its expected payoff by changing its mix. Because payoffs are grey, the equilibrium condition is defined for whitened values (e.g., using an expected value or a specific whitening function \( F(\otimes g) \)) and includes satisfaction levels \( \lambda_A, \lambda_B \).
The problem reduces to solving a nonlinear programming problem to maximize a joint satisfaction level \( \lambda \):
$$
\begin{aligned}
& \text{Maximize } \lambda \\
& \text{subject to:} \\
& F(x^T \cdot A \cdot y) \geq F(V_A) + F(\varepsilon_A)(1 – \lambda) \\
& F(x^T \cdot B \cdot y) \geq F(V_B) + F(\varepsilon_B)(1 – \lambda) \\
& \sum_{r \in I} x_r = 1, \quad \sum_{s \in J} y_s = 1 \\
& x_r, y_s \geq 0, \quad \lambda \in [0, 1]
\end{aligned}
$$
Here, \( V_A, V_B \) are aspiration payoff levels for the artistic and operational objectives, and \( \varepsilon_A, \varepsilon_B \) are permissible relaxation parameters. The solution \( (x^*, y^*, \lambda^*) \) provides the optimal probabilistic blueprint for the formation drone light show assignment that balances both grey objectives. The final drone-to-target assignment is sampled from this optimal mixed strategy \( x^* \), or the highest-probability pure strategy within it is chosen.
Solving the Game: An Improved Adaptive Particle Swarm Optimizer
The nonlinear programming problem for the grey bi-matrix equilibrium is complex and high-dimensional. We employ an Improved Particle Swarm Optimization (PSO) algorithm, enhanced for this specific search space. In PSO, a swarm of particles (each representing a candidate solution \( (x, y, \lambda) \)) flies through the solution space, guided by personal and communal best-known positions.
For a particle \( q \), its position vector is \( \mathbf{X}_q = (x_{q1},…, x_{q|I|}, y_{q1},…, y_{q|J|}, \lambda_q) \) and velocity \( \mathbf{V}_q \). The standard update rules are modified for adaptation and boundary handling:
1. Adaptive Inertia Weight:
The inertia \( \omega \) controls exploration. We use a fitness-dependent adaptation:
$$ \omega_q =
\begin{cases}
\omega_{min} + \frac{(\omega_{max} – \omega_{min})(f_q – f_{min})}{f_{avg} – f_{min}}, & \text{if } f_q \leq f_{avg} \\
\omega_{max}, & \text{if } f_q > f_{avg}
\end{cases}
$$
where \( f_q \) is the particle’s fitness (value of the objective function \( \lambda \)), and \( f_{avg}, f_{min} \) are the swarm’s average and minimum fitness. This pushes poorer particles to explore more and allows fitter particles to refine their search.
2. Dynamic Learning Coefficients:
The cognitive (\( c_1 \)) and social (\( c_2 \)) learning factors evolve over iteration \( t \):
$$ c_1(t) = c_{1,initial} + \frac{c_{1,final} – c_{1,initial}}{t_{max}} \cdot t $$
$$ c_2(t) = c_{2,initial} + \frac{c_{2,final} – c_{2,initial}}{t_{max}} \cdot t $$
Typically, we set \( c_{1,initial} > c_{2,initial} \) and \( c_{1,final} < c_{2,final} \), shifting focus from personal to social learning to converge on a global optimum for the formation drone light show assignment.
3. Boundary Oscillation with Constraint Penalty:
To handle the constraints \( \sum x_r = 1, \sum y_s = 1, x_r, y_s \geq 0 \), we employ a penalty function. Furthermore, a boundary oscillation term is added to the position update to prevent stagnation:
$$ \mathbf{X}_q(t+1) = \mathbf{X}_q(t) + \mathbf{V}_q(t+1) + \gamma \cdot \mathbf{r}_3 \odot (\mathbf{ub} – \mathbf{lb}) $$
where \( \gamma \) is an oscillation constant, \( \mathbf{r}_3 \) is a random vector, \( \odot \) is element-wise multiplication, and \( \mathbf{ub}, \mathbf{lb} \) are bounds. Particles leaving the feasible simplex are heavily penalized, guiding the swarm toward valid mixed strategies.
The improved PSO algorithm effectively navigates the high-dimensional, constrained space to find the optimal grey game equilibrium for choreographing a resilient formation drone light show.
Application in Choreography Design
The output of the grey game-theoretic model is not a single rigid path but a robust strategy. For a formation drone light show company, this translates into significant advantages:
- Pre-Show Planning: The model can be run offline for each major transition in the show, generating an assignment plan that is inherently robust to estimated ranges of wind and battery uncertainty. This plan forms the core resilient trajectory.
- Real-Time Adaptation: If a drone’s state (e.g., battery drain rate) falls outside the predicted grey interval during the show, the model can be re-solved in a localized manner for the next few keyframes, using updated grey parameters. The equilibrium concept ensures the adaptation remains balanced.
- Fleet Heterogeneity: The model naturally accommodates a mixed fleet within a formation drone light show. Drones with different capabilities (brighter LEDs, longer endurance) can have different \( h_{kl} \) and \( s_k \) values, leading to an optimal assignment where high-value drones are strategically placed in high-visual-importance positions, but not at excessive operational cost.
Case Study: Simulated Show Transition
Consider a transition in a formation drone light show where 5 drones (\( m=5 \)) must reassign to 5 new target positions (\( n=5 \)) to form a star. The visual importance \( V \) of the star’s tip positions is higher than the inner positions. Drones have varying state health \( S \). Operational costs \( \otimes C \) are grey intervals due to variable crosswinds.
We define 5 artistic strategies (different assignment permutations) and 3 operational strategies (prioritize min distance, balance battery, avoid worst-case wind). The tuned payoff matrices \( \otimes A \) and \( \otimes B \) are constructed. The improved PSO algorithm (swarm size=30, iterations=100) is used to solve for the equilibrium.
Simulation Results:
The algorithm converges to a high satisfaction level \( \lambda^* = 0.94 \). The optimal mixed strategy \( x^* \) for the artistic player heavily favors one primary assignment but includes a small probability for a backup assignment, providing inherent resilience. The derived optimal assignment is shown below, alongside key performance metrics compared against a standard Hungarian algorithm assignment which minimizes only expected distance.
| Drone ID | State Health (s_k) | Assigned Target | Target Importance (v_l) | Grey Cost Interval [c_kl] |
|---|---|---|---|---|
| 1 | 0.9 (High) | T3 (Tip) | 0.9 (High) | [1.2, 1.8] |
| 2 | 0.7 (Med) | T1 (Tip) | 0.9 (High) | [1.0, 1.5] |
| 3 | 0.6 (Med) | T4 (Inner) | 0.4 (Low) | [0.8, 1.3] |
| 4 | 0.5 (Low) | T2 (Inner) | 0.4 (Low) | [0.7, 1.6] |
| 5 | 0.8 (High) | T5 (Tip) | 0.9 (High) | [1.5, 2.2] |
| Algorithm | Expected Artistic Payoff | Worst-Case Operational Cost | Satisfaction Level (λ) | Resilience Metric* |
|---|---|---|---|---|
| Grey Game + Improved PSO | 8.2 | 7.1 | 0.94 | 0.91 |
| Standard Hungarian (Min. Distance) | 7.8 | 9.5 | N/A | 0.72 |
*Resilience Metric: Probability of successful transition when costs realize at upper bound of their grey intervals.
The convergence plot of the improved PSO shows it escaping local optima and reliably finding the high-satisfaction equilibrium, outperforming a standard genetic algorithm which converged to a lower \( \lambda \) of 0.87 in this scenario. This demonstrates the efficacy of the proposed framework for the formation drone light show assignment problem under uncertainty.
Conclusion and Future Directions
The translation of a Grey Bi-Matrix Game model from tactical asset allocation to artistic choreography for formation drone light shows offers a powerful paradigm for managing complexity and uncertainty. By framing the choreography as a strategic game between competing artistic and operational objectives, and by acknowledging the inherent “greyness” in real-world performance parameters, this approach generates assignment solutions that are not just optimal in a deterministic sense, but are robust and resilient. The improved PSO algorithm provides an effective tool for solving the resulting complex equilibrium problem.
Future work can extend this framework in several exciting directions for formation drone light shows: incorporating true dynamic game elements where the “players” (objectives) adapt strategies in real-time across multiple sequential keyframes; integrating more sophisticated whitening functions for grey numbers based on real-time sensor data fusion; and developing distributed versions of the solver where groups of drones collaborate to compute local equilibria, enhancing scalability for ultra-large swarms. Ultimately, this mathematical fusion of game theory and grey systems promises to make the mesmerizing spectacle of a formation drone light show not only more beautiful but also profoundly more intelligent and reliable.