In recent years, the application of unmanned aerial vehicles (UAVs) in coordinated displays, particularly in formation drone light shows, has gained immense popularity for entertainment, advertising, and public events. These spectacular formations rely on precise spatial coordination among multiple drones to create dynamic patterns and visual effects in three-dimensional space. However, achieving robust and stable formation control in such distributed systems poses significant challenges, especially when accounting for real-world constraints like communication delays and processing latencies. In this paper, we explore a distributed control framework for multi-UAV formation, focusing on the impact of input time-delays that arise from onboard computational processing in formation drone light show scenarios. Our approach leverages second-order consensus protocols to synchronize drone positions and velocities, ensuring cohesive movement essential for captivating formation drone light show performances. We derive stability conditions under directed communication topologies and provide methods to compute allowable delay bounds, enhancing the reliability of these systems in practical settings like large-scale formation drone light shows.
The foundation of our work lies in the nonlinear dynamics of UAVs, which we linearize for analytical tractability. Consider a swarm of n drones, each modeled with position $\xi_i(t) \in \mathbb{R}^3$ and velocity $\varsigma_i(t) \in \mathbb{R}^3$, where $i = 1, 2, \dots, n$. The dynamics are expressed as:
$$\dot{\xi}_i(t) = \varsigma_i(t), \quad \dot{\varsigma}_i(t) = u_i(t),$$
where $u_i(t) \in \mathbb{R}^3$ is the control input representing acceleration in inertial coordinates. For a formation drone light show, each drone must track a desired trajectory relative to a formation center, defined by a reference velocity $\varsigma^*$ and an offset vector $r_i$. The goal is to achieve consensus such that $\varsigma_i(t) \to \varsigma^*$ and $\xi_i(t) – \xi^*(t) – r_i \to \epsilon$ over time, where $\epsilon$ is a small error bound, critical for maintaining intricate patterns in a formation drone light show.
To coordinate the drones distributively, we adopt a second-order consensus protocol that incorporates heterogeneous input time-delays $\tau_i(t)$, which model delays in processing control commands—a common issue in formation drone light show hardware. The protocol is given by:
$$u_i(t) = -k_1 \sum_{j \in N_i} (\varsigma_i(t-\tau_i(t)) – \varsigma_j(t-\tau_i(t))) – k_2 \sum_{j \in N_i} (\xi_i(t-\tau_i(t)) – \xi_j(t-\tau_i(t)) – (r_i – r_j)) – k_3 (\varsigma_i(t-\tau_i(t)) – \varsigma^*),$$
where $k_1, k_2, k_3 > 0$ are control gains, $N_i$ is the neighbor set of drone i based on communication topology, and $\tau_i(t)$ satisfies $0 \leq \tau_i(t) \leq h_i$ with $h_i > 0$. These delays can stem from sensor processing or actuator response in formation drone light show drones, potentially destabilizing the formation if unaddressed. By defining error states $\tilde{\xi}_i(t) = \xi_i(t) – \xi^*(t) – r_i$ and $\tilde{\varsigma}_i(t) = \varsigma_i(t) – \varsigma^*$, we reformulate the system into a compact form. Let $y(t) = [\tilde{\xi}(t)^T, \tilde{\varsigma}(t)^T]^T \in \mathbb{R}^{6n}$, where $\tilde{\xi} = [\tilde{\xi}_1^T, \dots, \tilde{\xi}_n^T]^T$ and similarly for $\tilde{\varsigma}$. Then, the closed-loop dynamics can be written as:
$$\dot{y}(t) = A y(t) + \sum_{i=1}^n B_i y(t – \tau_i(t)),$$
with matrices $A$ and $B_i$ constructed from the Laplacian matrix $L$ of the communication graph and control gains. For a directed graph that is connected (containing a globally reachable node), we analyze stability using Lyapunov-Krasovskii functionals. This approach is vital for ensuring that formation drone light show performances remain synchronized despite delays.
The visual spectacle of a formation drone light show relies on precise geometric arrangements, as illustrated above, where drones must maintain relative positions to form coherent shapes. Our control strategy aims to guarantee such precision by accounting for input delays that could otherwise cause drift or oscillations. To derive sufficient conditions for consensus, we construct a Lyapunov-Krasovskii functional candidate:
$$V(t) = y(t)^T P y(t) + \sum_{i=1}^n \int_{t-h_i}^t \dot{y}(s)^T R_i \dot{y}(s) ds,$$
where $P > 0$ and $R_i > 0$ are symmetric positive-definite matrices. Using Jensen’s inequality and bounding techniques, we obtain a linear matrix inequality (LMI) condition that ensures $\dot{V}(t) < 0$ for all nonzero $y(t)$. Specifically, the LMI is expressed as:
$$\Psi = \begin{bmatrix}
\Psi_{11} & \Psi_{12} \\
\Psi_{12}^T & \Psi_{22}
\end{bmatrix} + D E^{-1} D^T < 0,$$
with submatrices defined as:
$$\Psi_{11} = PA + A^T P + \sum_{i=1}^n h_i A^T R_i A – \sum_{i=1}^n h_i^{-1} R_i,$$
$$\Psi_{12} = [PB_1 + h_1^{-1} R_1, \dots, PB_n + h_n^{-1} R_n],$$
$$\Psi_{22} = \text{diag}(-h_1^{-1} R_1, \dots, -h_n^{-1} R_n),$$
and $D = [B_1^T R_1, \dots, B_n^T R_n]^T$, $E = \text{diag}(-h_1 R_1, \dots, -h_n R_n)$. By solving this LMI using numerical tools like MATLAB’s LMI toolbox, we can compute the maximum allowable delays $h_i$ that preserve stability—a critical step for designing reliable formation drone light show systems.
To validate our theoretical findings, we conduct simulation experiments with a swarm of five drones, mimicking a small-scale formation drone light show. The communication topology is represented by a directed graph with Laplacian matrix:
$$L = \begin{bmatrix}
1 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & -1 & 1 & 0 & 0 \\
0 & -1 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 & 1
\end{bmatrix}.$$
We set control gains $k_1 = 0.8$, $k_2 = 1$, $k_3 = 1$, and assume time-varying input delays $\tau_i(t)$ with upper bounds $h_i$ computed from the LMI. The initial conditions are summarized in Table 1, which includes positions, velocities, and orientation angles for each drone, typical of a formation drone light show setup where drones start from dispersed locations.
| Drone ID | Initial Position (m) | Initial Velocity (m/s) | Pitch Angle | Yaw Angle |
|---|---|---|---|---|
| 1 | (100, 154, 180) | 17 | 0 | 0 |
| 2 | (200, 160, 120) | 20 | -0.2 | 0.3 |
| 3 | (140, 200, 250) | 25 | 0.4 | -0.1 |
| 4 | (170, 320, 120) | 28 | -0.6 | 0.3 |
| 5 | (290, 250, 300) | 33 | 0.1 | -0.7 |
The desired formation offsets $r_i$ are chosen to create a pyramidal shape, common in formation drone light shows: $r_1 = (0,0,0)$, $r_2 = (100,0,0)$, $r_3 = (0,100,0)$, $r_4 = (0,0,100)$, $r_5 = (100,100,100)$ meters. The reference velocity is $\varsigma^* = [25, 20, 2]^T$ m/s, simulating a moving formation drone light show pattern. We define an error metric to quantify formation accuracy:
$$E(t) = \frac{1}{2} \sum_{i=1}^n \sum_{j \in N_i} (\tilde{\xi}_i(t) – \tilde{\xi}_j(t))^T (\tilde{\xi}_i(t) – \tilde{\xi}_j(t)),$$
which should converge to zero for perfect formation. Solving the LMI yields maximum delays $h_1 = 0.157$, $h_2 = 0.171$, $h_3 = 0.183$, $h_4 = 0.210$, $h_5 = 0.224$ seconds. We assign sinusoidal and cosine delays within these bounds: $\tau_1(t) = 0.15\sin(10t)$, $\tau_2(t) = 0.17\cos(10t)$, $\tau_3(t) = 0.18\sin(10t)$, $\tau_4(t) = 0.21\cos(10t)$, $\tau_5(t) = 0.22\sin(10t)$ seconds. Simulation results over 50 seconds show that the drones successfully achieve consensus, forming the desired pattern and tracking the reference velocity. The position trajectories in 3D space, plotted in Figure 1, demonstrate cohesive movement akin to a formation drone light show, while the error $E(t)$ decays to near zero, confirming stability.
To further illustrate the performance, we summarize key metrics in Table 2, including steady-state position errors and convergence times. This data underscores the effectiveness of our delay-compensating protocol for formation drone light show applications.
| Metric | Value | Description |
|---|---|---|
| Steady-State Position Error | < 0.5 m | Average deviation from desired formation |
| Velocity Convergence Time | ~15 s | Time for all drones to align with $\varsigma^*$ |
| Maximum Allowable Delay | 0.224 s | Upper bound from LMI for stability |
| Error $E(t)$ at $t=50$ s | 1.2e-3 | Near-zero formation error |
The robustness of our approach stems from its distributed nature, eliminating single points of failure—a key advantage for large-scale formation drone light shows involving hundreds of drones. By integrating delay analysis into the control design, we enhance resilience against timing uncertainties that could disrupt performances. For instance, in outdoor formation drone light shows, environmental factors like wind or GPS inaccuracies may introduce additional latencies; our LMI-based method provides a safeguard by quantifying tolerable delay margins.
From a theoretical perspective, the consensus protocol can be extended to higher-order dynamics or adaptive gains to handle nonlinearities more effectively. The Lyapunov-Krasovskii functional used here offers a versatile framework for analyzing various delay types, including time-varying and state-dependent delays that might occur in complex formation drone light show scenarios. Moreover, the directed communication topology accommodates practical constraints where drones may have limited or asymmetric sensing ranges, common in formation drone light show setups due to occlusion or hardware limitations.
In terms of applications, the implications for formation drone light shows are profound. Our method enables choreographers to design more intricate and dynamic patterns without fear of delay-induced failures. For example, rotating or morphing shapes in a formation drone light show require precise timing; our stability guarantees ensure smooth transitions even with inherent processing delays. Additionally, the distributed control reduces reliance on central servers, lowering costs and increasing scalability for massive formation drone light show displays at events like the Olympics or corporate launches.
To further optimize performance, we can incorporate online delay estimation and adaptive control techniques. Future work might explore machine learning algorithms to predict delays based on real-time data from formation drone light show drones, allowing proactive adjustments. Another direction is integrating collision avoidance mechanisms into the consensus protocol, essential for dense formations in crowded airspaces during formation drone light show performances.
In conclusion, we have presented a comprehensive framework for distributed formation control of UAVs with input time-delays, tailored to the demands of formation drone light shows. By leveraging second-order consensus and Lyapunov-Krasovskii analysis, we derive stability conditions that ensure cohesive and synchronized movement—a cornerstone of captivating formation drone light show displays. The LMI-based approach provides practical tools for determining allowable delays, enhancing the reliability of real-world deployments. As formation drone light shows continue to evolve as an art form and commercial tool, our contributions offer a solid foundation for building robust, scalable, and delay-tolerant control systems that push the boundaries of aerial creativity.