The evolution of multi-drone systems has transitioned from theoretical research to breathtaking real-world applications, with the formation drone light show standing as a quintessential example. This spectacular display relies on the precise, coordinated motion of numerous drones to create complex, dynamic shapes and patterns in the night sky. The underlying technology, however, extends far beyond entertainment. The core challenge of maintaining a stable geometric configuration while navigating and performing a collective task is fundamental to advanced multi-agent operations. This article delves into the sophisticated control strategies that enable such coordinated behavior, focusing on a distributed framework for drone formations tasked with tracking dynamic ground targets in complex, obstacle-laden environments—a direct technological progenitor to the mesmerizing formation drone light show.
The coordinated flight in a formation drone light show is not merely about holding positions; it involves dynamic reconfiguration, smooth trajectory generation, and robust response to environmental factors. Similarly, for missions like surveillance, search-and-rescue, or environmental monitoring, a formation must not only track a moving target but also maintain its structural integrity while avoiding obstacles. This requires a fusion of control philosophies. Leader-follower approaches, while simple, lack robustness to single-point failures. This work, therefore, advocates for a hybrid methodology that synergizes the structural rigor of virtual structure control with the reactive adaptability of behavior-based control. The proposed framework ensures drones are distributed uniformly on a virtual sphere around a target or a virtual leader, creating a stable polyhedral formation reminiscent of the intricate structures seen in a large-scale formation drone light show.
1. Drone Dynamical Model and Feedback Linearization
To design effective high-level formation controllers, a clear understanding of the drone’s dynamics is essential. While a full six-degree-of-freedom model is complex, for formation control purposes, we often model each drone as a controllable point mass in three-dimensional space. Consider a formation comprising \(N\) drones. Let the position of the \(n\)-th drone be denoted by \((x_n, y_n, z_n)\) in a global coordinate system. A common simplified dynamic model can be expressed using second-order kinematics with damping:
$$
\begin{align}
M \ddot{x}_n &= u_{x_n} – k \dot{x}_n \\
M \ddot{y}_n &= u_{y_n} – k \dot{y}_n \\
M \ddot{z}_n &= u_{z_n} – k \dot{z}_n
\end{align}
$$
Here, \(M\) represents an effective mass, \(k\) is a damping coefficient, and \((u_{x_n}, u_{y_n}, u_{z_n})\) are the virtual control inputs (accelerations) for the \(n\)-th drone. These virtual inputs are derived from the actual drone’s autopilot or lower-level attitude controller. For instance, they can be related to high-level commands like desired velocity or force. This double-integrator model is feedback linearized, making it suitable for designing trajectory and formation control laws, which are then translated into lower-level actuator commands. This abstraction is key to scalable control in a formation drone light show, where managing the physics of each individual drone is delegated to its local controller, and the fleet-level logic focuses on geometry and coordination.
| Symbol | Description | Typical Role |
|---|---|---|
| \( (x_n, y_n, z_n) \) | Position of drone \(n\) | State variable |
| \( (\dot{x}_n, \dot{y}_n, \dot{z}_n) \) | Velocity of drone \(n\) | State variable |
| \( (u_{x_n}, u_{y_n}, u_{z_n}) \) | Virtual acceleration control input | Control command |
| \(M\) | Effective mass/inertia coefficient | System parameter |
| \(k\) | Linear damping coefficient | System parameter |
| \(N\) | Total number of drones in formation | Formation parameter |
2. Hybrid Formation Control: Virtual Structure Meets Behavioral Strategy
The cornerstone of a precise formation drone light show or any coordinated fleet is the formation-keeping strategy. Our proposed method is a distributed hybrid approach. The virtual structure concept provides a global geometric template—a rigid or semi-rigid shape that the entire formation should emulate. Here, we define this structure as a sphere. The behavioral component provides the local interaction rules that allow each drone to fulfill its role within that structure autonomously.
2.1 The Virtual Sphere and Convergence Control
The primary control objective is to drive and maintain all drones onto the surface of a virtual sphere of desired radius \(r_d\), centered at a point \(\mathbf{p}_c = (x_c, y_c, z_c)\). This center can be a physical leader drone or a calculated virtual point. We define a Lyapunov-inspired candidate function for each drone based on its distance from this sphere:
$$ V_n = \frac{1}{4} \left[ (x_n – x_c)^2 + (y_n – y_c)^2 + (z_n – z_c)^2 – r_d^2 \right]^2 $$
A control law that guarantees convergence to the sphere surface is designed as the gradient of this function:
$$
\begin{align}
f_{x_n}^{sphere} &= -k_s (x_n – x_c) \left[ (x_n – x_c)^2 + (y_n – y_c)^2 + (z_n – z_c)^2 – r_d^2 \right] \\
f_{y_n}^{sphere} &= -k_s (y_n – y_c) \left[ (x_n – x_c)^2 + (y_n – y_c)^2 + (z_n – z_c)^2 – r_d^2 \right] \\
f_{z_n}^{sphere} &= -k_s (z_n – z_c) \left[ (x_n – x_c)^2 + (y_n – y_c)^2 + (z_n – z_c)^2 – r_d^2 \right]
\end{align}
$$
where \(k_s > 0\) is a positive gain. This control component acts as an attractive/repulsive force from the spherical shell, driving the drone’s position error \( (r_n^2 – r_d^2) \) to zero, where \(r_n = \|\mathbf{p}_n – \mathbf{p}_c\|\).
2.2 Inter-Agent Repulsion for Uniform Spacing
The sphere convergence control alone could lead to all drones clustering at a single point on the sphere. To achieve a uniform, well-spaced distribution akin to the vertices of a polyhedron—a critical aesthetic in a formation drone light show—we introduce a behavioral repulsive force between drones. Modeling drones as particles with like charges, the repulsive force on drone \(n\) from drone \(i\) is:
$$ \mathbf{F}_{ni}^{rep} = k_r \frac{q_n q_i}{r_{ni}^2} \hat{\mathbf{r}}_{ni} $$
where \(k_r>0\), \(q_n\) is a virtual charge, \(r_{ni} = \|\mathbf{p}_n – \mathbf{p}_i\|\), and \(\hat{\mathbf{r}}_{ni}\) is the unit vector from drone \(i\) to drone \(n\). The total repulsive force on drone \(n\) is the vector sum from all other drones:
$$ \mathbf{F}_n^{rep} = k_r q_n \sum_{i=1, i\neq n}^{N} \frac{q_i}{r_{ni}^2} \hat{\mathbf{r}}_{ni} $$
2.3 Combined Distributed Control Law
The total formation-keeping control input for each drone is the superposition of the sphere convergence force and the inter-agent repulsion force:
$$
\begin{align}
u_{x_n}^{form} &= F_{n,x}^{rep} + f_{x_n}^{sphere} \\
u_{y_n}^{form} &= F_{n,y}^{rep} + f_{y_n}^{sphere} \\
u_{z_n}^{form} &= F_{n,z}^{rep} + f_{z_n}^{sphere}
\end{align}
$$
This decentralized law ensures that each drone, using only the position of the formation center and its neighbors, will settle into a stable equilibrium. At equilibrium, the sphere convergence force balances the net repulsive force from peers, resulting in a regular polyhedral arrangement (e.g., a tetrahedron for 4 drones, a cube for 8) on the sphere’s surface. This self-organizing property is vital for a formation drone light show where drones may enter or exit the performance, requiring automatic reconfiguration.
3. Tracking a Dynamic Ground Target
A static formation is only the first step. The true challenge, whether for surveillance or a moving formation drone light show sequence, is to have the entire formation track a moving target while maintaining its configuration. We assume the target’s ground position \(\mathbf{p}_t(t) = (x_t(t), y_t(t))\) is known or estimated, and the formation should orbit it at a constant altitude \(z_{form}\) and orbital radius \(r_{orbit}\).
The key is to control the motion of the formation center \(\mathbf{p}_c = (x_c, y_c, z_{form})\). We define the ground distance between the formation center and the target as \( \rho(t) = \sqrt{(x_c – x_t)^2 + (y_c – y_t)^2} \). The objective is to drive \(\rho\) to \(r_{orbit}\) and maintain a constant relative angular speed. We design a control law for the formation center’s velocity in the ground plane. Let the desired relative speed around the target be \(v_d\). A Lyapunov-based design yields the following velocity vector for the formation center:
$$
\begin{align}
\dot{x}_c &= \frac{v_d}{\rho^2 + r_{orbit}^2} \left[ -(x_c – x_t)(\rho^2 – r_{orbit}^2) + (y_c – y_t)(2 \rho r_{orbit}) \right] \\
\dot{y}_c &= \frac{v_d}{\rho^2 + r_{orbit}^2} \left[ -(y_c – y_t)(\rho^2 – r_{orbit}^2) – (x_c – x_t)(2 \rho r_{orbit}) \right]
\end{align}
$$
This control law has two intuitive components modulated by the distance error \((\rho^2 – r_{orbit}^2)\):
1. A radial component proportional to \(-(x_c – x_t)(\rho^2 – r_{orbit}^2)\) that drives \(\rho\) towards \(r_{orbit}\).
2. A tangential component proportional to \(\pm(y_c – y_t)(2 \rho r_{orbit})\) that generates the orbiting motion.
The stability of this approach can be shown by analyzing the derivative of a Lyapunov function \(V_t = (\rho^2 – r_{orbit}^2)^2\), proving that the formation center converges to and remains on the desired orbit around the moving target. The individual drones then use the updated \(\mathbf{p}_c(t)\) in their local formation-keeping law from Section 2, resulting in the entire polyhedral formation gracefully tracking and orbiting the target—a dynamic maneuver central to any advanced formation drone light show narrative.
4. Smooth Obstacle Avoidance for the Formation
Operating in real-world environments necessitates robust obstacle avoidance. For a formation drone light show over a city or for drones in a forest, the formation must deform and navigate around obstacles without breaking apart. We model obstacles as 3D axis-aligned boxes and derive a 2D elliptical potential field in the horizontal plane for navigation.
4.1 Elliptical Field and Tangent Calculation
Given an obstacle centered at \((x_o, y_o)\) with half-lengths \(l_x\) and \(l_y\), the tightest enclosing ellipse is:
$$ \frac{(x – x_o)^2}{2l_x^2} + \frac{(y – y_o)^2}{2l_y^2} = 1 $$
For a drone at \((x_u, y_u)\), we find the two tangent lines from the drone’s position to this ellipse. The direction vectors of these tangents, \(\chi_1\) and \(\chi_2\), provide the two smoothest possible avoidance paths. The tangent points \((x_{t,i}, y_{t,i})\) on the ellipse can be found analytically by solving the system defined by the ellipse equation and the condition that the line from the drone to the point is tangent to the ellipse. The avoidance direction angle is then:
$$ \chi_i = \arctan\left( \frac{y_u – y_{t,i}}{x_u – x_{t,i}} \right), \quad i \in \{1, 2\} $$
4.2 Path Selection Strategy
Each drone must choose between the clockwise or counter-clockwise tangent path. A consistent, distributed decision is based on the relative bearing \(\phi\) of the obstacle center from the drone:
$$ \phi = \arctan2\left( y_o – y_u, x_o – x_u \right) $$
By comparing the tangent direction angles \(\chi_i\) to the bearing \(\phi\), a simple rule ensures all drones tend to choose the same side of the obstacle, preserving formation coherence:
| Condition | Selected Tangent | Resulting Maneuver |
|---|---|---|
| \(\phi \leq \chi_1\) (assuming \(\chi_1 < \chi_2\)) | Direction \(\chi_1\) | Counter-clockwise avoidance |
| \(\phi > \chi_1\) | Direction \(\chi_2\) | Clockwise avoidance |
4.3 Integrated Avoidance Control
During obstacle avoidance, the drone’s velocity command is overridden or blended with the tangent direction. A potential field repulsion term, proportional to the inverse of the distance to the ellipse, can be added to provide a safety margin. The final velocity vector for a drone during avoidance becomes a weighted sum of the formation/tracking command and the obstacle avoidance command. Once the obstacle is cleared, the pure formation control law regains authority, and the drones seamlessly reconfigure back to their nominal positions on the virtual sphere. This ability to deform and recover is what allows a formation drone light show to create the illusion of shapes flowing through and around imaginary obstacles in the sky.
5. Simulation and Performance Analysis
The efficacy of the proposed distributed framework is validated through comprehensive simulation. Parameters are set as follows: \(M=10\), \(k=1\), \(k_s=5\), \(k_r=0.2\), \(r_d=2\), \(v_d=12\), \(r_{orbit}=15\).
5.1 Formation Assembly and Static Target Tracking
Four drones with random initial positions successfully converge to a stable tetrahedral formation on the virtual sphere centered at the target. Subsequently, the formation center control law engages, driving the entire assembly to orbit the static ground target at the specified radius and altitude. The drones maintain their relative positions throughout the maneuver, demonstrating stable formation-keeping during collective motion.
5.2 Dynamic Target Tracking
The target is given a constant velocity \(\mathbf{v}_t = (2, -2)\). The formation center controller successfully tracks the moving target, guiding the formation into a stable orbiting pattern around it. The individual drones continuously adjust their positions relative to the moving center, maintaining the polyhedral shape. This demonstrates the controller’s ability to handle dynamic tracking scenarios, a prerequisite for following a pre-programmed path in a formation drone light show or a live target in surveillance.
5.3 Obstacle Avoidance and Reconfiguration
A large cuboidal obstacle is placed in the formation’s path during a tracking mission. As the formation approaches, each drone calculates the tangent directions to the obstacle’s elliptical footprint and selects the avoidance path based on the rule in Table 2. The formation smoothly deforms as drones navigate along their chosen tangents. Crucially, after passing the obstacle, the sphere convergence and repulsion forces guide all drones back to their nominal positions on the sphere, fully re-establishing the original formation geometry without external intervention. This robust split-and-rejoin capability is critical for the reliability of any formation drone light show operating in non-ideal airspace.
6. Conclusion and Future Directions
This article presented a comprehensive distributed control framework for multi-drone formations, drawing direct inspiration from the requirements for complex aerial displays like a formation drone light show. By hybridizing virtual structure and behavior-based control, the strategy enables drones to self-organize into stable, uniformly spaced polyhedral configurations on a virtual sphere. The integrated Lyapunov-based tracker allows this formation to smoothly orbit dynamic ground targets. Furthermore, the novel tangent-based obstacle avoidance algorithm provides a method for the formation to navigate cluttered environments cohesively, deforming and recovering its shape autonomously.
The principles demonstrated here form the technological bedrock for the stunning synchronized choreography of a formation drone light show. Future work will focus on scaling this approach to very large numbers of drones, addressing limited communication ranges through distributed consensus protocols, and integrating more realistic dynamics and wind disturbance models. Additionally, extending the framework to handle heterogeneous drones with different capabilities will unlock even more complex and adaptive multi-agent applications, pushing the boundaries of what is possible in both artistic performance and autonomous systems operations.