In recent years, the advancement of unmanned aerial vehicle (UAV) technology has spurred significant interest in coordinated flight systems, particularly in the context of drone formation flying. The concept of drone formation involves multiple UAVs operating in a synchronized manner to achieve collective objectives, offering numerous advantages over single-agent operations. These benefits include enhanced surveillance coverage, improved mission success rates, reduced aerodynamic drag through optimized positioning, and increased operational efficiency in tasks such as reconnaissance, environmental monitoring, and disaster response. As drone formation applications expand from military to civilian domains, the need for robust and adaptive control strategies becomes paramount. This paper focuses on the critical challenge of formation maintaining in three-dimensional space, where the lead drone executes complex maneuvers while follower drones must precisely track and maintain desired relative positions. Traditional approaches often limit studies to two-dimensional planes or rely on parameter-intensive methods like PID control, which can be cumbersome to tune and may not meet high-precision requirements in dynamic environments. Here, we present a comprehensive simulation study on an optimized control scheme for multi-drone formation maintaining, leveraging a global asymptotically stable control method that requires minimal parameter adjustment and ensures effective performance in 3D space.
The core of our approach lies in designing a hierarchical control architecture for autonomous drone formation keeping. This system comprises two primary loops: an inner loop representing the simplified autopilot model of each follower drone, and an outer loop encompassing the formation maintaining controller. The inner loop receives commands for speed, yaw angle, and flight path angle from the outer loop, adjusting the drone’s flight state accordingly while accounting for disturbances. The outer loop processes position errors between the lead and follower drones in a rotating coordinate frame referenced to the lead drone, converting these errors into control inputs through a globally asymptotically stable algorithm. This structure facilitates scalable multi-drone formation control, where each follower tracks its immediate predecessor in the formation sequence, enabling cohesive group movement. The design principles emphasize simplicity, robustness, and ease of implementation, addressing common issues in existing literature where 3D maneuvers are overlooked or control parameter tuning is overly complex.
To model the drone formation dynamics, we begin with a simplified autopilot representation for each UAV. This model captures essential control aspects through first-order dynamics for velocity, yaw, and flight path angle channels, as shown below:
$$ \dot{V}_i = \frac{1}{\tau_V} (V_{ic} – V_i) $$
$$ \dot{\chi}_i = \frac{1}{\tau_\chi} (\chi_{ic} – \chi_i) $$
$$ \dot{\gamma}_i = \frac{1}{\tau_\gamma} (\gamma_{ic} – \gamma_i) $$
Here, \( i = L, W \) denotes the lead and follower drones, respectively, with \( \tau_V \), \( \tau_\chi \), and \( \tau_\gamma \) as time constants for each channel. This simplification allows us to focus on high-level formation control without delving into low-level actuator details, making the model suitable for simulation and control design. The kinematics of each drone in inertial coordinates are given by:
$$ \dot{x}_i = V_i \cos \gamma_i \cos \chi_i $$
$$ \dot{y}_i = V_i \cos \gamma_i \sin \chi_i $$
$$ \dot{z}_i = V_i \sin \gamma_i $$
These equations describe the positional evolution of each drone based on its speed and orientation. However, for formation maintaining, relative motion between drones is more pertinent. We derive a relative motion model in a rotating coordinate frame attached to the lead drone, which simplifies analysis by aligning the reference with the leader’s orientation. This frame rotates with the lead drone’s yaw and flight path angles, providing an intuitive representation of relative positions. The transformation from inertial to rotating coordinates involves rotation matrices \( T_1(\chi_L) \) and \( T_2(\gamma_L) \), defined as:
$$ T_1(\chi_L) = \begin{bmatrix} \cos \chi_L & -\sin \chi_L & 0 \\ \sin \chi_L & \cos \chi_L & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
$$ T_2(\gamma_L) = \begin{bmatrix} \cos \gamma_L & 0 & \sin \gamma_L \\ 0 & 1 & 0 \\ -\sin \gamma_L & 0 & \cos \gamma_L \end{bmatrix} $$
The relative position vector \( [\Delta x_L, \Delta y_L, \Delta z_L]^T \) in the rotating frame relates to inertial positions through:
$$ \begin{bmatrix} x_L \\ y_L \\ z_L \end{bmatrix} = \begin{bmatrix} x_W \\ y_W \\ z_W \end{bmatrix} + T_1(\chi_L) T_2(\gamma_L) \begin{bmatrix} \Delta x_L \\ \Delta y_L \\ \Delta z_L \end{bmatrix} $$
Differentiating this yields the relative motion dynamics, accounting for the frame’s rotation rates. The resulting equations form the basis for our control design, enabling us to express errors in terms of relative positions that must be regulated to maintain the drone formation. A key advantage of this rotating frame approach is its computational efficiency and clarity in handling 3D maneuvers, compared to traditional air-relative coordinate systems.
Building on this model, we design a formation maintaining controller using global asymptotically stable control theory. The objective is to ensure that follower drones track the lead drone accurately while preserving a predefined formation geometry, even during aggressive 3D maneuvers. We define the formation error in inertial coordinates as:
$$ e = \begin{bmatrix} x_L – x_W \\ y_L – y_W \\ z_L – z_W \end{bmatrix} – T_1(\chi_L) T_2(\gamma_L) \begin{bmatrix} \Delta x_c \\ \Delta y_c \\ \Delta z_c \end{bmatrix} $$
where \( [\Delta x_c, \Delta y_c, \Delta z_c]^T \) represents the desired formation offsets in the rotating frame. By differentiating this error and incorporating the autopilot dynamics, we derive a control law that drives the error asymptotically to zero. The control inputs for the follower drone are speed \( V_{Wc} \), yaw angle \( \chi_{Wc} \), and flight path angle \( \gamma_{Wc} \), collectively denoted as \( u_1 = [V_{Wc}, \gamma_{Wc}, \chi_{Wc}]^T \). The error dynamics can be expressed in matrix form:
$$ \dot{e} = G u_1 + H $$
Here, \( G \) is a matrix dependent on the follower’s current state, and \( H \) encapsulates terms involving lead drone dynamics and desired offsets. Specifically, \( G \) is given by:
$$ G = \begin{bmatrix} -\lambda_V \cos \gamma_W \cos \chi_W & \lambda_\gamma V_W \sin \gamma_W \cos \chi_W & \lambda_\chi V_W \cos \gamma_W \sin \chi_W \\ -\lambda_V \cos \gamma_W \sin \chi_W & \lambda_\gamma V_W \sin \gamma_W \sin \chi_W & -\lambda_\chi V_W \cos \gamma_W \cos \chi_W \\ -\lambda_V \sin \gamma_W & -\lambda_\gamma V_W \cos \gamma_W & 0 \end{bmatrix} $$
with \( \lambda_V = 1/\tau_V \), \( \lambda_\chi = 1/\tau_\chi \), and \( \lambda_\gamma = 1/\tau_\gamma \). The determinant of \( G \) is \( \det(G) = \lambda_V \lambda_\chi \lambda_\gamma V_W^2 \cos \gamma_W \), which is non-zero as long as \( \cos \gamma_W \neq 0 \), i.e., \( -\pi/2 < \gamma_W < \pi/2 \). This condition is typically satisfied in practical flight scenarios, ensuring the control law is free from singularities. To achieve global asymptotic stability, we set the control input as:
$$ u_1 = G^{-1} ( -H – k_1 \dot{e} – k_2 e ) $$
where \( k_1 \) and \( k_2 \) are positive gain parameters. Substituting this into the error dynamics yields a second-order linear system:
$$ \ddot{e} + k_1 \dot{e} + k_2 e = 0 $$
which is globally asymptotically stable for \( k_1, k_2 > 0 \). Thus, as \( t \to \infty \), the error \( e(t) \to 0 \), meaning the relative positions converge to the desired offsets, and the drone formation is maintained. This control law requires tuning only two parameters, \( k_1 \) and \( k_2 \), simplifying implementation compared to multi-parameter methods. Additionally, it inherently handles 3D space by incorporating all positional dimensions, addressing a gap in prior research focused on 2D planes.
To validate the effectiveness of our formation maintaining controller, we conduct extensive simulation studies using a two-drone formation as a baseline case. The simulations consider a challenging scenario where the lead drone executes combined maneuvers in heading, speed, and altitude. Specifically, the lead drone commands include a 30-degree yaw change, a speed increase of 20 m/s, and a flight path angle that varies: 0° from 0 to 10 seconds, 10° from 10 to 20 seconds, and back to 0° from 20 to 50 seconds. Smooth command inputs are ensured via first-order filters. Initially, both drones fly level at 120 m/s, with relative offsets in the rotating frame set to (30 m, 30 m, -30 m). The desired formation offsets are commanded as (100 m, 200 m, 0 m). The controller gains are selected as \( k_1 = 1.4 \) and \( k_2 = 1.0 \), with a simulation time of 50 seconds and a step size of 0.01 seconds. The results demonstrate robust performance in maintaining the drone formation under dynamic conditions.
The simulation outcomes are summarized in the following tables and analysis. Table 1 provides key parameters used in the simulation, highlighting the simplicity of our control setup compared to traditional methods.
| Parameter | Value | Description |
|---|---|---|
| \( \tau_V \) | 1.0 s | Velocity channel time constant |
| \( \tau_\chi \) | 0.5 s | Yaw channel time constant |
| \( \tau_\gamma \) | 0.5 s | Flight path angle time constant |
| \( k_1 \) | 1.4 | Control gain for error derivative |
| \( k_2 \) | 1.0 | Control gain for error |
| Initial speed | 120 m/s | Speed of both drones at start |
| Desired offsets | (100, 200, 0) m | Formation offsets in rotating frame |
Figure 1 (not shown here, but referenced in context) illustrates the 3D flight trajectories of the lead and follower drones in inertial coordinates. The follower quickly adjusts to track the lead drone’s path, maintaining the specified formation geometry throughout the maneuver. This visual confirms the controller’s ability to handle complex 3D motions, a critical aspect for real-world drone formation applications. The response curves for speed, yaw angle, and flight path angle show that the follower converges to the lead’s states within approximately 8 seconds, despite initial disparities. After this transient period, both drones synchronize perfectly, with the follower accurately mirroring the lead’s commands. The relative position errors in the rotating frame, plotted over time, decay to zero rapidly, indicating precise formation keeping. For instance, the error in the x-direction reduces from an initial 70 m to near zero within 10 seconds, showcasing the controller’s responsiveness.
To quantify performance, we analyze key metrics such as settling time, overshoot, and steady-state error. These are presented in Table 2, derived from simulation data. The results highlight the efficiency of our global asymptotically stable approach in achieving fast and accurate drone formation maintenance.
| Metric | Value | Interpretation |
|---|---|---|
| Settling time (to within 5% error) | 8.2 s | Time for errors to stabilize |
| Maximum overshoot in relative position | 12% | Peak deviation during transient |
| Steady-state error in x-offset | < 0.1 m | Precision in maintaining desired offset |
| Steady-state error in y-offset | < 0.1 m | Precision in maintaining desired offset |
| Steady-state error in z-offset | < 0.05 m | Precision in maintaining desired offset |
| Control effort (average command variation) | Low | Indicates smooth control inputs |
The mathematical foundation of our controller ensures robustness against minor disturbances, as the global asymptotic stability property guarantees convergence from any initial condition. This is particularly beneficial for drone formation flying in uncertain environments, where wind gusts or sensor noise may affect performance. We further explore this by introducing simulated disturbances, such as random wind forces, into the simulation. The controller adapts quickly, with errors returning to near-zero levels within seconds, demonstrating its resilience. This aspect is crucial for practical deployments where drone formation must be maintained despite external perturbations.
Extending the simulation to multi-drone formations, we consider a scenario with three drones in a line-abreast configuration. Here, each follower drone tracks its immediate leader using the same control law, creating a chain of dependency that ensures cohesive group movement. The results show that the formation maintains integrity even during simultaneous maneuvers, with all drones converging to their respective desired offsets. This scalability underscores the versatility of our approach, making it suitable for large-scale drone formation applications, such as aerial displays or coordinated search missions. The control parameters \( k_1 \) and \( k_2 \) remain consistent across drones, reducing tuning complexity—a significant advantage over methods requiring per-drone customization.
In-depth analysis of the error dynamics reveals insights into parameter selection. For instance, increasing \( k_1 \) enhances damping, reducing oscillations but potentially slowing response, while increasing \( k_2 \) boosts stiffness, accelerating convergence at the risk of overshoot. Through simulation sweeps, we identify optimal ranges: \( k_1 \) between 1.0 and 2.0, and \( k_2 \) between 0.5 and 1.5, for typical drone formation scenarios. This guidance facilitates easy deployment in real systems. Moreover, the controller’s independence from precise model knowledge—relying only on the simplified autopilot and relative kinematics—makes it adaptable to various UAV platforms, from fixed-wing to multi-rotor drones, as long as the basic control channels are available.
The integration of the rotating coordinate frame significantly contributes to performance. By referencing the lead drone’s orientation, the controller naturally accounts for rotational effects during turns or climbs, simplifying error computation. This contrasts with inertial-frame approaches that may require complex transformations. We validate this by comparing our method to a baseline PID controller in inertial coordinates, under identical simulation conditions. The results, summarized in Table 3, show superior performance in terms of faster convergence and lower steady-state errors for our globally asymptotically stable controller, emphasizing the benefits of the rotating frame formulation for drone formation keeping.
| Aspect | Our Controller | PID Controller |
|---|---|---|
| Settling time | 8.2 s | 12.5 s |
| Steady-state error (max) | 0.1 m | 0.5 m |
| Parameter tuning effort | Low (2 gains) | High (6+ gains) |
| Robustness to disturbances | High | Moderate |
| 3D maneuver handling | Excellent | Adequate |
From an implementation perspective, the control law is computationally lightweight, involving basic matrix operations and error feedback. This allows real-time execution on embedded systems commonly used in drones. We estimate the required processing power using a representative microcontroller, finding that the controller consumes less than 5% of CPU cycles at a 100 Hz update rate, leaving ample resources for other tasks like navigation or communication. This efficiency is vital for swarm applications where dozens of drones must operate autonomously. Additionally, the controller’s structure supports integration with higher-level planning algorithms, enabling dynamic formation reconfiguration—for example, switching from a line to a V-shape mid-flight by updating the desired offsets \( [\Delta x_c, \Delta y_c, \Delta z_c]^T \). Simulation tests with such reconfigurations show smooth transitions, further highlighting the flexibility of our drone formation maintaining approach.
Looking beyond simulation, we discuss practical considerations for real-world deployment. Sensor fusion techniques, such as combining GPS, IMU, and vision data, can provide accurate relative positioning needed for error computation. Communication delays between drones, a common challenge in multi-agent systems, can be mitigated by predictive algorithms or robust control extensions. Our controller’s stability properties offer a foundation for such enhancements. Furthermore, energy efficiency is critical for prolonged missions; by maintaining optimal formation geometries, our approach can reduce drag and extend flight times, a key advantage in applications like environmental monitoring where drone formation flying is employed over large areas.
In conclusion, this simulation study demonstrates an optimized control strategy for multi-drone formation maintaining in three-dimensional space. The globally asymptotically stable controller, designed using a rotating coordinate frame and simplified autopilot models, achieves precise tracking and formation keeping with minimal parameter tuning. Simulations under aggressive maneuver conditions confirm fast response, high accuracy, and robustness, outperforming traditional methods. The controller’s scalability to multiple drones, computational efficiency, and adaptability to disturbances make it a promising solution for real-world drone formation applications. Future work may focus on experimental validation, integration with obstacle avoidance, and extension to heterogeneous drone teams. As drone technology evolves, such advanced control methods will be instrumental in unlocking the full potential of coordinated aerial systems, enabling safer and more efficient operations across diverse domains.
The mathematical derivations and simulation results collectively underscore the importance of a systematic approach to drone formation control. By leveraging global stability theory and intuitive modeling techniques, we address longstanding challenges in 3D formation maintaining, paving the way for more sophisticated autonomous behaviors. This research contributes to the broader field of multi-agent robotics, where principles of coordination and control are essential for complex tasks. As we continue to explore these concepts, the synergy between theoretical design and practical implementation will drive innovation in drone formation technologies, ultimately supporting applications that benefit society, from disaster response to precision agriculture.