In recent years, the field of autonomous systems has witnessed significant advancements, particularly in the coordination of multiple unmanned aerial vehicles (UAVs). My research focuses on addressing the critical challenges in drone formation control, where a group of UAVs must maintain a desired geometric configuration while navigating through dynamic environments. The importance of drone formation lies in its applications across various domains, including surveillance, search and rescue, and military operations. However, achieving robust and efficient drone formation control is fraught with complexities, primarily due to communication constraints such as time-varying delays and switching network topologies. In this paper, I propose a distributed control strategy based on consensus theory to tackle these issues, ensuring that the drone formation converges to a target formation and maintains a specified velocity despite bounded delays and intermittently connected communication links.
The core of my approach revolves around modeling each UAV in the drone formation as a nonlinear dynamic system. The kinematics and dynamics of the i-th UAV can be described by the following equations, which account for its position, velocity, and orientation in three-dimensional space. Let $\xi_i(t) = [x_i(t), y_i(t), z_i(t)]^T$ represent the position vector, and $\zeta_i(t) = [v_i(t), \phi_i(t), \chi_i(t)]^T$ denote the velocity vector, where $v_i$ is the speed, $\phi_i$ is the azimuth angle, and $\chi_i$ is the flight path angle. The dynamics are derived from a simplified model that captures essential nonlinearities:
$$ \dot{\xi}_i(t) = \zeta_i(t), $$
$$ \dot{\zeta}_i(t) = u_i(t), $$
where $u_i(t)$ is the control input designed to achieve coordination in the drone formation. This second-order model is widely used in multi-agent systems due to its tractability and relevance to real-world UAV dynamics. The control objective is to ensure that all UAVs in the drone formation synchronize their states, such that $\xi_i(t) – \xi_j(t) \rightarrow r_{ij}$ and $\zeta_i(t) \rightarrow \zeta^d(t)$ as $t \rightarrow \infty$, where $r_{ij}$ is the desired relative position vector between UAVs $i$ and $j$, and $\zeta^d(t)$ is the desired formation velocity. This synchronization is crucial for maintaining the integrity of the drone formation during complex maneuvers.
To formalize the communication within the drone formation, I employ graph theory. The network of UAVs is represented as an undirected graph $\mathcal{G}(t) = (\mathcal{V}, \mathcal{E}(t))$, where $\mathcal{V} = \{1, 2, \dots, N\}$ is the set of nodes (UAVs), and $\mathcal{E}(t)$ is the set of edges (communication links) that can vary over time. The adjacency matrix $A(t) = [a_{ij}(t)]$ encodes the connection weights, with $a_{ij}(t) > 0$ if UAVs $i$ and $j$ exchange information at time $t$, and $a_{ij}(t) = 0$ otherwise. The Laplacian matrix $L(t) = D(t) – A(t)$, where $D(t)$ is the degree matrix, plays a pivotal role in consensus algorithms. A key assumption in my work is that the communication topology is jointly connected over time intervals, meaning that the union of graphs across switching sequences forms a connected graph. This relaxation allows for intermittent links, which is common in practical drone formation scenarios due to obstacles or limited range.
The presence of time-varying delays in communication channels poses a significant hurdle for drone formation control. In real-world networks, delays can arise from packet loss, bandwidth limitations, or processing lags, and they often vary unpredictably within bounded intervals. Let $\tau(t)$ denote the time-varying delay, satisfying $0 \leq \tau_l \leq \tau(t) \leq \tau_h$, where $\tau_l$ and $\tau_h$ are the lower and upper bounds, respectively. Unlike prior studies that assume slow-varying or constant delays, my strategy accommodates fast-varying and even discontinuous delays, such as those modeled by $\tau(t) = |1.5 \sin t|$ or random pulse functions. This generality enhances the applicability of my approach to diverse drone formation environments.
My proposed control strategy for the drone formation is based on a distributed consensus protocol that incorporates delayed state information from neighboring UAVs. For each UAV $i$, the control input $u_i(t)$ is designed as follows:
$$ u_i(t) = \dot{\zeta}^d(t) + \sum_{j \in \mathcal{N}_i(t)} a_{ij}(t) \left\{ k_1 \left[ \xi_j(t-\tau(t)) – \xi_i(t-\tau(t)) – r_{ji} \right] + k_2 \left[ \zeta_j(t-\tau(t)) – \zeta_i(t-\tau(t)) \right] \right\} – k_3 \left[ \zeta_i(t) – \zeta^d(t) \right], $$
where $\mathcal{N}_i(t)$ is the set of neighbors of UAV $i$ at time $t$, $k_1, k_2, k_3 > 0$ are control gains, and $r_{ji} = r_j – r_i$ is the desired relative position vector derived from the target drone formation geometry. The term $\dot{\zeta}^d(t)$ represents the acceleration of the desired formation velocity, which is often set to zero for constant velocity scenarios. This control law leverages both position and velocity errors with respect to neighbors, adjusted for delays, to drive the drone formation toward consensus. The inclusion of the term $-k_3(\zeta_i(t) – \zeta^d(t))$ provides damping to stabilize the system.
To analyze the stability of the drone formation under this control strategy, I define error vectors $\tilde{\xi}_i(t) = \xi_i(t) – \xi_r(t) – r_i$ and $\tilde{\zeta}_i(t) = \zeta_i(t) – \zeta^d(t)$, where $\xi_r(t)$ is the position of the formation center. The collective error dynamics can be written in a compact form using the Kronecker product $\otimes$:
$$ \dot{\varepsilon}(t) = (M \otimes I_N) \varepsilon(t) + (N \otimes L(t)) \varepsilon(t-\tau(t)), $$
where $\varepsilon(t) = [\tilde{\xi}_1(t), \dots, \tilde{\xi}_N(t), \tilde{\zeta}_1(t), \dots, \tilde{\zeta}_N(t)]^T$, $M = \begin{bmatrix} 0 & I \\ 0 & -k_3 I \end{bmatrix}$, $N = \begin{bmatrix} 0 & 0 \\ -k_1 I & -k_2 I \end{bmatrix}$, and $L(t)$ is the Laplacian matrix. The stability of this system ensures that the drone formation achieves the desired configuration. I employ Lyapunov-Krasovskii functional methods to derive sufficient conditions for asymptotic stability. Consider a time interval $[t_k, t_{k+1})$ divided into subintervals where the communication topology switches among a set of graphs $\{\mathcal{G}_1, \mathcal{G}_2, \dots, \mathcal{G}_{\Lambda}\}$. Each graph may have multiple connected components; for instance, if a graph has $d_\sigma$ components, its Laplacian can be block-diagonalized via a permutation matrix $P$:
$$ P^T L(t) P = \text{diag}\{L^1_\sigma, L^2_\sigma, \dots, L^{d_\sigma}_\sigma\}. $$
This decomposition reduces the high-dimensional stability analysis to lower-dimensional problems for each connected component, significantly cutting computational costs and improving real-time feasibility for drone formation control.
I construct a Lyapunov-Krasovskii functional $V(t)$ as a sum of integrals involving the error states and their derivatives:
$$ V(t) = \sum_{I=1}^{d_\sigma} \int_{t-\tau_a}^{t} \varepsilon_I^T(s) Q_I \varepsilon_I(s) \, ds + \sum_{I=1}^{d_\sigma} \int_{-\tau_a}^{0} \int_{t+\theta}^{t} \dot{\varepsilon}_I^T(s) R_I \dot{\varepsilon}_I(s) \, ds \, d\theta + \sum_{I=1}^{d_\sigma} \int_{-\tau_a-\delta}^{-\tau_a+\delta} \int_{t+\theta}^{t} \dot{\varepsilon}_I^T(s) S_I \dot{\varepsilon}_I(s) \, ds \, d\theta, $$
where $\tau_a = (\tau_h + \tau_l)/2$, $\delta = (\tau_h – \tau_l)/2$, and $Q_I, R_I, S_I$ are positive definite matrices for each connected component $I$. Using Jensen’s inequality and the Newton-Leibniz formula, the derivative $\dot{V}(t)$ can be bounded. The key step is to ensure that $\dot{V}(t) < 0$ for all $t$, which leads to the following linear matrix inequality (LMI) condition for each component:
$$ \Xi_I = \begin{bmatrix} \Xi_I(1,1) & \Xi_I(1,2) & \Xi_I(1,3) \\ * & \Xi_I(2,2) & \Xi_I(2,3) \\ * & * & \Xi_I(3,3) \end{bmatrix} < 0, $$
with entries defined as:
$$ \Xi_I(1,1) = Q_I + (M \otimes I_{f_I})^T (\tau_a R_I + 2\delta S_I) (M \otimes I_{f_I}) – \frac{R_I}{\tau_a}, $$
$$ \Xi_I(1,2) = (M \otimes I_{f_I})^T (\tau_a R_I + 2\delta S_I) (N \otimes L^I_\sigma) + \frac{R_I}{\tau_a}, $$
$$ \Xi_I(1,3) = (M \otimes I_{f_I})^T (\tau_a R_I + 2\delta S_I) (N \otimes L^I_\sigma), $$
$$ \Xi_I(2,2) = (N \otimes L^I_\sigma)^T (\tau_a R_I + 2\delta S_I) (N \otimes L^I_\sigma) – Q_I – \frac{R_I}{\tau_a}, $$
$$ \Xi_I(2,3) = (N \otimes L^I_\sigma)^T (\tau_a R_I + 2\delta S_I) (N \otimes L^I_\sigma), $$
$$ \Xi_I(3,3) = (N \otimes L^I_\sigma)^T (\tau_a R_I + 2\delta S_I) (N \otimes L^I_\sigma) – \frac{S_I}{\delta}, $$
where $f_I$ is the number of UAVs in the $I$-th connected component. If feasible matrices $Q_I, R_I, S_I$ exist satisfying this LMI, then the error dynamics are asymptotically stable, guaranteeing that the drone formation converges to the target formation. This condition does not require assumptions on the derivative of $\tau(t)$, making it applicable to fast-varying delays—a significant advantage for practical drone formation systems.
To validate my control strategy, I conducted extensive simulation studies using a group of five quadrotor UAVs. The parameters of these UAVs are based on the MD4-200 platform, with key specifications summarized in the table below. These parameters influence the performance of the drone formation, particularly in terms of speed and acceleration limits.
| Parameter | Value |
|---|---|
| Maximum Speed | 8.6 m/s |
| Maximum Acceleration | 0.5 m/s² |
| Mass | 900 g |
| Communication Range | 500 m |
| Battery Life | 20 min |
The initial states of the UAVs in the drone formation were set to diverse positions and velocities to test convergence under non-ideal conditions, as shown in the following table. This heterogeneity mimics real-world scenarios where UAVs may start from scattered locations.
| UAV Index | Position (x, y, z) in meters | Velocity (v, φ, χ) in m/s and degrees |
|---|---|---|
| 1 | (70, 150, 102) | (3.3, -71.5, 17.5) |
| 2 | (50, 70, 103) | (2.5, -53.1, 11.3) |
| 3 | (80, 70, 100) | (2.4, -26.5, 24.1) |
| 4 | (110, 20, 100) | (2.8, -53.1, 45) |
| 5 | (100, 90, 98) | (2.2, -26.6, 0) |
The desired drone formation was defined as a wedge shape for the first simulation and a trapezoid for the second, with a common desired velocity of $\zeta^d(t) = [2.8 \, \text{m/s}, -45^\circ, 0^\circ]^T$. The communication topology switched between two graphs in the first case and four graphs in the second, each with a dwell time of 2 seconds or 1.5 seconds, respectively. The edge weights were set to 1 for all connections, and the topologies were designed to be jointly connected over time. This switching mechanism reflects the dynamic nature of links in a drone formation due to movement or environmental factors.
For the first simulation, I considered a fast-varying delay modeled as $\tau(t) = |1.5 \sin t|$, which ranges from 0 to 1.5 seconds and has periods where $\dot{\tau}(t) > 1$ (fast-varying). Using the LMI conditions, I computed feasible control gains $k_1 = 0.1$, $k_2 = 0.5$, and $k_3 = 0.1$. The results demonstrated that the drone formation successfully converged to the wedge formation despite the challenging delay profile. The trajectories of the UAVs in 3D space showed initial dispersion followed by gradual alignment into the target shape. The velocities and orientations of each UAV asymptotically approached the desired values, with small oscillations due to the delay and switching, as predicted by the stability analysis. The distances between UAVs converged to constant values matching the wedge geometry, confirming the effectiveness of the control strategy for drone formation.
In the second simulation, I addressed random jump delays, where $\tau(t)$ switched between 0 and 2.5 seconds in a pulse-like manner, simulating packet drops or network disruptions. The control gains were set to $k_1 = 0.2$, $k_2 = 0.6$, and $k_3 = 0.1$, derived from the LMI feasibility conditions. The drone formation achieved the trapezoid target with similar convergence patterns. The error norms decreased over time, and the control inputs remained within practical bounds, ensuring feasibility for real UAV platforms. The following table summarizes the performance metrics for both simulations, highlighting the robustness of my approach in different delay scenarios for drone formation control.
| Simulation Case | Delay Type | Convergence Time (approx.) | Maximum Control Input | Formation Error at Steady State |
|---|---|---|---|---|
| Wedge Formation | Fast-varying ($\tau(t) = |1.5 \sin t|$) | 30 seconds | 0.8 m/s² | < 0.1 m |
| Trapezoid Formation | Random jump (0-2.5 s) | 35 seconds | 1.0 m/s² | < 0.15 m |
The convergence time is influenced by the delay bounds and switching frequency; larger delays or more frequent topology changes can slow down the drone formation’s response. However, the stability guarantees ensure that convergence will occur eventually, which is critical for safety-critical applications. My control strategy’s distributed nature means that each UAV only requires local information from neighbors, making it scalable for larger drone formations. The use of LMIs for gain tuning provides a systematic design procedure, though it may require offline computation for complex topologies.
To further illustrate the mathematical formulation, consider the error dynamics for a connected component with $f_I$ UAVs. The state vector $\varepsilon_I(t)$ evolves according to:
$$ \dot{\varepsilon}_I(t) = (M \otimes I_{f_I}) \varepsilon_I(t) + (N \otimes L^I_\sigma) \varepsilon_I(t-\tau(t)). $$
The Lyapunov-Krasovskii functional $V(t)$ is constructed to capture the effects of delays, and its derivative is bounded using integral inequalities. For instance, applying Jensen’s inequality to the term $-\int_{t-\tau_a}^{t} \dot{\varepsilon}_I^T(s) R_I \dot{\varepsilon}_I(s) \, ds$ yields:
$$ -\int_{t-\tau_a}^{t} \dot{\varepsilon}_I^T(s) R_I \dot{\varepsilon}_I(s) \, ds \leq -\frac{1}{\tau_a} \left[ \varepsilon_I(t) – \varepsilon_I(t-\tau_a) \right]^T R_I \left[ \varepsilon_I(t) – \varepsilon_I(t-\tau_a) \right]. $$
This bound is crucial for deriving the LMI condition $\Xi_I < 0$. The flexibility of this approach allows it to handle various delay patterns without requiring knowledge of $\dot{\tau}(t)$, which is often unavailable in practical drone formation networks. Moreover, the decomposition into connected components reduces the computational burden; for a drone formation with $N$ UAVs, solving LMIs for components with fewer nodes is more efficient than dealing with the full $N$-dimensional system.
In terms of implementation, the control law can be discretized for digital platforms. Assuming a sampling period $T_s$, the discrete-time version for UAV $i$ becomes:
$$ u_i[k] = \dot{\zeta}^d[k] + \sum_{j \in \mathcal{N}_i[k]} a_{ij}[k] \left\{ k_1 \left[ \xi_j[k-d[k]] – \xi_i[k-d[k]] – r_{ji} \right] + k_2 \left[ \zeta_j[k-d[k]] – \zeta_i[k-d[k]] \right] \right\} – k_3 \left[ \zeta_i[k] – \zeta^d[k] \right], $$
where $k$ is the time index, and $d[k]$ represents the discrete delay in samples. This formulation is compatible with standard microcontroller units on UAVs, enabling real-time drone formation control. The gains $k_1, k_2, k_3$ can be tuned via the LMI conditions adapted for discrete-time systems, though I focus on continuous-time analysis for brevity.
The robustness of my strategy extends to uncertainties in UAV dynamics. If the actual dynamics deviate from the nominal model due to wind gusts or payload variations, the control law can be augmented with adaptive terms. For example, let $\Delta_i(t)$ represent bounded uncertainties; then the modified dynamics are $\dot{\zeta}_i(t) = u_i(t) + \Delta_i(t)$. Using robust control techniques, such as sliding mode control, the drone formation can maintain stability. However, my current work assumes nominal dynamics to highlight the core contributions regarding delays and topology.
Comparing my approach with existing methods, such as leader-follower or virtual structure approaches, the consensus-based strategy offers greater flexibility and scalability for drone formation. Leader-follower methods rely on a single leader, whose failure can disrupt the entire formation, whereas my distributed approach ensures redundancy. Virtual structure methods require a predefined shape, which may not adapt well to dynamic environments, but my strategy allows online reconfiguration by adjusting the desired relative positions $r_{ij}$. These advantages make consensus-based control ideal for autonomous drone formation in unpredictable scenarios.
Future directions for this research include extending the framework to heterogeneous drone formations, where UAVs have different dynamics or capabilities. Additionally, incorporating obstacle avoidance and collision prevention mechanisms would enhance the practicality of the drone formation control system. Another avenue is to explore event-triggered communication to reduce network load, where UAVs only transmit data when necessary, further improving efficiency in large-scale drone formations.
In conclusion, I have presented a comprehensive study on distributed control for drone formation under bounded time-varying delays and switching topologies. The proposed control strategy, rooted in consensus theory, ensures asymptotic convergence to target formations and desired velocities. Through Lyapunov-Krasovskii analysis, I derived LMI-based sufficient conditions that accommodate fast-varying and jump delays without restricting delay derivatives. The decomposition into connected components lowers computational complexity, facilitating real-time implementation. Simulation results validated the effectiveness for both wedge and trapezoid formations, demonstrating robustness against diverse delay patterns. This work contributes to the advancement of autonomous drone formation systems, with potential applications in surveillance, logistics, and entertainment. As drone technology evolves, such control strategies will be pivotal for enabling reliable and scalable multi-UAV operations.