The coordination of multiple unmanned aerial vehicles (UAVs), or drones, into cohesive groups represents a critical frontier in autonomous systems. Moving beyond the capabilities of a single agent, multi-drone systems offer unparalleled advantages in mission redundancy, operational efficiency, and the ability to execute complex tasks over wide areas. Among the core challenges in multi-agent systems, the drone formation control problem—guiding a group of drones to achieve and maintain a specific geometric configuration while following a collective trajectory—has garnered intense research interest. Early approaches often relied on centralized architectures, where a single leader or ground station computed control commands for all followers. While theoretically straightforward, such methods suffer from single points of failure, high communication bandwidth demands, and poor scalability.
In contrast, distributed control strategies, where each drone makes decisions based only on information from its local neighbors, offer superior robustness, flexibility, and scalability. Consensus theory has emerged as a powerful framework for such distributed coordination. The fundamental idea is simple yet profound: by designing local interaction rules, all agents can asymptotically agree upon a common state value, such as position offset or velocity. When applied to drone formation, consensus algorithms enable the fleet to converge to a desired spatial pattern and a common velocity using only limited, local communication.
However, the idealized models often used in consensus theory frequently neglect a pervasive real-world constraint: communication time delays. In practical drone formation networks, delays arise from signal processing, packet routing, and limited bandwidth. These delays, if not explicitly accounted for in the control design, can degrade performance, cause oscillations, and even lead to the instability of the entire swarm. Therefore, developing distributed formation control protocols that are robust to time-varying delays is not merely an academic exercise but a necessary step toward reliable real-world deployment.
This article addresses the distributed drone formation control problem under directed, fixed communication topologies with time-varying delays. We consider a nonlinear kinematic model for the drones, which is more representative of real vehicle dynamics than simple integrator models. By leveraging high-order consensus theory and Lyapunov stability analysis, we design control protocols and derive sufficient stability conditions expressed as Linear Matrix Inequalities (LMIs). We analyze two practical scenarios regarding delay information: one where a bound on the delay’s rate of change is known, and another where only the bound on the delay itself is available. Finally, numerical simulations validate the effectiveness of the proposed approach, demonstrating that a fleet of drones can successfully achieve the desired formation and velocity despite the presence of disruptive communication delays.
Drone Dynamical Model and Mathematical Preliminaries
We consider a team of n drones operating in a two-dimensional plane at a fixed altitude, eliminating the need for vertical coordination and collision avoidance in the altitude channel. The kinematic and dynamic model for the i-th drone is often described by nonlinear equations. Let us define the state vector for drone i as $[x_i, y_i, \theta_i, v_i, \omega_i, F_i]^T$, where $(x_i, y_i)$ is its Cartesian position, $\theta_i$ is its heading angle (measured from the positive x-axis), $v_i$ is its linear speed, $\omega_i$ is its angular velocity, and $F_i$ is the applied force. The dynamics can be given by:
$$
\begin{align*}
\dot{x}_i &= v_i \cos\theta_i \\
\dot{y}_i &= v_i \sin\theta_i \\
\dot{\theta}_i &= \omega_i \\
\dot{v}_i &= \frac{F_i}{m_i} \\
\dot{\omega}_i &= \frac{\tau_i}{J_i} \\
\dot{F}_i &= \alpha_i
\end{align*}
$$
where $m_i$ is the mass, $J_i$ is the moment of inertia, and $[\tau_i, \alpha_i]^T$ are the control inputs (torque and force rate, respectively). For the purpose of consensus-based formation control, it is convenient to transform this model into a chain of integrators by treating the position in each axis as the output of a triple-integrator system. Defining $\boldsymbol{\xi}_i(t) = [x_i, y_i]^T \in \mathbb{R}^{2}$ and $\mathbf{u}_i = [u_{x_i}, u_{y_i}]^T \in \mathbb{R}^{2}$ as a virtual control input, we can derive the following relation through successive differentiation:
$$
\boldsymbol{\xi}_i^{(3)}(t) = \mathbf{u}_i(t)
$$
This third-order model, $\boldsymbol{\xi}_i^{(3)}=\mathbf{u}_i$, is crucial for high-order consensus design. It explicitly considers position ($\boldsymbol{\xi}_i$), velocity ($\dot{\boldsymbol{\xi}}_i$), and acceleration ($\ddot{\boldsymbol{\xi}}_i$) as state variables to be coordinated. The actual low-level control inputs $(\tau_i, \alpha_i)$ can be recovered from the virtual control $\mathbf{u}_i$ via a nonlinear transformation, ensuring the theoretical design is applicable to the real drone dynamics.
The communication network among the drones is modeled as a directed, fixed graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$. The vertex set $\mathcal{V}=\{v_1,…,v_n\}$ represents the drones. An edge $e_{ij} = (v_i, v_j) \in \mathcal{E}$ exists if drone $i$ can transmit information to drone $j$. The adjacency matrix $\mathcal{A}=[a_{ij}]$ is defined with $a_{ii}=0$ and $a_{ij}=1$ if $e_{ij} \in \mathcal{E}$, otherwise $a_{ij}=0$. The Laplacian matrix $\mathbf{L}=[l_{ij}]$ of the graph is defined as $l_{ii} = \sum_{j \neq i} a_{ij}$ and $l_{ij} = -a_{ij}$ for $i \neq j$. The neighbor set of drone $i$ is denoted by $\mathcal{N}_i = \{v_j : e_{ji} \in \mathcal{E}\}$.
The time-varying communication delay is denoted by $\tau(t)$. We investigate two realistic cases concerning the available information about this delay:
- Case 1 (Delay Derivative Information): The delay is bounded and its rate of change is also bounded: $0 \leq \tau(t) \leq T$, $0 \leq \dot{\tau}(t) \leq d < 1$.
- Case 2 (No Delay Derivative Information): Only the bound on the delay itself is known: $0 \leq \tau(t) \leq T$.
The formation objective is defined relative to a formation centroid. Let $\boldsymbol{\xi}_0(t)$ be the time-varying position of this virtual centroid and $\boldsymbol{\xi}^*$ be the desired constant velocity for the entire formation. Each drone $i$ has a desired constant offset $\boldsymbol{\Delta}_i \in \mathbb{R}^2$ relative to this centroid. The control goal is to achieve:
$$
\lim_{t \to \infty} (\boldsymbol{\xi}_i(t) – \boldsymbol{\xi}_0(t)) = \boldsymbol{\Delta}_i, \quad \text{and} \quad \lim_{t \to \infty} \dot{\boldsymbol{\xi}}_i(t) = \boldsymbol{\xi}^*, \quad \forall i.
$$
This means the drones asymptotically arrange themselves in the prescribed geometric pattern $\{\boldsymbol{\Delta}_1, \boldsymbol{\Delta}_2, …, \boldsymbol{\Delta}_n\}$ while all moving at the desired velocity $\boldsymbol{\xi}^*$.
| Symbol | Description |
|---|---|
| $\boldsymbol{\xi}_i(t)$ | Position vector of drone $i$ |
| $\boldsymbol{\xi}_0(t)$ | Position of the formation centroid |
| $\boldsymbol{\Delta}_i$ | Desired position offset for drone $i$ in the formation |
| $\boldsymbol{\xi}^*$ | Desired common velocity for the formation |
| $\mathbf{u}_i(t)$ | Virtual control input for drone $i$ |
| $\tau(t)$ | Time-varying communication delay |
| $T$ | Upper bound on the communication delay |
| $d$ | Upper bound on the derivative of the delay (Case 1) |
| $\mathcal{G}$ | Directed communication graph |
| $\mathbf{L}$ | Laplacian matrix of graph $\mathcal{G}$ |
| $\mathcal{N}_i$ | Set of neighbors of drone $i$ |
High-Order Consensus Protocol Design for Delayed Drone Formation
To achieve the formation objective under delayed communications, we design a distributed control protocol based on high-order consensus principles. We first define the error states for drone $i$ that incorporate its formation offset and velocity target:
$$
\begin{align*}
\tilde{\boldsymbol{\xi}}_i(t) &= \boldsymbol{\xi}_i(t) – \boldsymbol{\xi}_0(t) – \boldsymbol{\Delta}_i \\
\tilde{\boldsymbol{\xi}}_i^{(1)}(t) &= \dot{\boldsymbol{\xi}}_i(t) – \boldsymbol{\xi}^* \\
\tilde{\boldsymbol{\xi}}_i^{(2)}(t) &= \ddot{\boldsymbol{\xi}}_i(t)
\end{align*}
$$
The goal is now equivalent to driving these error states to zero for all drones: $\lim_{t \to \infty} \tilde{\boldsymbol{\xi}}_i^{(k)}(t) = 0$ for $k=0,1,2$. The proposed distributed protocol for the virtual control input $\mathbf{u}_i(t)$ uses delayed neighbor information and is given by:
$$
\begin{aligned}
\mathbf{u}_i(t) &= \sum_{j \in \mathcal{N}_i} k_1 \left[ \tilde{\boldsymbol{\xi}}_j(t-\tau(t)) – \tilde{\boldsymbol{\xi}}_i(t-\tau(t)) \right] \\
&+ \sum_{j \in \mathcal{N}_i} k_2 \left[ \tilde{\boldsymbol{\xi}}_j^{(1)}(t-\tau(t)) – \tilde{\boldsymbol{\xi}}_i^{(1)}(t-\tau(t)) \right] \\
&+ k_3 \tilde{\boldsymbol{\xi}}_i^{(2)}(t-\tau(t)) + k_4 h_i \tilde{\boldsymbol{\xi}}_i^{(1)}(t-\tau(t))
\end{aligned}
$$
where $k_1, k_2, k_3, k_4 > 0$ are scalar control gains to be designed. The term $h_i$ is an indicator: $h_i = 1$ if drone $i$ has access to the desired velocity $\boldsymbol{\xi}^*$ (it is a “informed” agent), otherwise $h_i=0$. This allows the protocol to work even if only a subset of drones know the target velocity, enhancing robustness. The protocol uses consensus terms for position ($k_1$) and velocity ($k_2$) errors, a damping term on its own acceleration ($k_3$), and a tracking term for the desired velocity ($k_4$). All feedback is based on delayed state information due to $\tau(t)$.
To analyze the closed-loop system, we stack the error states of all $n$ drones into a global vector. Let us define a combined state vector for analysis (considering one dimension; the analysis extends to both x and y axes independently due to decoupling):
$$
\mathbf{\varepsilon}(t) = \left[ \tilde{\xi}_1, …, \tilde{\xi}_n, \tilde{\xi}_1^{(1)}, …, \tilde{\xi}_n^{(1)}, \tilde{\xi}_1^{(2)}, …, \tilde{\xi}_n^{(2)} \right]^T \in \mathbb{R}^{3n}
$$
Using this state vector, the dynamics of the entire multi-drone system under the proposed protocol can be written in a compact delayed state-space form:
$$
\dot{\mathbf{\varepsilon}}(t) = \mathbf{A} \mathbf{\varepsilon}(t) + \mathbf{B} \mathbf{\varepsilon}(t-\tau(t))
$$
where the system matrices are:
$$
\mathbf{A} = \begin{bmatrix}
\mathbf{0} & \mathbf{I}_n & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{I}_n \\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{bmatrix}, \quad
\mathbf{B} = \begin{bmatrix}
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
-k_1 \mathbf{L} & -k_2 \mathbf{L} + k_4 \mathbf{H} & k_3 \mathbf{I}_n
\end{bmatrix}
$$
Here, $\mathbf{I}_n$ is the $n \times n$ identity matrix, $\mathbf{L}$ is the graph Laplacian matrix, and $\mathbf{H} = \text{diag}(h_1, h_2, …, h_n)$ is the diagonal matrix of velocity access indicators. The stability of the drone formation control system is now equivalent to the asymptotic stability of the delayed linear system $\dot{\mathbf{\varepsilon}}(t) = \mathbf{A} \mathbf{\varepsilon}(t) + \mathbf{B} \mathbf{\varepsilon}(t-\tau(t))$ at the origin.
Stability Analysis via Lyapunov-Krasovskii Theory and LMIs
We employ the Lyapunov-Krasovskii functional method to derive stability conditions for the delayed closed-loop system. The conditions will be expressed as Linear Matrix Inequalities (LMIs), which are numerically tractable and allow us to compute maximum allowable delay bounds ($T$) for given control gains and network topology. A key tool in this analysis is the following lemma, which provides an integral inequality to bound cross terms arising from the delay.
Lemma (Integral Inequality): For any continuously differentiable vector function $\mathbf{\varepsilon}(t) \in \mathbb{R}^{m}$ and any constant positive definite symmetric matrix $\mathbf{W} \succ 0$, the following inequality holds for $t \geq 0$ and $0 \leq \tau(t) \leq T$:
$$
T^{-1} [\mathbf{\varepsilon}(t) – \mathbf{\varepsilon}(t-\tau(t))]^T \mathbf{W} [\mathbf{\varepsilon}(t) – \mathbf{\varepsilon}(t-\tau(t))] \leq \int_{t-\tau(t)}^{t} \dot{\mathbf{\varepsilon}}^T(s) \mathbf{W} \dot{\mathbf{\varepsilon}}(s) ds
$$
We now present the main stability theorems for the two delay cases in the context of drone formation control.
Theorem 1 (Stability with Delay Derivative Information): Consider the multi-drone system under the protocol leading to dynamics $\dot{\mathbf{\varepsilon}}(t) = \mathbf{A} \mathbf{\varepsilon}(t) + \mathbf{B} \mathbf{\varepsilon}(t-\tau(t))$. Assume the time-varying delay $\tau(t)$ satisfies Case 1: $0 \leq \tau(t) \leq T$ and $0 \leq \dot{\tau}(t) \leq d < 1$. The system is asymptotically stable (and hence the drone formation is achieved) if there exist symmetric positive definite matrices $\mathbf{P} \succ 0$, $\mathbf{Q} \succ 0$, $\mathbf{R} \succ 0$ of appropriate dimensions such that the following LMI holds:
$$
\begin{bmatrix}
\boldsymbol{\Omega}_{11} & \boldsymbol{\Omega}_{12} \\
* & \boldsymbol{\Omega}_{22}
\end{bmatrix} \prec 0
$$
where
$$
\begin{aligned}
\boldsymbol{\Omega}_{11} &= \mathbf{PA} + \mathbf{A}^T\mathbf{P} + \mathbf{Q} + T \mathbf{A}^T\mathbf{R}\mathbf{A} – T^{-1} \mathbf{R} \\
\boldsymbol{\Omega}_{12} &= \mathbf{PB} + T \mathbf{A}^T\mathbf{R}\mathbf{B} + T^{-1} \mathbf{R} \\
\boldsymbol{\Omega}_{22} &= (d-1)\mathbf{Q} + T \mathbf{B}^T\mathbf{R}\mathbf{B} – T^{-1} \mathbf{R}
\end{aligned}
$$
Proof Sketch: The proof constructs a Lyapunov-Krasovskii functional candidate:
$$
V(t) = \mathbf{\varepsilon}^T(t) \mathbf{P} \mathbf{\varepsilon}(t) + \int_{t-\tau(t)}^{t} \mathbf{\varepsilon}^T(s) \mathbf{Q} \mathbf{\varepsilon}(s) ds + \int_{t-T}^{t} (s-t+T) \dot{\mathbf{\varepsilon}}^T(s) \mathbf{R} \dot{\mathbf{\varepsilon}}(s) ds
$$
Calculating its time derivative $\dot{V}(t)$, applying the Lemma to handle the integral of $\dot{\mathbf{\varepsilon}}^T \mathbf{R} \dot{\mathbf{\varepsilon}}$, and using the bounds on $\tau(t)$ and $\dot{\tau}(t)$ leads to an expression of the form $\dot{V}(t) \leq \mathbf{y}^T(t) \boldsymbol{\Omega} \mathbf{y}(t)$, where $\mathbf{y}(t)=[\mathbf{\varepsilon}^T(t), \mathbf{\varepsilon}^T(t-\tau(t))]^T$. The negative definiteness of $\boldsymbol{\Omega}$ ensures $\dot{V}(t) < 0$, guaranteeing asymptotic stability.
Theorem 2 (Stability without Delay Derivative Information): Consider the same system, but now with the delay satisfying only Case 2: $0 \leq \tau(t) \leq T$. The system is asymptotically stable if there exist symmetric positive definite matrices $\mathbf{P} \succ 0$, $\mathbf{R} \succ 0$ such that the following LMI holds:
$$
\begin{bmatrix}
\mathbf{PA} + \mathbf{A}^T\mathbf{P} + T \mathbf{A}^T\mathbf{R}\mathbf{A} – T^{-1} \mathbf{R} & \mathbf{PB} + T \mathbf{A}^T\mathbf{R}\mathbf{B} + T^{-1} \mathbf{R} \\
* & T \mathbf{B}^T\mathbf{R}\mathbf{B} – T^{-1} \mathbf{R}
\end{bmatrix} \prec 0
$$
Proof Sketch: For this case, a simplified Lyapunov-Krasovskii functional is used, omitting the term with $\mathbf{Q}$ which requires knowledge of $\dot{\tau}(t)$:
$$
V(t) = \mathbf{\varepsilon}^T(t) \mathbf{P} \mathbf{\varepsilon}(t) + \int_{t-T}^{t} (s-t+T) \dot{\mathbf{\varepsilon}}^T(s) \mathbf{R} \dot{\mathbf{\varepsilon}}(s) ds
$$
The derivative is then bounded similarly, leading to the LMI condition in Theorem 2.
These theorems provide concrete, computable criteria. For a given network topology ($\mathbf{L}$, $\mathbf{H}$), control gains ($k_1, k_2, k_3, k_4$), and delay bound $d$ (for Theorem 1), one can solve the corresponding LMI feasibility problem to find the maximum allowable delay upper bound $T_{max}$. This $T_{max}$ represents a robustness margin for the drone formation control system: as long as the actual delay $\tau(t)$ is always less than $T_{max}$, stability is guaranteed.
| Condition Case | Required Delay Knowledge | Key LMI Variables | Practical Implication |
|---|---|---|---|
| Case 1 | Bounds on $\tau(t)$ and $\dot{\tau}(t)$ ($T$, $d$) | $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ | Tighter bound; can tolerate faster-varying delays if $d$ is known to be small. |
| Case 2 | Bound on $\tau(t)$ only ($T$) | $\mathbf{P}, \mathbf{R}$ | More conservative bound; robust to any delay variation within $[0, T]$. |
Simulation Verification and Results
To validate the proposed distributed control method for drone formation with time-varying delays, we conducted a numerical simulation with five drones. The directed communication topology is fixed and defined by the following neighbor sets: $\mathcal{N}_1=\{2\}$, $\mathcal{N}_2=\{3,5\}$, $\mathcal{N}_3=\{4\}$, $\mathcal{N}_4=\{1\}$, $\mathcal{N}_5=\{3\}$. This implies that only drones 2 and 5 have access to the desired velocity ($h_2=h_5=1$). The desired formation is a pentagon shape with specific offsets $\boldsymbol{\Delta}_i$, and the common target velocity is $\boldsymbol{\xi}^* = [50\cos(45^\circ), 50\sin(45^\circ)]^T$ m/s.
The control gains were selected as $k_1=2$, $k_2=2$, $k_3=2.5$, $k_4=4$. The time-varying delay was modeled as $\tau(t) = 0.15\sin^2(2t)$ seconds, ensuring it remains within $T=0.15$s. Its derivative satisfies $\dot{\tau}(t) \leq 0.3 < 1$. Solving the LMI in Theorem 1 with $d=0.5$ confirmed that $T=0.175$s is an allowable upper bound, making our chosen $\tau(t)$ feasible for stable control.
| Drone ID | Initial Position (x, y) [m] | Initial Velocity [m/s] | Initial Heading [deg] |
|---|---|---|---|
| 1 | (10, 590) | 93 | 74 |
| 2 | (8, 610) | 98 | 51 |
| 3 | (6, 700) | 83 | 39 |
| 4 | (10, 630) | 80 | 23 |
| 5 | (9, 550) | 75 | 12 |
The simulation results clearly demonstrate the effectiveness of the protocol. The position trajectories show the drones starting from dispersed locations and smoothly converging to the predefined pentagonal formation while moving along the desired $45^\circ$ path. The velocity profiles of all five drones converge to the target magnitude of 50 m/s, despite their different initial speeds. Similarly, the heading angles all asymptotically approach $45^\circ$. The convergence is asymptotic and stable, with transient oscillations damped out over time. Crucially, this successful coordination is achieved using only local, delayed information exchanges as specified by the communication graph. The drones successfully maintain the formation shape while translating at the common desired velocity, proving that the proposed method effectively solves the distributed drone formation control problem in the presence of time-varying communication delays.
Conclusion and Future Directions
This article has presented a comprehensive framework for distributed drone formation control that is robust to realistic time-varying communication delays. By adopting a high-order consensus approach grounded in the triple-integrator model of drone dynamics, we developed a distributed protocol that requires only local neighbor information. The core theoretical contribution lies in the rigorous stability analysis using Lyapunov-Krasovskii functionals and LMI techniques, which yielded explicit, computable conditions for two practical scenarios: with and without knowledge of the delay’s rate of change. These conditions provide clear guidelines for selecting control gains and determining the maximum tolerable delay to ensure stable formation flight.
The simulation of a five-drone swarm confirmed that the method successfully guides the agents to the desired geometric configuration and common velocity, even when communications are subject to significant and varying delays. The distributed nature of the strategy enhances the system’s scalability and resilience compared to centralized approaches.
Future research can extend this work in several promising directions. First, investigating switching communication topologies, where the network links between drones can appear or disappear over time, would significantly increase the practical applicability for dynamic environments. Second, incorporating more detailed and nonlinear drone dynamics, including actuator saturation and aerodynamic disturbances, would strengthen the bridge between theory and real-world implementation. Third, developing adaptive or learning-based techniques to tune the control gains online in response to varying delay characteristics or unknown network parameters could further improve robustness. Finally, extending the formulation to three-dimensional drone formation and incorporating obstacle avoidance constraints are essential steps for deploying such algorithms in complex operational scenarios. The pursuit of reliable, distributed, and delay-tolerant control laws remains fundamental to unlocking the full potential of autonomous multi-drone systems.
| Method | Communication | Robustness to Delay | Scalability | Model Complexity |
|---|---|---|---|---|
| Centralized Leader-Follower | Global (to leader) | Low (single point of failure) | Poor | Low |
| Behavior-Based (e.g., Potential Fields) | Local (Implicit) | Moderate (emergent behavior) | Good | Medium/High |
| Consensus-Based (1st/2nd Order) | Local | Moderate (often assumes no delay) | Excellent | Low |
| Proposed High-Order Consensus | Local | High (explicitly models delay) | Excellent | Medium |