In this work, I present a comprehensive study on bearing-based edge event-triggered formation control for multi-drone systems operating under a leader-follower architecture. The primary objective is to address two critical challenges encountered in practical drone technology: unknown time-varying velocities and formation scaling adjustments. The proposed approach leverages relative bearing measurements and a novel distributed velocity observer that communicates only when necessary, thereby significantly reducing the communication burden while maintaining formation accuracy. I first introduce the mathematical model of the drone technology system and the necessary graph theory preliminaries. Then, I develop the edge event-triggered velocity observer and prove its asymptotic stability using Lyapunov theory. Subsequently, I design the bearing-based formation controller that ensures the tracking errors are uniformly ultimately bounded. Extensive simulation experiments, including software-in-the-loop (SITL) tests with Gazebo and PX4, validate the effectiveness of the proposed strategies. The results demonstrate that the control scheme not only achieves precise formation maneuvering under time-varying velocities and scaling commands but also exhibits remarkable communication efficiency through event-triggered updates.
1. Introduction
Unmanned aerial vehicle (UAV) swarms, often referred to as multi-drone systems, play a pivotal role in modern drone technology applications such as surveillance, search and rescue, and agricultural monitoring. Formation control is a fundamental capability that enables a group of drones to maintain a desired geometric configuration while navigating. Among various formation control approaches, bearing-based methods have attracted significant attention because they rely on relative bearing measurements, which are easily obtained from onboard cameras and are invariant to translation and scaling. However, most existing bearing-based controllers assume continuous communication or periodic sampling, which is impractical for resource-constrained drone technology. Event-triggered control offers an efficient alternative by updating control inputs only when a predefined condition is violated, thus reducing the number of actuator updates and communication messages. In this paper, I focus on a specific type of event-triggered mechanism: the edge event-triggered scheme, where each communication link (edge) in the interaction graph decides independently when to transmit information. This approach is particularly suitable for large-scale drone networks because it decentralizes the decision-making process. The main contributions of this work are threefold. First, I propose a distributed velocity observer that uses edge event-triggered communication to estimate the unknown time-varying velocities of the leaders. Second, I design a bearing-based formation controller that simultaneously achieves formation tracking and scaling by dynamic leader coordination. Third, I provide rigorous stability analysis using Lyapunov methods and validate the theoretical results through both numerical simulations and SITL experiments on a realistic drone technology platform.
2. Problem Formulation and Preliminaries
2.1 Drone Technology Model
Consider a multi-drone system consisting of n drones. The dynamics of each drone i are modeled as a double integrator:
$$ \dot{\boldsymbol{p}}_i(t) = \boldsymbol{v}_i(t), \quad \dot{\boldsymbol{v}}_i(t) = \boldsymbol{u}_i(t) $$
where \(\boldsymbol{p}_i \in \mathbb{R}^3\) denotes the position, \(\boldsymbol{v}_i \in \mathbb{R}^3\) the velocity, and \(\boldsymbol{u}_i \in \mathbb{R}^3\) the control input of the i-th drone. The actual drone technology platform (e.g., quadrotor) uses an inner-outer loop structure; here I assume that the inner attitude loop is fast enough so that the outer loop can be treated as a double integrator. The set of drones is partitioned into leaders and followers: the leaders generate a desired formation trajectory, while the followers adjust their motions to maintain the formation.
2.2 Graph Theory and Bearing Vectors
The interaction topology among drones is modeled by an undirected graph \(\mathcal{G} = (\mathcal{V}, \mathcal{E})\), where \(\mathcal{V} = \{1,2,\dots,n\}\) is the node set. let \(\mathcal{V}_l = \{1,\dots,n_l\}\) be the set of leaders and \(\mathcal{V}_f = \{n_l+1,\dots,n\}\) the set of followers. Edges between followers are undirected, while edges from a leader to a follower are directed (the follower receives information). For an edge \((i,j) \in \mathcal{E}\), define the relative position \(\boldsymbol{e}_{ij} = \boldsymbol{p}_j – \boldsymbol{p}_i\) and the unit bearing vector \(\boldsymbol{g}_{ij} = \boldsymbol{e}_{ij} / \|\boldsymbol{e}_{ij}\|\) (provided \(\|\boldsymbol{e}_{ij}\|>0\)). The orthogonal projection matrix onto the nullspace of \(\boldsymbol{g}_{ij}\) is given by \(\boldsymbol{P}_{\boldsymbol{g}_{ij}} = \boldsymbol{I}_d – \boldsymbol{g}_{ij}\boldsymbol{g}_{ij}^\top\). For 2D formations, \(d=2\).
2.3 Control Objective
Let the desired formation be defined by a set of constant bearing vectors \(\{\boldsymbol{g}_{ij}^*\}_{(i,j)\in\mathcal{E}}\). The goal is to design control laws for the follower drones such that, as time progresses, the actual bearing vectors converge to the desired ones, the velocities of the followers track the leaders’ velocities, and the formation can be scaled dynamically. In addition, the control signals should be updated only at discrete time instants determined by an event-triggered condition to reduce communication.
3. Edge Event-Triggered Distributed Velocity Observer
3.1 Observer Design
I propose a distributed velocity observer that estimates the unknown time-varying velocities of the leaders using only relative bearing information and intermittent communication. For each edge \((i,j)\), let \(t_k^{ij}\) denote the k-th triggering instant. The observer for drone i computes a local estimate \(\hat{\boldsymbol{v}}_i(t)\) as follows:
$$ \dot{\hat{\boldsymbol{v}}}_i(t) = a_1 \sum_{j \in \mathcal{N}_i} a_{ij} \left( \hat{\boldsymbol{v}}_j(t) – \hat{\boldsymbol{v}}_i(t) \right) – a_2 \sum_{j \in \mathcal{N}_i} a_{ij} \text{sgn}\left( \hat{\boldsymbol{v}}_i(t) – \boldsymbol{v}_j(t_k^{ij}) \right) $$
where \(a_1>0, a_2>0\) are design parameters, and \(\boldsymbol{v}_j(t_k^{ij})\) is the velocity of drone j transmitted at the last trigger time of edge \((i,j)\). Define the estimation error \(\tilde{\boldsymbol{v}}_i = \hat{\boldsymbol{v}}_i – \boldsymbol{v}_i\). The measurement error for edge \((i,j)\) is \(\boldsymbol{q}_{ij} = \hat{\boldsymbol{v}}_i – \boldsymbol{v}_i(t_k^{ij})\) and the edge error is \(\boldsymbol{r}_{ij} = \boldsymbol{v}_j(t_k^{ij}) – \boldsymbol{v}_i(t_k^{ij})\). The triggering condition for edge \((i,j)\) is:
$$ f_{ij}(t) = \|\boldsymbol{q}_{ij}\|^2 + \frac{a_1}{a_2} \|\boldsymbol{q}_{ij}\|^2 – \frac{2a_1}{a_2} \|\boldsymbol{r}_{ij}\|^2 – d_1 > 0 $$
with a positive constant \(d_1\). The next trigger instant is defined as \(t_{k+1}^{ij} = \inf\{t > t_k^{ij} : f_{ij}(t) > 0\}\). This mechanism ensures that communication events occur only when the estimation quality degrades.
3.2 Stability Analysis
I consider the Lyapunov function \(V_v = \frac{1}{2}\sum_{i=1}^n \tilde{\boldsymbol{v}}_i^\top \tilde{\boldsymbol{v}}_i\). Its derivative along the observer dynamics, after using Young’s inequality and the triggering condition, leads to:
$$ \dot{V}_v \leq -a_1 \lambda_{\min}(\mathcal{L}) V_v + \beta $$
where \(\mathcal{L}\) is the Laplacian matrix of the graph, \(\beta\) is a bounded term related to the triggering threshold. Since \(\beta\) is constant, \(V_v\) converges exponentially to a small neighborhood of zero. Thus, the velocity estimation errors are uniformly ultimately bounded, and the bound can be made arbitrarily small by choosing appropriate observer gains.
3.3 Parameter Selection for Observer
| Parameter | Symbol | Value | Description |
|---|---|---|---|
| Coupling gain 1 | \(a_1\) | 10 | Drives consensus among estimates |
| Coupling gain 2 | \(a_2\) | 0.1 | Robustness to unknown velocities |
| Trigger threshold | \(d_1\) | 0.01 | Controls triggering frequency |
4. Bearing-Based Formation Controller
4.1 Controller Design Using Bearing Error
Define the bearing error for edge \((i,j)\) as \(\boldsymbol{e}_{g,ij} = \boldsymbol{g}_{ij} – \boldsymbol{g}_{ij}^*\), and the velocity tracking error \(\boldsymbol{e}_{v,i} = \hat{\boldsymbol{v}}_i – \boldsymbol{v}_i\). I construct a virtual control signal:
$$ \boldsymbol{u}_{v,i} = -k_1 \boldsymbol{J}_i^\top \boldsymbol{\tilde{g}} $$
where \(\boldsymbol{J} = \text{diag}(\boldsymbol{P}_{\boldsymbol{g}_{k}}) \cdot (\boldsymbol{H} \otimes \boldsymbol{I}_d)\) is the Jacobian matrix of the bearing vector stack with respect to the positions, and \(\boldsymbol{\tilde{g}}\) is the stack of all bearing errors. Let \(\boldsymbol{\alpha}_i = \boldsymbol{e}_{v,i} – \boldsymbol{u}_{v,i}\). Then the final control input for drone i is designed as:
$$ \boldsymbol{u}_i = -k_2 \boldsymbol{\alpha}_i – \boldsymbol{\dot{u}}_{v,i} + \dot{\hat{\boldsymbol{v}}}_i + \boldsymbol{J}_i^\top \boldsymbol{\tilde{g}} $$
where \(k_2>0\) is another gain. The combination of these terms ensures that the overall Lyapunov function candidate \(V_\alpha = V_g + \frac{1}{2}\sum_i \boldsymbol{\alpha}_i^\top \boldsymbol{\alpha}_i\) (with \(V_g = \frac{1}{2}\sum_{(i,j)\in\mathcal{E}} \boldsymbol{e}_{g,ij}^\top \boldsymbol{e}_{g,ij}\)) satisfies:
$$ \dot{V}_\alpha \leq -\eta V_\alpha + \delta $$
with \(\eta = \min\{k_1, \beta\}\) and \(\delta = \frac{1}{2}\lambda_{\max}(\boldsymbol{J}\boldsymbol{J}^\top)\|\tilde{\boldsymbol{v}}\|^2\). Provided that the observer error \(\tilde{\boldsymbol{v}}\) is bounded, the formation errors are uniformly ultimately bounded. In particular, if the observer converges, the formation errors converge to zero.
4.2 Controller Parameters
| Parameter | Symbol | Value | Description |
|---|---|---|---|
| Virtual control gain | \(k_1\) | 10 | Convergence speed for bearing error |
| Damping gain | \(k_2\) | 15 | Stabilizes velocity tracking |
5. Simulation Results
5.1 Numerical Simulations
I conducted two sets of numerical simulations: formation scaling and time-varying velocity tracking. The drone team consists of two leaders (drones 1 and 2) and two followers (drones 3 and 4). The desired formation is a square with unit side length. Below is the summary of initial conditions and leader velocity profiles.
| Drone | Initial Position (m) | Role |
|---|---|---|
| 1 | (8,8) | Leader |
| 2 | (8,0) | Leader |
| 3 | (3,12) | Follower |
| 4 | (-2,-2) | Follower |
| Method | Follower 3 | Follower 4 |
|---|---|---|
| Edge event-triggered (proposed) | 221 | 216 |
| State event-triggered | 276 | 281 |
| Time event-triggered | 263 | 275 |
The results show that the proposed edge event-triggered scheme reduces the number of communication events by approximately 20% compared to the state-based and time-based approaches, while maintaining comparable formation accuracy. The bearing errors converge to near zero, and the velocity estimates accurately track the leaders’ velocities. The trigger instants are more frequent during transient phases and become sparse when the system reaches steady state, confirming the efficiency of the mechanism.
5.2 Software-in-the-Loop (SITL) Simulation
To further validate the proposed control strategy in a realistic drone technology environment, I implemented the algorithm in a SITL framework using Gazebo, PX4 autopilot, and ROS. The same four-drone formation scenario was used, but with leader velocities varying as a sinusoidal function to test time-varying tracking. The simulation ran for 90 s. The formation was successfully maintained: the geometric shape remained intact, the bearing errors were bounded within \([-0.05, 0.05]\) rad, and the velocity estimates converged to the actual velocities after an initial transient. The number of edge event triggers per follower was approximately 450, while the state-based method required over 600 triggers. This demonstrates that the edge event-triggered observer significantly reduces communication load without degrading performance.
6. Conclusion
In this paper, I have developed a bearing-based edge event-triggered formation control strategy for multi-drone systems subject to unknown time-varying velocities and formation scaling. The key contributions include a distributed velocity observer that uses only intermittent edge-level communication, a bearing formation controller that adjusts scaling via leader coordination, and rigorous Lyapunov-based stability proofs. The numerical and SITL simulations confirm that the proposed drone technology solution achieves accurate formation tracking while reducing communication events by up to 20% compared to conventional event-triggered schemes. Future work will extend the results to 3D formations, consider communication delays, and incorporate obstacle avoidance, further enhancing the applicability of edge event-triggered control in advanced drone technology.