We address the problem of bearing-based formation control for multi-rotor unmanned aerial vehicles (UAVs), a crucial topic for China drone swarming technology in complex environments. The goal is to achieve formation tracking with unknown time-varying velocities and dynamic formation scaling, while significantly reducing communication overhead. In this work, we propose a novel distributed control strategy that combines an edge-event-triggered velocity observer with a bearing-based formation controller. The system adopts a leader–follower architecture, where leaders define the desired trajectory and formation scale, and followers use only relative bearing measurements to maintain the formation shape. Our approach guarantees asymptotic stability in velocity estimation and ultimately bounded formation errors, as demonstrated through Lyapunov analysis and comprehensive simulations. The results confirm the practicality of the method for real-time China drone operations, especially in scenarios requiring limited communication bandwidth.
1. Introduction
Formation control of multi-UAV systems has attracted considerable attention in both military and civilian domains, including surveillance, search-and-rescue, and agricultural mapping. For China drone industry, scalable and communication-efficient formation algorithms are essential to deploy large swarms in GPS-denied or contested environments. Among existing formation strategies, bearing-based approaches have distinct advantages: they are invariant to translation and rotation, require only local bearing measurements (often from monocular cameras), and naturally support formation scaling. However, most prior works assume continuous communication or constant leader velocities, which limits their applicability to real digital control systems.
To overcome these limitations, we introduce an edge-event-triggered mechanism that decides when each edge (communication link) should transmit velocity information. This dramatically reduces the number of data exchanges while preserving control performance. We further design a distributed velocity observer that allows each follower to estimate the leader’s unknown time-varying velocity using only intermittent communication. The observer is coupled with a bearing-only controller that corrects formation errors and adapts to scaling changes commanded by the leaders. This integrated framework is, to the best of our knowledge, the first to combine edge-event-triggered communication, bearing formation scaling, and velocity estimation for second-order UAV dynamics.
2. Problem Formulation
Consider a multi-UAV system comprising n agents, among which nl are leaders (indices 1 to nl) and nf are followers (indices nl+1 to n). Each agent obeys the double-integrator dynamics in 2D (extension to 3D is straightforward):
$$
\begin{aligned}
\dot{\mathbf{p}}_i(t) &= \mathbf{v}_i(t),\\
\dot{\mathbf{v}}_i(t) &= \mathbf{u}_i(t),\quad i \in \{1,\dots,n\},
\end{aligned}
$$
where \(\mathbf{p}_i\in\mathbb{R}^2\) is the position, \(\mathbf{v}_i\in\mathbb{R}^2\) the velocity, and \(\mathbf{u}_i\in\mathbb{R}^2\) the control input. The target formation shape is defined by a set of desired bearing vectors \(\mathbf{g}_{ij}^*\) for each edge \((i,j)\) in the fixed undirected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\). The leaders move with unknown time-varying velocity \(\mathbf{v}_l(t)\) and command a scaling factor \(\alpha(t)\) that uniformly expands or contracts the formation.
The only information available to a follower i is the relative bearing \(\mathbf{g}_{ij} = (\mathbf{p}_j – \mathbf{p}_i)/\|\mathbf{p}_j – \mathbf{p}_i\|\) to its neighbors \(j\in\mathcal{N}_i\), and the intermittent velocity estimate \(\hat{\mathbf{v}}_j(t)\) transmitted when an event triggers. The control objective is to realize the desired formation, i.e.,
$$
\mathbf{g}_{ij}(t) \rightarrow \mathbf{g}_{ij}^*,\quad \mathbf{v}_i(t) \rightarrow \alpha(t)\mathbf{v}_l(t),\quad \forall (i,j)\in\mathcal{E},
$$
while minimizing communication events. This paper focuses on China drone applications where communication bandwidth is precious and GPS signals may be degraded.
3. Edge-Event-Triggered Velocity Observer
To estimate the leader’s time-varying velocity without continuous communication, we designed a distributed observer that updates only when a local event condition is violated. Define the velocity estimation error as \(\tilde{\mathbf{v}}_i = \hat{\mathbf{v}}_i – \mathbf{v}_i\), and the measurement error for edge \((i,j)\) as \(\mathbf{q}_{ij} = \hat{\mathbf{v}}_i – \mathbf{v}_{ij}\) and \(\mathbf{r}_{ij} = \hat{\mathbf{v}}_i – \hat{\mathbf{v}}_j\). The observer dynamics are given by:
$$
\dot{\hat{\mathbf{v}}}_i = -a_1\sum_{j\in\mathcal{N}_i} a_{ij}\bigl(\hat{\mathbf{v}}_i – \hat{\mathbf{v}}_j\bigr) – a_2\sum_{j\in\mathcal{N}_i} a_{ij}\operatorname{sgn}\bigl(\hat{\mathbf{v}}_i – \hat{\mathbf{v}}_j\bigr),
$$
where \(a_1>0\), \(a_2>0\) are gains, and \(a_{ij}=a_{ji}\) is the adjacency weight. The event-triggering function for edge \((i,j)\) is defined as:
$$
f_{ij}(t) = \|\mathbf{q}_{ij}\|^2 + \|\mathbf{r}_{ij}\|^2 – \frac{3}{2}a_1^{-1}a_2\|\mathbf{q}_{ij}\|^2 – \frac{1}{2}a_2\|\mathbf{r}_{ij}\|^2 – d_1,
$$
with \(d_1>0\). An event is triggered at time \(t_k^{ij}\) when \(f_{ij}(t)>0\). At that instant, both agents exchange their current velocity estimates.
The stability of the observer is established via the Lyapunov function \(V_v = \frac12\sum_{i=1}^n \tilde{\mathbf{v}}_i^\top \tilde{\mathbf{v}}_i\). Under the event condition, its derivative satisfies:
$$
\dot{V}_v \leq – \alpha_1 V_v + \beta_1,
$$
leading to exponential convergence of \(\tilde{\mathbf{v}}_i\) to a bounded residual. The design parameters must satisfy \(a_2 > 2\lambda_{\min}(\mathbf{L})\) where \(\mathbf{L}\) is the Laplacian matrix of the graph. Theorem 1 in our earlier work (based on the original paper) provides the rigorous proof.
Table 1 summarizes the key symbols used in the observer.
| Symbol | Description |
|---|---|
| \(\hat{\mathbf{v}}_i\) | Estimated velocity of agent i |
| \(a_1, a_2\) | Observer gains |
| \(\mathbf{q}_{ij}\) | Measurement error of agent i w.r.t. edge |
| \(\mathbf{r}_{ij}\) | Relative estimation error between i and j |
| \(f_{ij}\) | Event-triggering function |
| \(d_1\) | Positive threshold constant |
4. Bearing-Based Formation Controller
Using the estimated velocities from the observer, we design a low-level controller that drives the bearing errors to zero while ensuring velocity tracking. Define the bearing error for edge \((i,j)\) as:
$$
\mathbf{e}_{ij}^g = \mathbf{g}_{ij} – \mathbf{g}_{ij}^*,
$$
and the velocity tracking error as \(\mathbf{e}_i^v = \hat{\mathbf{v}}_i – \mathbf{v}_i\). Stack these errors into vectors \(\mathbf{e}_g\) and \(\mathbf{e}_v\). The bearing Jacobian matrix \(\mathbf{J}\) relates the time derivative of the bearing errors to the velocities:
$$
\dot{\mathbf{e}}_g = \mathbf{J}\tilde{\mathbf{v}} + \mathbf{J}\hat{\mathbf{v}}.
$$
We introduce a virtual control \(\mathbf{u}_v = -k_1\mathbf{J}^\top\mathbf{e}_g\) and define an auxiliary error \(\mathbf{e}_\alpha = \mathbf{e}_v – \mathbf{u}_v\). The final control law for each follower is:
$$
\mathbf{u}_i = -k_2\mathbf{e}_v – \mathbf{J}^\top\mathbf{e}_g + \dot{\hat{\mathbf{v}}} + \dot{\mathbf{u}}_v,
$$
where \(k_1>0\) and \(k_2>0\) are controller gains that must satisfy \(k_2 > \lambda_{\max}(\overline{\mathbf{J}})/\bigl(2\lambda_{\min}(\overline{\mathbf{J}})\bigr)\) with \(\overline{\mathbf{J}} = \mathbf{J}^\top\mathbf{J}\). The closed-loop system yields a Lyapunov function \(V_\alpha = V_g + \frac12 \mathbf{e}_\alpha^\top\mathbf{e}_\alpha\) whose derivative is bounded by:
$$
\dot{V}_\alpha \leq -\eta V_\alpha + \delta,
$$
with \(\eta = \min\{k_1,\; k_2 – \lambda_{\max}(\overline{\mathbf{J}})/(2\lambda_{\min}(\overline{\mathbf{J}}))\}\) and \(\delta = \frac12\lambda_{\max}(\overline{\mathbf{J}})\|\tilde{\mathbf{v}}\|^2\). This proves that bearing errors and velocity tracking errors are uniformly ultimately bounded. Theorem 2 details the convergence condition.
Table 2 lists the controller parameters and their roles.
| Parameter | Meaning | Typical value |
|---|---|---|
| \(k_1\) | Bearing error gain | 10 |
| \(k_2\) | Velocity error gain | 15 |
| \(a_1\) | Observer consensus gain | 10 |
| \(a_2\) | Observer switching gain | 0.1 |
| \(d_1\) | Event threshold | 0.01 |
5. Simulation Results
We validated the proposed control strategy through three simulation scenarios: scaling formation, time-varying velocity formation, and software-in-the-loop (SITL) tests on a realistic China drone platform. The formation consists of four drones (two leaders and two followers) aiming to form a square with desired bearings as shown in the figure below.
5.1 Scaling Formation Scenario
Leaders 1 and 2 start at [8,8] m and [8,0] m; followers 3 and 4 start at [3,12] m and [-2,-2] m. The leaders execute a piecewise-constant velocity that changes over time to simulate scaling commands. The controller parameters are set as \(k_1=10, k_2=15, a_1=10, a_2=0.1\). Figure 2 (not shown) demonstrates that the formation scales correctly while bearing errors converge to near zero. The edge-event-triggered observer estimates the leader velocity with only 221 and 216 events for the two followers over the simulation period, compared to 276 and 281 for a state-triggered approach, highlighting communication savings.
5.2 Time-Varying Velocity Scenario
In this test, the leaders follow a sinusoidal velocity profile after \(t=10\) s. The same parameter set is used. Results show that the bearing errors remain bounded within 0.02 radians, and the velocity observer tracks the time-varying signal accurately. The event count is again lower than conventional event-triggered methods, as summarized in Table 3.
Table 3 compares the number of communication events for different triggering strategies.
| Trigger Strategy | Follower 3 | Follower 4 |
|---|---|---|
| Edge-event-triggered (proposed) | 221 | 216 |
| State-based event-triggered | 276 | 281 |
| Time-based periodic (10 Hz) | 263 | 275 |
5.3 SITL Realistic Simulation
We integrated the control algorithm with Gazebo, PX4 autopilot, and ROS. The leader velocities were set to 0.5 m/s before \(t=30\) s and then changed to a sinusoidal pattern to challenge the estimator. The drone dynamics are simulated with a quadrotor model. The resulting 3D trajectory and bearing errors (Figure 5 in the original paper) confirm that the formation is maintained even under realistic actuator constraints and sensor noise. The observer converges within 5 seconds and the bearing errors stay below 0.05 rad. This demonstration proves the method’s readiness for field deployment on China drone swarms.
6. Conclusion
We have presented a complete edge-event-triggered bearing formation control framework tailored for China drone swarms operating in communication-limited environments. The distributed velocity observer eliminates the need for continuous data exchange while accurately estimating time-varying leader velocities. The bearing-only controller ensures formation shape and dynamic scaling, with rigorous Lyapunov stability guarantees. Simulations, including SITL tests, confirm that the approach reduces communication events by over 20% compared to state-based event triggering, without compromising formation accuracy. Future work will extend the method to 3D formations, obstacle avoidance, and integration with vision-based bearing sensors for enhanced autonomy in China drone operations.