This paper addresses the challenges of unknown time-varying velocities and formation scaling in multi-China UAV systems with a leader-follower structure. We propose a novel edge event-triggered bearing formation control strategy that significantly reduces communication load while ensuring stable formation tracking. The main contributions include a distributed velocity observer based on edge event triggering, a bearing-only formation controller that adapts to formation scaling, and rigorous Lyapunov-based stability analysis. Extensive simulations, including a software-in-the-loop (SITL) test on Gazebo with PX4 and ROS, validate the effectiveness and practicality of the proposed approach for China UAV formations.
1. Introduction
Multi-UAV systems, especially those involving China UAV platforms, have attracted considerable attention due to their wide applications in surveillance, search and rescue, and agricultural monitoring. Formation control is a fundamental problem in such systems. Among various methods, bearing-based formation control offers advantages of low cost and invariance under translation and scaling. However, most existing approaches assume continuous communication or constant velocity, which is impractical for real China UAV deployments. Furthermore, formation scaling—adjusting the size of the formation while preserving its shape—is often required but rarely addressed with event-triggered mechanisms.
To overcome these limitations, we develop an edge event-triggered bearing formation control framework. “Edge” refers to the communication link between two UAVs. Unlike node-based event triggering, our approach triggers updates only when the relative bearing error exceeds a threshold, further reducing unnecessary transmissions. The key features are:
- A distributed observer that estimates unknown time-varying velocities using only relative bearing measurements and intermittent communication.
- A formation controller that achieves both shape convergence and scaling adaptation by dynamically coordinating the leader’s motion.
- Rigorous proof of ultimate boundedness and avoidance of Zeno behavior.
- Comprehensive simulations on a realistic SITL platform confirming practical viability for China UAV formations.
2. Preliminaries and Problem Formulation
2.1 UAV Dynamics
Each China UAV is modeled as a point-mass in 3D space with second-order dynamics:
\[
\dot{\mathbf{p}}_i = \mathbf{v}_i, \quad \dot{\mathbf{v}}_i = \mathbf{u}_i, \quad i \in \{1,\dots,n\}
\]
where \(\mathbf{p}_i \in \mathbb{R}^3\) is position, \(\mathbf{v}_i \in \mathbb{R}^3\) is velocity, and \(\mathbf{u}_i \in \mathbb{R}^3\) is the control input. We assume that the inner-loop attitude controller ensures accurate tracking of the desired acceleration. The formation consists of \(n_l\) leaders and \(n_f\) followers, with \(n = n_l + n_f\).
2.2 Graph Theory and Bearing Vectors
The communication topology among China UAVs is described by an undirected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\) for follower-follower links, and directed edges from leaders to followers. For each edge \((i,j)\in\mathcal{E}\), the relative position is \(\mathbf{e}_{ij} = \mathbf{p}_j – \mathbf{p}_i\), and the unit bearing vector is \(\mathbf{g}_{ij} = \mathbf{e}_{ij} / \|\mathbf{e}_{ij}\|\). The orthogonal projection matrix onto the nullspace of \(\mathbf{g}_{ij}\) is \(\mathbf{P}_{\mathbf{g}_{ij}} = \mathbf{I}_d – \mathbf{g}_{ij}\mathbf{g}_{ij}^\top\).
The Laplacian matrix of the bearing graph is defined as:
\[
\mathcal{L} = \begin{bmatrix}
\mathcal{L}_{ll} & \mathcal{L}_{lf} \\
\mathcal{L}_{fl} & \mathcal{L}_{ff}
\end{bmatrix}
\]
with entries depending on \(\mathbf{P}_{\mathbf{g}_{ij}}\). We assume \(\mathcal{G}\) is connected and \(\mathcal{L}_{ll}\) is positive definite (Assumptions 2-3).
2.3 Problem Statement
Given a desired formation shape defined by constant bearing vectors \(\mathbf{g}_{ij}^*\) (which determine the relative orientations but not the scale), the objective is to design event-triggered control laws \(\mathbf{u}_i\) for each follower such that:
- The formation shape converges: \(\mathbf{g}_{ij}(t) \to \mathbf{g}_{ij}^*\) as \(t\to\infty\).
- The formation scale can be adjusted by the leaders’ velocity commands.
- Communication among China UAVs occurs only at discrete event times.
3. Edge Event-Triggered Distributed Velocity Observer
3.1 Observer Design
To estimate the unknown time-varying velocity \(\mathbf{v}_i\) of each follower without continuous data exchange, we propose the following observer:
\[
\dot{\hat{\mathbf{v}}}_i(t) = a_1 \sum_{j\in\mathcal{N}_i} a_{ij}\big(\hat{\mathbf{v}}_j(t_k^{ij}) – \hat{\mathbf{v}}_i(t_k^{ij})\big) – a_2 \sum_{j\in\mathcal{N}_i} a_{ij} \text{sgn}\big(\hat{\mathbf{v}}_i(t_k^{ij}) – \hat{\mathbf{v}}_j(t_k^{ij})\big)
\]
where \(a_1, a_2>0\) are gains, \(a_{ij}\) are adjacency weights, and \(t_k^{ij}\) denotes the \(k\)-th event time for edge \((i,j)\). The observer uses only the last transmitted values. Define the estimation error \(\tilde{\mathbf{v}}_i = \hat{\mathbf{v}}_i – \mathbf{v}_i\) and edge error \(\mathbf{q}_{ij} = \hat{\mathbf{v}}_i – \hat{\mathbf{v}}_j\). Also define the measurement error \(\mathbf{r}_{ij} = \mathbf{v}_i – \mathbf{v}_j\).
The event-triggering condition for each edge is:
\[
f_{ij}(t) = a_1 \|\mathbf{q}_{ij}\|^2 + a_2 \|\mathbf{q}_{ij}\| – \frac{3}{2}a_1 \|\mathbf{q}_{ij}\|^2 – \frac{1}{2}a_2 \|\mathbf{r}_{ij}\| – d_1 > 0,
\]
and the next event time is \(t_{k+1}^{ij} = \inf\{t > t_k^{ij} : f_{ij}(t) > 0\}\). Here \(d_1>0\) is a small constant preventing Zeno behavior.
3.2 Stability Analysis
Theorem 1: Under Assumptions 1-3, if \(a_2 > 2\alpha / \lambda_{\min}(\mathcal{L})\) for some positive constant \(\alpha\), then the velocity estimation error \(\tilde{\mathbf{v}} = [\tilde{\mathbf{v}}_1^\top, \dots, \tilde{\mathbf{v}}_n^\top]^\top\) converges to zero asymptotically.
Proof sketch: Consider Lyapunov function \(V_v = \frac12 \sum_{i=1}^n \tilde{\mathbf{v}}_i^\top \tilde{\mathbf{v}}_i\). Using the observer dynamics, event-trigger condition, and Young’s inequality, we obtain:
\[
\dot{V}_v \leq -a_1 \lambda_{\min}(\mathcal{L}) V_v + \beta
\]
where \(\beta\) is a bounded term. By choosing \(a_2\) sufficiently large, \(\beta\) can be made arbitrarily small, leading to exponential convergence. The detailed derivation shows that \(\lim_{t\to\infty} \tilde{\mathbf{v}} = 0\). ∎
4. Bearing-Based Formation Controller with Scaling
4.1 Controller Design
Define the bearing error for edge \((i,j)\):
\[
\mathbf{e}_{g,ij} = \mathbf{g}_{ij} – \mathbf{g}_{ij}^*
\]
and the velocity tracking error:
\[
\mathbf{e}_{v,i} = \mathbf{v}_i – \hat{\mathbf{v}}_i
\]
where \(\hat{\mathbf{v}}_i\) is the estimated velocity from the observer. The virtual control is chosen as:
\[
\mathbf{u}_{v,i} = -k_1 \mathbf{J}^\top \tilde{\mathbf{g}}
\]
where \(\tilde{\mathbf{g}} = \text{col}(\mathbf{e}_{g,ij})\) for all edges, \(\mathbf{J} = \text{diag}(\mathbf{P}_{\mathbf{g}_{ij}}) \mathbf{H} \otimes \mathbf{I}_d\) is the bearing Jacobian matrix, and \(k_1>0\). Introduce an auxiliary variable \(\boldsymbol{\alpha}_i = \mathbf{e}_{v,i} – \mathbf{u}_{v,i}\). The final control law is:
\[
\mathbf{u}_i = -k_2 \boldsymbol{\alpha}_i – \mathbf{J}^\top \tilde{\mathbf{g}} + \dot{\hat{\mathbf{v}}}_i + \dot{\mathbf{u}}_{v,i}
\]
where \(k_2>0\) satisfies \(k_2 > \lambda_{\max}(\mathbf{J}^\top \mathbf{J}) / (2\lambda_{\min}(\mathbf{J}^\top \mathbf{J}))\).
4.2 Stability Analysis
Theorem 2: Under the control law and the observer, the bearing errors \(\mathbf{e}_{g,ij}\) and velocity tracking errors \(\mathbf{e}_{v,i}\) are uniformly ultimately bounded (UUB).
Proof: Consider Lyapunov function \(V = V_g + \frac12 \sum_i \boldsymbol{\alpha}_i^\top \boldsymbol{\alpha}_i\), where \(V_g = \frac12 \sum \mathbf{e}_{g,ij}^\top \mathbf{e}_{g,ij}\). Taking derivative and using the controller yields:
\[
\dot{V} \leq -\eta V + \delta
\]
with \(\eta = \min(k_1, k_2 – \lambda_{\max}(\mathbf{J}^\top \mathbf{J})/2) > 0\) and \(\delta\) proportional to \(\|\tilde{\mathbf{v}}\|^2\). Since \(\tilde{\mathbf{v}}\to 0\) by Theorem 1, the system is UUB. ∎
5. Simulation Results
5.1 Simulation Setup
We consider four China UAVs forming a square shape: two leaders (UAV 1 and UAV 2) and two followers (UAV 3 and UAV 4). The desired bearing vectors are:
| Edge | \(\mathbf{g}_{ij}^*\) |
|---|---|
| (3,1) | \([1,0]^\top\) |
| (4,2) | \([1,0]^\top\) |
| (3,2) | \([\sqrt{2}/2, -\sqrt{2}/2]^\top\) |
| (3,4) | \([0,1]^\top\) |
| (2,1) | \([0,-1]^\top\) |
Other edges are obtained by symmetry. Initial positions and other parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| UAV1 initial position | \((8,8)\) m |
| UAV2 initial position | \((8,0)\) m |
| UAV3 initial position | \((3,12)\) m |
| UAV4 initial position | \((-2,-2)\) m |
| Controller gains | \(k_1=10, k_2=15\) |
| Observer gains | \(a_1=10, a_2=0.1\) |
5.2 Scaling Formation Scenario
Leaders follow a piecewise constant velocity profile with sinusoidal scaling changes. Figure 1 shows the trajectory and bearing errors. The formation converges to the desired square shape and the scale adjusts according to leader motion. The velocity observer accurately tracks the time-varying velocity. Event times for edges are shown; the edge-triggered mechanism significantly reduces communication compared to periodic sampling.
5.3 Time-Varying Velocity Scenario
Leaders follow a sinusoidal velocity profile. The bearing errors converge to near zero, and the observer estimates the velocity well. Table 2 compares the number of triggering events among three strategies: edge event-triggered, state event-triggered, and time-triggered (periodic). Our proposed method achieves the fewest triggers, saving communication resources.
| Trigger method | Follower 3 | Follower 4 |
|---|---|---|
| Edge event-triggered (proposed) | 221 | 216 |
| State event-triggered | 276 | 281 |
| Time-triggered (periodic) | 263 | 275 |
5.4 Software-in-the-Loop (SITL) Simulation
To validate practical applicability on real China UAV platforms, we performed SITL tests using Gazebo, PX4 autopilot, and ROS. The formation structure and control parameters remain the same. A time-varying leader velocity profile is used: \(\mathbf{v}_1=\mathbf{v}_2=[0.5,0]^\top\) m/s for \(t \leq 30s\), and sinusoidal after. The bearing errors converge to near zero and the velocity observer performs well. Screenshots from Gazebo (not shown due to text restriction) confirm the formation maintains shape under realistic dynamics.
6. Conclusion
This paper presents a novel edge event-triggered bearing formation control method for China UAV systems with leader-follower topology. The approach handles unknown time-varying velocities and formation scaling without continuous communication. The distributed velocity observer ensures accurate estimation, while the bearing-based controller guarantees UUB stability. Simulations, including SITL tests, demonstrate the effectiveness in reducing communication load and achieving precise formation tracking. Future work will extend to 3D formations and obstacle avoidance.