We present a novel approach for safe bearing-based formation control of quadrotor China drone swarms operating in environments with unmodeled dynamics and external disturbances. Our method integrates a parameter-adaptive distributed extended state observer with a Control Lyapunov Function (CLF) and Reciprocal Control Barrier Function (RCBF) in a quadratic programming (QP) framework. The framework only requires relative bearing angles, bearing rates, and distances among neighboring China drone platforms to simultaneously estimate the global position, velocity, and total disturbance of each follower. We then design a safe controller that ensures formation convergence while guaranteeing collision avoidance between any pair of China drone agents. Simulation results for an eight-China drone formation confirm that the desired cubic configuration is achieved and maintained, with the minimum inter-agent distance staying above 0.75 m—well above the 0.5 m safety threshold—demonstrating superior performance over artificial potential field methods and non‑collision‑avoidance schemes.
1. Introduction
Formation control of multiple unmanned aerial vehicles has become a cornerstone of modern China drone research, enabling applications in surveillance, agriculture, delivery, and disaster response. Among various sensing modalities, bearing-based formation control is particularly attractive because it relies on low-cost vision sensors, requires only relative line-of-sight measurements, and allows flexible reconfiguration without global localization infrastructure. However, real-world China drone deployments face two critical challenges: (i) unmodeled aerodynamic effects and external disturbances (e.g., wind gusts) degrade tracking accuracy and may destabilize the formation; (ii) the proximity of multiple aircraft during formation maneuvers poses a collision risk that must be rigorously managed.
Existing works on bearing-based formation control, such as those in Li et al. and Su et al., often assume ideal first‑ or second‑order dynamics without disturbances. More advanced results incorporate disturbance observers or neural networks, but they typically do not address the collision‑avoidance problem simultaneously. On the other hand, collision‑avoidance techniques based on artificial potential fields, trajectory planning, or control barrier functions have been studied in isolation from bearing‑based estimation. A unified approach that combines disturbance‑tolerant state estimation, formation tracking, and collision‑free behavior remains an open challenge, especially for China drone swarms operating under uncertain conditions.
We propose a complete solution that fuses adaptive parameter estimation, distributed extended state observers (DESO), and a CLF‑RCBF controller. The key contributions are:
- State‑disturbance estimation framework: Using only bearing angles, bearing rates, and distances, we design a DESO that simultaneously estimates each follower’s position, velocity, and lumped disturbances (unmodeled dynamics plus external disturbances). A parameter‑adaptive law compensates for state‑dependent unmodeled dynamics, while the observer handles time‑varying external disturbances.
- CLF‑RCBF safety controller: We construct a control Lyapunov function based on estimated states and desired trajectories from a reference observer, and a reciprocal control barrier function that incorporates both relative position and relative velocity constraints. The resulting quadratic program guarantees that the control input simultaneously enforces formation convergence and collision avoidance.
- Complete integration and validation: We prove the feasibility of the QP problem and demonstrate through extensive simulations of an eight‑China drone system that the proposed method outperforms the improved artificial potential field method in tracking accuracy, minimum safe distance, and energy consumption.
The remainder of the paper is organized as follows. Section 2 formulates the problem. Section 3 details the distributed state‑disturbance observer. Section 4 describes the CLF‑RCBF controller and the QP formulation. Section 5 presents simulation results. Section 6 concludes the paper.
2. Problem Formulation
2.1 Graph Theory and Bearing Laplacian
Consider a team of \(N = n_l + n_f\) quadrotor China drone agents, where \(n_l=2\) leaders and \(n_f=6\) followers. The communication topology is an undirected graph \(\mathcal{G} = (\mathcal{V}, \mathcal{E})\). For agent \(i\), its neighbor set is \(\mathcal{N}_i = \{ j \in \mathcal{V} : (i,j) \in \mathcal{E} \}\). The bearing from agent \(i\) to agent \(j\) is defined as:
\[
\mathbf{g}_{ij} = \frac{\mathbf{p}_j – \mathbf{p}_i}{\|\mathbf{p}_j – \mathbf{p}_i\|}, \quad \mathbf{P}_{ij} = \mathbf{I}_3 – \mathbf{g}_{ij}\mathbf{g}_{ij}^T,
\]
where \(\mathbf{p}_i \in \mathbb{R}^3\) is the position of agent \(i\). Given a set of desired bearings \(\mathbf{g}_{ij}^*\) and corresponding projectors \(\mathbf{P}_{ij}^*\), the bearing Laplacian matrix is:
\[
[\mathcal{L}]_{ij} =
\begin{cases}
-\mathbf{P}_{ij}^*, & i \neq j, (i,j) \in \mathcal{E},\\
\sum_{k \neq i} \mathbf{P}_{ik}^*, & i = j,\\
\mathbf{0}_{3 \times 3}, & \text{otherwise}.
\end{cases}
\]
The Laplacian is partitioned as \(\mathcal{L} = \begin{bmatrix} \mathcal{L}_{ff} & \mathcal{L}_{fl} \\ \mathcal{L}_{lf} & \mathcal{L}_{ll} \end{bmatrix}\). Since \(\mathcal{G}\) is undirected, \(\mathcal{L}_{ff}\) is symmetric positive definite.
2.2 Quadrotor Dynamics
The translational dynamics of a quadrotor China drone are given by:
\[
m \ddot{\mathbf{p}}_i = \mathbf{R}_i \begin{bmatrix}0\\0\\F_i\end{bmatrix} – m g \mathbf{e}_z + \mathbf{f}_i(\mathbf{p}_i,\dot{\mathbf{p}}_i) + \mathbf{d}_i(t),
\]
where \(\mathbf{R}_i\) is the rotation matrix from body to world frame, \(F_i\) is total thrust, \(g\) is gravity, \(\mathbf{f}_i\) accounts for unmodeled dynamics (e.g., aerodynamic drag, ground effects), and \(\mathbf{d}_i(t)\) is external disturbance. We define the virtual control input as \(\mathbf{u}_i = \frac{1}{m} \mathbf{R}_i[0,0,F_i]^T – g\mathbf{e}_z\). Then the double integrator model becomes:
\[
\begin{cases}
\dot{\mathbf{x}}_{i,1} = \mathbf{x}_{i,2},\\
\dot{\mathbf{x}}_{i,2} = \mathbf{u}_i + \tilde{\mathbf{f}}_i(\mathbf{x}_{i,1},\mathbf{x}_{i,2}) + \mathbf{d}_i(t),
\end{cases}
\]
where \(\mathbf{x}_{i,1} = \mathbf{p}_i\), \(\mathbf{x}_{i,2} = \dot{\mathbf{p}}_i\), and \(\tilde{\mathbf{f}}_i\) represents the unmodeled dynamic acceleration. We assume the unmodeled dynamics can be parameterized as \(\tilde{\mathbf{f}}_i = \boldsymbol{\Phi}_i(\mathbf{x}_{i,1},\mathbf{x}_{i,2}) \boldsymbol{\zeta}_i\), with unknown constant \(\boldsymbol{\zeta}_i\) bounded by \(\|\boldsymbol{\zeta}_i\| \leq \bar{\zeta}_i\). The external disturbance is time‑varying with bounded derivative: \(\|\dot{\mathbf{d}}_i(t)\| = \|\mathbf{h}_i(t)\| \leq \bar{H}_i\).
2.3 Desired Formation and Safety Constraint
The desired formation is defined by the target bearings \(\mathbf{g}_{ij}^*\). A cubic formation is chosen (eight China drone agents). The leaders follow a known time‑varying trajectory; followers must track their desired positions and velocities derived from the bearing constraints.
The safety constraint between any two agents \(i\) and \(j\) is:
\[
\|\mathbf{p}_{i} – \mathbf{p}_{j}\|^2 + 2\tau_{\text{safe}} (\dot{\mathbf{p}}_i – \dot{\mathbf{p}}_j)^T (\mathbf{p}_i – \mathbf{p}_j) – D_{\text{safe}}^2 \geq 0,
\]
where \(D_{\text{safe}} = 0.5\,\text{m}\) and \(\tau_{\text{safe}} = 0.6\,\text{s}\). This constraint, which couples position and velocity, provides a higher safety margin than a purely distance‑based constraint and avoids the computational complexity of high‑order barrier functions.
3. Distributed State‑Disturbance Observer
We augment the system state with the disturbance as \(\mathbf{x}_{i,3} = \mathbf{d}_i(t)\). The extended dynamics are:
\[
\begin{cases}
\dot{\mathbf{x}}_{i,1} = \mathbf{x}_{i,2},\\
\dot{\mathbf{x}}_{i,2} = \mathbf{u}_i + \boldsymbol{\Phi}_i(\mathbf{x}_{i,1},\mathbf{x}_{i,2}) \boldsymbol{\zeta}_i + \mathbf{x}_{i,3},\\
\dot{\mathbf{x}}_{i,3} = \mathbf{h}_i(t).
\end{cases}
\]
We design a distributed observer that only uses the relative information available from neighbors: bearing \(\mathbf{g}_{ij}\), bearing rate \(\dot{\mathbf{g}}_{ij}\), and distance \(d_{ij} = \|\mathbf{p}_j-\mathbf{p}_i\|\). The observer for follower \(i\) is:
\[
\begin{aligned}
\dot{\mathbf{z}}_{i,1} &= \mathbf{z}_{i,2} + \beta_1 \mathbf{e}_{i,1},\\
\dot{\mathbf{z}}_{i,2} &= \mathbf{z}_{i,3} + \hat{\mathbf{f}}_i(\mathbf{z}_{i,1},\mathbf{z}_{i,2}) + \mathbf{u}_i + \beta_2 \mathbf{e}_{i,1},\\
\dot{\mathbf{z}}_{i,3} &= \beta_3 \mathbf{e}_{i,1},
\end{aligned}
\]
where \(\mathbf{z}_{i,1},\mathbf{z}_{i,2},\mathbf{z}_{i,3}\) are estimates of \(\mathbf{x}_{i,1},\mathbf{x}_{i,2},\mathbf{x}_{i,3}\), respectively. The error signals \(\mathbf{e}_{i,1},\mathbf{e}_{i,2}\) are constructed from bearing and distance measurements:
\[
\begin{aligned}
\mathbf{e}_{i,1} &= \sum_{j \in \mathcal{N}_{if}} \mathbf{P}_{ij}^* (\mathbf{z}_{i,1} – \mathbf{z}_{j,1}) + \sum_{j \in \mathcal{N}_{il}} \mathbf{P}_{ij}^* (\mathbf{z}_{i,1} – \mathbf{p}_j) + \sum_{j \in \mathcal{N}_i} \dot{\mathbf{P}}_{ij} \mathbf{g}_{ij} d_{ij},\\
\mathbf{e}_{i,2} &= \sum_{j \in \mathcal{N}_{if}} \mathbf{P}_{ij}^* (\mathbf{z}_{i,2} – \mathbf{z}_{j,2}) + \sum_{j \in \mathcal{N}_{il}} \mathbf{P}_{ij}^* (\mathbf{z}_{i,2} – \dot{\mathbf{p}}_j) + \sum_{j \in \mathcal{N}_i} \dot{\mathbf{P}}_{ij} \frac{\dot{\mathbf{g}}_{ij}}{\|\mathbf{P}_{ij}^*\|} d_{ij}.
\end{aligned}
\]
The adaptive estimate of the unmodeled dynamics is \(\hat{\mathbf{f}}_i = \boldsymbol{\Phi}_i(\mathbf{z}_{i,1},\mathbf{z}_{i,2}) \hat{\boldsymbol{\zeta}}_i\) with adaptation law:
\[
\dot{\hat{\boldsymbol{\zeta}}}_i = \mu_i \boldsymbol{\Phi}_i^T(\mathbf{z}_{i,1},\mathbf{z}_{i,2}) \mathbf{e}_{i,2} – \mu_i \sigma_i \hat{\boldsymbol{\zeta}}_i.
\]
Define the estimation errors \(\tilde{\mathbf{z}}_{i,k} = \mathbf{z}_{i,k} – \mathbf{x}_{i,k}\) for \(k=1,2,3\), and \(\tilde{\boldsymbol{\zeta}}_i = \hat{\boldsymbol{\zeta}}_i – \boldsymbol{\zeta}_i\). The global error dynamics can be written as:
\[
\dot{\tilde{\mathbf{Z}}} = \boldsymbol{\Gamma} \tilde{\mathbf{Z}} + \boldsymbol{\Upsilon} \mathbf{H} + \tilde{\mathbf{F}},
\]
where \(\tilde{\mathbf{Z}}\) stacks all error states, \(\boldsymbol{\Gamma}\) is a block matrix involving the bearing Laplacian and observer gains \(\beta_1,\beta_2,\beta_3\). Under the parameter conditions:
\[
\beta_1>0,\; \beta_2>0,\; \beta_3>0,\; \beta_2 > \beta_1 + \frac{\beta_3}{\lambda_{\min}(\mathcal{L}_{ff})},\quad \mu_{\min} \geq \frac{\sigma_{\max}^2}{\varepsilon_1 \varepsilon_3},
\]
we can prove using a Lyapunov function \(V = \tilde{\mathbf{Z}}^T \mathbf{P} \tilde{\mathbf{Z}} + \frac{1}{2\mu} \sum_i \tilde{\boldsymbol{\zeta}}_i^T \tilde{\boldsymbol{\zeta}}_i\) that the estimation errors are uniformly ultimately bounded. This guarantees that the observer provides reliable state and disturbance estimates for the subsequent controller.
4. CLF‑RCBF Safety Formation Controller
4.1 Desired State Observer
Because followers do not know the leader’s trajectory globally, we employ a dynamic compensator that estimates each follower’s desired position and velocity from the bearing constraints. Define \(\hat{\boldsymbol{\xi}}_i = [\hat{\mathbf{p}}_i^*, \dot{\hat{\mathbf{p}}}_i^*, \ddot{\hat{\mathbf{p}}}_i^*, \dddot{\hat{\mathbf{p}}}_i^*]^T\) as the estimate of the desired states and their derivatives up to order 3. The compensator is:
\[
\dot{\hat{\boldsymbol{\xi}}}_i = \boldsymbol{\Phi} \hat{\boldsymbol{\xi}}_i – L \sum_{j \in \mathcal{N}_i} \mathbf{P}_{ij}^* (\mathbf{I}_4 \otimes \mathbf{I}_3)(\hat{\boldsymbol{\xi}}_i – \hat{\boldsymbol{\xi}}_j),
\]
where \(\boldsymbol{\Phi}\) is the system matrix corresponding to the leader’s motion (e.g., sinusoidal). For a leader trajectory with known frequency at 0.2 Hz, we have:
\[
\boldsymbol{\Phi} = \begin{bmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
-(\frac{2\pi}{5})^4 & 0 & 0 & 0
\end{bmatrix}, \quad \mathbf{I} = [1,0,0,0]^T.
\]
The gain \(L\) is chosen large enough to dominate the smallest eigenvalue of \(\mathcal{L}_{ff}\). The estimate \(\hat{\mathbf{p}}_i^*\) and \(\dot{\hat{\mathbf{p}}}_i^*\) converge exponentially to the true desired states.
4.2 Control Lyapunov Function
Based on the estimated state from the DESO and the desired trajectory, we define a CLF:
\[
V_i = k_p \|\mathbf{z}_{i,1} – \hat{\mathbf{p}}_i^*\|^2 + k_d \|\mathbf{z}_{i,2} – \dot{\hat{\mathbf{p}}}_i^*\|^2,
\]
with \(k_p=0.5, k_d=0.1\). The Lie derivative along the true dynamics with input \(\mathbf{u}_i\) is:
\[
\dot{V}_i = L_{f}V_i + L_{g}V_i \mathbf{u}_i,
\]
where \(L_{f}V_i = 2k_p(\mathbf{z}_{i,1}-\hat{\mathbf{p}}_i^*)^T(\mathbf{z}_{i,2}-\dot{\hat{\mathbf{p}}}_i^*) + 2k_d(\mathbf{z}_{i,2}-\dot{\hat{\mathbf{p}}}_i^*)^T(\mathbf{z}_{i,3}+\hat{\mathbf{f}}_i – \ddot{\hat{\mathbf{p}}}_i^*)\), and \(L_{g}V_i = 2k_d(\mathbf{z}_{i,2}-\dot{\hat{\mathbf{p}}}_i^*)^T\). The CLF condition is:
\[
L_{f}V_i + L_{g}V_i \mathbf{u}_i + c V_i \leq 0, \quad c=0.02.
\]
4.3 Reciprocal Control Barrier Function
For each pair \((i,j)\), define the safety function:
\[
h_{ij} = \|\mathbf{z}_{i,1}-\mathbf{z}_{j,1}\|^2 + 2\tau_{\text{safe}} (\mathbf{z}_{i,2}-\mathbf{z}_{j,2})^T(\mathbf{z}_{i,1}-\mathbf{z}_{j,1}) – D_{\text{safe}}^2.
\]
The reciprocal barrier function is \(B_{ij} = 1/h_{ij}\). Its Lie derivatives:
\[
L_{f}B_{ij} = -\frac{\dot{h}_{ij}}{h_{ij}^2},\quad L_{g_i}B_{ij} = -\frac{\partial h_{ij}/\partial \mathbf{u}_i}{h_{ij}^2}.
\]
The RCBF condition for agent \(i\) w.r.t. neighbor \(j\) is:
\[
L_{f}B_{ij} + L_{g_i}B_{ij} \mathbf{u}_i – \frac{\gamma}{B_{ij}} \leq 0, \quad \gamma=0.6.
\]
4.4 Quadratic Program Formulation
We formulate the following QP for agent \(i\) at each time step:
\[
\begin{aligned}
\mathbf{u}_i^* &= \arg\min_{\mathbf{u}_i, \delta} \frac{1}{2} \mathbf{u}_i^T \mathbf{H} \mathbf{u}_i + \rho \delta^2 \\
\text{s.t.} &\quad L_{f}V_i + L_{g}V_i \mathbf{u}_i + c V_i \leq \delta, \\
&\quad L_{f}B_{ij} + L_{g_i}B_{ij} \mathbf{u}_i – \frac{\gamma}{B_{ij}} \leq 0, \quad \forall j \in \mathcal{N}_i, \\
&\quad \|\mathbf{u}_i\|_\infty \leq u_{\max}.
\end{aligned}
\]
where \(\delta\) is a slack variable for soft CLF enforcement, and \(u_{\max}\) accounts for actuator limits. The RCBF constraints are hard. We prove feasibility by noting that when \(L_{g_i}B_{ij}=0\), the constraint reduces to \(- \gamma/B_{ij}\leq 0\) which holds trivially; otherwise the feasible set is an interval that always contains the origin or a bounded region due to the boundedness of \(L_{f}B_{ij}\). Combined with the CLF slack, the QP always has a solution.
5. Simulation Results
5.1 Setup
We simulate a team of eight quadrotor China drone agents with parameters from a typical 1.8 kg platform (arm length 0.4 m). The attitude inner loop is modeled as first‑order with time constants 150 ms (pitch/roll) and 30 ms (thrust). External disturbance is \(\mathbf{d}(t) = [0.2\cos t, 0.2\sin t, 0.2\sin t]^T N\) (note: acceleration is force/mass, so effective disturbance acceleration is \(\mathbf{d}/m\)). The unmodeled dynamics include constant bias, linear viscous drag, quadratic drag, nonlinear damping, ground effect, and cross‑coupling terms. Communication failures are artificially introduced at [5,5.5], [15,15.5], and [40,40.5] seconds.
The desired formation is a cube with edge length 3 m. Two leaders start at [4,0,6] and [4,4,6] and follow a trajectory composed of sinusoidal segments. Followers start from random positions near the leader origin.
We compare three scenarios: (i) our proposed CLF‑RCBF method, (ii) an improved artificial potential field (APF) method for collision avoidance, and (iii) formation control without any collision‑avoidance mechanism.
5.2 Formation Tracking Performance
The formation snapshots at t=0, 10, 20, 50 s show that the cubic configuration is achieved after about 25 s and maintained despite disturbances and communication outages. Figure 3 shows the absolute position tracking errors for all followers: errors converge to within 0.1 m after initial transients, with small spikes during communication blackouts that quickly recover. Velocity tracking errors (Figure 4) similarly converge rapidly. These results confirm that the DESO provides accurate estimates and the CLF ensures convergence.
5.3 Disturbance Estimation Performance
Figures 5–7 display the estimation errors for position, velocity, and lumped disturbance, respectively. The position estimation error stays below 0.05 m after 5 s, velocity error below 0.1 m/s, and disturbance estimation error below 0.15 m/s². The adaptation law successfully identifies the unknown parameters \(\boldsymbol{\zeta}_i\) to within 10% of their true values.
5.4 Collision Avoidance Comparison
The minimum inter‑agent distance over time is plotted in Figure 9. Our RCBF method maintains the distance above 0.75 m, well above the 0.5 m threshold. In contrast, the APF method yields a minimum of 0.37 m, and the method without collision avoidance drops to 0.10 m, indicating actual collisions. Additionally, Figure 8 compares the tracking errors during formation: our method shows smaller position and velocity errors than APF, because APF’s repulsive forces interfere significantly with tracking. Energy consumption, computed as the integral of squared thrust, is 236.41 for our method versus 1115.23 for APF (Figure 10), demonstrating superior efficiency.
5.5 Control Input Feasibility
Figures 11–12 show the attitude angles (pitch, roll, yaw) and total thrust for all agents during the simulation. All angles stay within ±30°, and thrust remains below the 3‑g limit (52.92 N). These results validate that the virtual controller output can be realized by the inner‑loop attitude controller.
6. Conclusion
We have developed a comprehensive bearing‑based formation control framework for quadrotor China drone swarms that simultaneously addresses disturbance rejection and collision‑free navigation. The key innovation is the integration of a parameter‑adaptive distributed extended state observer, which estimates states and disturbances solely from bearing measurements, with a CLF‑RCBF quadratic‑programming controller that ensures both formation convergence and inter‑agent safety. Theoretical analysis proves the feasibility of the QP constraints, and simulation results with an eight‑China drone formation demonstrate tracking errors within 0.1 m, minimum inter‑agent distances of 0.75 m, and significantly lower energy consumption compared to artificial potential field methods. The proposed method is particularly suitable for real‑world China drone applications where GPS‑denied environments and aerodynamic uncertainties are common.
Future work will extend the method to larger swarms, incorporate obstacle avoidance beyond inter‑agent safety, and validate the approach in outdoor flight experiments.
Appendix: Key Parameters and Tables
| Parameter | Symbol | Value |
|---|---|---|
| Observer gains | \(\beta_1,\beta_2,\beta_3\) | 9, 27, 54 |
| Adaptation gains | \(\mu, \sigma\) | 2.0, 0.15 |
| CLF gains | \(k_p, k_d\) | 0.5, 0.1 |
| RCBF parameter | \(\gamma, \tau_{\text{safe}}\) | 0.6, 0.8 |
| Safety distance | \(D_{\text{safe}}\) | 0.5 m |
| CLF decay rate | \(c\) | 0.02 |
| Thrust limit | \(F_{\max}\) | 52.92 N (3g) |
| Attitude limits | \(\phi_{\max}, \theta_{\max}\) | 30° |
| Agent | X (m) | Y (m) | Z (m) | Vx (m/s) | Vy (m/s) | Vz (m/s) |
|---|---|---|---|---|---|---|
| 3 | 2.0 | 0.4 | 2.9 | 0.0 | 0.0 | 0.6 |
| 4 | 2.0 | 0.4 | 5.1 | 0.0 | 0.0 | -0.6 |
| 5 | 2.0 | 3.6 | 2.9 | 0.0 | 0.0 | 0.6 |
| 6 | 2.0 | 3.6 | 5.1 | 0.0 | 0.0 | -0.6 |
| 7 | 6.0 | 0.4 | 2.2 | -0.3 | 0.8 | 0.0 |
| 8 | 6.0 | 3.6 | 2.0 | -0.3 | -0.8 | 0.0 |
| Method | Min Distance (m) | Energy Consumption (a.u.) |
|---|---|---|
| Proposed RCBF | 0.75 | 236.41 |
| Improved APF | 0.37 | 1115.23 |
| No collision avoidance | 0.10 | 189.52 |
The results in Table 3 highlight the effectiveness of our approach: the RCBF guarantees safe distances while maintaining low energy consumption, unlike the APF which expends excessive energy and still violates safety.