We propose a novel safe control framework that integrates adaptive parameter estimation, a distributed extended state observer (ESO), and a reciprocal control barrier function (RCBF) for quadrotor unmanned aerial vehicle (UAV) swarms performing bearing-based formation control under both unmodeled dynamics and external disturbances. Our scheme addresses two critical challenges: (1) state and disturbance estimation using only relative bearing angles, bearing rates, and distances, and (2) collision avoidance among swarm members while maintaining formation tracking. The leader–follower architecture is employed, where each follower estimates its own global position, velocity, and the lumped disturbance (including unmodeled dynamics and external wind gusts) through a fusion of a parameter adaptation law and a distributed ESO. Based on the estimated states, a Control Lyapunov Function (CLF) is constructed for formation tracking, while a RCBF that considers both relative position and relative velocity ensures inter‑agent safety. The CLF and RCBF constraints are then incorporated into a quadratic programming (QP) solver to obtain optimal control inputs in real time. We provide a theoretical proof of the feasibility of the QP problem. Extensive 8‑UAV formation simulations demonstrate that the swarm can successfully form and maintain a cubic formation under time‑varying disturbances and communication interruptions. The minimum inter‑agent distance remains at 0.75 m, well above the safety threshold of 0.5 m, outperforming an improved artificial potential field method (0.37 m) and a baseline without collision avoidance (0.10 m). The results highlight the effectiveness of our approach in merging accurate disturbance compensation, formation precision, and collision safety, showcasing its potential for real‑world UAV swarms in complex environments.
Keywords: quadrotor; bearing-based formation control; adaptive estimation; distributed extended state observer; UAV collision avoidance; drone technology; safety‑critical control; control barrier function.
1. Introduction
Multi‑UAV formation flight has attracted significant attention in both civilian and military applications, such as search‑and‑rescue, surveillance, and cooperative transportation. Among various formation control paradigms, bearing‑based formation control has emerged as a promising approach due to its low sensor requirements and ease of reconfiguration. However, practical implementation of bearing‑based formation control for quadrotor UAVs faces two major obstacles: (1) the presence of unmodeled dynamics and external disturbances that degrade tracking performance, and (2) the risk of collisions among swarm members during formation maneuvering. In this work, we aim to develop a comprehensive solution that integrates disturbance estimation and collision‑avoidance control within a bearing‑only framework, thereby advancing the state of the art in safe drone technology.
Existing studies on bearing‑based formation control have primarily focused on kinematic or ideal dynamic models, neglecting the effects of disturbances and safety constraints. For instance, Li et al. proposed a finite‑time scaling formation control for first‑order systems, while Su et al. extended the idea to second‑order systems with time‑varying velocities. Erskine et al. applied model predictive control to quadrotor bearing formations, but their method did not explicitly handle disturbances. Ding et al. incorporated a dynamic surface control with disturbance attenuation, yet collision avoidance was not considered. These works highlight the need for a unified framework that simultaneously addresses estimation, disturbance rejection, and safety.
To handle disturbances, various techniques have been proposed, including extended state observers (ESOs), fuzzy observers, and neural networks. However, most existing methods either treat all disturbances as a lumped term without separating internal and external components, or are not designed for bearing‑only measurements. In contrast, we combine a parameter adaptive law (for state‑dependent unmodeled dynamics) with a distributed ESO (for time‑varying external disturbances) that uses only relative bearing information and distances. This hybrid estimator provides accurate estimates of position, velocity, and the total disturbance, enabling effective feedforward compensation.
For collision avoidance, control barrier functions (CBFs) have gained popularity due to their formal safety guarantees. The reciprocal CBF (RCBF) introduced by Ames et al. ensures forward invariance of a safe set, while high‑order CBFs handle higher‑relative‑degree constraints. However, existing CBF‑based collision‑avoidance methods for UAVs often consider only position‑based safety, neglecting velocity terms, which can lead to overly conservative or unsafe behavior. We design a RCBF that incorporates both relative position and relative velocity, providing a balanced trade‑off between safety and performance. Moreover, we embed the RCBF constraints into a QP problem together with a CLF for formation tracking, ensuring that the control inputs simultaneously satisfy stability and safety requirements.
The main contributions of this paper are:
- We propose a state‑disturbance estimation framework that fuses a parameter adaptation method and a distributed extended state observer using only bearing angles, bearing rates, and relative distances. This framework obtains the follower’s position, velocity, and the combined effect of unmodeled dynamics and external disturbances.
- We design a CLF‑RCBF‑based safe controller for bearing formation control. The RCBF barrier condition considers both relative position and relative velocity, reducing interference with control performance while guaranteeing safety.
- We integrate the estimation framework with the safe controller to form a complete bearing‑based collision‑avoidance formation control scheme for quadrotor UAVs under internal and external disturbances. Theoretical analysis and simulation results validate the effectiveness of the proposed scheme.
The remainder of this paper is organized as follows. Section 2 introduces preliminaries including graph theory, quadrotor dynamics, and reciprocal control barrier functions. Section 3 presents the main approach: the distributed ESO with adaptive parameter identification, the formation tracking CLF, the collision‑avoidance RCBF, and the QP formulation. Section 4 provides simulation results and comparisons. Section 5 concludes the paper.
2. Preliminaries
2.1 Graph Theory and Bearing Constraints
We consider a swarm of $N$ quadrotor UAVs consisting of $n_l$ leaders and $n_f$ followers. The communication topology is described by an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}=\{1,\dots,N\}$ is the node set and $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ is the edge set representing bidirectional information exchange. For UAV $i$, its neighbor set is $\mathcal{N}_i = \{j\in\mathcal{V}:(i,j)\in\mathcal{E}\}$. The bearing between UAV $i$ and $j$ is defined as
$$ g_{ij} = \frac{p_i – p_j}{\|p_i – p_j\|}, $$
where $p_i\in\mathbb{R}^3$ is the position of UAV $i$. The orthogonal projection matrix for this bearing is
$$ P_{ij} = I_3 – g_{ij}g_{ij}^{\mathrm{T}}, $$
with $I_3$ the $3\times3$ identity matrix.
Given a set of desired bearings $g_{ij}^*$, the bearing Laplacian matrix $\mathcal{L}$ is constructed as in Eq. (4) of the original paper. For a minimally rigid graph, the submatrix $\mathcal{L}_{ff}$ is positive definite and symmetric. This property is crucial for the convergence of the distributed observer.
2.2 Quadrotor Dynamics
The quadrotor’s translational dynamics in the world frame can be expressed as
$$ m \ddot{p}_i = R(\phi_i,\theta_i,\psi_i) \begin{bmatrix}0\\0\\F_i\end{bmatrix} – \begin{bmatrix}0\\0\\mg\end{bmatrix} + f_i(p_i,\dot{p}_i) + d_i(t), $$
where $R$ is the rotation matrix, $F_i$ the total thrust, $g$ gravity, $f_i$ the unmodeled dynamics (e.g., aerodynamic drag, ground effect), and $d_i(t)$ the external time‑varying disturbance. By defining the desired acceleration $u_i = \frac{R}{m}[0,0,F_i]^{\mathrm{T}} – [0,0,g]^{\mathrm{T}}$, we obtain the simplified control‑oriented model:
$$ \dot{x}_{i,1} = x_{i,2}, \quad \dot{x}_{i,2} = u_i + f_i(x_{i,1},x_{i,2}) + d_i(t), $$
where $x_{i,1}=p_i$ and $x_{i,2}=\dot{p}_i$. This model captures the essential second‑order dynamics with additive disturbances.
2.3 Reciprocal Control Barrier Function (RCBF)
Let the safe set for UAV $i$ and $j$ be defined as
$$ \mathcal{S}_{ij} = \{ (z_{i,1},z_{i,2}) \mid h_{ij}(z_{i,1},z_{i,2}) \ge 0 \}, $$
with
$$ h_{ij} = \|z_{i,1}-z_{j,1}\|^2 – D_{\mathrm{safe}}^2 + \tau \big( (z_{i,1}-z_{j,1})^{\mathrm{T}}(z_{i,2}-z_{j,2}) \big), $$
where $z_{i,1}$ and $z_{i,2}$ are the estimated position and velocity of UAV $i$, $D_{\mathrm{safe}}$ the minimum safe distance, and $\tau$ a safety time parameter. The RCBF is defined as $B_{ij} = 1/h_{ij}$. The barrier condition is
$$ \dot{B}_{ij} + \gamma B_{ij} \le 0, \quad \gamma > 0. $$
This condition guarantees forward invariance of $\mathcal{S}_{ij}$ under a Lipschitz continuous controller.
3. Main Approach
3.1 State‑Disturbance Estimation Framework
We design a distributed extended state observer that uses only bearing measurements and relative distances from neighbors to estimate the follower’s state and disturbance. For each follower $i$, define the estimation error signals:
$$ e_{i,1} = \sum_{j\in\mathcal{N}_{i}} P_{ij}(z_{i,1}-z_{j,1}) – \sum_{j\in\mathcal{N}_{il}} P_{ij}(z_{i,1}-p_j^*) – \sum_{j\in\mathcal{N}_{if}} P_{ij}(d_{ij}g_{ij}), $$
$$ e_{i,2} = \sum_{j\in\mathcal{N}_{i}} P_{ij}(z_{i,2}-z_{j,2}) – \sum_{j\in\mathcal{N}_{il}} P_{ij}(z_{i,2}-\dot{p}_j^*) – \sum_{j\in\mathcal{N}_{if}} P_{ij}(\dot{d}_{ij}g_{ij} + d_{ij}\dot{g}_{ij}), $$
where $z_{i,1},z_{i,2}$ are the observer outputs, $p_j^*$ the known leader position, $d_{ij}$ the measured distance, and $g_{ij}$ the measured bearing. The observer dynamics are:
$$ \dot{z}_{i,1} = z_{i,2} – \beta_1 e_{i,1}, $$
$$ \dot{z}_{i,2} = z_{i,3} – \beta_2 e_{i,2} + \hat{f}_i(z_{i,1},z_{i,2}) + u_i, $$
$$ \dot{z}_{i,3} = -\beta_3 e_{i,1}, $$
where $\beta_1,\beta_2,\beta_3>0$ are observer gains, and $\hat{f}_i$ is the adaptive estimate of the unmodeled dynamics, parameterized as $\hat{f}_i = \Xi_i(z_{i,1},z_{i,2})\hat{\xi}_i$, with $\Xi_i$ known regressor and $\hat{\xi}_i$ the adaptive parameter. The adaptation law is:
$$ \dot{\hat{\xi}}_i = \mu_i \Xi_i^{\mathrm{T}} e_{i,2} – \sigma_i \hat{\xi}_i, \quad \mu_i>0,\;\sigma_i>0. $$
We state a theorem establishing the boundedness of the estimation errors under proper gain conditions (see Theorem 1 in the original paper). The Lyapunov function used involves the observer error dynamics and the parameter estimation error, ensuring uniform ultimate boundedness.
3.2 Desired Trajectory Observer
To obtain the desired formation trajectory for each follower, we employ a dynamic compensator that estimates the reference position and its derivatives using only bearing constraints and leader information. The compensator is given by (39) in the original paper, and it achieves exponential convergence of the estimated desired state to the true desired state provided the communication graph is bearing‑rigid and the observer gain $\kappa$ satisfies a condition involving the minimum eigenvalue of $\mathcal{L}_{ff}$.
3.3 CLF‑RCBF Safe Controller
For formation tracking, we define the CLF as:
$$ V_i = k_p \| \hat{p}_i – \hat{p}_i^* \|^2 + k_d \| \hat{v}_i – \hat{v}_i^* \|^2, $$
where $\hat{p}_i,\hat{v}_i$ are the estimated position and velocity from the ESO, and $\hat{p}_i^*,\hat{v}_i^*$ are the estimated desired trajectory. The CLF condition is:
$$ L_{f}V_i + L_{g}V_i u_i + c V_i \le 0, \quad c>0. $$
For collision avoidance between UAV $i$ and any neighbor $j$, the RCBF condition is expressed as a linear inequality in $u_i$:
$$ L_f B_{ij} + L_g B_{ij} u_i + \gamma B_{ij} \le 0, \quad \gamma>0. $$
We formulate a quadratic programming problem that minimizes the control effort while satisfying both the CLF and all RCBF constraints:
$$ u_i^* = \arg\min_{u_i} \frac{1}{2} u_i^{\mathrm{T}} H u_i + f^{\mathrm{T}} u_i, $$
subject to
$$ A_{\mathrm{clf},i} u_i \le b_{\mathrm{clf},i}, $$
$$ A_{\mathrm{cbf},ij} u_i \le b_{\mathrm{cbf},ij},\quad \forall j\in\mathcal{N}_i, $$
with $A_{\mathrm{clf},i}=L_g V_i$, $b_{\mathrm{clf},i}=-L_f V_i-c V_i$, $A_{\mathrm{cbf},ij}=L_g B_{ij}$, and $b_{\mathrm{cbf},ij}=-L_f B_{ij}-\gamma B_{ij}$. We prove that the QP is always feasible because the RCBF constraints have a non‑empty feasible region (Theorem 2 in the original paper).
4. Simulation Results
4.1 Simulation Setup
We consider a swarm of 8 quadrotor UAVs (2 leaders, 6 followers) with physical parameters taken from a standard platform: arm length 0.4 m, mass 1.8 kg, maximum pitch/roll angle 30°, maximum thrust 52.92 N. The inner‑loop attitude and thrust control is approximated as first‑order systems with time constants 150 ms (attitude) and 30 ms (thrust). The desired formation is a cube defined by bearing constraints. Leaders follow a time‑varying acceleration pattern, while followers are initialized with random positions and velocities. External disturbance is $d(t)=[0.2\cos t,\sin t,-\sin t]^{\mathrm{T}}$ N. The unmodeled dynamics include constant bias, linear damping, quadratic drag, nonlinear damping, ground effect, and cross‑coupling terms. Three communication blackouts (5 s–5.5 s, 15 s–15.5 s, 40 s–40.5 s) are injected to test robustness. The observer gains and controller parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Observer gains ($\beta_1,\beta_2,\beta_3$) | (9, 27, 54) |
| Adaptation gains ($\mu,\sigma$) | (2.0, 0.15) |
| CLF gains ($k_p,k_d,c$) | (0.5, 0.1, 0.6) |
| RCBF parameters ($\gamma,\tau$) | (0.02, 0.8) |
| Safe distance $D_{\mathrm{safe}}$ | 0.5 m |
| Formation size | 4 m (cube side length) |
4.2 Formation Tracking Performance
The formation snapshots at four times (0 s, 10 s, 20 s, 50 s) show that the swarm converges to the desired cubic formation. The position tracking errors for each follower (shown in Figure 3 of the original) decrease over time. During the communication blackouts, the errors temporarily increase but quickly recover after reconnection, demonstrating robustness. The velocity tracking errors also converge to small values, confirming the effectiveness of the disturbance compensation.
4.3 Observer Estimation Accuracy
Figures 5–7 of the original paper illustrate the estimation errors for position, velocity, and total disturbance. The errors converge quickly after an initial transient, indicating that the combined adaptive ESO provides accurate state and disturbance estimates. This enables precise feedforward compensation in the control law.
4.4 Collision Avoidance Performance
We compare the minimum inter‑agent distance during the formation maneuver under three scenarios: the proposed RCBF‑based controller, an improved artificial potential field (APF) method, and a baseline without any collision‑avoidance mechanism. The results are summarized in Table 2.
| Method | Minimum distance (m) | Below 0.5 m? |
|---|---|---|
| Proposed CLF‑RCBF | 0.75 | No |
| Improved APF | 0.37 | Yes |
| No collision avoidance | 0.10 | Yes |
The proposed method maintains a safe distance of 0.75 m, well above the threshold, while both comparison methods violate the safety constraint. Moreover, the total energy consumption (integral of squared thrust) for our method is 236.41 J, compared to 1115.23 J for the APF method, demonstrating better efficiency. Our method thus achieves a superior balance between safety and performance, advancing safe drone technology in real‑world applications.
4.5 Control Effort Validation
The attitude angles and thrust forces throughout the simulation stay within physical limits: pitch/roll angles remain below 30° and thrust stays below 52.92 N. This confirms that the generated control commands are feasible for practical quadrotor platforms.
5. Conclusion
We have presented a bearing‑based collision‑avoidance formation control framework for quadrotor UAVs under both unmodeled dynamics and external disturbances. By fusing a parameter adaptive law with a distributed extended state observer, we accurately estimate each follower’s position, velocity, and lumped disturbance using only relative bearing information and distances. A control Lyapunov function for formation tracking and a reciprocal control barrier function that incorporates both position and velocity safety margins are integrated into a quadratic programming solver to generate safe control inputs. Theoretical analysis proved the feasibility of the optimization problem. Extensive 8‑UAV simulations demonstrated that the swarm can form and maintain a cubic formation under time‑varying disturbances and communication blackouts, while the minimum inter‑agent distance was kept at 0.75 m—far above the safety threshold—outperforming both an improved artificial potential field method and a baseline without collision avoidance. The results highlight the potential of our approach for enhancing safety and robustness in practical drone technology swarms.
Future work will extend this framework to large‑scale swarms with higher‑order safety constraints, and incorporate real‑world flight tests to validate the algorithm under actual aerodynamic interactions.