In this work, we aim to address the obstacle avoidance problem of quadrotor unmanned aerial vehicle (UAV) formations in complex environments, specifically focusing on the practical deployment of China UAV swarms. We propose a novel distributed formation obstacle avoidance method that integrates a consensus algorithm with a smooth bounded repulsion function. The approach is validated through extensive simulations, demonstrating its effectiveness in both static and dynamic irregular obstacle scenarios.
Introduction
With the rapid development of unmanned aerial vehicle technology, China UAV systems have gained widespread attention in both civilian and military applications. The formation control of multiple UAVs is a key enabler for complex missions such as search and rescue, surveillance, and cooperative transportation. However, existing methods often assume simple-shaped obstacles or require precise geometric knowledge, which is unrealistic in practice. Moreover, the coupling between formation keeping and obstacle avoidance remains a challenge, especially when external disturbances and model uncertainties are present.
To overcome these limitations, we design a new distributed controller that combines a consensus-based formation protocol with a repulsion function defined on the minimum distance between the UAV and the obstacle surface. This approach allows China UAV to avoid irregular obstacles more efficiently without being trapped in local minima. Additionally, an adaptive integral backstepping method is employed to track the desired trajectory obtained from the formation controller, ensuring robustness against external disturbances.
System Model
We consider a group of n quadrotor UAVs, each modeled by the following Newton-Euler equations. For the i-th UAV:
$$
\dot{P}_i = V_i
$$
$$
m\dot{V}_i = U_i^\Gamma R(\Theta_i) \mathbf{t}_3 – mg\mathbf{t}_3 + \mathbf{d}_i^V
$$
$$
\dot{\Theta}_i = \Pi(\Theta_i) \Omega_i
$$
$$
J_i \dot{\Omega}_i = -\Omega_i \times J_i \Omega_i + G_i^a + U_i^T + \mathbf{d}_i^\Omega
$$
where the symbols and parameters used in our China UAV model are defined below.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Mass of UAV | $m$ | $0.28$ | kg |
| Arm length | $l$ | $0.23$ | m |
| Moment of inertia (x) | $J_{ix}$ | $5.67 \times 10^{-3}$ | kg·m² |
| Moment of inertia (y) | $J_{iy}$ | $5.67 \times 10^{-3}$ | kg·m² |
| Moment of inertia (z) | $J_{iz}$ | $8.34 \times 10^{-3}$ | kg·m² |
| Motor inertia | $J_{ri}$ | $6.60 \times 10^{-5}$ | kg·m² |
| Lift coefficient | $k_b$ | $9.23 \times 10^{-5}$ | N·s² |
| Drag coefficient | $k_d$ | $3.03 \times 10^{-5}$ | N·m·s² |
Distributed Formation Control with Obstacle Avoidance
Consensus-Based Formation Protocol
We adopt a virtual leader approach. The dynamics of each UAV (including the virtual leader 0) are simplified as double integrators:
$$
\dot{x}_i = v_i, \quad \dot{v}_i = u_i, \quad i = 0,1,\dots,n
$$
The communication topology among the China UAV formation is represented by a weighted undirected graph. The consensus-based formation control law without obstacle avoidance is:
$$
\begin{aligned}
u_i &= \alpha\beta\left[(x_0 – x_i – l_i) + (v_0 – v_i)\right] \\
&\quad + \beta \sum_{j=1}^{n} a_{ij} \left[(x_j – x_i – l_i + l_j) + (v_j – v_i)\right]
\end{aligned}
$$
where $\alpha=1$ if the i-th UAV can receive information from the virtual leader, otherwise 0; $\beta>0$ is the gain; $l_i$ is the desired relative position of UAV i with respect to the leader.
Repulsion Function for Inter-UAV Collision Avoidance
To prevent collisions among China UAVs, we define a repulsion function based on the Euclidean distance between two UAVs. The prohibited zone has radius $r_u$ and the avoidance zone has outer radius $R_u$. The repulsion potential is:
$$
U_{ij}(x_{ij}) = \left\{\begin{array}{ll}
\frac{R_u^2 – x_{ij}^2}{(R_u^2 – r_u^2)(x_{ij}^2 – r_u^2)} , & r_u < x_{ij} \le R_u \\
0, & x_{ij} > R_u
\end{array}\right.
$$
The corresponding repulsive force is:
$$
F_{ij} = \nabla U_{ij}(x_{ij}) = \left\{
\begin{array}{ll}
\frac{4(R_u^2 – r_u^2)(x_{ij} – r_u^2)}{(R_u^2 – r_u^2)^2 (x_{ij}^2 – r_u^2)^3} (x_i – x_j), & r_u < x_{ij} \le R_u \\
0, & x_{ij} > R_u
\end{array}
\right.
$$
Obstacle Detection and Avoidance for Irregular Obstacles
A key contribution of this work is a method for China UAV to detect and avoid irregular obstacles based on the minimum distance between the UAV and the obstacle surface. Instead of modeling obstacles as spheres or cylinders, we continuously update the closest point on the obstacle boundary as the reference for the repulsion field. The repulsion function for obstacle b is:
$$
U_{i b}(x_{i b}) = \left\{\begin{array}{ll}
\frac{R_o^2 – x_{i b}^2}{(R_o^2 – r_o^2)(x_{i b}^2 – r_o^2)}, & r_o < x_{i b} \le R_o \\
0, & x_{i b} > R_o
\end{array}\right.
$$
where $x_{i b}$ is the distance from UAV i to the nearest obstacle point, $r_o$ and $R_o$ are the inner (prohibited) and outer (detection) radii for obstacles. The gradient yields the repulsive force $F_{i b}$.
Integrated Formation Avoidance Controller
The final control law for each China UAV integrates the consensus term with the repulsion forces:
$$
\begin{aligned}
u_i^z &= \alpha\beta\left[(x_0 – x_i – l_i) + (v_0 – v_i)\right] \\
&\quad + \beta \sum_{j=1}^{n} a_{ij} \left[(x_j – x_i – l_i + l_j) + (v_j – v_i)\right] \\
&\quad + \beta_1 \sum_{j=1}^{n} \nabla U_{ij}(x_{ij}) + \beta_1 \sum_{b=1}^{M} \nabla U_{i b}(x_{i b})
\end{aligned}
$$
where $\beta_1>0$ is the repulsion gain. This design guarantees connectivity preservation, collision avoidance, and relative distance stabilization.
Adaptive Integral Backstepping Trajectory Tracking
Once the desired trajectory $P_i^d$ is generated by the formation avoidance controller, we implement an adaptive integral backstepping method to track it. Define the position tracking error:
$$
e_{1i} = P_i^d – P_i
$$
The virtual control law is:
$$
\mu_{1i} = \dot{P}_i^d + k_{1i}e_{1i} + \gamma_{1i}\zeta_{1i},\quad \dot{\zeta}_{1i} = e_{1i}
$$
The position control law is then:
$$
\eta_i = m(g \mathbf{t}_3 + \mathbf{e}_1 + k_{2i}e_{2i} – \dot{\mu}_{1i} – \hat{\mathbf{d}}_i^V)
$$
where $\hat{\mathbf{d}}_i^V$ is the adaptive estimate of external disturbance, updated by $\dot{\hat{\mathbf{d}}}_i^V = \lambda_{1i} e_{2i}$. The stability is Lyapunov-based.
Similarly, attitude control is designed using the same methodology. The desired roll and pitch angles are derived from the virtual control inputs as:
$$
U_i^\Gamma = m\sqrt{U_{ix}^2 + U_{iy}^2 + (U_{iz} + g)^2}
$$
$$
\phi_i^d = \arcsin\left( \frac{m (U_{ix}\sin\psi_i^d – U_{iy}\cos\psi_i^d)}{U_i^\Gamma}\right)
$$
$$
\theta_i^d = \arctan\left( \frac{U_{ix}\cos\psi_i^d + U_{iy}\sin\psi_i^d}{U_{iz} + g}\right)
$$
Numerical Simulations and Performance Evaluation
We performed two simulation scenarios to validate the proposed method for China UAV formations. The parameters used are: $k_{1i}=8$, $k_{2i}=6$, $k_{3i}=8$, $k_{4i}=6$, $\gamma_{1i}=2$, $\gamma_{2i}=2$, $\lambda_{1i}=0.5$, $\lambda_{2i}=0.5$, $\tau=0.05$, $\beta=2$, $\beta_1=0.03$. The formation consists of four follower UAVs with the communication topology shown in the following figure.
Scenario 1: Static Irregular Obstacles
The virtual leader follows a sinusoidal trajectory. All four China UAVs start from different positions and form the desired diamond shape. At t=50s, the formation switches to a tighter configuration. The obstacle is an irregular shape approximated by a polygon. The 3D avoidance trajectories obtained by our method and the traditional artificial potential field (APF) method are compared. Table 2 summarizes the performance.
| Method | Total avoidance time (s) | Maximum avoidance speed (m/s) | Maximum trajectory tracking error (m) |
|---|---|---|---|
| Proposed method | 22.3 | 1.78 | 0.72 |
| Traditional APF | 27.8 | 3.49 | 2.43 |
Our method results in shorter avoidance time, lower peak speed, and significantly smaller tracking error. The velocity profiles are smoother, indicating less aggressive maneuvers.
Scenario 2: Dynamic Obstacles
Three moving obstacles are introduced. The China UAV formation must adjust its shape continuously. The performance comparison is given in Table 3.
| Method | Total avoidance time (s) | Maximum avoidance speed (m/s) | Maximum trajectory tracking error (m) |
|---|---|---|---|
| Proposed method | 24.2 | 1.46 | 0.72 |
| Traditional APF | 31.5 | 1.98 | 1.45 |
Again, the proposed method outperforms the traditional APF. The inter-UAV distance remains above the safety threshold of 0.7 m throughout the entire simulation, ensuring collision-free operation.
Conclusion
We have presented a distributed formation obstacle avoidance control strategy for quadrotor China UAVs. The method synergistically integrates a consensus-based formation protocol with a smooth repulsion function defined on the minimum distance to irregular obstacles. An adaptive integral backstepping controller ensures accurate trajectory tracking despite external disturbances. Simulation results demonstrate that our approach yields smoother avoidance paths, faster recovery, and lower tracking errors compared to traditional artificial potential field methods. The proposed technique is well-suited for practical China UAV missions in complex cluttered environments.