We present a comprehensive study on the distributed robust collision avoidance formation tracking control problem for multiple fixed-wing UAVs subject to input saturation, velocity constraints, external disturbances, and safety constraints. This work is motivated by the practical need for fixed-wing UAV swarms to execute complex missions such as search, surveillance, and cargo delivery in cluttered environments while maintaining strict safety margins. The nonholonomic nature, underactuation, and minimum speed requirement of fixed-wing UAVs make the control design particularly challenging. To address these challenges, we propose a novel framework that integrates a sliding-mode-based nominal formation tracking controller with high-order control barrier functions (HOCBF) to achieve both precise tracking and provable collision avoidance. The key contributions include: (i) a distributed finite-time observer that provides each agent with estimates of the leader’s states despite bounded disturbances; (ii) a nominal control law that satisfies input and velocity constraints and ensures asymptotic convergence of formation errors; (iii) the construction of robust HOCBF constraints that explicitly account for input disturbances and actuator saturation; and (iv) a local quadratic programming (QP) formulation that minimally modifies the nominal control to enforce safety. Extensive simulation results demonstrate the effectiveness of the proposed method compared to existing approaches such as potential field methods and distributed MPC, highlighting superior computational efficiency and robustness.
1. Introduction
Fixed-wing UAVs are widely used in both military and civilian applications due to their long endurance and high cruising speed. However, controlling multiple fixed-wing UAVs in formation while avoiding collisions with each other and with static obstacles is a challenging problem. The dynamics are nonholonomic and underactuated, and the presence of external disturbances further complicates the design. Existing methods, such as artificial potential fields, often lead to chattering and cannot explicitly handle input constraints. Model predictive control (MPC) can handle constraints but is computationally heavy, especially for real-time applications. Control barrier functions (CBF) have emerged as a powerful tool to enforce safety constraints with low computational burden. In this work, we extend the CBF framework to fixed-wing UAVs by designing high-order CBFs (HOCBF) that ensure collision avoidance while respecting velocity and input saturation under bounded disturbances. We develop a distributed architecture where each UAV uses local information from neighbors and a finite-time observer to estimate the leader’s states. The nominal controller is designed using sliding mode techniques to guarantee asymptotic convergence of formation errors. Then, a QP problem is solved at each time step to blend the nominal control with safety constraints. The resulting control law is robust, computationally efficient, and provably safe. This paper provides a systematic solution to the multi-fixed-wing UAV safe formation control problem, filling a gap in the literature.
2. Problem Formulation
Consider a group of \(N\) fixed-wing UAVs (followers) and one virtual leader flying at the same altitude. The kinematic and dynamic model of the \(i\)-th fixed-wing UAV is given by:
\[
\dot{x}_i = v_i \cos\theta_i,\quad \dot{y}_i = v_i \sin\theta_i,\quad \dot{v}_i = a_i + \delta_{a_i},\quad \dot{\theta}_i = \omega_i + \delta_{\omega_i},
\]
where \((x_i, y_i)\) is the position, \(v_i\) is the linear velocity, \(\theta_i\) is the heading angle, \(a_i\) and \(\omega_i\) are the control inputs (linear acceleration and angular velocity), and \(\delta_{a_i}\) and \(\delta_{\omega_i}\) are bounded disturbances satisfying \(|\delta_{a_i}| \le \delta_a^m\) and \(|\delta_{\omega_i}| \le \delta_\omega^m\). The following constraints must be satisfied:
\[
0 < v_{\min} \le v_i(t) \le v_{\max},\quad |a_i(t)| \le a_m,\quad |\omega_i(t)| \le \omega_m.
\]
The leader’s dynamics are defined similarly with known bounds on its inputs. The communication graph among followers plus the leader contains a directed spanning tree rooted at the leader. The formation tracking error for each follower is defined as \(e_i^x = x_0 – x_i – d_i^x\), \(e_i^y = y_0 – y_i – d_i^y\), \(e_i^v = v_0 – v_i\), \(e_i^\theta = \theta_0 – \theta_i\), where \((d_i^x, d_i^y)\) is the desired offset. The control objective is to drive all tracking errors to zero while avoiding collisions with other fixed-wing UAVs and obstacles, and respecting the constraints. The safety condition between two fixed-wing UAVs \(i\) and \(j\) is \(\|\Delta p_{ij}\| > D_s\), where \(D_s\) is the safety distance, and similarly for obstacles.
3. Preliminaries
3.1 Graph Theory
The communication topology among the \(N\) followers is described by an undirected graph \(\mathcal{G} = \{\mathcal{V}, \mathcal{E}\}\) with adjacency matrix \(A = [a_{ij}]\). The leader is node 0. The overall graph \(\bar{\mathcal{G}}\) contains a directed spanning tree rooted at the leader. The Laplacian matrix \(L\) and the diagonal matrix \(A_0 = \text{diag}(a_{10}, …, a_{N0})\) are used to define the matrix \(H = L + A_0\). Under Assumption 1, there exists a positive vector \(\eta\) such that \(\Psi = \frac{1}{2}( \text{diag}(\eta) H + H^T \text{diag}(\eta))\) is positive definite.
3.2 High-Order Control Barrier Functions
Consider a nonlinear system \(\dot{x} = f(x) + g(x)u\). For a constraint \(h(x) > 0\) with relative degree \(m\), we define a sequence of functions \(\psi_0 = h(x)\), \(\psi_k = \dot{\psi}_{k-1} + \alpha_k(\psi_{k-1})\), where \(\alpha_k\) are class \(\mathcal{K}\) functions. The set \(C_k = \{x | \psi_{k-1}(x) > 0\}\) is forward invariant if there exists a control law satisfying:
\[
L_f^m h(x) + L_g L_f^{m-1} h(x) u + S(h(x)) + \alpha_m(\psi_{m-1}(x)) > 0.
\]
This inequality defines the admissible control set. In this work, we construct HOCBF of relative degree 2 for both inter-UAV collision avoidance and UAV-obstacle collision avoidance, while explicitly considering input disturbances.
4. Nominal Formation Tracking Control Law
4.1 Distributed Finite-Time Observer
Since not all followers have direct access to the leader’s states, each follower maintains a virtual leader and updates it using a finite-time observer. The observer for follower \(i\) estimates the leader’s position error, velocity, acceleration, heading, and angular rate. The observer dynamics are:
\[
\begin{aligned}
\dot{\hat{e}}_i^x &= \hat{v}_i^0 \cos\hat{\theta}_i^0 – v_i\cos\theta_i – \alpha \operatorname{sgn}\Big(\sum_{j=0}^N a_{ij}[(\hat{e}_i^x – \hat{e}_j^x) + (x_i – x_j) + d_{ij}^x]\Big),\\
\dot{\hat{v}}_i^0 &= -\beta_{v1} \Big|\sum_{j=0}^N a_{ij}(\hat{v}_i^0 – \hat{v}_j^0)\Big|^{1/2} \operatorname{sgn}\big(\sum_{j=0}^N a_{ij}(\hat{v}_i^0 – \hat{v}_j^0)\big) + \hat{a}_i^0,\\
\dot{\hat{\theta}}_i^0 &= -\beta_{\theta1} \Big|\sum_{j=0}^N a_{ij}(\hat{\theta}_i^0 – \hat{\theta}_j^0)\Big|^{1/2} \operatorname{sgn}\big(\sum_{j=0}^N a_{ij}(\hat{\theta}_i^0 – \hat{\theta}_j^0)\big) + \hat{\omega}_i^0,
\end{aligned}
\]
with appropriate update laws for \(\hat{a}_i^0\) and \(\hat{\omega}_i^0\). Under Assumptions 1 and 2, the observer converges in finite time to the true leader states (Theorem 2). This provides each agent with accurate local information for control.
4.2 Nominal Control Design
Using the estimated leader states, we transform the formation tracking problem into a local tracking problem. Define the transformed error coordinates:
\[
\begin{bmatrix} x_{ei} \\ y_{ei} \\ \theta_{ei} \end{bmatrix} =
\begin{bmatrix} \cos\theta_i & \sin\theta_i & 0 \\ -\sin\theta_i & \cos\theta_i & 0 \\ 0 & 0 & 1 \end{bmatrix}
\begin{bmatrix} e_i^x \\ e_i^y \\ e_i^\theta \end{bmatrix}.
\]
The dynamics become:
\[
\dot{x}_{ei} = v_0 \cos\theta_{ei} – v_i + (\omega_i + \delta_{\omega_i}) y_{ei},\quad
\dot{y}_{ei} = -(\omega_i + \delta_{\omega_i}) x_{ei} + v_0 \sin\theta_{ei},\quad
\dot{\theta}_{ei} = \omega_0 – \omega_i – \delta_{\omega_i}.
\]
We propose the following nominal control law:
\[
v_i = v_0 + \phi_i,\quad \dot{\phi}_i = -\phi_i + \frac{\alpha_1 x_{ei}}{\sqrt{1 + x_{ei}^2 + y_{ei}^2}},\quad
\omega_i = \omega_0 + \frac{k_3 v_0 \big( y_{ei} \cos\frac{\theta_{ei}}{2} – x_{ei} \sin\frac{\theta_{ei}}{2} \big)}{\sqrt{1 + x_{ei}^2 + y_{ei}^2}} + (\delta_\omega^m + \varepsilon_\omega) \operatorname{sgn}\big(\sin\frac{\theta_{ei}}{2}\big).
\]
Using a Lyapunov function, we prove that the formation errors converge to zero (Lemma 5). To implement this control with acceleration inputs, we design a sliding surface \(s_i = v_i – v_0 – \phi_i\) and set the linear acceleration as:
\[
a_i = a_0 – \phi_i + \frac{\alpha_1 x_{ei}}{\sqrt{1 + x_{ei}^2 + y_{ei}^2}} – (\delta_a^m + \varepsilon_a) \operatorname{sgn} s_i.
\]
This ensures finite-time convergence of the sliding surface (Lemma 6). The overall nominal controller (combining observer estimates and above laws) achieves formation tracking while respecting input and velocity constraints after a finite transient (Theorem 3). The parameters \(\alpha_1, k_3, \varepsilon_a, \varepsilon_\omega\) are chosen to guarantee the constraints are satisfied.
5. HOCBF-based Collision Avoidance Constraints
5.1 Inter-UAV Collision Avoidance
To avoid collisions between two fixed-wing UAVs \(i\) and \(j\), we define the safety function \(h_{ij} = \|\Delta p_{ij}\| – D_s\) with relative degree 2. The HOCBF is constructed by defining:
\[
\psi^{ij}_0 = h_{ij},\quad \psi^{ij}_1 = \dot{\psi}^{ij}_0 + k_4 \psi^{ij}_0,
\]
where \(k_4 > 0\). To handle input disturbances and ensure feasibility under velocity constraints, we replace \(\psi^{ij}_1\) with a sufficient condition:
\[
\bar{\psi}^{ij}_1 = v_i (1 – |\theta_i – \theta_{ij}^d|) + v_j (1 – |\theta_j – \theta_{ji}^d|) + k_4 (\|\Delta p_{ij}\| – D_s) > 0,
\]
where \(\theta_{ij}^d = \text{atan2}(y_j – y_i, x_j – x_i)\). Then the second-order HOCBF condition \(\dot{\bar{\psi}}^{ij}_1 + k_5 \bar{\psi}^{ij}_1 > 0\) yields a linear inequality in the control inputs \((a_i, \omega_i)\) and \((a_j, \omega_j)\). After accounting for disturbance bounds, we obtain a decoupled constraint:
\[
E_{ij} u_i \le c_{ij},\quad E_{ji} u_j \le c_{ji},
\]
with \(u_i = [a_i, \omega_i]^\top\), \(E_{ij} = [ -1 + |\theta_i – \theta_{ij}^d|,\; v_i \operatorname{sgn}(\theta_i – \theta_{ij}^d) ]\), and \(c_{ij}\) given by:
\[
c_{ij} = v_i \operatorname{sgn}(\theta_i-\theta_{ij}^d) \dot{\theta}_{ij}^d – \delta_a^m(1-|\theta_i-\theta_{ij}^d|) – \delta_\omega^m v_i + k_4 v_i \frac{\Delta p_{ij}^\top}{\|\Delta p_{ij}\|} \begin{bmatrix} \cos\theta_i \\ \sin\theta_i \end{bmatrix} + \frac{k_5}{2} v_i (1-|\theta_i-\theta_{ij}^d|) + \frac{k_5}{2} v_j (1-|\theta_j-\theta_{ji}^d|) + \frac{k_4 k_5}{2}(\|\Delta p_{ij}\|-D_s).
\]
Each UAV only needs to consider neighbors within a certain distance to reduce computational load.
5.2 UAV-Obstacle Collision Avoidance
For a static obstacle \(O_l\), define \(h_{iO_l} = \|\Delta p_{iO_l}\| – D_{sl}\). Similar derivations lead to the HOCBF condition:
\[
E_{iO_l} u_i \le c_{iO_l},
\]
where
\[
c_{iO_l} = v_i \operatorname{sgn}(\theta_i-\theta_{iO_l}^d) \dot{\theta}_{iO_l}^d – \delta_a^m(1-|\theta_i-\theta_{iO_l}^d|) – \delta_\omega^m v_i + k_4 v_i \frac{\Delta p_{iO_l}^\top}{\|\Delta p_{iO_l}\|} \begin{bmatrix} \cos\theta_i \\ \sin\theta_i \end{bmatrix} + k_5 v_i (1-|\theta_i-\theta_{iO_l}^d|) + k_4 k_5(\|\Delta p_{iO_l}\|-D_{sl}).
\]
Again, only obstacles within a certain range are considered.
6. Robust Collision Avoidance Formation Control via QP
At each time step, each fixed-wing UAV \(i\) solves the following quadratic programming problem to obtain the actual control input \(u_i^* = [a_i^*, \omega_i^*]^\top\):
\[
\begin{aligned}
u_i^* &= \arg\min_{u_i \in \mathbb{R}^2} \| u_i – u_{i,\text{nom}} \|^2 \\
\text{s.t.}\quad & E_{ij} u_i \le c_{ij}, \quad \forall j \in \mathcal{N}_i^c, \\
& E_{iO_l} u_i \le c_{iO_l}, \quad \forall O_l \in \mathcal{N}_i^O, \\
& \left[ v_i – \frac{v_{\max}+v_{\min}}{2},\; 0 \right] u_i \le – \left| v_i – \frac{v_{\max}+v_{\min}}{2} \right| \delta_a^m + \frac{k_6}{2} h_i, \\
& |a_i| \le a_m,\quad |\omega_i| \le \omega_m,
\end{aligned}
\]
where \(u_{i,\text{nom}}\) is the nominal control from Section 4, \(k_6 > 0\), and \(h_i = \left(\frac{v_{\max}-v_{\min}}{2}\right)^2 – \left(v_i – \frac{v_{\max}+v_{\min}}{2}\right)^2\) is a control barrier function ensuring the velocity constraint. The constraints are all linear, making the QP easy to solve. Theorem 4 states that if the initial conditions satisfy the safety sets and velocity bounds, then the proposed control law guarantees collision avoidance and constraint satisfaction for all time.
7. Simulation Results
We simulate a group of 5 follower fixed-wing UAVs and 1 leader with communication topology as given in the original paper. The parameters are: \(a_m = 6\,\text{m/s}^2\), \(\omega_m = 3\,\text{rad/s}\), \(v_{\min}=0.5\,\text{m/s}\), \(v_{\max}=3\,\text{m/s}\), \(D_s = 2\,\text{m}\), obstacle radius \(3\,\text{m}\), disturbances \(\delta_{a_i}=0.5\sin t\), \(\delta_{\omega_i}=0.1\cos t\). The nominal controller parameters are \(\alpha_1=0.5\), \(k_3=2\), \(\varepsilon_a=0.1\), \(\varepsilon_\omega=0.1\). The HOCBF parameters are \(k_4=1\), \(k_5=2\), \(k_6=1\). The results demonstrate that all fixed-wing UAVs successfully track the formation while avoiding obstacles and maintaining safe distances. The control inputs always stay within bounds. Table 1 summarizes a comparison with three other methods: (i) HOCBF without disturbance robustness (this method fails to avoid collision), (ii) potential field method (causes chattering and violates input constraints), and (iii) distributed MPC (achieves all objectives but with much higher computation time). Our method achieves low computation time (47.6 s for the whole simulation) and satisfies all constraints.
| Control Law | Collision Avoidance | Input & Velocity Constraints Satisfied | Computation Time (s) |
|---|---|---|---|
| Proposed Robust HOCBF | Yes | Yes | 47.6 |
| HOCBF without disturbance robustness [33] | No | Yes | – |
| Potential function method [34] | Yes | No | 72.1 |
| Distributed MPC [35] | Yes | Yes | 5219.0 |
The evolution of the minimum distance between each fixed-wing UAV and its nearest object shows that the distances always exceed the safety threshold \(D_s\). The velocity profiles and control inputs are all within the specified bounds after a short transient. The distributed finite-time observer converges to the true leader states within about 5 seconds. These results validate the theoretical analysis.
8. Conclusion
We have developed a distributed robust control framework for multi-fixed-wing UAV formation tracking with guaranteed collision avoidance and constraint satisfaction. By combining sliding-mode-based nominal control with high-order control barrier functions, we handle input disturbances, velocity constraints, and safety in a unified manner. The local QP formulation is computationally efficient, making it suitable for real-time implementation. Simulation comparisons demonstrate the superiority of our method over potential field and MPC approaches. Future work will extend the method to three-dimensional maneuvers, dynamic obstacles, and hardware experiments on actual fixed-wing UAV platforms.