In recent years, the rapid advances in control theory, navigation technology, and material science have significantly propelled the development of unmanned aerial vehicles (UAVs). Among various types, the fixed-wing drone stands out due to its long endurance, high speed, and large payload capacity. However, single fixed-wing drone systems often suffer from high cost, long development cycles, and poor fault tolerance. To overcome these limitations, the concept of fixed-wing drone swarms has emerged. By coordinating multiple low-cost fixed-wing drone platforms, the swarm can accomplish complex missions such as reconnaissance, surveillance, and search-and-rescue with enhanced robustness and flexibility. A fundamental problem in fixed-wing drone swarms is formation tracking control, which aims to drive the swarm to follow a desired geometric configuration while tracking a virtual leader trajectory.
Existing formation control strategies include consensus-based and leader-follower approaches. Many works achieve finite-time convergence, but the settling time bound depends on initial conditions. This is undesirable in practical scenarios where the swarm must form a specific configuration within a strict deadline, e.g., collision avoidance or time-critical rendezvous. To address this, the fixed-time control concept guarantees that the convergence time is uniformly bounded regardless of initial states. However, most fixed-time results focus on simple integrator dynamics, while fixed-wing drone swarms exhibit nonlinear kinematics and actuator uncertainties. In this paper, we propose a distributed fixed-time formation tracking controller for a group of fixed-wing drones subject to unmodeled dynamics and external disturbances. We leverage a non-singular fixed-time terminal sliding mode to achieve both the reaching and sliding phases with a priori known convergence upper bound. The control law includes a robust term to compensate for perturbations. Stability is proven via Lyapunov theory, and numerical simulations validate the effectiveness.
Problem Formulation
Consider a swarm of \(N\) fixed-wing drones. The 3-degree-of-freedom (3-DOF) kinematic model in the inertial frame (North-East-Down) is described by:
\[
\begin{aligned}
\dot{x}_i &= V_i \cos\gamma_i \cos\chi_i, \\
\dot{y}_i &= V_i \cos\gamma_i \sin\chi_i, \\
\dot{z}_i &= V_i \sin\gamma_i, \\
\dot{V}_i &= (V_{ci} – V_i)/\tau_V, \\
\dot{\chi}_i &= (\chi_{ci} – \chi_i)/\tau_\chi, \\
\dot{\gamma}_i &= (\gamma_{ci} – \gamma_i)/\tau_\gamma,
\end{aligned}
\]
where \(p_i = [x_i, y_i, z_i]^T\) is the position, \(V_i\) the ground speed, \(\chi_i\) the heading angle, \(\gamma_i\) the flight path angle. \(V_{ci}, \chi_{ci}, \gamma_{ci}\) are the corresponding control inputs; \(\tau_V, \tau_\chi, \tau_\gamma\) are time constants.
We simplify the model to a double-integrator form by defining the virtual acceleration input \(u_i \in \mathbb{R}^3\) and lumped disturbance \(d_i\) including actuator unmodeled dynamics and uncertainties:
\[
\dot{p}_i = v_i, \quad \dot{v}_i = u_i + d_i,
\]
where \(v_i = [\dot{x}_i, \dot{y}_i, \dot{z}_i]^T\). The relationship between the original control inputs \((V_{ci},\chi_{ci},\gamma_{ci})\) and \(u_i\) is given by nonlinear algebraic equations (see, e.g., Dong et al., 2020). We assume the disturbance is bounded: \(\|d_i\| \leq d_{\max}\).
The communication topology among the fixed-wing drones is modeled as an undirected graph \(\mathcal{G} = (\mathcal{V},\mathcal{E},A)\). Let \(\mathcal{V} = \{1,\dots,N\}\), adjacency matrix \(A = [a_{ij}]\) with \(a_{ij}=1\) if drone \(i\) receives information from drone \(j\), else 0. Each drone also may receive data from a virtual leader; let \(B = \operatorname{diag}\{b_1,\dots,b_N\}\) where \(b_i=1\) if drone \(i\) can access the leader. Define \(H = L + B\) where \(L\) is the Laplacian matrix of \(\mathcal{G}\). If the leader-connected graph is connected, \(H\) is positive definite.
We define the desired formation for each fixed-wing drone as \(p_{di} = p_d + h_{pi}\), \(v_{di} = v_d + h_{vi}\), where \(p_d, v_d\) are the leader’s position and velocity, and \(h_{pi}, h_{vi}\) encode the desired relative formation. The tracking errors are:
\[
e_{pi} = p_i – p_{di}, \quad e_{vi} = v_i – v_{di}.
\]
The error dynamics become:
\[
\dot{e}_{pi} = e_{vi}, \quad \dot{e}_{vi} = u_i – \dot{v}_{di} + d_i,
\]
with \(\dot{v}_{di} = u_{di}\) bounded. Define the aggregated variables:
\[
e_{1i} = \sum_{j\in N_i} a_{ij}( e_{pi} – e_{pj}) + b_i e_{pi}, \quad e_{2i} = \sum_{j\in N_i} a_{ij}( e_{vi} – e_{vj}) + b_i e_{vi}.
\]
Clearly \(\dot{e}_{1i} = e_{2i}\). In vector form, \(\mathbf{e}_1 = (H\otimes I_3)\mathbf{e}_p\), \(\mathbf{e}_2 = (H\otimes I_3)\mathbf{e}_v\). Our goal is to design \(u_i\) such that \(e_{pi}\) and \(e_{vi}\) converge to a small neighborhood of zero within a fixed time independent of initial conditions.
Controller Design
Fixed-Time Sliding Surface
We adopt a non-singular fixed-time terminal sliding mode (Zuo, 2015). For a scalar variable \(x\) with \(\dot{x} = y\), define the sliding surface:
\[
s = y + f_s(x),
\]
where
\[
f_s(x) =
\begin{cases}
\alpha_1 \operatorname{sig}^{p_1}(x) + \beta_1 \operatorname{sig}^{g_1}(x), & \text{if } \bar{s}=0 \text{ or } (\bar{s}\neq0, |x|>\mu), \\
\alpha_2 x + \beta_2 \operatorname{sig}^2(x), & \text{if } \bar{s}\neq0, |x|\leq\mu,
\end{cases}
\]
with \(\bar{s} = y + \alpha_1 \operatorname{sig}^{p_1}(x) + \beta_1 \operatorname{sig}^{g_1}(x)\). Parameters satisfy \(\alpha_1,\beta_1>0\), \(0<p_1<10\) small, and \(\alpha_2,\beta_2\) are chosen to ensure smoothness at \(x=\pm\mu\). When \(s=\bar{s}=0\), the state \(x\) converges to zero in fixed time bounded by:</p_1<1
\[
T_{ft} = \frac{1}{\alpha_1(1-p_1)} + \frac{1}{\beta_1(g_1-1)}.
\]
For the fixed-wing drone swarm, we design the sliding variable for each drone as:
\[
s_i = e_{2i} + b_i f_{si}(e_{pi}) + \sum_{j=1}^N l_{ij} f_{sj}(e_{pj}),
\]
where \(l_{ij}\) are elements of the Laplacian matrix, and \(f_{si}\) follows the same structure as above but applied componentwise. In compact form:
\[
\mathbf{s} = H \big[ \mathbf{e}_2 + f_s(\mathbf{e}_1) \big].
\]
Control Law and Stability Analysis
We propose the distributed control law for the \(i\)-th fixed-wing drone:
\[
u_i = -l_1 \operatorname{sig}^{p_2}(s_i) – l_2 \operatorname{sig}^{g_2}(s_i) – \psi_i – \dot{f}_{si}(e_{pi}),
\]
where \(l_1,l_2>0\), \(0<p_2<1<g_2\), \(\dot{f}_{si}\)="" and="" as:
\[
\dot{f}_{si,m}(e_{pi,m}) =
\begin{cases}
\alpha_1 p_1 |e_{pi,m}|^{p_1-1} e_{vi,m} + \beta_1 g_1 |e_{pi,m}|^{g_1-1} e_{vi,m}, & \text{if } \bar{s}_{i,m}=0 \text{ or } (|e_{pi,m}|>\mu), \\
\alpha_2 e_{vi,m} + 2\beta_2 e_{pi,m} e_{vi,m}, & \text{if } \bar{s}_{i,m}\neq0 \text{ and } |e_{pi,m}|\leq\mu.
\end{cases}
\]
The robust term \(\psi_i = [\psi_{i1},\psi_{i2},\psi_{i3}]^T\) is designed to compensate disturbances:
\[
\psi_{i,m} = \kappa_i \tanh\!\left(\frac{3k_u \kappa_i s_{i,m}}{\zeta}\right),
\]
with \(\kappa_i \geq d_{\max}+\|u_{di}\|\), \(k_u>0\), \(\zeta>0\) small. This term satisfies:
\[
s_i^T(d_i – u_{di}) – s_i^T\psi_i \leq \zeta.
\]
Theorem 1. Consider the fixed-wing drone swarm system with unmodeled dynamics and bounded disturbances, under the undirected and connected communication graph (leader-connected). With the sliding surface and control law defined above, the following holds:
- The sliding manifold \(\mathbf{s}\) converges to a region \(\Omega_1\) within a fixed time \(T_1\) bounded by:
\[
T_1 = \frac{2}{\tau_m(1-p_2)} + \frac{2}{\delta_m(g_2-1)},
\]
\[
\tau_m = l_1 \left(\frac{2}{\lambda_{\max}(H_2)}\right)^{\frac{p_2+1}{2}}, \quad \delta_m = l_2 (3N)^{\frac{1-g_2}{2}}\left(\frac{2}{\lambda_{\max}(H_2)}\right)^{\frac{g_2+1}{2}},
\]
and \(\Omega_1 = \{ \mathbf{s} : \|\mathbf{s}\| \leq \Delta_1 \}\) with
\[
\Delta_1 = \min\left\{ \frac{2}{\lambda_{\min}(H_2)} \left( \frac{N\zeta}{\tau_m} \right)^{\frac{1}{p_2+1}}, \frac{2}{\lambda_{\min}(H_2)} \left( \frac{N\zeta}{\delta_m} \right)^{\frac{1}{g_2+1}} \right\}.
\]
- Consequently, the tracking errors \(e_{pi}\) and \(e_{vi}\) converge to a small region \(\Omega_2\) within a fixed total time \(T_t = T_1 + T_2\), where
\[
T_2 = \frac{1}{\alpha_1(1-p_1)} + \frac{1}{\beta_1(g_1-1)}.
\]
The region \(\Omega_2\) is characterized by:
\[
|e_{pi,m}| \leq \max\left\{ \mu, \min\left( \left( \frac{\|H_2\|\Delta_1}{\alpha_1} \right)^{1/p_1}, \left( \frac{\|H_2\|\Delta_1}{\beta_1} \right)^{1/g_1} \right) \right\}, \quad
|e_{vi,m}| \leq \|H_2\|\Delta_1 + \alpha_1 \Delta_{e_p}^{p_1} + \beta_1 \Delta_{e_p}^{g_1}.
\]
The proof uses the Lyapunov function \(V = \frac12 \mathbf{s}^T H_2 \mathbf{s}\) with \(H_2 = H^{-1}\). Differentiating and applying the control law leads to \(\dot{V} \leq -\tau_m V^{\frac{p_2+1}{2}} – \delta_m V^{\frac{g_2+1}{2}} + N\zeta\). Fixed-time convergence follows from Lemma 2 of (Polyakov, 2012). Once \(\mathbf{s}\) enters \(\Omega_1\), the sliding dynamics ensure the error states converge to \(\Omega_2\) within \(T_2\).
Numerical Simulation
We validate the proposed controller through simulations on a swarm of three fixed-wing drones. The parameters are listed in Tables I and II. The communication topology is a complete graph among drones, and drone 2 is connected to the virtual leader. The leader maintains constant velocity \(v_d = [22.63, 22.63, 0]^T\) m/s from initial position \([0,0,100]^T\) m. The desired formation is an equilateral triangle in the horizontal plane with side \(20\) m, as given in Table III.
| State | Drone 1 | Drone 2 | Drone 3 |
|---|---|---|---|
| Position [m] | [75.89, -136.09, 55.19] | [-1.72, 1.54, 60.03] | [158.58, -229.71, 129.04] |
| Speed [m/s] | 24.86 | 30.68 | 25.47 |
| Heading [rad] | 0.81 | 0.88 | 0.87 |
| Flight path angle [rad] | 0.07 | 0.06 | 0.04 |
| Drone | \(h_{px}\) [m] | \(h_{py}\) [m] | \(h_{pz}\) [m] | \(h_{v}\) [m/s] |
|---|---|---|---|---|
| 1 | -17.39 | -9.88 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 |
| 3 | -9.88 | -17.39 | 0 | 0 |
| Param | Value | Param | Value |
|---|---|---|---|
| \(\alpha_1\) | 0.15 | \(\beta_1\) | 0.15 |
| \(p_1\) | 0.6 | \(g_1\) | 9/7 |
| \(l_1\) | 0.5 | \(l_2\) | 0.5 |
| \(p_2\) | 0.6 | \(g_2\) | 9/7 |
| \(\mu\) | 0.1 | \(k_u\) | 0.2785 |
The disturbance is set as \(d_i = 0.001[\sin(t), \cos(t), -\sin(t)]^T\). Figure 1 shows the time histories of speed, heading, and flight path angles. All three fixed-wing drones converge to the desired leader states within approximately 14 s, demonstrating fixed-time convergence. Figure 2 plots the position tracking errors \(e_{pi}\) and velocity tracking errors \(e_{vi}\); they decrease to negligibly small values before 14 s. Figure 3 presents the 3D trajectories of the swarm; initially scattered, the fixed-wing drones form a precise triangular formation by the final time. A top view of the final formation is shown in Figure 4, confirming the desired geometry.
The convergence time predicted by our theoretical bound (with the chosen parameters) is \(T_1 + T_2 \approx 2.73 + 16.67 = 19.4\) s, which is conservative but of the same order as the observed 14 s. The discrepancy arises from the worst-case analysis; the actual performance is faster. The robust term effectively compensates for the bounded disturbances, and the sliding mode is maintained within a small region. The results verify that the swarm of fixed-wing drones can achieve formation tracking within a fixed time independent of initial conditions.
Conclusion
In this paper, we addressed the distributed fixed-time formation tracking control problem for a swarm of fixed-wing drones subject to actuator unmodeled dynamics and uncertainties. By employing a non-singular fixed-time terminal sliding surface, we designed a control law that guarantees the sliding variables converge to a bounded region within a fixed time, and subsequently the position and velocity tracking errors converge to a small neighborhood of zero within a fixed total time that does not depend on initial states. The robust term ensures disturbance rejection. Lyapunov analysis provided rigorous stability guarantees. Simulation results on a three-fixed-wing drone swarm confirmed that the swarm achieves the desired formation within the predicted fixed time, validating the effectiveness and practicality of the approach. Future work will extend the method to time-varying formation and obstacle avoidance scenarios, and consider directed communication topologies and actuator saturation.
</p_2<1