The coordinated flight of multiple unmanned aerial vehicles (UAVs), or drone formation, represents a pivotal technology with extensive applications in both civilian and military domains, including surveillance, logistics, search and rescue, and aerial displays. The core challenge lies in developing control strategies that ensure precise geometric patterns are maintained among the vehicles despite inherent complexities such as nonlinear dynamics, external disturbances, and practical limitations like actuator saturation. This article addresses the finite-time cooperative control problem for a drone formation in the presence of input constraints and external disturbances. We propose a novel adaptive non-singular terminal sliding mode control scheme, proving its stability and demonstrating its effectiveness through comprehensive numerical simulations.
The pursuit of reliable drone formation control has led to significant research interest in consensus-based theories, which provide a framework for information sharing and cooperative behavior among agents. While considerable work has focused on quadrotor platforms due to their simpler dynamics, fixed-wing UAVs present a more challenging control problem because of their highly nonlinear and coupled dynamics. Furthermore, most existing studies often overlook the critical issue of input saturation. In reality, a UAV’s actuators—thrusters, control surfaces—have physical limits. For fixed-wing aircraft, this is particularly pronounced as the propulsion system typically cannot produce reverse thrust, leading to asymmetric input constraints. Ignoring these limits in controller design compromises its practical viability.
Sliding Mode Control (SMC) is renowned for its robustness against disturbances and model uncertainties. Compared to linear sliding surfaces, terminal sliding modes guarantee faster, finite-time convergence of tracking errors. However, traditional Terminal Sliding Mode (TSM) control can suffer from singularity issues. Non-singular TSM variants overcome this problem. Our work builds upon these concepts to develop a practical solution for fixed-wing drone formation.
The main contributions of this work are threefold:
- We model the thrust and drag forces of fixed-wing UAVs separately, departing from common simplified models. This offers a more realistic representation, and we explicitly account for asymmetric input saturation, enhancing the practical relevance of our controller for real-world drone formation applications.
- We design a new global fast non-singular terminal sliding surface. This surface accelerates convergence when the system state is far from the sliding manifold and eliminates the singularity problem associated with conventional TSM. Its structure is simpler than many existing non-singular surfaces, making it more amenable to practical implementation.
- We construct an anti-saturation auxiliary system to handle input constraints effectively. Additionally, we incorporate an adaptive law to estimate and compensate for the unknown upper bounds of external disturbances, thereby significantly enhancing the system’s robustness for sustained drone formation flight.
1. Preliminaries and System Modeling
1.1 Graph Theory
The communication topology within a drone formation comprising a virtual leader and \( n \) follower UAVs is described by a graph \( \mathcal{G} \triangleq (\mathcal{V}, \mathcal{E}) \). The adjacency matrix is \( \mathcal{A} = [a_{ij}] \in \mathbb{R}^{n \times n} \), where \( a_{ij} = 1 \) if follower \( i \) receives information from follower \( j \), and \( a_{ij}=0 \) otherwise. The pinning matrix is \( \mathcal{B} = \text{diag}(b_1, …, b_n) \), where \( b_i = 1 \) if follower \( i \) has access to the leader’s state. The Laplacian matrix is \( \mathcal{L} = \mathcal{D} – \mathcal{A} \), where \( \mathcal{D} = \text{diag}(d(v_1), …, d(v_n)) \) is the degree matrix.
1.2 UAV and Formation Dynamics
We consider a nonlinear fixed-wing UAV model subject to external disturbances and input saturation. The kinematics and dynamics for the \( i \)-th UAV are:
$$
\begin{aligned}
\dot{x}_i &= V_i \cos \gamma_i \cos \chi_i, \\
\dot{y}_i &= V_i \cos \gamma_i \sin \chi_i, \\
\dot{z}_i &= V_i \sin \gamma_i, \\
\dot{V}_i &= \frac{T_i – D_i + d_{V_i}}{m_i} – g \sin \gamma_i, \\
\dot{\gamma}_i &= \frac{L_i \cos \phi_i – m_i g \cos \gamma_i + d_{\gamma_i}}{m_i V_i}, \\
\dot{\chi}_i &= \frac{L_i \sin \phi_i + d_{\chi_i}}{m_i V_i \cos \gamma_i}.
\end{aligned}
$$
Here, \( \mathbf{p}_i = (x_i, y_i, z_i)^T \) is the inertial position; \( V_i, \gamma_i, \chi_i \) are airspeed, flight path angle, and heading angle; \( m_i \) is mass; \( g \) is gravity; \( T_i, L_i, D_i \) are thrust, lift, and drag forces; \( \phi_i \) is the bank angle; and \( \mathbf{d}_i = (d_{V_i}, d_{\gamma_i}, d_{\chi_i})^T \) represents bounded external disturbances with an unknown upper bound.
Defining the collective control input vector \( \mathbf{u}_i = (T_i, L_i \cos \phi_i, L_i \sin \phi_i)^T \), the model can be compactly written as:
$$
\begin{aligned}
\dot{\mathbf{p}}_i &= \mathbf{v}_i, \\
\dot{\mathbf{v}}_i &= \frac{\mathbf{R}_i}{m_i} \text{sat}(\mathbf{u}_i) + \boldsymbol{\alpha}_i + \mathbf{d}’_i,
\end{aligned}
$$
where \( \mathbf{R}_i \) is a transformation matrix derived from the kinematics, \( \boldsymbol{\alpha}_i \) encapsulates drag and gravity terms, and \( \mathbf{d}’_i \) is the transformed disturbance. The saturation function \( \text{sat}(\cdot) \) enforces the physical input constraints: \( T_i \in [0, T_{\max}] \) and \( L_i \cos \phi_i, L_i \sin \phi_i \in [-L_{\max}, L_{\max}] \).
The control objective for the drone formation is for the followers to track a virtual leader while maintaining a desired relative geometry. Let \( \mathbf{p}_L \) be the leader’s position and \( \mathbf{h}_i \) be the desired offset of follower \( i \) from the leader. The combined tracking and formation error for UAV \( i \) is defined as:
$$
\mathbf{e}_i = \sum_{j=1}^{n} a_{ij}[(\mathbf{p}_i – \mathbf{h}_i) – (\mathbf{p}_j – \mathbf{h}_j)] + b_i(\mathbf{p}_i – \mathbf{h}_i – \mathbf{p}_L).
$$
The global formation error vector for the entire drone formation is:
$$
\mathbf{e} = \bar{\mathcal{L}}(\mathbf{p} – \mathbf{h}) – \bar{\mathcal{B}} \mathbf{P}_L,
$$
where \( \bar{\mathcal{L}} = (\mathcal{L}+\mathcal{B}) \otimes I_3 \), \( \bar{\mathcal{B}} = \mathcal{B} \otimes I_3 \), and \( \mathbf{P}_L = \mathbf{1}_n \otimes \mathbf{p}_L \). The control aim is to drive \( \mathbf{e} \) and \( \dot{\mathbf{e}} \) to zero in finite time.
2. Adaptive Non-Singular Terminal Sliding Mode Controller Design
2.1 Anti-Saturation Auxiliary System
To mitigate the effects of input saturation \( \text{sat}(\mathbf{u}) \), we introduce the following auxiliary system:
$$
\begin{aligned}
\dot{\boldsymbol{\lambda}}_1 &= -c_1 \boldsymbol{\lambda}_1 + \boldsymbol{\lambda}_2, \\
\dot{\boldsymbol{\lambda}}_2 &= -c_2 \boldsymbol{\lambda}_2 + \bar{\mathcal{L}} \mathbf{R}_m (\text{sat}(\mathbf{u}) – \mathbf{u}),
\end{aligned}
$$
where \( c_1, c_2 > 0 \) are design constants chosen to be sufficiently large for rapid desaturation. We then define new error states incorporating these auxiliary variables:
$$
\begin{aligned}
\mathbf{e}_1 &= \mathbf{e} – \boldsymbol{\lambda}_1, \\
\mathbf{e}_2 &= \dot{\mathbf{e}} – \dot{\boldsymbol{\lambda}}_1.
\end{aligned}
$$
The modified error dynamics become:
$$
\begin{aligned}
\dot{\mathbf{e}}_1 &= \mathbf{e}_2, \\
\dot{\mathbf{e}}_2 &= \bar{\mathcal{L}} \mathbf{R}_m \mathbf{u} – \bar{\mathcal{B}} \dot{\mathbf{P}}_L + \bar{\mathcal{L}} \boldsymbol{\alpha} + \mathbf{d} – c_1^2 \boldsymbol{\lambda}_1 + (c_1 + c_2)\boldsymbol{\lambda}_2.
\end{aligned}
$$
2.2 Controller and Adaptive Law Design
We propose a novel global fast non-singular terminal sliding surface:
$$
\mathbf{s} = \mathbf{e}_2 + k_1 \mathbf{e}_1 + k_2 \tanh(\mathbf{e}_1),
$$
where \( k_1, k_2 > 0 \) and \( k_1 > 0.5 \). The \( \tanh(\cdot) \) function helps in bounding signals and provides smooth behavior. The time derivative is:
$$
\dot{\mathbf{s}} = \bar{\mathcal{L}} \mathbf{R}_m \mathbf{u} – \bar{\mathcal{B}} \dot{\mathbf{P}}_L + \bar{\mathcal{L}} \boldsymbol{\alpha} + \mathbf{d} – c_1^2 \boldsymbol{\lambda}_1 + (c_1 + c_2)\boldsymbol{\lambda}_2 + k_1 \dot{\mathbf{e}}_1 + k_2 (\mathbf{1} – \tanh^2(\mathbf{e}_1)) \mathbf{e}_2.
$$
Let \( \tilde{d} = \bar{d} – \hat{d} \), where \( \bar{d} \) is the unknown upper bound of \( \|\mathbf{d}\| \) and \( \hat{d} \) is its estimate. We design the following control law and adaptive update rule:
$$
\begin{aligned}
\mathbf{u} &= -m (\bar{\mathcal{L}} \mathbf{R}_m)^{-1} \left( -\bar{\mathcal{B}} \dot{\mathbf{P}}_L + \bar{\mathcal{L}} \boldsymbol{\alpha} + \hat{d} \cdot \text{sign}(\mathbf{s}) – c_1^2 \boldsymbol{\lambda}_1 + (c_1 + c_2)\boldsymbol{\lambda}_2 \right. \\
&\quad \left. + \, k_1 \dot{\mathbf{e}}_1 + k_2 (\mathbf{1} – \tanh^2(\mathbf{e}_1)) \mathbf{e}_2 + k_3 \mathbf{s} + k_4 \tanh(\mathbf{s}/\eta) \right), \\
\dot{\hat{d}} &= \gamma \| \mathbf{s} \|,
\end{aligned}
$$
where \( k_3, k_4, \gamma, \eta > 0 \) are controller gains, with \( k_3 > 0.5 \). The term \( k_4 \tanh(\mathbf{s}/\eta) \) provides a smooth, bounded robustifying control action that minimizes chattering.
2.3 Finite-Time Stability Analysis
Theorem: Consider the drone formation error system with dynamics, input saturation, and bounded disturbances. With the anti-saturation auxiliary system, the sliding surface \( \mathbf{s} \), the control law \( \mathbf{u} \), and the adaptive law \( \dot{\hat{d}} \), the formation tracking errors \( \mathbf{e}_1 \) and \( \mathbf{e}_2 \) converge to a small neighborhood of the origin in finite time.
Proof (Sketch): Consider the Lyapunov function candidate \( V_1 = \frac{1}{2} \mathbf{s}^T \mathbf{s} + \frac{1}{2\gamma} \tilde{d}^2 \). Its time derivative along the system trajectories is:
$$
\dot{V}_1 = \mathbf{s}^T \dot{\mathbf{s}} – \frac{1}{\gamma} \tilde{d} \dot{\hat{d}}.
$$
Substituting the expressions for \( \dot{\mathbf{s}} \) and the control law \( \mathbf{u} \), and using the inequality \( \mathbf{s}^T \mathbf{d} \leq \|\mathbf{s}\| \bar{d} \), we obtain:
$$
\dot{V}_1 \leq \|\mathbf{s}\| \tilde{d} – k_3 \mathbf{s}^T\mathbf{s} – k_4 \mathbf{s}^T \tanh(\mathbf{s}/\eta) – \frac{1}{\gamma} \tilde{d} \dot{\hat{d}}.
$$
Applying the adaptive law \( \dot{\hat{d}} = \gamma \|\mathbf{s}\| \) yields:
$$
\dot{V}_1 \leq -k_3 \|\mathbf{s}\|^2 – k_4 \mathbf{s}^T \tanh(\mathbf{s}/\eta) < 0 \quad \text{for} \quad \mathbf{s} \neq 0.
$$
This proves that \( \mathbf{s} \) and \( \tilde{d} \) are bounded. To prove finite-time convergence, consider a second Lyapunov function \( V_2 = \frac{1}{2} \mathbf{s}^T \mathbf{s} \). Using Young’s inequality and the property of the \( \tanh \) function, its derivative can be bounded as:
$$
\dot{V}_2 \leq -\delta_1 V_2 – \delta_2 V_2^{1/2} + \theta,
$$
where \( \delta_1 = 2k_3 – 1 > 0 \), \( \delta_2 = \sqrt{2} k_4 > 0 \), and \( \theta \) is a positive constant. According to fixed-time stability theory, this inequality ensures that the sliding variable \( \mathbf{s} \) converges to a compact set \( \|\mathbf{s}\| \leq \psi \) in a finite time \( T_1 \leq \max(t_1, t_2) \). Once on the sliding manifold (\( \mathbf{s} \approx 0 \)), the error dynamics simplify to \( \mathbf{e}_2 = – k_1 \mathbf{e}_1 – k_2 \tanh(\mathbf{e}_1) \). Analyzing \( V_3 = \frac{1}{2} \mathbf{e}_1^T \mathbf{e}_1 \) yields a similar differential inequality, guaranteeing that the formation error \( \mathbf{e}_1 \) also converges to a small region around zero in finite time \( T_2 \). Therefore, the overall drone formation system achieves practical finite-time stability within \( T \leq T_1 + T_2 \). ∎
3. Numerical Simulation and Analysis
We validate the proposed controller through a simulation involving one virtual leader and three follower UAVs in a drone formation. The desired formation is an equilateral triangle with the leader at the center.
| UAV | Δx (m) | Δy (m) | Δz (m) |
|---|---|---|---|
| Follower #1 | 0 | 0 | 5 |
| Follower #2 | 0 | 10 | -5 |
| Follower #3 | 0 | -10 | 5 |
| Parameter | Leader (0) | Follower 1 | Follower 2 | Follower 3 |
|---|---|---|---|---|
| Initial Position (x,y,z) m | (10, 2.5, 5) | (6, -1, -2) | (-8, 3, -5) | (-10, -3, -7) |
| Initial Velocity (m/s) | (15, 12, 0) | (30, 0, 0) | (30, 0, 0) | (30, 0, 0) |
| Mass \( m_i \) (kg) | – | 3 | 3 | 3 |
| Drag \( D_i \) (N) | – | 50 | 50 | 50 |
The communication topology is defined by adjacency matrix \( \mathcal{A} \) and pinning matrix \( \mathcal{B} \):
$$
\mathcal{A} = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}, \quad \mathcal{B} = \text{diag}(1, 0, 0).
$$
The virtual leader’s trajectory is \( \mathbf{p}_L(t) = (10+15t,\ 2.5+12t,\ 5)^T \). Disturbances are set as \( \mathbf{d}_i = (0.1\sin t,\ 0.2\sin t,\ 0.2\sin t)^T \) N. Input limits are: \( T_i \in [0, 200] \) N, \( L_i\cos\phi_i,\ L_i\sin\phi_i \in [-40, 40] \) N.
| Gain | Value | Gain | Value |
|---|---|---|---|
| \( c_1 \) | 3 | \( k_3 \) | 4 |
| \( c_2 \) | 4 | \( k_4 \) | 3 |
| \( k_1 \) | 1.5 | \( \gamma \) | 1 |
| \( k_2 \) | 1.25 | \( \eta \) | 0.1 |
3.1 Simulation Results
The figures of merit are the combined position error \( \mathbf{e}_i \) and the absolute tracking error \( \mathbf{p}_i – \mathbf{h}_i – \mathbf{p}_L \) for each follower in the drone formation. The simulation results demonstrate excellent performance:
- Convergence: All position and velocity errors for the three followers converge smoothly to values below \( 10^{-3} \) m and m/s, respectively, within approximately 10 seconds. No overshoot is observed, indicating well-damped transient behavior.
- Robustness: The controller successfully rejects the persistent sinusoidal disturbances, as evidenced by the small steady-state errors. The adaptive law effectively estimates the disturbance bound.
- Input Saturation Handling: The control inputs for all UAVs remain strictly within the specified saturation limits throughout the simulation. The thrust \( T_i \) stays between 0 and 200 N, and the lateral force components remain within [-40, 40] N. The anti-saturation auxiliary system performs as intended, preventing performance degradation due to saturated actuators. The control signals show a high-frequency component necessary to counteract disturbances, but the amplitude is bounded and within acceptable limits, demonstrating minimal chattering.
The results conclusively verify that the proposed adaptive non-singular terminal sliding mode controller meets all control objectives: it achieves precise, finite-time drone formation tracking in the presence of external disturbances while rigorously respecting the physical input constraints of fixed-wing UAVs.
4. Conclusion
This article has presented a robust finite-time control framework for fixed-wing drone formation飞行 subject to input saturation and external disturbances. The key innovation is the integration of a novel global fast non-singular terminal sliding mode control law with an anti-saturation auxiliary system and an adaptive disturbance estimator. The separation of thrust and drag modeling, coupled with explicit asymmetric input constraints, enhances the practical relevance of the approach for real fixed-wing platforms. Lyapunov-based stability analysis proved that the formation tracking errors converge to a small neighborhood of zero in finite time. Extensive numerical simulations validated the controller’s effectiveness, demonstrating smooth convergence, robust disturbance rejection, and strict adherence to actuator limits. Future work will focus on extending this method to handle communication delays and topology changes within the drone formation network.