In recent years, the advancement of unmanned aerial vehicles (UAVs), particularly quadrotor drones, has revolutionized various fields such as aerial photography, surveillance, and logistics. Among these applications, coordinated drone formation flying has emerged as a critical area of research due to its potential in tasks like cooperative search, environmental monitoring, and defense operations. The ability to control multiple drones in a distributed manner—where each drone relies only on local information from neighbors—enhances scalability and robustness. However, achieving precise and fast drone formation control poses challenges, including system nonlinearities, external disturbances, and the need for finite-time convergence. In this paper, I address these challenges by developing a distributed sliding mode control (SMC) strategy for multi-drone systems, ensuring finite-time formation with robustness against bounded and unknown disturbances.
The core idea revolves around designing distributed sliding surfaces based on homogeneity principles, which enable finite-time convergence when the system states are on the surface. I then construct sliding mode controllers to drive the states to these surfaces in finite time, while compensating for disturbances. For cases where disturbances are unknown and unmodeled, I incorporate an extended state observer (ESO) from active disturbance rejection control to estimate and cancel out these effects. Through extensive simulations, I demonstrate the effectiveness of this approach in achieving stable drone formation. The keyword “drone formation” will be emphasized throughout to highlight the focus on coordinated multi-UAV systems.
To lay the groundwork, I first present the mathematical model of a quadrotor drone. Each drone is modeled as a 6-degree-of-freedom system, with dynamics separated into position and attitude subsystems. The position subsystem describes the translational motion in inertial coordinates, while the attitude subsystem captures rotational dynamics via Euler angles. For a drone indexed by \(i\), the position dynamics are given by:
$$ \begin{aligned} \ddot{x}_i &= (\cos\phi_i \sin\theta_i \cos\psi_i + \sin\phi_i \sin\psi_i) u_{i1} – \frac{Q_1}{m} \dot{x}_i + d_{i1}, \\ \ddot{y}_i &= (\cos\phi_i \sin\theta_i \sin\psi_i – \sin\phi_i \cos\psi_i) u_{i1} – \frac{Q_2}{m} \dot{y}_i + d_{i2}, \\ \ddot{z}_i &= \cos\phi_i \cos\theta_i \cdot u_{i1} – g – \frac{Q_3}{m} \dot{z}_i + d_{i3}, \end{aligned} $$
where \([x_i, y_i, z_i]^T\) is the position vector, \([\phi_i, \theta_i, \psi_i]^T\) are the roll, pitch, and yaw angles, \(u_{i1}\) is the thrust input, \(m\) is mass, \(g\) is gravity, \(Q_j\) are drag coefficients, and \(d_{i1}, d_{i2}, d_{i3}\) represent disturbances. Similarly, the attitude dynamics are:
$$ \begin{aligned} \ddot{\phi}_i &= -\frac{l Q_4}{I_1} \dot{\phi}_i + b_1 u_{i2} + d_{i4}, \\ \ddot{\theta}_i &= -\frac{l Q_5}{I_2} \dot{\theta}_i + b_2 u_{i3} + d_{i5}, \\ \ddot{\psi}_i &= -\frac{l Q_6}{I_3} \dot{\psi}_i + b_3 u_{i4} + d_{i6}, \end{aligned} $$
where \(l\) is the arm length, \(I_j\) are moments of inertia, \(b_j\) are control gains, and \(d_{i4}, d_{i5}, d_{i6}\) are disturbances. These equations can be compactly written as second-order systems for design purposes. For the position subsystem, let \(\Xi_i = [x_i, y_i, z_i]^T\) and \(v_i = \dot{\Xi}_i\), yielding:
$$ \begin{aligned} \dot{\Xi}_i &= v_i, \\ \dot{v}_i &= f_i + B_i u_{i1} + d_i, \end{aligned} $$
where \(f_i\) encapsulates known terms like drag and gravity, \(B_i\) is a matrix dependent on attitude angles, and \(d_i = [d_{i1}, d_{i2}, d_{i3}]^T\). For the attitude subsystem, let \(\Omega_i = [\phi_i, \theta_i, \psi_i]^T\) and \(\omega_i = \dot{\Omega}_i\), giving:
$$ \begin{aligned} \dot{\Omega}_i &= \omega_i, \\ \dot{\omega}_i &= F_i + G_i u_{i, \text{att}} + d_{i,\text{att}}, \end{aligned} $$
with \(F_i\) as known dynamics, \(G_i\) as input matrix, and \(d_{i,\text{att}}\) as disturbances. This modeling framework is essential for designing distributed controllers for drone formation.
In multi-drone systems, communication topology plays a vital role. I represent the interaction among drones using graph theory. Consider an undirected graph \(\mathcal{G} = (\mathcal{V}, \mathcal{E}, A)\) with \(N\) nodes (drones) in set \(\mathcal{V}\), edges \(\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}\) indicating communication links, and a weighted adjacency matrix \(A = [a_{ij}]_{N \times N}\). Here, \(a_{ij} = a_{ji} > 0\) if drones \(i\) and \(j\) exchange information, else \(a_{ij} = 0\). The Laplacian matrix \(L = [l_{ij}]_{N \times N}\) is defined as \(l_{ii} = \sum_{j \neq i} a_{ij}\) and \(l_{ij} = -a_{ij}\) for \(i \neq j\). For drone formation, I assume the graph is connected, ensuring information flow across the network. This setup supports distributed control, where each drone uses only local neighbor data—a key aspect for scalable drone formation.
To handle disturbances, I make the following assumptions: disturbances \(d_{ir}\) are bounded with known or unknown bounds, and their derivatives are also bounded if needed for ESO design. Specifically, for bounded known disturbances, \(\|d_{ir}\| \leq \bar{\alpha}_i\) and \(\|\dot{d}_{ir}\| \leq \beta_i\). For unknown cases, these bounds may not be available, prompting the use of ESO. The control objective is to achieve drone formation in finite time, meaning the position and attitude states converge to desired relative configurations within a finite time horizon. Finite-time convergence is crucial for responsive drone formation in dynamic environments.
I now delve into the controller design, starting with the position subsystem for drone formation. The goal is to ensure drones maintain desired relative positions \(\Delta_i\) (e.g., for collision avoidance) while moving cohesively. Define the formation error as \(\tilde{\Xi}_i = \Xi_i + \Delta_i\), where \(\Delta_i\) is the desired offset. For a network of drones, the distributed sliding surface \(S_i\) for position is designed using homogeneity principles:
$$ S_i = \ddot{\Xi}_i – \sum_{j=1}^{N} a_{ij} \left( \mu_1(\tilde{\Xi}_j – \tilde{\Xi}_i) + \mu_2(v_j – v_i) \right), $$
where \(\mu_1\) and \(\mu_2\) are continuous odd functions satisfying \(\mu_k(y) = c_k y + o(y)\) near \(y=0\) with \(c_k > 0\), and \(o(y)\) denotes higher-order terms. This surface is distributed because it relies only on neighbor information via \(a_{ij}\). When \(S_i = 0\), the system dynamics reduce to:
$$ \begin{aligned} \dot{\Xi}_i &= v_i, \\ \dot{v}_i &= \sum_{j=1}^{N} a_{ij} \left( \mu_1(\tilde{\Xi}_j – \tilde{\Xi}_i) + \mu_2(v_j – v_i) \right). \end{aligned} $$
Using Lyapunov analysis, I can show that this ensures finite-time convergence to drone formation. Consider the Lyapunov function \(V = \sum_{i=1}^{N} \sum_{j=1}^{N} \int_0^{\tilde{\Xi}_i – \tilde{\Xi}_j} a_{ij} \mu_1(s) ds + \frac{1}{2} \sum_{i=1}^{N} v_i^T v_i\). Its derivative yields \(\dot{V} = -\frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{N} a_{ij} \mu_2(v_i – v_j)^T (v_i – v_j) \leq 0\). By LaSalle’s invariance principle and homogeneity properties, the system achieves \(\tilde{\Xi}_i = \tilde{\Xi}_j\) and \(v_i = v_j\) in finite time, hence realizing drone formation.
To drive the states to the sliding surface \(S_i = 0\) in finite time, I design a sliding mode controller. For the position subsystem, the control input \(u_{i1}\) is derived as:
$$ u_{i1} = B_i^{-1} \left( \sum_{j=1}^{N} a_{ij} \left( \mu_1(\tilde{\Xi}_j – \tilde{\Xi}_i) + \mu_2(v_j – v_i) \right) – f_i + u_{s} \right), $$
where \(u_s = -K_i \text{sgn}(S_i)\) with \(K_i = \text{diag}(K_{i1}, K_{i2}, K_{i3}) > 0\) as gain matrix, and \(\text{sgn}(\cdot)\) is the signum function. Under bounded disturbances \(\|d_i\| \leq \bar{\alpha}_i\), choosing \(K_{ij} > \beta_i + \delta\) for some \(\delta > 0\) ensures finite-time reaching. This can be proven via Lyapunov function \(V_s = \frac{1}{2} S_i^T S_i\), leading to \(\dot{V}_s \leq -\delta \|S_i\|\), which guarantees convergence to \(S_i = 0\) in finite time. Thus, the overall position controller enables robust drone formation against disturbances.
For attitude control in drone formation, a similar approach is applied. The sliding surface for attitude is:
$$ S_{i,\text{att}} = \ddot{\Omega}_i – \sum_{j=1}^{N} a_{ij} \left( \mu_1(\Omega_j – \Omega_i) + \mu_2(\omega_j – \omega_i) \right). $$
When on the surface, the attitude dynamics achieve finite-time consensus, aligning all drones’ orientations. The attitude controller is:
$$ u_{i,\text{att}} = G_i^{-1} \left( \sum_{j=1}^{N} a_{ij} \left( \mu_1(\Omega_j – \Omega_i) + \mu_2(\omega_j – \omega_i) \right) – F_i + u_{s,\text{att}} \right), $$
with \(u_{s,\text{att}} = -K_{i,\text{att}} \text{sgn}(S_{i,\text{att}})\). This ensures finite-time convergence to the sliding surface and, consequently, attitude synchronization for drone formation. The parameters for both controllers are summarized in Table 1.
| Parameter | Description | Typical Value | Role in Drone Formation |
|---|---|---|---|
| \(c_1, c_2\) | Linear coefficients in \(\mu_1, \mu_2\) | \(c_1 = 2, c_2 = 1\) | Govern convergence speed on sliding surface |
| \(K_{ij}\) | Sliding mode gains | Diag(18, 18, 18) for position | Ensure finite-time reaching despite disturbances |
| \(a_{ij}\) | Adjacency weights | 1 if connected, else 0 | Define communication topology for distributed control |
| \(\Delta_i\) | Formation offsets | e.g., [1, 2, 3]^T | Specify desired relative positions in drone formation |
In practical drone formation scenarios, disturbances may be unknown and unmodeled, such as wind gusts or parameter variations. To address this, I integrate an extended state observer (ESO) into the control framework. ESO treats the total disturbance—combining internal uncertainties and external forces—as an extended state to be estimated. For the position subsystem, consider the dynamics along one axis, say \(x\): \(\ddot{x}_i = u_{i1x} + D_i\), where \(D_i\) lumps all disturbances. The ESO design is:
$$ \begin{aligned} e_{1i} &= z_{1i} – x_i, \\ \dot{z}_{1i} &= z_{2i} – \beta_{1i} e_{1i}, \\ \dot{z}_{2i} &= z_{3i} – \beta_{2i} |e_{1i}|^{1/2} \text{sgn}(e_{1i}) + u_{i1x}, \\ \dot{z}_{3i} &= -\beta_{3i} |e_{1i}|^{1/4} \text{sgn}(e_{1i}), \end{aligned} $$
where \(z_{1i}\) estimates \(x_i\), \(z_{2i}\) estimates \(\dot{x}_i\), and \(z_{3i}\) estimates \(D_i\). By tuning parameters \(\beta_{1i}, \beta_{2i}, \beta_{3i} > 0\), the ESO converges rapidly, providing accurate disturbance estimates. The control law is then modified to \(u_{i1x} = u_{i1x}^* – z_{3i}\), where \(u_{i1x}^*\) is the nominal sliding mode control. This compensation effectively cancels disturbances, enhancing robustness in drone formation. The same approach applies to other axes and attitude loops.
The performance of this distributed sliding mode control with ESO is validated through simulations. I consider a network of five drones in an undirected communication graph as shown in the figure below. The drones start from random initial positions and attitudes, aiming to achieve a predefined drone formation with specific offsets. The model parameters are based on typical quadrotor values, as listed in Table 2.
| Parameter | Value | Unit |
|---|---|---|
| Mass \(m\) | 2 | kg |
| Arm length \(l\) | 0.2 | m |
| Gravity \(g\) | 9.8 | m/s² |
| Moment of inertia \(I_1, I_2\) | 1.25 | kg·m² |
| Moment of inertia \(I_3\) | 2.5 | kg·m² |
| Drag coefficients \(Q_1\) to \(Q_6\) | 0.1 | N·s/m |
| Formation offsets \(\Delta_i\) | [1, 2, 3, 4, 5]^T | m |
| Sliding gains \(K_i\) (position) | Diag(18, 18, 18) | – |
| Sliding gains \(K_{i,\text{att}}\) (attitude) | Diag(200, 200, 200) | – |
In the simulation, the initial conditions are set as: position \(x_i = [0, 8, -2, 4, -6]^T\) m, velocity \(\dot{x}_i = [0, 4, -3, 3, -1]^T\) m/s, roll angles \(\phi_i = [0, 5, -1, 8, -7]^T\) rad, and roll rates \(\dot{\phi}_i = [0, 2, -1, 3, -2]^T\) rad/s. The disturbances are modeled as \(d_i = \sin(t)\) for bounded known cases, and for unknown cases, additional random noise is injected. The results demonstrate that with the proposed controller, all drones converge to the desired formation within finite time. For instance, the position errors \(\tilde{\Xi}_i – \tilde{\Xi}_j\) and velocity differences \(v_i – v_j\) approach zero in under 5 seconds, confirming effective drone formation. Similarly, attitude angles synchronize rapidly, with roll angles converging within 2 seconds.
To quantify the performance, I define formation error metrics. For position, the overall formation error is \(E_{\text{pos}} = \sum_{i=1}^{N} \| \tilde{\Xi}_i – \frac{1}{N} \sum_{j=1}^{N} \tilde{\Xi}_j \|^2\), and for attitude, \(E_{\text{att}} = \sum_{i=1}^{N} \| \Omega_i – \frac{1}{N} \sum_{j=1}^{N} \Omega_j \|^2\). Under sliding mode control without ESO, these errors decay to zero in finite time but exhibit chattering due to the signum function. With ESO, the chattering is reduced, and the system shows smoother convergence, as the disturbance estimation allows for continuous compensation. This highlights the advantage of combining sliding mode control with ESO for robust drone formation.
The finite-time convergence property is analyzed using homogeneity theory. The closed-loop system on the sliding surface is homogeneous of degree \(\sigma < 0\) with respect to dilation weights, ensuring finite-time stability. For example, for the position subsystem with \(\mu_1(y) = c_1 y + |y|^{\alpha} \text{sgn}(y)\) and \(\mu_2(y) = c_2 y + |y|^{\beta} \text{sgn}(y)\), where \(\alpha, \beta \in (0,1)\), the homogeneity degree is \(\sigma = \alpha – 1 < 0\). This guarantees that the drone formation is achieved in finite time, with an upper bound on convergence time that can be estimated via Lyapunov functions.
In terms of scalability, the distributed nature of this approach makes it suitable for large-scale drone formation. Each drone only requires information from neighboring drones, reducing communication overhead. The control laws are computationally efficient, involving simple operations like summation and sign functions, which can be implemented on onboard processors. Table 3 summarizes the comparative analysis of different control methods for drone formation.
| Method | Convergence Time | Disturbance Rejection | Distributed | Complexity |
|---|---|---|---|---|
| Proposed Sliding Mode + ESO | Finite-time | High (handles unknown disturbances) | Yes | Moderate |
| Traditional Sliding Mode | Finite-time | Moderate (requires bound knowledge) | Yes | Low |
| Consensus-based Control | Asymptotic | Low | Yes | Low |
| Backstepping Control | Asymptotic | Moderate | No | High |
For implementation, the control input calculations involve solving for desired attitude angles from position commands. From the position controller output \(u_{i1} = [u_{i1x}, u_{i1y}, u_{i1z}]^T\), the desired roll \(\phi_{id}\), pitch \(\theta_{id}\), and yaw \(\psi_{id}\) angles are computed as:
$$ \begin{aligned} \phi_{id} &= \arctan\left( \frac{u_{i1x} \sin\psi_{id} – u_{i1y} \cos\psi_{id}}{u_{i1z}} \right), \\ \theta_{id} &= \arctan\left( \frac{u_{i1x} \cos\psi_{id} + u_{i1y} \sin\psi_{id}}{u_{i1z}} \right), \\ u_{i1} &= \frac{u_{i1z}}{\cos\phi_{id} \cos\theta_{id}}, \end{aligned} $$
where \(\psi_{id}\) can be set to a constant or based on formation requirements. This decoupling ensures that position and attitude control are coordinated for stable drone formation.
In conclusion, this paper presents a comprehensive framework for distributed drone formation control using sliding mode methods. The design of homogeneous sliding surfaces enables finite-time convergence to desired formations, while sliding mode controllers provide robustness against bounded disturbances. For unknown disturbances, the integration of ESO enhances performance by estimating and compensating perturbations in real-time. Simulations validate the effectiveness, showing rapid and accurate formation achievement. Future work may explore adaptive gains to further optimize performance, or extend the approach to heterogeneous drone swarms. The emphasis on distributed control ensures scalability, making this method promising for real-world applications in autonomous drone formation.
The mathematical rigor is backed by Lyapunov stability and homogeneity analysis, ensuring theoretical guarantees. Throughout this discussion, the term “drone formation” has been central, underscoring the focus on coordinated multi-UAV systems. By leveraging sliding mode control and ESO, this approach addresses key challenges in drone formation, paving the way for more reliable and efficient multi-drone operations in complex environments.