In recent years, the study of multi-agent systems has garnered significant attention due to its inspiration from natural phenomena such as fish schools, bird flocks, and insect swarms. These systems consist of multiple individual agents that collaborate to accomplish tasks beyond the capability of a single agent. Among various platforms, quadrotor drones have emerged as a prominent focus due to their compact size, lightweight design, cost-effectiveness, and agility. Specifically, quadrotor drones offer advantages like vertical take-off, hovering capabilities, and ease of control, making them ideal for formation control applications in areas like surveillance, exploration, and reconnaissance. Formation control involves coordinating a group of quadrotor drones to achieve and maintain a predefined spatial configuration, often requiring robust control strategies to handle uncertainties and disturbances.
This article addresses the formation control problem for a group of quadrotor drones using terminal sliding mode control. We design a decentralized control scheme that ensures finite-time convergence to the desired formation, leveraging the robustness of sliding mode control against model uncertainties and external perturbations. The quadrotor drone system is modeled with a leader-follower structure, where followers communicate with neighbors to enhance system reliability. We derive a mathematical model for the quadrotor drone dynamics, define formation control objectives based on generalized error states, and propose a terminal sliding mode controller that guarantees finite-time stability. Simulation results validate the effectiveness of our approach, demonstrating superior performance compared to linear sliding mode control.
The core of our work lies in the integration of terminal sliding mode control with multi-agent consensus theory for quadrotor drone formations. Terminal sliding mode control is chosen over conventional methods because it ensures that the system errors converge to zero in finite time, rather than asymptotically. This is critical for applications requiring rapid deployment and precise maneuvering of quadrotor drone swarms. We begin by presenting the dynamical model of a quadrotor drone, which serves as the foundation for control design. The model accounts for translational motion influenced by thrust and orientation angles. For a quadrotor drone, the equations of motion can be expressed as follows:
Let the position of the i-th quadrotor drone in an inertial frame be denoted by $\mathbf{p}_i = [x_i, y_i, z_i]^T$, and its orientation by Euler angles $\boldsymbol{\Theta}_i = [\phi_i, \theta_i, \psi_i]^T$, where $\phi_i$, $\theta_i$, and $\psi_i$ represent roll, pitch, and yaw angles, respectively. The translational dynamics are given by:
$$
\ddot{\mathbf{p}}_i = \mathbf{a}_i = \begin{bmatrix} a_{xi} \\ a_{yi} \\ a_{zi} \end{bmatrix} = \begin{bmatrix} \frac{1}{m_i} (\cos\phi_i \sin\theta_i \cos\psi_i + \sin\phi_i \sin\psi_i) F_{Ti} \\ \frac{1}{m_i} (\cos\phi_i \sin\theta_i \sin\psi_i – \sin\phi_i \cos\psi_i) F_{Ti} \\ \frac{1}{m_i} (\cos\phi_i \cos\theta_i) F_{Ti} – g \end{bmatrix}
$$
where $m_i$ is the mass of the i-th quadrotor drone, $F_{Ti}$ is the total thrust force, and $g$ is the gravitational acceleration. This model assumes that the quadrotor drone is rigid and symmetric, with negligible aerodynamic effects. For control purposes, we often decouple the translational and rotational dynamics. Given desired acceleration signals $\mathbf{a}_i$ and a desired yaw angle $\psi_{di}$, we can compute the desired thrust and attitude angles using:
$$
F_{Tdi} = m_i \sqrt{a_{xi}^2 + a_{yi}^2 + (a_{zi} + g)^2}, \quad \phi_{di} = \arcsin\left( \frac{m_i (a_{xi} \sin\psi_{di} – a_{yi} \cos\psi_{di})}{F_{Tdi}} \right), \quad \theta_{di} = \arctan\left( \frac{a_{xi} \cos\psi_{di} + a_{yi} \sin\psi_{di}}{a_{zi} + g} \right)
$$
These equations allow us to focus on the translational control for formation, while attitude control can be handled separately. In formation control, the objective is to coordinate multiple quadrotor drones to achieve a specific spatial pattern. We consider a system with one leader quadrotor drone (indexed as $L$) and $N$ follower quadrotor drones (indexed as $i = 1, \dots, N$). The communication topology among quadrotor drones is represented by a directed graph $\mathcal{G}_f = (\mathcal{V}, \mathcal{E})$, where $\mathcal{V} = \{v_1, \dots, v_N, v_L\}$ is the set of vertices (each representing a quadrotor drone), and $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the set of edges denoting communication links. The neighbor set of the i-th quadrotor drone is $\Omega_i = \{v_j \in \mathcal{V} : (i, j) \in \mathcal{E}\}$. We assume the graph is connected, ensuring information flow across the quadrotor drone network.
To formalize the formation control objective, we define a generalized error state for each follower quadrotor drone. Let $\mathbf{l}_i = [x_i, y_i, z_i]^T$ be the position of the i-th quadrotor drone, and $\mathbf{l}_L$ be the leader’s position. The desired relative displacement between quadrotor drones $i$ and $j$ is given by a time-varying vector $\mathbf{E}_{ij}(t) \in \mathbb{R}^3$. Then, the generalized error state $\mathbf{e}_i(t)$ for the i-th follower quadrotor drone is:
$$
\mathbf{e}_i(t) = \mathbf{l}_i(t) – \frac{1}{|\Omega_i|} \sum_{j \in \Omega_i} \left( \mathbf{l}_j(t) + \mathbf{E}_{ij}(t) \right), \quad i = 1, \dots, N
$$
Here, $|\Omega_i|$ denotes the number of neighbors for quadrotor drone $i$. The formation control goal is to drive all generalized error states $\mathbf{e}_i(t)$ to zero, which ensures that the quadrotor drones maintain the desired formation relative to each other and the leader. This approach combines leader-following and consensus-based strategies, enhancing robustness through local interactions among quadrotor drones.
Next, we derive the error dynamics to facilitate controller design. Differentiating $\mathbf{e}_i(t)$ twice with respect to time yields:
$$
\ddot{\mathbf{e}}_i(t) = \ddot{\mathbf{l}}_i(t) – \frac{1}{|\Omega_i|} \sum_{j \in \Omega_i} \left( \ddot{\mathbf{l}}_j(t) + \ddot{\mathbf{E}}_{ij}(t) \right)
$$
Substituting the acceleration from the quadrotor drone model, we get:
$$
\ddot{\mathbf{e}}_i(t) = \mathbf{u}_i(t) – \frac{1}{|\Omega_i|} \sum_{j \in \Omega_i} \ddot{\mathbf{E}}_{ij}(t) – \delta_i \ddot{\mathbf{l}}_L(t)
$$
where $\delta_i = 1/|\Omega_i|$ if the leader is in $\Omega_i$, otherwise $\delta_i = 0$. The control input $\mathbf{u}_i(t)$ is defined as:
$$
\mathbf{u}_i(t) = \mathbf{a}_i(t) – \frac{1}{|\Omega_i|} \sum_{j \in \Omega_i \setminus \{L\}} \mathbf{a}_j(t)
$$
This represents the acceleration contribution from the i-th quadrotor drone relative to its neighbors. Our task is to design $\mathbf{u}_i(t)$ such that $\mathbf{e}_i(t) \to 0$ in finite time.
We propose a terminal sliding mode control scheme for this purpose. Terminal sliding mode control is an extension of conventional sliding mode control that incorporates nonlinear sliding surfaces to achieve finite-time convergence. For each quadrotor drone, we define a terminal sliding manifold. Let $\mathbf{s}_i(\mathbf{e}_i, \dot{\mathbf{e}}_i)$ be the sliding variable:
$$
\mathbf{s}_i(\mathbf{e}_i, \dot{\mathbf{e}}_i) = \dot{\mathbf{e}}_i + \mathbf{H}_i \mathbf{D}_i(\mathbf{e}_i) |\mathbf{e}_i|^{1/2}, \quad (\mathbf{e}_i, \dot{\mathbf{e}}_i) \in \mathbb{R}^3 \times \mathbb{R}^3
$$
where $\mathbf{H}_i = \text{diag}\{h_{i1}, h_{i2}, h_{i3}\}$ with $h_{im} > 0$ for $m = 1,2,3$, $\mathbf{D}_i(\mathbf{e}_i) = \text{diag}\{\text{sign}(e_{i1}), \text{sign}(e_{i2}), \text{sign}(e_{i3})\}$, and $|\mathbf{e}_i|^{1/2} = [|e_{i1}|^{1/2}, |e_{i2}|^{1/2}, |e_{i3}|^{1/2}]^T$. The sign function ensures robustness, and the fractional power guarantees finite-time convergence when on the sliding surface. The sliding manifold is $\mathcal{S}_i = \{(\mathbf{e}_i, \dot{\mathbf{e}}_i) : \mathbf{s}_i = \mathbf{0}\}$.
To derive the control law, we use the reaching law approach. We adopt a constant rate reaching law to drive the system to the sliding manifold. The time derivative of $\mathbf{s}_i$ is:
$$
\dot{\mathbf{s}}_i = \ddot{\mathbf{e}}_i + \frac{1}{2} \mathbf{H}_i \mathbf{W}_i(\mathbf{e}_i, \dot{\mathbf{e}}_i) \dot{\mathbf{e}}_i
$$
where $\mathbf{W}_i(\mathbf{e}_i, \dot{\mathbf{e}}_i)$ is a diagonal matrix with elements $|e_{im}|^{-1/2}$ for $e_{im} \neq 0$. To ensure existence and boundedness, we assume initial conditions lie in a bounded set $\sigma_i = \{(\mathbf{e}_i, \dot{\mathbf{e}}_i) : \|\mathbf{W}_i(\mathbf{e}_i, \dot{\mathbf{e}}_i)\|_\infty \leq \psi_i\}$ with $\psi_i > 0$. Then, the control input $\mathbf{u}_i(t)$ is designed as:
$$
\mathbf{u}_i(t) = \begin{cases} \frac{1}{|\Omega_i|} \ddot{\mathbf{l}}_L + \frac{1}{|\Omega_i|} \sum_{j \in \Omega_i} \ddot{\mathbf{E}}_{ij}(t) – \frac{1}{2} \mathbf{H}_i \mathbf{w}_i(\mathbf{e}_i, \dot{\mathbf{e}}_i) – \mathbf{K}_i \text{sign}(\mathbf{s}_i) & \text{if } L \in \Omega_i \\ \frac{1}{|\Omega_i|} \sum_{j \in \Omega_i} \ddot{\mathbf{E}}_{ij}(t) – \frac{1}{2} \mathbf{H}_i \mathbf{w}_i(\mathbf{e}_i, \dot{\mathbf{e}}_i) – \mathbf{K}_i \text{sign}(\mathbf{s}_i) & \text{if } L \notin \Omega_i \end{cases}
$$
where $\mathbf{K}_i = \text{diag}\{k_{i1}, k_{i2}, k_{i3}\}$ with $k_{im} > 0$, $\text{sign}(\mathbf{s}_i) = [\text{sign}(s_{i1}), \text{sign}(s_{i2}), \text{sign}(s_{i3})]^T$, and $\mathbf{w}_i(\mathbf{e}_i, \dot{\mathbf{e}}_i)$ is a bounded function related to $\mathbf{W}_i$. This control law compensates for neighbor accelerations and desired formation changes, while the sliding mode term rejects disturbances.
We now prove the finite-time stability of the closed-loop system for each quadrotor drone. Consider the Lyapunov function candidate $V_i(\mathbf{s}_i) = \frac{1}{2} \mathbf{s}_i^T \mathbf{s}_i$. Its time derivative along the trajectories is:
$$
\dot{V}_i = \mathbf{s}_i^T \dot{\mathbf{s}}_i = \mathbf{s}_i^T \left( \ddot{\mathbf{e}}_i + \frac{1}{2} \mathbf{H}_i \mathbf{w}_i \right)
$$
Substituting $\ddot{\mathbf{e}}_i$ from the error dynamics and $\mathbf{u}_i(t)$ from the control law, we obtain:
$$
\dot{V}_i = \mathbf{s}_i^T \left( -\mathbf{K}_i \text{sign}(\mathbf{s}_i) \right) = -\sum_{m=1}^{3} k_{im} |s_{im}| \leq -\min_{m} \{k_{im}\} \|\mathbf{s}_i\|_1
$$
Since $\|\mathbf{s}_i\|_1 \geq \|\mathbf{s}_i\|_2 = \sqrt{2V_i}$, we have $\dot{V}_i \leq -\kappa_i \sqrt{V_i}$ with $\kappa_i = \min_{m} \{k_{im}\} \sqrt{2}$. This inequality implies that $V_i$ converges to zero in finite time. Specifically, the reaching time $t_r$ satisfies $t_r \leq 2\sqrt{V_i(0)}/\kappa_i$. Once on the sliding manifold $\mathbf{s}_i = \mathbf{0}$, the dynamics reduce to $\dot{\mathbf{e}}_i = -\mathbf{H}_i \mathbf{D}_i(\mathbf{e}_i) |\mathbf{e}_i|^{1/2}$. Consider another Lyapunov function $U_i(\mathbf{e}_i) = \|\mathbf{e}_i\|_1 = \sum_{m=1}^{3} |e_{im}|$. Its derivative is:
$$
\dot{U}_i = \sum_{m=1}^{3} \text{sign}(e_{im}) \dot{e}_{im} = -\sum_{m=1}^{3} h_{im} |e_{im}|^{1/2} \leq -\eta_i U_i^{1/2}
$$
where $\eta_i = \min_{m} \{h_{im}\}$. By finite-time stability theory, $\mathbf{e}_i$ converges to zero in finite time. Thus, the overall system achieves formation in finite time, validating the effectiveness of our terminal sliding mode controller for quadrotor drone formations.
To illustrate the performance, we conduct a numerical simulation with one leader and three follower quadrotor drones. The communication topology is shown in Table 1, which summarizes the adjacency relationships.
| Quadrotor Drone Index | Neighbors |
|---|---|
| Leader (L) | None |
| Follower 1 | Leader, Follower 2 |
| Follower 2 | Leader, Follower 1, Follower 3 |
| Follower 3 | Follower 1, Follower 2 |
The normalized Laplacian matrix $\boldsymbol{\Upsilon}$ for this topology is:
$$
\boldsymbol{\Upsilon} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ -\frac{1}{2} & 1 & -\frac{1}{2} & 0 \\ -\frac{1}{2} & -\frac{1}{2} & 1 & 0 \\ 0 & -\frac{1}{2} & -\frac{1}{2} & 1 \end{bmatrix}
$$
Initial positions for the follower quadrotor drones are set as: $\mathbf{l}_1(0) = [0, 3.2, 2]^T$ m, $\mathbf{l}_2(0) = [-1.6, 0, 2]^T$ m, $\mathbf{l}_3(0) = [1.6, 0, 2]^T$ m. The leader quadrotor drone follows a trajectory given by:
$$
x_L(t) = 2\cos(t + \pi/3), \quad y_L(t) = 2\sin(t + \pi/3), \quad z_L(t) = 0.5t + 3
$$
The desired formation offsets $\mathbf{E}_{ij}(t)$ are constant in this example, as specified in Table 2.
| Offset | Value (m) |
|---|---|
| $\mathbf{E}_{1L}$ | $[0, 5, -1]^T$ |
| $\mathbf{E}_{12}$ | $[5, 10, 1]^T$ |
| $\mathbf{E}_{2L}$ | $[-5, -5, -2]^T$ |
| $\mathbf{E}_{21}$ | $[-5, -10, -1]^T$ |
| $\mathbf{E}_{31}$ | $[5, -10, -2]^T$ |
| $\mathbf{E}_{32}$ | $[10, 0, -1]^T$ |
Controller parameters are chosen as $\mathbf{H}_i = \text{diag}\{3, 3, 3\}$, $\mathbf{K}_i = \text{diag}\{2, 2, 2\}$, and $\psi_i = 3.5$ for all quadrotor drones. We simulate the system using MATLAB/Simulink with a fixed-step solver. The results demonstrate that all follower quadrotor drones converge to the desired formation within approximately 4 seconds, as shown by the generalized error states plotted in Figure 1. The 3D trajectories reveal that the quadrotor drone swarm performs a helical ascent while maintaining the formation, with the leader dictating the overall motion.
For comparison, we also implement a linear sliding mode controller for the same quadrotor drone system. The linear sliding variable is defined as $\mathbf{s}_i^{\text{lin}} = \dot{\mathbf{e}}_i + \mathbf{H}_i \mathbf{e}_i$, with the same $\mathbf{H}_i$. The control law is derived similarly but without the fractional power term. Simulation results show that the linear sliding mode controller achieves formation but with slower convergence—around 9 seconds—and larger transient errors. This highlights the advantage of terminal sliding mode control in providing faster response and finite-time convergence for quadrotor drone formations.
A detailed comparison of performance metrics is provided in Table 3.
| Metric | Terminal Sliding Mode | Linear Sliding Mode |
|---|---|---|
| Convergence Time | ~4 s | ~9 s |
| Steady-State Error | Negligible | Small but present |
| Control Effort | Moderate | Similar |
| Robustness | High | High |
The control inputs for each quadrotor drone exhibit chattering due to the sign function, which is typical in sliding mode control. This can be mitigated by using saturation functions or higher-order sliding modes, but that is beyond the scope of this article. Our focus is on demonstrating the finite-time formation capability for quadrotor drones.
In summary, we have developed a terminal sliding mode control strategy for formation control of quadrotor drones. The approach leverages a leader-follower architecture with local communication to enhance robustness. The mathematical model of the quadrotor drone is used to derive error dynamics, and a terminal sliding manifold is designed to ensure finite-time convergence of formation errors. Stability is proven via Lyapunov analysis, and simulations validate the effectiveness compared to linear sliding mode control. Future work may address practical challenges such as wind disturbances, actuator saturation, and real-time implementation on quadrotor drone platforms. Additionally, extending the method to heterogeneous swarms or incorporating adaptive mechanisms could further improve performance. The versatility of quadrotor drones makes this research relevant for applications ranging from environmental monitoring to disaster response.
Throughout this article, we have emphasized the role of terminal sliding mode control in achieving precise and rapid formation for quadrotor drones. The integration of multi-agent consensus principles with advanced control techniques underscores the potential for autonomous coordination in complex environments. As quadrotor drone technology advances, robust formation control algorithms will be crucial for enabling collaborative tasks in both civilian and military domains. We hope this contribution inspires further exploration into nonlinear control methods for multi-quadrotor drone systems.