The coordination and control of multiple unmanned aerial vehicles (UAVs) have emerged as a critical area of research, significantly enhancing mission efficiency and operational scope. Among various platforms, the quadrotor drone has gained prominence due to its mechanical simplicity, vertical take-off and landing (VTOL) capability, and agile maneuverability. Formations of quadrotor drone units can accomplish complex tasks such as surveillance, search and rescue, and payload delivery more effectively than a single agent. However, the practical deployment of a quadrotor drone swarm is fraught with challenges stemming from inherent system nonlinearities, strong coupling between translational and rotational dynamics, susceptibility to external environmental disturbances like wind gusts, and the potential for actuator faults due to component wear or damage.
This article presents a holistic control framework designed to ensure the stable and precise formation flying of a quadrotor drone swarm in the presence of these compounded challenges. The core of our approach lies in a two-layer strategy: first, ensuring the robust stability of each individual quadrotor drone (the formation member) against lumped disturbances, and second, orchestrating the coordinated motion of the entire swarm. For the first objective, we propose a novel control synthesis integrating a Compensation Function Observer (CFO) with an Inverse Hyperbolic Tangent Sliding Mode Controller (IHTSMC). For the second objective, we adopt a well-established leader-follower topology and design a formation-keeping controller based on sliding mode theory. Extensive simulation studies validate the superiority of the proposed method over conventional approaches, demonstrating enhanced disturbance rejection, faster convergence, and precise formation tracking.
1. Modeling the Quadrotor Drone Dynamics with Disturbances and Faults
We begin by establishing the nonlinear dynamic model of a quadrotor drone. We define an earth-fixed inertial frame $\{O_G: X_G, Y_G, Z_G\}$ and a body-fixed frame $\{O_b: x_b, y_b, z_b\}$ attached to the drone’s center of mass. The state variables are the position $(x, y, z)$ and the Euler angles $(\phi, \theta, \psi)$ representing roll, pitch, and yaw, respectively. The control inputs are the collective thrust $U_1$ and the torque inputs $U_2, U_3, U_4$ generated by varying the speeds of the four rotors.
Considering external disturbances $d_i$, actuator faults, and linear drag effects, the nonlinear dynamics under the standard assumptions (rigid body, symmetric structure, negligible blade flapping) are given by:
$$
\begin{aligned}
\ddot{x} &= \frac{(\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi)}{m} U_1 – \frac{k_x}{m} \dot{x} + d_1, \\
\ddot{y} &= \frac{(\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi)}{m} U_1 – \frac{k_y}{m} \dot{y} + d_2, \\
\ddot{z} &= \frac{\cos\phi \cos\theta}{m} U_1 – g – \frac{k_z}{m} \dot{z} + d_3, \\
\ddot{\phi} &= \frac{I_{yy} – I_{zz}}{I_{xx}} \dot{\theta} \dot{\psi} + \frac{U_2}{I_{xx}} – \frac{k_{\phi}}{I_{xx}} \dot{\phi} + \frac{J_r}{I_{xx}} \dot{\theta} \Omega + d_4, \\
\ddot{\theta} &= \frac{I_{zz} – I_{xx}}{I_{yy}} \dot{\phi} \dot{\psi} + \frac{U_3}{I_{yy}} – \frac{k_{\theta}}{I_{yy}} \dot{\theta} + \frac{J_r}{I_{yy}} \dot{\phi} \Omega + d_5, \\
\ddot{\psi} &= \frac{I_{xx} – I_{yy}}{I_{zz}} \dot{\phi} \dot{\theta} + \frac{U_4}{I_{zz}} – \frac{k_{\psi}}{I_{zz}} \dot{\psi} + d_6.
\end{aligned}
$$
Here, $m$ is the mass, $g$ is gravity, $I_{xx}, I_{yy}, I_{zz}$ are moments of inertia, $k_{(\cdot)}$ are drag coefficients, $J_r$ is the rotor inertia, and $\Omega$ is the overall propeller speed. To account for actuator faults, we model the actual control input $U_r^{actual}$ as:
$$
U_r^{actual} = \delta_\epsilon U_r^{cmd} + f_r, \quad r=1,2,3,4; \epsilon=z,\phi,\theta,\psi
$$
where $0 < \delta_\epsilon \le 1$ is an effectiveness factor and $f_r$ is a bias fault. A healthy actuator corresponds to $\delta_\epsilon=1, f_r=0$. For this study, we focus on the bias fault scenario where $\delta_\epsilon=1$ but $f_r \neq 0$. The combined effect of external disturbances $d_i$, model uncertainties, and the bias fault $f_r$ is treated as a lumped disturbance $\Gamma$ for each channel, which our controller must actively reject.
2. Core Control Design: The CFO-IHTSMC for a Single Quadrotor Drone
The control problem is decomposed into an outer position loop and an inner attitude loop, following a backstepping-like procedure. The key innovation is the integration of a high-precision observer for lumped disturbances with a robust, chatter-mitigated sliding mode controller.
2.1 Compensation Function Observer (CFO) Design
Conventional Extended State Observers (ESOs) are limited in their estimation accuracy and convergence rate for higher-order disturbances. The CFO addresses this by incorporating a pure integration and compensation mechanism, effectively becoming a Type-III system (compared to ESO’s Type-I), leading to zero steady-state estimation error for disturbances with bounded third derivatives.
Consider the generic dynamics of a single channel (e.g., roll angle $\phi$):
$$\dot{x}_1 = x_2, \quad \dot{x}_2 = f + g u + \Gamma.$$
Here, $f$ represents known coupling terms, $g$ is the control gain, $u$ is the control input, and $\Gamma$ is the lumped disturbance. The CFO is designed as follows:
$$
\begin{aligned}
\dot{z}_1 &= z_2, \\
\dot{z}_2 &= L e + z_3 + \hat{\Gamma} + g u, \\
\dot{z}_3 &= \lambda L e, \\
\hat{\Gamma} &= L e + z_3.
\end{aligned}
$$
In this formulation, $z_1$ and $z_2$ estimate the states $x_1$ and $x_2$, respectively. $\hat{\Gamma}$ is the estimate of the lumped disturbance $\Gamma$. The vector $e = [x_1 – z_1, \quad x_2 – z_2]^T$ is the estimation error. $L = [l_2, l_1]$ and $\lambda$ are tunable observer gains. The term $\lambda L e$ acts as a compensation function, enabling faster and more accurate convergence of $\hat{\Gamma}$ to $\Gamma$. We design such a CFO for each of the six channels (three positional, three rotational) of the quadrotor drone.
2.2 Inverse Hyperbolic Tangent Sliding Mode Control (IHTSMC)
Sliding Mode Control (SMC) is renowned for its robustness against matched uncertainties. However, its discontinuous control law causes chattering. We combine the recursive structure of backstepping control with a continuous SMC law that uses a hyperbolic tangent function to approximate the sign function, effectively suppressing chattering while maintaining robustness.
The design proceeds step-by-step. For the x-position subsystem, define the tracking error $e_x = x_d – x$. Choose a sliding surface:
$$s_x = \dot{e}_x + c_x e_x, \quad c_x > 0.$$
The first Lyapunov function is $V_1 = \frac{1}{2}e_x^2$. Its derivative is $\dot{V}_1 = e_x \dot{e}_x = -c_x e_x^2 + e_x s_x$. To stabilize this, we proceed to the second step with a new Lyapunov function $V_2 = V_1 + \frac{1}{2}s_x^2$. Taking its derivative and substituting the dynamics yields:
$$
\dot{V}_2 = -c_x e_x^2 + e_x s_x + s_x(\ddot{x}_d + c_x \dot{e}_x + \frac{k_x}{m}\dot{x} – u_{1x} – \Gamma_x).
$$
We now design the virtual control law $u_{1x}$ for the x-position loop. Crucially, we incorporate the disturbance estimate $\hat{\Gamma}_x$ from the CFO and use the $\tanh$ function for chatter reduction:
$$
u_{1x} = \ddot{x}_d + c_x \dot{e}_x + \frac{k_x}{m}\dot{x} + e_x + \eta_x s_x + \lambda_x \tanh\left(\frac{s_x}{\gamma_x}\right) – \hat{\Gamma}_x.
$$
The parameters $\eta_x > 0, \lambda_x > 0, \gamma_x > 0$ are controller gains. The term $\lambda_x \tanh(s_x/\gamma_x)$ provides a smooth, high-gain feedback near the sliding surface $s_x=0$. Substituting this control law and using the property $s_x \cdot \tanh(s_x/\gamma_x) \ge 0$, we can prove that $\dot{V}_2 \le -G s_x^2$ for some positive $G$, guaranteeing the convergence of both $e_x$ and $s_x$ to zero. The same design procedure is applied to the $y$ and $z$ channels:
$$
\begin{aligned}
u_{1y} &= \ddot{y}_d + c_y \dot{e}_y + \frac{k_y}{m}\dot{y} + e_y + \eta_y s_y + \lambda_y \tanh\left(\frac{s_y}{\gamma_y}\right) – \hat{\Gamma}_y, \\
u_{1z} &= \ddot{z}_d + c_z \dot{e}_z + \frac{k_z}{m}\dot{z} + e_z + \eta_z s_z + \lambda_z \tanh\left(\frac{s_z}{\gamma_z}\right) – \hat{\Gamma}_z.
\end{aligned}
$$
The virtual controls $u_{1x}, u_{1y}, u_{1z}$ are then used to compute the desired total thrust $U_1$ and the desired roll ($\phi_d$) and pitch ($\theta_d$) angles:
$$
\begin{aligned}
U_1 &= m \frac{u_{1z}}{\cos\phi_d \cos\theta_d}, \\
\phi_d &= \arctan\left( \frac{\cos\psi_d \cdot u_{1x} + \sin\psi_d \cdot u_{1y}}{u_{1z}} \right), \\
\theta_d &= \arctan\left( \frac{\cos\theta_d (\sin\psi_d \cdot u_{1x} – \cos\psi_d \cdot u_{1y})}{u_{1z}} \right).
\end{aligned}
$$
Finally, the attitude controllers for $\phi, \theta, \psi$ are designed using the same CFO-IHTSMC principle:
$$
\begin{aligned}
U_2 &= I_{xx}\left( \ddot{\phi}_d + c_\phi \dot{e}_\phi + e_\phi + \eta_\phi s_\phi + \lambda_\phi \tanh\left(\frac{s_\phi}{\gamma_\phi}\right) – \hat{\Gamma}_\phi \right), \\
U_3 &= I_{yy}\left( \ddot{\theta}_d + c_\theta \dot{e}_\theta + e_\theta + \eta_\theta s_\theta + \lambda_\theta \tanh\left(\frac{s_\theta}{\gamma_\theta}\right) – \hat{\Gamma}_\theta \right), \\
U_4 &= I_{zz}\left( \ddot{\psi}_d + c_\psi \dot{e}_\psi + e_\psi + \eta_\psi s_\psi + \lambda_\psi \tanh\left(\frac{s_\psi}{\gamma_\psi}\right) – \hat{\Gamma}_\psi \right).
\end{aligned}
$$
This completes the robust controller for a single quadrotor drone. The CFO provides a precise estimate of the lumped disturbance, and the IHTSMC actively cancels it while driving the tracking error to zero with minimal chattering.
3. Formation Control: Leader-Follower Strategy with Sliding Mode Coordination
To orchestrate multiple quadrotor drone units into a stable formation, we employ the leader-follower strategy. One drone is designated as the leader, which tracks a predefined trajectory. The other drones are followers, tasked with maintaining a specific relative position and orientation with respect to the leader.
We simplify the model to 2D planar formation for clarity. The kinematics for drone $j$ (leader $L$ or follower $F$) are:
$$
\begin{aligned}
\dot{x}_j &= v_{xj} \cos\psi_j – v_{yj} \sin\psi_j, \\
\dot{y}_j &= v_{xj} \sin\psi_j + v_{yj} \cos\psi_j, \\
\dot{\psi}_j &= \Omega_j.
\end{aligned}
$$
The relative position errors between the leader and a follower in the leader’s body frame are defined as:
$$
\begin{aligned}
d_x^{LF} &= -(x_L – x_F)\cos\psi_L – (y_L – y_F)\sin\psi_L + d_f \cos(\psi_L – \psi_F), \\
d_y^{LF} &= (x_L – x_F)\sin\psi_L – (y_L – y_F)\cos\psi_L – d_f \sin(\psi_L – \psi_F).
\end{aligned}
$$
Here, $d_f$ is the distance from the drone’s center of rotation to a propeller. The formation control errors are then:
$$
e_x^{LF} = d_x^{LF}_d – d_x^{LF}, \quad e_y^{LF} = d_y^{LF}_d – d_y^{LF}, \quad e_\psi^{LF} = \psi_F – \psi_L.
$$
Differentiating these errors leads to a state-space model:
$$\dot{\mathbf{e}} = E(\mathbf{e}) + Q(\mathbf{e}) \mathbf{u}_F,$$
where $\mathbf{e} = [e_x^{LF}, e_y^{LF}, e_\psi^{LF}]^T$ and $\mathbf{u}_F = [v_{xF}, v_{yF}, \Omega_F]^T$ are the follower’s control velocities.
We design an integral sliding surface for the formation controller:
$$\mathbf{s}_{L-F} = \mathbf{e} + \Upsilon \int \mathbf{e} \, dt,$$
where $\Upsilon$ is a positive definite gain matrix. The formation control law for the follower is designed using a sliding mode approach, again employing the $\tanh$ function for smoothness:
$$
\mathbf{u}_F = Q(\mathbf{e})^{-1} \left( -E(\mathbf{e}) – \zeta_1 \mathbf{s}_{L-F}^{1/2} \circ \tanh\left(\frac{\mathbf{s}_{L-F}}{\gamma_f}\right) – \zeta_2 \int \tanh\left(\frac{\mathbf{s}_{L-F}}{\gamma_f}\right) dt – \Upsilon \mathbf{e} \right).
$$
The operator $\circ$ denotes element-wise multiplication. $\zeta_1, \zeta_2, \gamma_f$ are positive controller parameters. This control law ensures that the formation errors $\mathbf{e}$ converge to zero, meaning the follower maintains the desired relative pose. If the leader experiences disturbances, its internal CFO-IHTSMC stabilizes it, and the follower’s formation controller adjusts accordingly to preserve the formation geometry. This hierarchical approach ensures robust formation flight for the quadrotor drone swarm.
4. Simulation Analysis and Performance Validation
We validate the proposed control framework (CFO-IHTSMC) through comprehensive numerical simulations in MATLAB/Simulink. We compare its performance against two other algorithms: 1) ESO-based IHTSMC (ESO-IHTSMC), and 2) CFO-based conventional SMC with a sign function (CFO-SMC). The simulation involves a leader and two followers forming a triangular shape. The parameters for the quadrotor drone model and controllers are listed below.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Mass, $m$ | 1.2 kg | Gravity, $g$ | 9.81 m/s² |
| $I_{xx}$ | 9.1e-3 N·m·s² | $I_{yy}$ | 9.6e-3 N·m·s² |
| $I_{zz}$ | 1.89e-2 N·m·s² | Drag Coeff. $k_x, k_y, k_z$ | 1.2e-2 N·s/m |
| Observer Gain $l_1$ | 54 | Observer Gain $l_2$ | 432 |
| Observer Parameter $\lambda$ | 8 | Controller Gains $c_{(\cdot)}$ | 1.5 |
| Controller Gains $\eta_{x,y,z}$ | 1.2 | Controller Gains $\eta_{\phi,\theta,\psi}$ | 1.5 |
| Controller Gains $\lambda_{(\cdot)}$ | 3.0 | Smoothing Factors $\gamma_{(\cdot)}$ | 0.5 |
The leader is commanded to follow a helical trajectory: $x_d = 3\cos(t), y_d = 3\sin(t), z_d = 2 + 0.5t$, with a constant desired yaw $\psi_d=0.5$ rad. At time $t = 8$ seconds, we inject lumped disturbances (simulating combined wind gusts and actuator bias faults) into all channels of the leader drone. For example, the x-channel disturbance is:
$$\Gamma_x = \begin{cases} 0, & t < 8 \\ 0.1\sin(3\pi t + 0.2) + \sin(3\pi t), & t \ge 8 \end{cases}.$$
Similar sinusoidal disturbances are added to other channels.
4.1 Single Agent Performance (Leader)
The tracking performance for the leader’s position and attitude is shown in the analysis. The proposed CFO-IHTSMC demonstrates superior performance:
| Metric | CFO-IHTSMC (Proposed) | ESO-IHTSMC | CFO-SMC |
|---|---|---|---|
| Convergence Speed | Fastest (~1.5s for x, ~1.9s for z) | Moderate | Slowest (~4s for x) |
| Steady-State Error | Smallest, virtually zero | Small | Larger |
| Disturbance Rejection (at t=8s) | Excellent. Minimal deviation, fastest recovery. | Good. Noticeable deviation, slower recovery. | Fair. Largest deviation, slow recovery with oscillations. |
| Control Chattering | Negligible (smooth $\tanh$ function) | Negligible | Pronounced (discontinuous sign function) |
| Overall Robustness | Highest | Moderate | Lower |
The CFO’s estimation capability is a key factor. When comparing the CFO and ESO, the CFO demonstrates faster and more accurate estimation of the lumped disturbance $\Gamma_x$. The CFO converges to the true disturbance profile within approximately 2 seconds after the disturbance onset, while the ESO takes nearly 4 seconds and shows larger estimation errors during transients. This superior estimation directly feeds into the control law, enabling more precise cancellation.
The 3D trajectory tracking plot clearly shows that the quadrotor drone under CFO-IHTSMC adheres most closely to the desired helical path, both during the initial transient and after the disturbance injection, validating the single-agent controller’s robustness.
4.2 Formation Flight Performance
For the formation flight test, two followers are tasked with maintaining a fixed offset of -6 meters in both x and y directions relative to the leader. The relative distance and velocity errors between the leader and followers are critical metrics.
| Metric | Result | Interpretation |
|---|---|---|
| Relative Distance Error Convergence | Errors converge to zero within ~3.09 seconds and remain zero. | The formation is established quickly and maintained precisely. |
| Relative Velocity Error Convergence | Errors converge to zero concurrently with distance errors. | The followers achieve and maintain perfect velocity matching with the leader. |
| Formation Stability Under Disturbance | When the leader is perturbed at t=8s, the formation errors show only minuscule, momentary spikes before returning to zero. | The formation controller is highly robust. The leader’s internal stabilization (CFO-IHTSMC) and the follower’s sliding mode coordination work synergistically to reject formation disruptions. |
The simulation results conclusively demonstrate the effectiveness of the integrated framework. The proposed CFO-IHTSMC provides a high level of robustness and precision for individual quadrotor drone control against compound disturbances. When this robust single-agent controller is embedded within a leader-follower formation architecture powered by a smooth sliding mode coordinator, the entire swarm achieves stable, accurate, and resilient formation flight. The use of the hyperbolic tangent function successfully mitigates the chattering problem typically associated with SMC, leading to smoother control signals suitable for practical quadrotor drone applications.
5. Conclusion and Future Work
This article presented a comprehensive control solution for the challenging problem of quadrotor drone swarm formation flight under external disturbances and actuator faults. The proposed method hinges on two pillars:
- Robust Single-Agent Control (CFO-IHTSMC): A Compensation Function Observer was developed to achieve high-fidelity, fast estimation of lumped disturbances (encompassing couplings, wind, and faults). An Inverse Hyperbolic Tangent Sliding Mode Controller was then synthesized to actively compensate for these estimated disturbances. The backstepping structure ensures systematic stability, while the smooth $\tanh$ function eliminates chattering.
- Coordinated Formation Control: A leader-follower strategy was implemented, where a sliding mode-based formation controller for followers ensures that relative position and orientation errors converge to zero, maintaining the desired swarm geometry even when the leader is perturbed.
Simulation studies confirmed that the proposed framework outperforms comparable methods (ESO-IHTSMC, CFO-SMC) in terms of convergence speed, tracking accuracy, disturbance rejection, and control smoothness. The formation flight tests verified that the swarm can establish and maintain a precise formation robustly.
Future research will focus on extending this work to more complex and realistic scenarios. This includes investigating the control strategy under different types of simultaneous actuator and sensor faults, modeling and compensating for more realistic 3D turbulent wind fields, and implementing the framework on a distributed communication network. Addressing time delays and packet loss in inter-drone communication will be crucial for scaling the solution to large-scale quadrotor drone swarms operating in real-world, complex environments.