In recent years, the deployment of multiple quadrotor drones in formation has garnered significant attention due to its potential to enhance operational efficiency in various applications, such as surveillance, search and rescue, and payload delivery. However, the practical implementation of quadrotor drone formations is fraught with challenges, including susceptibility to external environmental disturbances, system coupling effects, parameter uncertainties, and actuator failures. These factors can severely degrade performance and even lead to system instability. Therefore, developing robust control strategies that ensure both individual quadrotor drone stability and coordinated formation flight is of paramount importance. In this work, we address these challenges by proposing a novel control framework that integrates a compensation function observer (CFO) with an inverse hyperbolic tangent sliding mode control (IHTSMC) approach for quadrotor drone formation control. The methodology is built upon a leader-follower strategy to achieve cooperative flight, ensuring that the formation can effectively execute commands from a control center even under adverse conditions.
The core innovation lies in the design of a compensation function observer that provides high-precision estimation of lumped disturbances—encompassing external干扰, coupling terms, and actuator faults—for each quadrotor drone in the formation. By leveraging this estimation, we develop an inverse sliding mode controller that compensates for these disturbances, thereby enhancing the robustness of individual quadrotor drones. Furthermore, to mitigate the chattering phenomenon inherent in traditional sliding mode control, we employ a hyperbolic tangent function as a smooth approximation of the sign function. For formation coordination, a leader-follower architecture is adopted, where a sliding mode-based协同controller ensures that follower quadrotor drones maintain desired relative positions and velocities with respect to the leader. Through comprehensive simulation studies, we demonstrate the superiority of our proposed CFO-IHTSMC method over alternative approaches, such as CFO-based sliding mode control (CFO-SMC) and extended state observer-based inverse hyperbolic tangent sliding mode control (ESO-IHTSMC), in terms of tracking accuracy, convergence speed, disturbance rejection, and overall formation stability.
The remainder of this article is organized as follows. First, we present the dynamic modeling of a quadrotor drone, incorporating actuator faults and external disturbances. Next, we detail the design of the compensation function observer for both position and attitude subsystems. Subsequently, we derive the inverse hyperbolic tangent sliding mode controller for individual quadrotor drone control, followed by the design of the formation协同controller based on the leader-follower strategy. We then provide extensive simulation results and analysis, including parameter settings, comparative experiments, and formation flight performance. Finally, we conclude with a discussion of the findings and potential future research directions.
Dynamic Modeling of Quadrotor Drones with Actuator Faults and Disturbances
To formulate the control problem, we first establish a dynamic model for a quadrotor drone. The quadrotor drone is treated as a rigid body with symmetrical structure, where the center of gravity coincides with the geometric center. We define an inertial frame \( O_G(X_G, Y_G, Z_G) \) and a body frame \( O_b(x_b, y_b, z_b) \). The quadrotor drone is equipped with four rotors that generate lift forces \( F_1, F_2, F_3, F_4 \) and angular velocities \( \omega_1, \omega_2, \omega_3, \omega_4 \). The orientation is described by roll (\( \phi \)), pitch (\( \theta \)), and yaw (\( \psi \)) angles. The dynamics of the quadrotor drone, considering actuator faults and external disturbances, can be derived using Newton-Euler equations as follows:
$$ \ddot{x} = \frac{\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi}{m} (\delta_1 U_{1a} + f_1) – \frac{k_x}{m} \dot{x} + d_1 $$
$$ \ddot{y} = \frac{\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi}{m} (\delta_1 U_{1a} + f_1) – \frac{k_y}{m} \dot{y} + d_2 $$
$$ \ddot{z} = \frac{\cos\phi \cos\theta}{m} (\delta_1 U_{1a} + f_1) – g – \frac{k_z}{m} \dot{z} + d_3 $$
$$ \ddot{\phi} = \frac{I_{yy} – I_{zz}}{I_{xx}} \dot{\theta} \dot{\psi} + \frac{(\delta_\phi U_{2a} + f_2)}{I_{xx}} – \frac{k_\phi}{I_{xx}} \dot{\phi} + \frac{J_r}{I_{xx}} \dot{\theta} \lambda + d_4 $$
$$ \ddot{\theta} = \frac{I_{zz} – I_{xx}}{I_{yy}} \dot{\phi} \dot{\psi} + \frac{(\delta_\theta U_{3a} + f_3)}{I_{yy}} – \frac{k_\theta}{I_{yy}} \dot{\theta} + \frac{J_r}{I_{yy}} \dot{\phi} \lambda + d_5 $$
$$ \ddot{\psi} = \frac{I_{xx} – I_{yy}}{I_{zz}} \dot{\phi} \dot{\theta} + \frac{(\delta_\psi U_{4a} + f_4)}{I_{zz}} – \frac{k_\psi}{I_{zz}} \dot{\psi} + d_6 $$
where \( x, y, z \) represent the position coordinates; \( \phi, \theta, \psi \) are the attitude angles; \( m \) is the mass; \( g \) is gravitational acceleration; \( I_{xx}, I_{yy}, I_{zz} \) are moments of inertia; \( k_x, k_y, k_z, k_\phi, k_\theta, k_\psi \) are aerodynamic damping coefficients; \( J_r \) is the rotor inertia; \( \lambda \) is the overall propeller speed difference; \( d_i \) ( \( i = 1, 2, 3, 4, 5, 6 \) ) denote external disturbances; \( U_{ra} \) ( \( r = 1,2,3,4 \) ) are the nominal control inputs; \( \delta_\epsilon \) ( \( \epsilon = z, \phi, \theta, \psi \) ) represent actuator effectiveness factors ( \( 0 < \delta_\epsilon \leq 1 \) ); and \( f_r \) are bias faults. The actuator fault model is given by:
$$ U_r = \delta_\epsilon U_{ra} + f_r $$
For this study, we focus on the case where \( \delta_\epsilon = 1 \), indicating the presence of bias faults. The lumped disturbance for each channel, combining coupling terms, external disturbances, and actuator faults, is defined as \( \tilde{f}_i \) for subsequent observer design.
Design of Compensation Function Observer for Quadrotor Drones
The compensation function observer is designed to accurately estimate the lumped disturbances in both position and attitude subsystems of the quadrotor drone. Compared to traditional extended state observers (ESO), the CFO offers higher estimation precision and faster convergence due to its increased system type. Consider the roll angle subsystem as an example. Let the state variables be \( x_{\phi 1} = \phi \) and \( x_{\phi 2} = \dot{\phi} \). The dynamics can be rewritten as:
$$ \dot{x}_{\phi 1} = x_{\phi 2} $$
$$ \dot{x}_{\phi 2} = \tilde{f}_\phi + g_\phi U_{2a} $$
where \( \tilde{f}_\phi \) encompasses the coupling term \( \frac{I_{yy} – I_{zz}}{I_{xx}} \dot{\theta} \dot{\psi} – \frac{k_\phi}{I_{xx}} \dot{\phi} + \frac{J_r}{I_{xx}} \dot{\theta} \lambda \), actuator fault effect \( g_\phi f_2 \), and external disturbance \( d_4 \), with \( g_\phi = 1/I_{xx} \). The CFO for this subsystem is designed as:
$$ \dot{z}_{\phi 1} = z_{\phi 2} $$
$$ \dot{z}_{\phi 2} = L e + z_{\phi 3} + \tilde{f}_\phi + b_\phi U_{2a} $$
$$ \dot{z}_{\phi 3} = \lambda L e $$
$$ \hat{\tilde{f}}_\phi = L e + z_{\phi 3} $$
Here, \( z_{\phi 1}, z_{\phi 2} \) are estimates of \( x_{\phi 1}, x_{\phi 2} \); \( \hat{\tilde{f}}_\phi \) is the estimate of \( \tilde{f}_\phi \); \( e = [e_{\phi 1}, e_{\phi 2}]^T = [x_{\phi 1} – z_{\phi 1}, x_{\phi 2} – z_{\phi 2}]^T \) is the error vector; \( L = [l_2, l_1] \) is a tunable gain matrix; \( \lambda \) is a known filter parameter; and \( b_\phi = g_\phi \). The CFO is a type-III system, which ensures zero steady-state error convergence for the lumped disturbance under the condition that \( \tilde{f}_\phi \) is a third-order infinitesimal function. Similarly, CFOs can be designed for other subsystems (i.e., position \( x, y, z \) and attitude \( \theta, \psi \)) using the same structure, enabling comprehensive disturbance estimation for the quadrotor drone.
The advantages of the CFO over ESO are summarized in Table 1, highlighting key aspects such as system type, estimation accuracy, and convergence properties.
| Feature | Compensation Function Observer (CFO) | Extended State Observer (ESO) |
|---|---|---|
| System Type | Type-III | Type-I |
| Estimation Precision | High (zero steady-state error achievable) | Moderate (steady-state error may exist) |
| Convergence Speed | Fast | Relatively slower |
| Complexity | Moderate (requires tuning of \( L \) and \( \lambda \)) | Moderate (requires tuning of gains) |
Inverse Hyperbolic Tangent Sliding Mode Controller Design for Quadrotor Drones
With the lumped disturbances estimated by the CFO, we now design an inverse hyperbolic tangent sliding mode controller to achieve robust trajectory tracking for each quadrotor drone. The control system is divided into an outer position loop and an inner attitude loop. The position loop generates desired attitude angles, which are then tracked by the attitude loop.
Position Loop Controller
For the position subsystem, define the tracking errors as \( e_x = x_d – x \), \( e_y = y_d – y \), and \( e_z = z_d – z \), where \( x_d, y_d, z_d \) are desired trajectories. Taking the x-channel as an example, we design a sliding manifold:
$$ s_x = \dot{e}_x + c_x e_x $$
where \( c_x > 0 \) is a constant. The Lyapunov function candidate is chosen as \( V_x = \frac{1}{2} e_x^2 + \frac{1}{2} s_x^2 \). By applying the inverse control methodology and incorporating the disturbance estimate \( \hat{\tilde{f}}_x \) from the CFO, the control law for the x-channel virtual input \( u_{1x} \) is derived as:
$$ u_{1x} = \ddot{x}_d + c_x \dot{e}_x + \frac{k_x}{m} \dot{x} + e_x + \eta_x s_x + \lambda_x \tanh\left(\frac{s_x}{\gamma_x}\right) – \hat{\tilde{f}}_x $$
where \( \eta_x > 0 \), \( \lambda_x > 0 \), and \( \gamma_x > 0 \) are controller gains, and \( \tanh(\cdot) \) is the hyperbolic tangent function used to reduce chattering. The stability is proven by showing that \( \dot{V}_x \leq 0 \) under the control law, ensuring that the tracking error converges to zero. Similarly, control laws for y and z channels are:
$$ u_{1y} = \ddot{y}_d + c_y \dot{e}_y + \frac{k_y}{m} \dot{y} + e_y + \eta_y s_y + \lambda_y \tanh\left(\frac{s_y}{\gamma_y}\right) – \hat{\tilde{f}}_y $$
$$ u_{1z} = \ddot{z}_d + c_z \dot{e}_z + \frac{k_z}{m} \dot{z} + e_z + \eta_z s_z + \lambda_z \tanh\left(\frac{s_z}{\gamma_z}\right) – \hat{\tilde{f}}_z $$
The actual total thrust \( U_{1a} \) and desired roll and pitch angles \( \phi_d, \theta_d \) are computed from these virtual inputs:
$$ U_{1a} = \frac{m u_{1z}}{\cos\phi_d \cos\theta_d} $$
$$ \phi_d = \arctan\left( \frac{\sin\psi_d u_{1x} – \cos\psi_d u_{1y}}{u_{1z}} \cos\theta_d \right) $$
$$ \theta_d = \arctan\left( \frac{\cos\psi_d u_{1x} + \sin\psi_d u_{1y}}{u_{1z}} \right) $$
where \( \psi_d \) is the desired yaw angle.
Attitude Loop Controller
For the attitude subsystem, define errors \( e_\phi = \phi_d – \phi \), \( e_\theta = \theta_d – \theta \), and \( e_\psi = \psi_d – \psi \). The sliding manifolds are:
$$ s_\phi = \dot{e}_\phi + c_\phi e_\phi, \quad s_\theta = \dot{e}_\theta + c_\theta e_\theta, \quad s_\psi = \dot{e}_\psi + c_\psi e_\psi $$
The control laws for roll, pitch, and yaw channels are designed as:
$$ u_2 = I_{xx} \left( \ddot{\phi}_d + c_\phi \dot{e}_\phi + e_\phi + \eta_\phi s_\phi + \lambda_\phi \tanh\left(\frac{s_\phi}{\gamma_\phi}\right) – \hat{\tilde{f}}_\phi \right) $$
$$ u_3 = I_{yy} \left( \ddot{\theta}_d + c_\theta \dot{e}_\theta + e_\theta + \eta_\theta s_\theta + \lambda_\theta \tanh\left(\frac{s_\theta}{\gamma_\theta}\right) – \hat{\tilde{f}}_\theta \right) $$
$$ u_4 = I_{zz} \left( \ddot{\psi}_d + c_\psi \dot{e}_\psi + e_\psi + \eta_\psi s_\psi + \lambda_\psi \tanh\left(\frac{s_\psi}{\gamma_\psi}\right) – \hat{\tilde{f}}_\psi \right) $$
where \( \hat{\tilde{f}}_\phi, \hat{\tilde{f}}_\theta, \hat{\tilde{f}}_\psi \) are the CFO estimates for the respective attitude channels. The stability of each attitude loop is similarly proven via Lyapunov analysis, guaranteeing that the attitude tracking errors converge to zero exponentially.
Formation Cooperative Control Based on Leader-Follower Strategy
For multi-quadrotor drone formation control, we adopt a leader-follower approach. The leader quadrotor drone tracks a desired trajectory using the individual CFO-IHTSMC controller, while follower quadrotor drones maintain specified relative positions and velocities with respect to the leader. Consider a formation of \( N \) quadrotor drones in a two-dimensional plane (extension to 3D is straightforward). The kinematics of the \( j \)-th quadrotor drone are:
$$ \dot{x}_j = v_{xj} \cos\psi_j – v_{yj} \sin\psi_j $$
$$ \dot{y}_j = v_{xj} \sin\psi_j + v_{yj} \cos\psi_j $$
$$ \dot{\psi}_j = \Omega_j $$
where \( x_j, y_j \) are positions, \( v_{xj}, v_{yj} \) are body-frame velocities, \( \psi_j \) is the heading angle, and \( \Omega_j \) is the angular rate. For a leader ( \( j = L \) ) and a follower ( \( j = F \) ), define the relative distance errors:
$$ e_{x}^{LF} = d_{x}^{d, LF} – d_{x}^{LF}, \quad e_{y}^{LF} = d_{y}^{d, LF} – d_{y}^{LF}, \quad e_{\psi}^{LF} = \psi_F – \psi_L $$
where \( d_{x}^{LF}, d_{y}^{LF} \) are actual relative distances, and \( d_{x}^{d, LF}, d_{y}^{d, LF} \) are desired formation distances. The error dynamics can be expressed in state-space form:
$$ \dot{\chi} = E(\chi) + Q(\chi) u^* $$
with \( \chi = [e_{x}^{LF}, e_{y}^{LF}, e_{\psi}^{LF}]^T \), \( u^* = [v_{xF}, v_{yF}, \Omega_F]^T \), and matrices \( E(\chi) \), \( Q(\chi) \) defined appropriately. To achieve formation tracking, we design an integral sliding manifold:
$$ S_{L-F} = \chi + \Upsilon \int \chi $$
where \( \Upsilon \) is an invertible gain matrix. The formation control law for the follower is derived as:
$$ u^* = Q(\chi)^{-1} \left( -E(\chi) – \zeta_1 S_{L-F}^{\frac{1}{2}} \tanh\left(\frac{S_{L-F}}{\gamma_{L-F}}\right) – \zeta_2 \int \tanh\left(\frac{S_{L-F}}{\gamma_{L-F}}\right) – \Upsilon \chi \right) $$
with gains \( \zeta_1, \zeta_2 > 0 \) and \( \gamma_{L-F} > 0 \). Using Lyapunov theory, we prove that this controller ensures convergence of the formation errors to zero, thus maintaining the desired formation geometry even when the leader quadrotor drone experiences disturbances.
Simulation Verification and Results Analysis
To validate the proposed CFO-IHTSMC method for quadrotor drone formation control, we conduct extensive simulations in MATLAB/Simulink. The simulation setup includes a leader quadrotor drone and two follower quadrotor drones. The parameters for the quadrotor drone model, CFO, and controllers are listed in Table 2.
| Parameter | Value |
|---|---|
| Mass \( m \) (kg) | 1.2 |
| Gravity \( g \) (m/s²) | 9.81 |
| Moments of Inertia \( I_{xx}, I_{yy}, I_{zz} \) (N·s²·rad⁻¹) | 9.1e-3, 9.6e-3, 1.89e-2 |
| Aerodynamic Damping \( k_x, k_y, k_z \) (N·s²·rad⁻¹) | 1.2e-2 |
| Gyroscopic Coefficients \( k_\phi, k_\theta, k_\psi \) (N·s²·rad⁻¹) | 2.4e-3 |
| Rotor Inertia \( J_r \) (N·s²·rad⁻¹) | 1.13 |
| CFO Gains \( l_1, l_2 \) | 54, 432 |
| CFO Filter \( \lambda \) | 8 |
| Controller Gains \( c_q \) ( \( q = x,y,z,\phi,\theta,\psi \) ) | 1.5 |
| Controller Gains \( \eta_p \) ( \( p = x,y,z \) ) | 1.2 |
| Controller Gains \( \eta_\alpha \) ( \( \alpha = \phi,\theta,\psi \) ) | 1.5 |
| Controller Gains \( \lambda_\chi \) ( \( \chi = x,y,z,\phi,\theta,\psi \) ) | 3 |
| Controller Gains \( \gamma_\varepsilon \) ( \( \varepsilon = x,y,z,\phi,\theta,\psi \) ) | 0.5 |
| Formation Control Gains \( \zeta_1, \zeta_2, \gamma_{L-F} \) | 2, 1, 0.5 |
The desired trajectory for the leader is a helical path: \( x_d = 3\cos(t) \), \( y_d = 3\sin(t) \), \( z_d = 2 + 0.5t \), with desired yaw \( \psi_d = 0.5 \) rad. The followers are tasked to maintain relative distances of \( d_x^{d, LF} = d_y^{d, LF} = -6 \) m from the leader. External disturbances and actuator bias faults are introduced at \( t = 8 \) s, modeled as sinusoidal signals with amplitudes and phases. For instance, the lumped disturbance in the x-position channel is set as:
$$ \tilde{f}_x = \begin{cases}
0 & t < 8 \\
0.1 \sin(3\pi t + 0.2) + \sin(3\pi t) & t \geq 8
\end{cases} $$
Similar disturbances are applied to other channels. We compare three control methods: (1) Proposed CFO-IHTSMC, (2) CFO-based sliding mode control (CFO-SMC) using a sign function, and (3) ESO-based inverse hyperbolic tangent sliding mode control (ESO-IHTSMC). The performance metrics include trajectory tracking error, convergence speed, disturbance rejection, and formation maintenance.
Results for Individual Quadrotor Drone Control
The trajectory tracking results for the leader quadrotor drone under the three methods are shown in Figures 1-3 for position channels and Figures 4-6 for attitude channels. The proposed CFO-IHTSMC demonstrates superior performance: it achieves faster convergence (within 1.5-2 seconds) to the desired trajectories compared to CFO-SMC (4 seconds) and ESO-IHTSMC (2.5 seconds). Moreover, when disturbances are applied at \( t = 8 \) s, the CFO-IHTSMC exhibits minimal deviation and rapid recovery, with tracking errors reduced by over 50% compared to the other methods. The hyperbolic tangent function effectively attenuates chattering, resulting in smoother control inputs. The estimation performance of CFO versus ESO is illustrated in Figure 7 for the x-position channel and Figure 8 for the roll channel. The CFO provides more accurate and faster disturbance estimation, with near-zero steady-state error, whereas ESO shows larger estimation lag and overshoot.
Results for Quadrotor Drone Formation Control
The formation flight performance is evaluated by analyzing the relative distance and velocity errors between the leader and followers. As shown in Figure 9, the relative distance errors converge to zero within 3.09 seconds and remain negligible throughout the simulation, even after disturbance injection. The relative velocity errors, depicted in Figure 10, also converge to zero rapidly, confirming that the formation controller maintains the desired geometry. The overall 3D formation trajectory is plotted in Figure 11, showcasing that the quadrotor drone formation accurately follows the helical path while preserving formation shape.
Discussion and Comparative Analysis
The simulation results unequivocally validate the efficacy of the proposed CFO-IHTSMC approach for quadrotor drone formation control. Key advantages are summarized in Table 3.
| Performance Metric | CFO-IHTSMC (Proposed) | CFO-SMC | ESO-IHTSMC |
|---|---|---|---|
| Tracking Accuracy | Highest (errors < 0.05 m/rad) | Moderate (errors ~ 0.1 m/rad) | Moderate (errors ~ 0.08 m/rad) |
| Convergence Speed | Fastest (1.5-2 s) | Slowest (4 s) | Medium (2.5 s) |
| Disturbance Rejection | Excellent (quick recovery) | Good (some oscillation) | Fair (slower recovery) |
| Chattering Suppression | Effective (smooth control) | Poor (significant chattering) | Effective (smooth control) |
| Formation Stability | High (errors ~ 0) | Medium (small deviations) | Medium (small deviations) |
The superior performance of CFO-IHTSMC stems from the synergistic combination of the high-precision disturbance estimation by CFO and the robust tracking capability of the inverse sliding mode controller with hyperbolic tangent smoothing. The CFO’s type-III structure enables accurate estimation of lumped disturbances, including actuator faults, which are then compensated in real-time. This is particularly crucial for quadrotor drone formations operating in uncertain environments. In contrast, CFO-SMC suffers from chattering due to the discontinuous sign function, leading to degraded performance, while ESO-IHTSMC has lower estimation accuracy, resulting in slower disturbance rejection.
Furthermore, the leader-follower formation controller, based on sliding mode theory, ensures that follower quadrotor drones precisely track the leader’s state, even when the leader experiences disturbances. This hierarchical control structure simplifies the coordination problem and scales well for larger quadrotor drone swarms. However, it is worth noting that the leader-follower approach may be susceptible to single-point failures if the leader quadrotor drone encounters severe faults. Future work could explore distributed consensus protocols to enhance fault tolerance.
Conclusion and Future Work
In this article, we have presented a comprehensive control framework for quadrotor drone formation flight under lumped disturbances, including external environmental干扰, system coupling, and actuator faults. The proposed method integrates a compensation function observer for high-fidelity disturbance estimation and an inverse hyperbolic tangent sliding mode controller for robust trajectory tracking. The formation coordination is achieved via a leader-follower strategy with a sliding mode-based协同controller. Simulation studies demonstrate that the CFO-IHTSMC approach outperforms alternative methods in terms of tracking precision, convergence speed, disturbance rejection, and formation stability. These findings underscore the potential of this method for real-world applications where quadrotor drone formations must operate reliably in challenging conditions.
For future research, several directions can be pursued. First, the current work assumes perfect communication among quadrotor drones; incorporating communication delays and packet losses into the formation controller would enhance practical applicability. Second, extending the control framework to handle more severe actuator faults, such as complete failure or time-varying faults, would improve fault tolerance. Third, experimental validation with physical quadrotor drone platforms is necessary to assess performance in real environments. Fourth, integrating adaptive mechanisms to automatically tune controller and observer parameters could further optimize performance across varying operational conditions. Lastly, exploring distributed formation control strategies that do not rely on a single leader could increase robustness and scalability for large-scale quadrotor drone swarms.
In summary, the CFO-IHTSMC method offers a promising solution for advanced quadrotor drone formation control, paving the way for more reliable and efficient multi-quadrotor drone systems in diverse applications. By continuously refining these control strategies, we can unlock the full potential of quadrotor drone formations for complex missions in dynamic and uncertain environments.