In recent years, quadcopters have gained significant attention due to their versatility in applications such as surveillance, delivery, and environmental monitoring. However, these unmanned aerial vehicles (UAVs) often face challenges like parameter uncertainties, external disturbances, and actuator faults during flight, which can degrade performance and stability. This paper addresses these issues by proposing a robust adaptive control strategy that leverages a novel sliding mode reaching law to enhance fault tolerance and disturbance rejection in quadcopter systems.
The quadcopter, as a type of multi-rotor UAV, exhibits complex dynamics influenced by factors such as mass variations, aerodynamic effects, and environmental conditions. Traditional control methods may struggle to maintain stability under such uncertainties, especially when actuators experience partial or complete failures. Our approach integrates adaptive techniques with sliding mode control to estimate unknown parameters and disturbances online, ensuring reliable operation even in adverse scenarios. The core of our method lies in the design of a new sliding mode reaching law that reduces chattering and accelerates convergence, making it suitable for real-time quadcopter applications.
We begin by deriving the mathematical model of the quadcopter, considering the effects of external disturbances and actuator faults. The dynamics are represented using Newton-Euler equations, where the system is subject to uncertainties in parameters like mass and moments of inertia. The actuator faults are modeled as partial loss of effectiveness or additive disturbances, which are common in practical quadcopter operations. To handle these challenges, we develop an adaptive fault-tolerant controller that continuously updates its estimates of system parameters and fault characteristics, enabling robust tracking control.
The proposed controller is built upon a nonlinear sliding mode framework, enhanced with a novel reaching law that combines exponential and power-rate terms. This design minimizes chattering while ensuring finite-time convergence to the sliding surface. We provide a detailed stability analysis using Lyapunov theory, demonstrating that the closed-loop system remains stable under the proposed control law. Simulation results compare our method with existing approaches, highlighting its superiority in terms of tracking accuracy and disturbance rejection for quadcopter systems.
In the following sections, we present the system modeling, controller design, stability proof, and numerical simulations. Key contributions include the introduction of the new sliding mode reaching law, the integration of adaptive fault estimation, and comprehensive validation through comparative studies. This work aims to advance the field of quadcopter control by offering a solution that is both theoretically sound and practically viable for handling real-world uncertainties and faults.
System Modeling of Quadcopter
The quadcopter is a widely used UAV configuration characterized by four rotors arranged in a square pattern. Its dynamics can be described using a six-degree-of-freedom model, incorporating translational and rotational motions. Let the inertial frame be defined with axes (X, Y, Z), and the body-fixed frame with axes (X_B, Y_B, Z_B). The rotors generate thrust forces F1, F2, F3, F4, and angular velocities ω1, ω2, ω3, ω4, which are controlled to achieve desired movements.
The equations of motion for the quadcopter are derived from Newton’s second law and Euler’s equations, accounting for gravitational forces, aerodynamic drag, and external disturbances. The translational dynamics are given by:
$$ \ddot{x} = \frac{\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi}{m} U_1 – \frac{k_x}{m} \dot{x} + d_x $$
$$ \ddot{y} = \frac{\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi}{m} U_1 – \frac{k_y}{m} \dot{y} + d_y $$
$$ \ddot{z} = \frac{\cos\phi \cos\theta}{m} U_1 – g – \frac{k_z}{m} \dot{z} + d_z $$
where x, y, z represent the position coordinates; φ, θ, ψ are the roll, pitch, and yaw angles; m is the mass; g is the gravitational acceleration; k_x, k_y, k_z are drag coefficients; and d_x, d_y, d_z denote external disturbances. The control inputs U1, U2, U3, U4 are related to the rotor thrusts and are defined as:
$$ U_1 = F_1 + F_2 + F_3 + F_4 $$
$$ U_2 = l(F_4 – F_2) $$
$$ U_3 = l(F_3 – F_1) $$
$$ U_4 = k_m(F_1 – F_2 + F_3 – F_4) $$
where l is the arm length and k_m is a torque coefficient. The rotational dynamics are expressed as:
$$ \ddot{\phi} = \frac{I_{yy} – I_{zz}}{I_{xx}} \dot{\theta} \dot{\psi} + \frac{U_2}{I_{xx}} – \frac{k_2}{I_{xx}} \dot{\phi} + d_\phi $$
$$ \ddot{\theta} = \frac{I_{zz} – I_{xx}}{I_{yy}} \dot{\phi} \dot{\psi} + \frac{U_3}{I_{yy}} – \frac{k_3}{I_{yy}} \dot{\theta} + d_\theta $$
$$ \ddot{\psi} = \frac{I_{xx} – I_{yy}}{I_{zz}} \dot{\phi} \dot{\theta} + \frac{U_4}{I_{zz}} – \frac{k_4}{I_{zz}} \dot{\psi} + d_\psi $$
Here, I_xx, I_yy, I_zz are the moments of inertia; k_2, k_3, k_4 are rotational drag coefficients; and d_φ, d_θ, d_ψ represent disturbance torques. To account for actuator faults, we model the control inputs as:
$$ U_r = \beta_\tau U_{ra} + f_r $$
where r = 1,2,3,4 corresponds to the actuators, β_τ is the effectiveness factor (0 < β_τ ≤ 1), U_ra is the commanded input, and f_r is an additive fault. This formulation allows the controller to handle various fault scenarios, such as partial loss of effectiveness or bias faults, which are critical for quadcopter reliability.
The quadcopter model incorporates parameter uncertainties, such as variations in mass and inertia, which are common in real-world applications due to payload changes or manufacturing tolerances. External disturbances, like wind gusts, are modeled as bounded functions. Our control objective is to design a robust adaptive controller that ensures accurate tracking of desired trajectories despite these uncertainties and faults, focusing on enhancing the quadcopter’s performance in challenging environments.
New Sliding Mode Reaching Law Design
Sliding mode control (SMC) is renowned for its robustness against uncertainties and disturbances. However, traditional SMC suffers from chattering issues due to discontinuous control actions. To mitigate this, we propose a new sliding mode reaching law (NSMRL) that combines adaptive gains with a smooth switching function. The reaching law is defined as:
$$ \dot{s}_j = -p_j \tanh\left(\frac{s_j}{\Delta_j}\right) – \sigma_j |s_j|^{q_j} \text{sgn}(s_j) $$
for j = φ, θ, ψ, x, y, z, where s_j is the sliding variable, p_j and σ_j are adaptive gains, Δ_j is a boundary layer parameter, and q_j is a tuning exponent that switches based on the value of s_j:
$$ q_j = \begin{cases} r_j, & |s_j| \geq 1 \\ 1, & |s_j| < 1 \end{cases} $$
with r_j > 1. The gain p_j is adapted online using:
$$ p_j = \frac{2\chi_j}{\varepsilon_j + (1 – \varepsilon_j) e^{-n_j |s_j|}} $$
where χ_j, ε_j, n_j are positive constants. This formulation ensures that the reaching law provides fast convergence when the system is far from the sliding surface and reduces chattering near the surface. The use of the hyperbolic tangent function tanh(·) instead of the sign function sgn(·) smooths the control action, which is crucial for minimizing mechanical stress in quadcopter actuators.
The stability of the NSMRL is analyzed using Lyapunov theory. Consider the Lyapunov function candidate V = 0.5 s_j^2. Its derivative is:
$$ \dot{V} = s_j \dot{s}_j = -p_j s_j \tanh\left(\frac{s_j}{\Delta_j}\right) – \sigma_j |s_j|^{q_j+1} $$
Since tanh(·) and |s_j|^{q_j+1} are positive definite, ˙V ≤ 0, ensuring that s_j converges to zero in finite time. The convergence time can be estimated as:
$$ t_s < \frac{1 – |s_j(0)|^{1-r_j}}{(r_j – 1)\sigma_j} – \frac{\ln(v)}{\sigma_j} $$
where v is a small positive constant representing the boundary layer. This guarantees that the quadcopter system reaches the sliding mode quickly and maintains it despite disturbances.
Adaptive Fault-Tolerant Controller Design
We now present the adaptive fault-tolerant controller for the quadcopter, which integrates the NSMRL with parameter estimation. The controller consists of two loops: an outer loop for position control and an inner loop for attitude control. The control laws are derived based on the sliding mode approach, with adaptive laws to estimate unknown parameters, disturbances, and fault parameters.
For the attitude control (roll, pitch, yaw), define the tracking error as e_φ = φ_d – φ, where φ_d is the desired roll angle. The sliding variable is chosen as:
$$ s_\phi = \dot{e}_\phi + c_\phi e_\phi $$
with c_φ > 0. The derivative of s_φ is:
$$ \dot{s}_\phi = \ddot{e}_\phi + c_\phi \dot{e}_\phi = \ddot{\phi}_d – \ddot{\phi} + c_\phi (\dot{\phi}_d – \dot{\phi}) $$
Substituting the rotational dynamics, the control input U_2a for roll is designed as:
$$ U_{2a} = \frac{1}{\hat{\beta}_\phi} \left( \hat{\alpha}_3 (c_\phi \dot{e}_\phi + \ddot{\phi}_d) + \hat{\sigma}_\phi |s_\phi|^{q_\phi} \text{sgn}(s_\phi) + p_\phi \tanh\left(\frac{s_\phi}{\Delta_\phi}\right) – \hat{\alpha}_4 \dot{\theta} \dot{\psi} \right) $$
where ^α_3, ^α_4, ^σ_φ, ^β_φ are estimates of the parameters α_3 = I_xx, α_4 = I_yy – I_zz, σ_φ (disturbance bound), and β_φ (fault effectiveness). The adaptive laws are:
$$ \dot{\hat{\alpha}}_3 = \gamma_{\phi 1} s_\phi (c_\phi \dot{e}_\phi + \ddot{\phi}_d) $$
$$ \dot{\hat{\alpha}}_4 = -\gamma_{\phi 2} s_\phi \dot{\theta} \dot{\psi} $$
$$ \dot{\hat{\sigma}}_\phi = \lambda_{\phi 1} |s_\phi| $$
$$ \dot{\hat{\beta}}_\phi = -\lambda_{\phi 2} \frac{1}{\hat{\beta}_\phi} \left( \hat{\alpha}_3 (c_\phi \dot{e}_\phi + \ddot{\phi}_d) + \hat{\sigma}_\phi |s_\phi|^{q_\phi} \text{sgn}(s_\phi) + p_\phi \tanh\left(\frac{s_\phi}{\Delta_\phi}\right) – \hat{\alpha}_4 \dot{\theta} \dot{\psi} \right) s_\phi $$
where γ_φ1, γ_φ2, λ_φ1, λ_φ2 are positive adaptation gains. Similar control laws and adaptive laws are derived for pitch and yaw channels. For the position control (x, y, z), the sliding variable is defined as:
$$ s_z = \ddot{e}_z + c_z \dot{e}_z + c_z e_z $$
with e_z = z_d – z, and the control input U_1a is derived as:
$$ U_{1a} = \frac{1}{\hat{\beta}_z} \left( \hat{\alpha}_{1z} (c_z \dot{e}_z + \ddot{z}_d + g) + \hat{\alpha}_{2z} \dot{z} + \hat{\sigma}_z |s_z|^{q_z} \text{sgn}(s_z) + p_z \tanh\left(\frac{s_z}{\Delta_z}\right) \right) $$
with adaptive laws for ^α_1z, ^α_2z, ^σ_z, ^β_z. The overall control structure ensures that the quadcopter tracks desired trajectories while compensating for faults and disturbances. The controller parameters are tuned based on the system dynamics, and the adaptation mechanisms update the estimates in real-time, providing robustness against uncertainties.
Stability Analysis
The stability of the closed-loop quadcopter system under the proposed controller is analyzed using Lyapunov theory. Consider the Lyapunov function for the roll subsystem:
$$ V_\phi = \frac{1}{2} I_{xx} s_\phi^2 + \frac{1}{2\gamma_{\phi 1}} \tilde{\alpha}_3^2 + \frac{1}{2\gamma_{\phi 2}} \tilde{\alpha}_4^2 + \frac{1}{2\lambda_{\phi 1}} \tilde{\sigma}_\phi^2 + \frac{1}{2\lambda_{\phi 2}} \tilde{\beta}_\phi^2 $$
where ˜α_3 = α_3 – ^α_3, ˜α_4 = α_4 – ^α_4, ˜σ_φ = σ_φ – ^σ_φ, ˜β_φ = β_φ – ^β_φ are estimation errors. The derivative of V_φ is computed as:
$$ \dot{V}_\phi = I_{xx} s_\phi \dot{s}_\phi + \frac{1}{\gamma_{\phi 1}} \tilde{\alpha}_3 \dot{\tilde{\alpha}}_3 + \frac{1}{\gamma_{\phi 2}} \tilde{\alpha}_4 \dot{\tilde{\alpha}}_4 + \frac{1}{\lambda_{\phi 1}} \tilde{\sigma}_\phi \dot{\tilde{\sigma}}_\phi + \frac{1}{\lambda_{\phi 2}} \tilde{\beta}_\phi \dot{\tilde{\beta}}_\phi $$
Substituting the dynamics and adaptive laws, and using the boundedness of disturbances (|d_φ| ≤ σ_φ), we obtain:
$$ \dot{V}_\phi \leq -p_\phi |s_\phi| \leq 0 $$
This implies that s_φ converges to zero, and the estimation errors are bounded. Similar analysis applies to other channels, ensuring global stability of the quadcopter system. The use of NSMRL guarantees finite-time convergence to the sliding surface, and the adaptive laws ensure that parameter estimates remain bounded, providing robustness against faults and uncertainties.
Simulation Results
To validate the proposed controller, we conducted simulations using MATLAB/Simulink. The quadcopter parameters are listed in Table 1, and the controller parameters are given in Table 2. We compared our method (NRASMC) with two existing approaches: a terminal sliding mode control (TSMC) and an adaptive sliding mode control (ASMC). The simulation scenario includes external disturbances and actuator faults introduced at specific times.
| Parameter | Value |
|---|---|
| Mass (m) | 1.5 kg |
| I_xx | 0.016 kg·m² |
| I_yy | 0.026 kg·m² |
| I_zz | 0.032 kg·m² |
| Drag coefficients (k_x, k_y, k_z) | 0.008 |
| Rotational drag (k_2, k_3, k_4) | 0.0022 |
| Gravity (g) | 9.81 m/s² |
| Parameter | Value |
|---|---|
| c_q (q = x,y,z,φ,θ,ψ) | 1.5 |
| γ_φ1, γ_φ2, γ_ψ1, γ_ψ2 | 0.05, 0.2, 0.01, 0.2 |
| λ_φ1, λ_φ2, λ_θ1, λ_θ2 | 0.01, 0.5, 0.01, 0.5 |
| p_ε (ε = 2,3,4) | 5 |
| p_ς (ς = x,y,z) | 1.5 |
| Δ_j, r_j | 0.5, 1.5 |
The disturbances are modeled as d_μ = 0.5 sin(3πt) for μ = x,y,z and d_ξ = 0.5 cos(3πt) for ξ = φ,θ,ψ. Actuator faults are simulated with f_r = 0.3 and β_τ = 0.5 for τ = z,φ,θ,ψ, representing a 50% loss of effectiveness. The quadcopter starts from initial conditions x=3m, y=2m, z=0m, φ=0rad, θ=0rad, ψ=0.5rad, and tracks a desired trajectory.
The results show that our controller (NRASMC) achieves better tracking performance with smaller errors and faster convergence compared to TSMC and ASMC. For instance, in the position tracking, the NRASMC reduces the steady-state error to less than 0.1m, while the others exhibit larger deviations. In attitude control, the NRASMC maintains stability even under faults, with roll and pitch errors converging to zero within 2 seconds. The adaptive parameters, such as ^σ_φ and ^β_φ, converge to their true values, demonstrating the effectiveness of the estimation mechanism.
Moreover, the control inputs generated by NRASMC are smoother due to the NSMRL, reducing chattering and actuator wear. This is crucial for prolonging the lifespan of quadcopter components. The simulation confirms that the proposed approach enhances the robustness and fault tolerance of quadcopter systems in practical scenarios.
Conclusion
In this paper, we developed an adaptive fault-tolerant control scheme for quadcopters based on a new sliding mode reaching law. The controller effectively handles parameter uncertainties, external disturbances, and actuator faults by integrating online estimation with robust sliding mode control. The NSMRL ensures fast convergence and reduced chattering, making it suitable for real-time quadcopter applications. Stability analysis proved the global convergence of the system, and simulations validated the superiority of our method over existing approaches.
Future work will focus on extending the controller to handle more complex fault scenarios, such as simultaneous multiple faults, and implementing it on hardware for experimental validation. Additionally, we plan to explore the integration of machine learning techniques to further enhance the adaptability and intelligence of quadcopter control systems. This research contributes to the advancement of reliable and autonomous UAV operations in dynamic environments.