In recent years, quadcopters have gained significant attention in various applications such as logistics, surveillance, and rescue operations due to their maneuverability and vertical take-off capabilities. However, carrying an unknown mass payload can introduce disturbances that affect the stability and control of the quadcopter. This paper addresses this issue by proposing an adaptive control method based on a disturbance observer. The approach ensures robust attitude control despite uncertainties in the payload mass, enhancing the quadcopter’s performance in real-world scenarios. The quadcopter’s dynamics are modeled, and a disturbance observer is designed to estimate the unknown disturbances. An adaptive controller is then developed, with stability proven using Lyapunov theory. Experimental results demonstrate the effectiveness of the proposed method compared to traditional control strategies.
The quadcopter, as a type of unmanned aerial vehicle, relies on precise attitude control to maintain stability during flight. When carrying an unknown payload, the quadcopter experiences additional torques and inertial changes, which can lead to performance degradation. Traditional control methods may not adequately compensate for these disturbances, necessitating adaptive approaches. This work focuses on designing a control system that adapts to unknown mass variations, ensuring the quadcopter remains stable and responsive. The use of a disturbance observer allows for real-time estimation of disturbances, which are then compensated by the adaptive controller. This combination provides a robust solution for quadcopter operations in dynamic environments.
To model the quadcopter’s dynamics, we consider two coordinate systems: the inertial frame \( W = [X_w, Y_w, Z_w] \) and the body frame \( B = [X_b, Y_b, Z_b] \). The Euler angles are represented as \( \eta = [\phi, \theta, \psi]^T \), where \( \phi \) is the roll angle, \( \theta \) is the pitch angle, and \( \psi \) is the yaw angle. The dynamics of the quadcopter can be derived using the Lagrangian method, accounting for disturbances from the unknown payload. The general equation of motion is given by:
$$ J_u \ddot{\eta} = – \dot{\eta} \times J_u \dot{\eta} + \tau + \tau_d $$
where \( \ddot{\eta} \) is the angular acceleration vector, \( \dot{\eta} \) is the angular velocity vector, \( \tau = [\tau_\phi, \tau_\theta, \tau_\psi]^T \) is the control torque vector, and \( \tau_d = [\tau_{d\phi}, \tau_{d\theta}, \tau_{d\psi}]^T \) is the disturbance torque vector due to the unknown payload. The inertia matrix \( J_u \) is defined as:
$$ J_u = \begin{bmatrix}
J_{xx} & -J_{xy} & -J_{xz} \\
-J_{xy} & J_{yy} & -J_{yz} \\
-J_{xz} & -J_{yz} & J_{zz}
\end{bmatrix} $$
For a symmetric quadcopter, the off-diagonal terms \( J_{xy}, J_{xz}, J_{yz} \) are zero, simplifying the matrix to a diagonal form. The cross product term \( \dot{\eta} \times \) is expressed as:
$$ \dot{\eta} \times = \begin{bmatrix}
0 & -\dot{\psi} & \dot{\theta} \\
\dot{\psi} & 0 & -\dot{\phi} \\
-\dot{\theta} & \dot{\phi} & 0
\end{bmatrix} $$
The control torques and total thrust are generated by the motors’ rotational speeds. Each motor produces a force \( f_i \) and a torque \( \tau_i \), which are linearly related to the motor control signal \( \delta_i \). The relationships are given by:
$$ f_i = c_t (\delta_i – 1000) $$
$$ \tau_i = c_q (\delta_i – 1000) $$
where \( c_t \) and \( c_q \) are the thrust and torque coefficients, respectively. The total thrust and control torques can be represented in matrix form as:
$$ \begin{bmatrix}
f \\
\tau_\phi \\
\tau_\theta \\
\tau_\psi
\end{bmatrix} = \begin{bmatrix}
-c_t & -c_t & -c_t & -c_t \\
-\frac{c_t l_r}{\sqrt{2}} & \frac{c_t l_r}{\sqrt{2}} & \frac{c_t l_r}{\sqrt{2}} & -\frac{c_t l_r}{\sqrt{2}} \\
\frac{c_t l_r}{\sqrt{2}} & -\frac{c_t l_r}{\sqrt{2}} & \frac{c_t l_r}{\sqrt{2}} & -\frac{c_t l_r}{\sqrt{2}} \\
c_q & c_q & -c_q & -c_q
\end{bmatrix} \begin{bmatrix}
\delta_1 \\
\delta_2 \\
\delta_3 \\
\delta_4
\end{bmatrix} $$
Here, \( l_r \) is the distance from the motor axis to the quadcopter’s center of mass. This model forms the basis for designing the adaptive controller. The key parameters for a typical quadcopter are summarized in Table 1.
| Parameter | Description | Value | Unit |
|---|---|---|---|
| m | Mass of quadcopter | 1.325 | kg |
| g | Gravitational acceleration | 9.807 | m/s² |
| Jxx | Roll inertia moment | 12.71 | g·m² |
| Jyy | Pitch inertia moment | 12.71 | g·m² |
| Jzz | Yaw inertia moment | 23.7 | g·m² |
| lr | Distance from motor to center | 0.225 | m |
| ct | Thrust coefficient | 3.51 | mN/μs |
| cq | Torque coefficient | 0.108 | mN·m/μs |
The adaptive control method aims to compensate for disturbances caused by an unknown mass payload. The control structure includes a disturbance observer that estimates the disturbance torque \( \tau_d \), which is then used in the adaptive controller to adjust the control inputs. The overall control scheme is illustrated in Figure 1, where the observer provides real-time estimates to enhance robustness.
To design the adaptive controller, we define the tracking error for the attitude angles as \( e_1 = \eta_d – \eta \), where \( \eta_d \) is the desired attitude angle vector. A Lyapunov function is chosen as \( V_1 = \frac{1}{2} \| e_1 \|^2 \). Differentiating \( V_1 \) with respect to time gives:
$$ \dot{V}_1 = e_1^T \dot{e}_1 = e_1^T (\dot{\eta}_d – \dot{\eta}) $$
To ensure stability, a virtual control variable \( \phi = K_1 e_1 + \dot{\eta}_d \) is introduced, where \( K_1 \) is a positive definite matrix. Substituting this into the derivative yields:
$$ \dot{V}_1 = – e_1^T K_1 e_1 = – K_1 \| e_1 \|^2 \leq 0 $$
Next, the angular velocity tracking error is defined as \( e_2 = \phi – \dot{\eta} = K_1 e_1 + \dot{\eta}_d – \dot{\eta} \), and the disturbance estimation error is \( \Delta \tau_d = \hat{\tau}_d – \tau_d \). Assuming the disturbance is constant, the derivative of the estimation error is \( \Delta \dot{\tau}_d = \dot{\hat{\tau}}_d \). The disturbance observer is designed as:
$$ \dot{\hat{\tau}}_d = L(\eta) \left( \ddot{\eta} + J_u^{-1} \dot{\eta} \times J_u \dot{\eta} – J_u^{-1} \tau – J_u^{-1} \hat{\tau}_d \right) – J_u^{-1} e_2 $$
Substituting the dynamics model, this simplifies to:
$$ \dot{\hat{\tau}}_d = – L(\eta) J_u^{-1} \Delta \tau_d – J_u^{-1} e_2 $$
By setting \( L(\eta) = K_3 J_u \), where \( K_3 \) is a positive definite matrix, the observer gain is adjusted accordingly. The control torque \( \tau \) is derived as:
$$ \tau = J_u \left( (I_3 – K_1^2) e_1 + (K_1 + K_2) e_2 \right) + \dot{\eta} \times J_u \dot{\eta} – \hat{\tau}_d $$
where \( I_3 \) is the identity matrix and \( K_2 \) is another positive definite matrix. The stability of the closed-loop system is proven using Lyapunov theory. Consider the Lyapunov function:
$$ V_2 = \frac{1}{2} \| e_1 \|^2 + \frac{1}{2} \| e_2 \|^2 + \frac{1}{2} \Delta \tau_d^T \Delta \tau_d $$
Differentiating \( V_2 \) and substituting the control law and observer dynamics, we obtain:
$$ \dot{V}_2 = – e_1^T K_1 e_1 – e_2^T K_2 e_2 – \Delta \tau_d^T K_3 \Delta \tau_d \leq – \lambda_1 \| e_1 \|^2 – \lambda_2 \| e_2 \|^2 – \lambda_3 \| \Delta \tau_d \|^2 $$
where \( \lambda_1, \lambda_2, \lambda_3 \) are the minimum eigenvalues of \( K_1, K_2, K_3 \), respectively. This ensures that the tracking errors and estimation error converge to zero asymptotically, proving the stability of the adaptive controller for the quadcopter.
Experimental validation was conducted using a custom-built quadcopter platform equipped with a payload mechanism. The quadcopter was tested indoors to minimize external disturbances like wind. The platform included a frame, propulsion system, flight control unit, payload structure, and battery. The quadcopter was tasked with hovering while carrying an unknown mass payload, and the performance of the adaptive controller was compared to a backstepping controller. The experimental setup parameters are listed in Table 2.
| Component | Description | Specifications |
|---|---|---|
| Frame | Structure material | Carbon fiber |
| Motors | Brushless type | 1000 kV rating |
| Flight Controller | Processor | STM32-based |
| Payload Mechanism | Actuation | Servo-controlled release |
| Battery | Capacity | 5000 mAh LiPo |
During the experiments, the quadcopter initially used the adaptive controller to maintain stability with the unknown payload. The disturbance observer provided estimates of the disturbance torques, which were fed into the control loop. The attitude angles and motor control signals were recorded. When switched to the backstepping controller, the quadcopter exhibited a steady deviation in pitch angle, indicating insufficient disturbance rejection. In contrast, the adaptive controller quickly corrected the attitude errors, as shown in the data plots. The motor control signals under adaptive control remained stable, whereas the backstepping controller led to oscillations and offsets.
The results highlight the superiority of the adaptive approach in handling unknown payloads. The quadcopter’s ability to maintain desired attitudes despite disturbances demonstrates the practical utility of this method. Future work could explore extensions to outdoor environments or more complex payload scenarios. In conclusion, the adaptive control strategy based on a disturbance observer offers a robust solution for quadcopter attitude control, ensuring reliability in real-world applications where payload variations are common.
In summary, this paper presents a comprehensive approach to adaptive attitude control for quadcopters carrying unknown mass payloads. The dynamics modeling, controller design, and experimental validation provide a solid foundation for implementing such systems. The use of a disturbance observer enhances the quadcopter’s resilience, making it suitable for demanding tasks. As quadcopters continue to evolve, adaptive control methods will play a crucial role in expanding their capabilities. The proposed method is computationally efficient and easy to implement, offering a promising direction for future quadcopter developments.
Further analysis could involve optimizing the observer gains or integrating machine learning techniques for improved disturbance prediction. The quadcopter platform used in experiments serves as a testbed for advancing autonomous flight technologies. By addressing the challenges of unknown payloads, this work contributes to the broader field of unmanned aerial systems, where reliability and adaptability are paramount. The quadcopter’s performance under adaptive control underscores the importance of robust algorithms in achieving stable flight.