In recent years, quadrotor drones have become increasingly prevalent in agricultural plant protection due to their advantages such as low-altitude operation, minimal drift, hovering capability, and remote control. However, during pesticide spraying missions, the gradual reduction in pesticide load leads to variations in the total mass and moment of inertia of the quadrotor drone, while external disturbances like wind gusts further compromise flight stability and control performance. Ensuring uniform and effective spraying requires robust flight control strategies that can adapt to these dynamic changes and disturbances. This paper addresses these challenges by proposing a novel control scheme that combines integral backstepping control with cascade RBF neural network PID control, enhancing the anti-disturbance capabilities of quadrotor drones operating under variable mass loads and wind conditions.
The quadrotor drone is an underactuated, highly coupled nonlinear system, making precise modeling and control inherently complex. Traditional control methods, such as PID, sliding mode, backstepping, and neural network controls, have been widely explored. Yet, when faced with model uncertainties and external interferences, a single control approach often falls short. Therefore, integrating multiple strategies to leverage their strengths is essential for achieving superior stability and robustness. For instance, prior research has combined extended state observers with backstepping control or employed integral backstepping to improve disturbance rejection, but many studies neglect the combined effects of variable mass loads and wind disturbances. In agricultural applications, the quadrotor drone’s payload decreases continuously during flight, altering its dynamics, while wind fields introduce additional forces and torques. This paper fills this gap by developing a comprehensive dynamics model that accounts for both variable mass loads and wind disturbances, followed by a dual-loop control architecture. The outer loop utilizes integral backstepping control for position and altitude tracking, while the inner loop employs cascade RBF neural network PID control for attitude stabilization, leveraging neural network self-learning to dynamically tune PID parameters.
To establish the foundation for control design, we first derive the dynamic model of the quadrotor drone. The quadrotor drone is modeled as a rigid body with symmetric structure, where the center of gravity coincides with the geometric center. Two coordinate systems are defined: the Earth-fixed frame \(E(OXYZ)\) and the body-fixed frame \(B(O_bX_bY_bZ_b)\). The transformation matrix from the body to Earth frame is given by:
$$ \mathbf{R} = \begin{bmatrix}
\cos\theta \cos\psi & \cos\psi \sin\phi \sin\theta – \sin\psi \cos\phi & \sin\psi \sin\phi + \cos\psi \cos\phi \sin\theta \\
\sin\psi \cos\theta & \sin\phi \sin\theta \sin\psi + \cos\theta \cos\psi & \sin\psi \cos\phi \sin\theta – \cos\psi \sin\phi \\
-\sin\theta & \cos\theta \sin\phi & \cos\theta \cos\phi
\end{bmatrix} $$
where \(\phi\), \(\theta\), and \(\psi\) represent the roll, pitch, and yaw angles, respectively. The position vector in the Earth frame is \(\mathbf{X} = [x, y, z]^T\). The quadrotor drone carries a symmetric variable mass load \(m_{\text{load}}\) attached below its center, which contributes additional moments of inertia. For a symmetric load, the incremental moments of inertia about the body axes are:
$$ I_{xl} = m_{\text{load}} d_z^2, \quad I_{yl} = m_{\text{load}} d_z^2, \quad I_{zl} = 0 $$
where \(d_z\) is the vertical distance from the load to the body plane \(O_bX_bY_b\). The total mass becomes \(m_z = m + m_{\text{load}}\), with \(m\) being the nominal mass of the quadrotor drone. The wind disturbance is modeled as a composite of basic wind, gust, gradual wind, and random wind components, expressed as:
$$ V_w = V_b + V_g + V_r + V_n $$
where \(V_b = k\) (constant), \(V_g\) is the gust profile, \(V_r\) is the gradual wind, and \(V_n\) is random wind. The detailed formulations are based on established aerodynamic models. The rotor dynamics under wind influence are analyzed, where the resultant force \(\mathbf{F}_R\) and torque \(\mathbf{M}_R\) on each rotor incorporate both thrust and wind-induced components. Using Newton-Euler formalism, the translational and rotational dynamics of the quadrotor drone are derived as follows:
$$ \begin{aligned}
\ddot{x} &= (\sin\psi \sin\phi + \cos\psi \cos\phi \sin\theta)\frac{U_1}{m + m_{\text{load}}} + D_1 \\
\ddot{y} &= (-\cos\psi \sin\phi + \sin\psi \cos\phi \sin\theta)\frac{U_1}{m + m_{\text{load}}} + D_1 \\
\ddot{z} &= (\cos\theta \cos\phi)\frac{U_1}{m + m_{\text{load}}} – g + D_1 \\
\ddot{\phi} &= \frac{I_y – I_z + m_{\text{load}} d_z^2}{I_x + m_{\text{load}} d_z^2} \dot{\theta} \dot{\psi} – \frac{J_{TP}}{I_x + m_{\text{load}} d_z^2} \dot{\theta} \Omega_r + \frac{U_2}{I_x + m_{\text{load}} d_z^2} + D_2 \\
\ddot{\theta} &= \frac{I_z – I_x – m_{\text{load}} d_z^2}{I_y + m_{\text{load}} d_z^2} \dot{\phi} \dot{\psi} + \frac{J_{TP}}{I_y + m_{\text{load}} d_z^2} \dot{\phi} \Omega_r + \frac{U_3}{I_y + m_{\text{load}} d_z^2} + D_3 \\
\ddot{\psi} &= \frac{I_x – I_y}{I_z} \dot{\phi} \dot{\theta} + \frac{U_4}{I_z} + D_4
\end{aligned} $$
Here, \(\mathbf{U} = [U_1, U_2, U_3, U_4]^T\) is the control input vector, defined by rotor speeds \(\Omega_i\) (\(i=1,2,3,4\)):
$$ \begin{bmatrix} U_1 \\ U_2 \\ U_3 \\ U_4 \end{bmatrix} = \begin{bmatrix}
\lambda (\Omega_1^2 + \Omega_2^2 + \Omega_3^2 + \Omega_4^2) \\
\lambda l (\Omega_4^2 – \Omega_2^2) \\
\lambda l (\Omega_3^2 – \Omega_1^2) \\
\mu (\Omega_2^2 + \Omega_4^2 – \Omega_1^2 – \Omega_3^2)
\end{bmatrix} $$
where \(\lambda\) and \(\mu\) are lift and drag coefficients, and \(l\) is the arm length. The disturbance vector \(\mathbf{D} = [D_1, D_2, D_3, D_4]^T\) captures wind effects, including forces and torques from wind on the rotors. This comprehensive model forms the basis for designing the control strategy to enhance the quadrotor drone’s performance under variable mass loads and wind disturbances.
The proposed control scheme employs a dual-loop structure. The outer loop manages translational motion (position and altitude) using integral backstepping control, which incorporates integral terms to improve steady-state accuracy and disturbance rejection. The inner loop handles rotational motion (attitude angles) via cascade RBF neural network PID control, where an RBF network dynamically adjusts PID parameters to adapt to uncertainties and disturbances. This combination leverages the robustness of integral backstepping and the adaptability of neural networks, ensuring stable flight for the quadrotor drone in challenging environments.
For the outer loop, integral backstepping is applied to the altitude dynamics. Define the altitude tracking error \(e_1 = z_d – z\) and its integral \(\Gamma_1 = \int e_1 dt\). Choose a Lyapunov function \(V_1 = \frac{1}{2} e_1^2 + \frac{1}{2} k_1 \Gamma_1^2\) with \(k_1 > 0\). Differentiating and introducing virtual controls, we derive the control law for \(U_1\):
$$ U_1 = \frac{( \ddot{z}_d + g + (1 + k_1 – c_1^2) e_1 + (c_1 + c_2) e_2 – k_1 c_1 \Gamma_1 – W_1 )(m + m_{\text{load}})}{\cos\theta \cos\phi} $$
where \(c_1\) and \(c_2\) are positive constants, \(e_2\) is an error term, and \(W_1\) represents disturbances. Similarly, virtual controls \(U_x\) and \(U_y\) are derived for horizontal positions \(x\) and \(y\), ensuring asymptotic tracking. The integral backstepping method enhances robustness by integrating error terms, reducing steady-state errors caused by wind disturbances and mass variations in the quadrotor drone.
For the inner loop, cascade RBF neural network PID control is implemented for attitude stabilization. The cascade structure consists of two PID controllers: an outer PID for angle control and an inner PID for angular rate control. The RBF neural network continuously tunes the parameters of the outer PID controller online. The RBF network has an input layer, a hidden layer with Gaussian activation functions, and an output layer. The network’s output is used to approximate the Jacobian of the system, enabling adaptive parameter adjustment. The incremental PID algorithm is employed:
$$ \Delta u(k) = k_p [e(k) – e(k-1)] + k_i e(k) + k_d [e(k) – 2e(k-1) + e(k-2)] $$
where \(e(k)\) is the error at step \(k\), and \(k_p\), \(k_i\), \(k_d\) are parameters tuned by the RBF network. The network adjusts these parameters based on a performance index \(E(k) = \frac{1}{2} e(k)^2\) using gradient descent. This allows the quadrotor drone to maintain precise attitude control despite wind gusts and changing inertia.
The RBF network’s learning process involves updating weights, centers, and widths. The Jacobian information is estimated as:
$$ \frac{\partial y(k)}{\partial \Delta u(k)} \approx \sum_{j=1}^m w_j h_j \frac{c_{ji} – x_1}{b_j^2} $$
where \(w_j\) are weights, \(h_j\) are hidden layer outputs, \(c_{ji}\) are centers, and \(b_j\) are widths. The parameter updates follow:
$$ \begin{aligned}
\Delta k_p &= \eta e(k) \frac{\partial y}{\partial \Delta u} x_c(1) \\
\Delta k_i &= \eta e(k) \frac{\partial y}{\partial \Delta u} x_c(2) \\
\Delta k_d &= \eta e(k) \frac{\partial y}{\partial \Delta u} x_c(3)
\end{aligned} $$
with learning rates \(\eta\). This adaptive mechanism ensures that the quadrotor drone’s control system remains responsive to dynamic changes, enhancing overall stability.
To validate the proposed control scheme, simulations are conducted using MATLAB/Simulink. The quadrotor drone parameters are listed in Table 1, wind model parameters in Table 2, and controller parameters in Tables 3 and 4. The initial conditions are set as \((x, y, z) = (0, 0, 0)\) and \((\phi, \theta, \psi) = (0, 0, 0)\). The desired trajectory is:
$$ \begin{aligned}
x_d &= 1 + 0.1 \sin(0.5t) \\
y_d &= 3 + 0.1 \sin(0.5t) \\
z_d &= 8 + 0.1 \sin(0.5t) \\
\psi_d &= 5
\end{aligned} $$
A variable mass load is applied from 6s to 18s, modeled as:
$$ m_{\text{load}} = \begin{cases}
0.5, & 0 \leq t < 6 \\
0.75 – \frac{1}{24}t, & 6 \leq t < 18 \\
0, & 18 \leq t
\end{cases} $$
Wind disturbances are introduced from 5.5s to 17s, with a composite wind profile as described earlier. The simulation runs for 30 seconds, comparing the proposed integral backstepping with cascade RBF neural network PID control against conventional PID control. The results demonstrate that the proposed method offers superior performance in terms of convergence speed, steady-state error, and disturbance rejection for the quadrotor drone.
| Parameter | Physical Meaning | Value | Unit |
|---|---|---|---|
| \(m\) | Mass of quadrotor | 1.2 | kg |
| \(g\) | Gravity acceleration | 9.81 | m/s² |
| \(l\) | Arm length | 0.24 | m |
| \(I_x\) | Moment of inertia about X-axis | 0.008 | kg·m² |
| \(I_y\) | Moment of inertia about Y-axis | 0.008 | kg·m² |
| \(I_z\) | Moment of inertia about Z-axis | 0.014 | kg·m² |
| \(J_{TP}\) | Rotor inertia | 1.1×10⁻⁴ | kg·m² |
| \(\rho\) | Air density | 1.29 | kg/m³ |
| \(A\) | Rotor area | 0.01 | m² |
| \(\mu\) | Drag coefficient | 1.1×10⁻⁶ | N·s² |
| \(\lambda\) | Lift coefficient | 5.3×10⁻⁵ | N·s² |
| Parameter | Physical Meaning | Value | Unit |
|---|---|---|---|
| \(k\) | Basic wind speed | 6.8 | m/s |
| \(V_{gmax}\) | Gust peak speed | 1 | m/s |
| \(V_{rmax}\) | Gradual wind peak speed | 1 | m/s |
| \(V_{nmax}\) | Random wind peak speed | 1 | m/s |
| \(t_{g1}\) | Gust start time | 5.5 | s |
| \(t_{g2}\) | Gust end time | 17 | s |
| \(t_{r1}\) | Gradual wind start time | 6 | s |
| \(t_{r2}\) | Gradual wind end time | 9 | s |
| \(t_{r3}\) | Gradual wind duration | 8 | s |
| Parameter | Value | Parameter | Value | Parameter | Value |
|---|---|---|---|---|---|
| \(C_1\) | 8 | \(C_2\) | 4 | \(C_3\) | 8 |
| \(C_4\) | 4 | \(C_5\) | 8 | \(C_6\) | 4 |
| \(K_1\) | 0.001 | \(K_2\) | 0.001 | \(K_3\) | 0.001 |
| Parameter | Physical Meaning | Value |
|---|---|---|
| \(\eta\) | Learning rate for neural network | 0.57 |
| \(\alpha\) | Momentum factor for neural network | 0.25 |
| \(e_{kp}\) | Learning rate for P parameter | 10 |
| \(e_{ki}\) | Learning rate for I parameter | 0.25 |
| \(e_{kd}\) | Learning rate for D parameter | 0.01 |
The simulation outcomes highlight the effectiveness of the proposed control strategy for the quadrotor drone. In attitude control, the cascade RBF neural network PID method yields smaller steady-state errors and reduced fluctuations under wind disturbances compared to conventional PID. The quadrotor drone’s roll, pitch, and yaw angles exhibit smoother transitions and less oscillation, indicating enhanced disturbance rejection. For position tracking, the integral backstepping control achieves faster convergence: the settling times for \(x\), \(y\), and \(z\) directions are approximately 2.2s, 2.5s, and 2.6s, respectively, whereas conventional PID requires 2.5s, 3s, and 24s. Moreover, under variable mass loads and wind gusts, the proposed scheme maintains smaller tracking errors and minimal overshoot, demonstrating robust performance for the quadrotor drone in dynamic agricultural environments.
In conclusion, this paper addresses the critical challenges of variable mass loads and wind disturbances in quadrotor drones used for agricultural plant protection. By developing a comprehensive dynamics model and proposing a novel control scheme integrating integral backstepping and cascade RBF neural network PID, we enhance the quadrotor drone’s flight stability and anti-disturbance capabilities. The dual-loop control architecture ensures precise position and attitude tracking, with the neural network providing adaptive parameter tuning to cope with uncertainties. Simulation results confirm that the proposed method outperforms conventional PID in terms of convergence speed, steady-state accuracy, and robustness, making it a viable solution for real-world quadrotor drone applications. Future work may involve hardware implementation and field testing to further validate the approach under diverse environmental conditions.
The quadrotor drone’s adaptability to changing payloads and external winds is crucial for efficient agricultural operations. This study contributes to advancing control methodologies for unmanned aerial systems, emphasizing the importance of integrated approaches in handling nonlinearities and disturbances. As quadrotor drones continue to evolve, such robust control strategies will play a pivotal role in expanding their utility across various sectors, including precision agriculture, surveillance, and logistics. The proposed framework offers a foundation for further research on intelligent control systems for autonomous quadrotor drones operating in complex and unpredictable environments.