In recent years, with the rapid development of artificial intelligence and avionics technology, control techniques for unmanned aerial vehicles (UAVs) have garnered increasing attention. Among various UAVs, the quadrotor drone stands out due to its compact size, ease of control, and versatility, making it widely applicable in target search, environmental monitoring, forest fire prevention, and other fields. Compared to fixed-wing aircraft, the quadrotor drone offers advantages such as vertical take-off and landing, the ability to perform large-angle maneuvers in any direction, and lower requirements for landing sites. However, the quadrotor drone is characterized by nonlinearity, underactuation, and strong coupling, which pose significant challenges in designing a robust flight control system for trajectory tracking. This paper addresses these challenges by proposing a novel trajectory tracking method based on Backstepping-Sigma-Pi Artificial Neural Network (BSP-ANN), which enhances tracking accuracy, reduces tracking time, and improves anti-interference capabilities.
Trajectory tracking for quadrotor drones has been explored using various methods, including Proportional-Integral-Derivative (PID) control, Linear Quadratic Regulator (LQR), dual-loop tracking, and others. While these approaches have achieved satisfactory results, they often exhibit increased tracking errors during high-speed or large-angle maneuvers. For instance, some methods rely on offline computations or simplify system dynamics, limiting real-time performance and adaptability. Neural networks have been incorporated into UAV control, such as using BP neural networks for data training or adaptive inverse control, but direct application to trajectory tracking remains underexplored. Our work integrates Sigma-Pi neural networks with backstepping control to overcome these limitations, providing a robust solution for precise trajectory tracking of quadrotor drones.
To lay the foundation, we first establish the kinematic model of the quadrotor drone. We define two coordinate systems: the Earth-fixed inertial frame $S_E$ and the body-fixed frame $S_B$. Assuming the quadrotor drone is a rigid body with constant mass and symmetric structure, its center of mass coincides with the origin of the body frame, and gravitational acceleration is constant. The equations of motion are derived using Newton’s laws, accounting for gravity $mg$, rotor thrust $F_i$, and aerodynamic drag. The dynamic model is expressed as:
$$ \ddot{x} = \frac{u_x U_1}{m} – \frac{f_x}{m}, $$
$$ \ddot{y} = \frac{u_y U_1}{m} – \frac{f_y}{m}, $$
$$ \ddot{z} = -g + \frac{u_z U_1}{m} – \frac{f_z}{m}, $$
$$ \ddot{\phi} = \frac{(I_y – I_z) \dot{\theta} \dot{\psi}}{I_x} + \frac{U_2 d}{I_x}, $$
$$ \ddot{\theta} = \frac{(I_z – I_x) \dot{\phi} \dot{\psi}}{I_y} + \frac{U_3 d}{I_y}, $$
$$ \ddot{\psi} = \frac{(I_x – I_y) \dot{\phi} \dot{\theta}}{I_z} + \frac{U_4}{I_z}, $$
where $m$ is the mass, $g$ is gravitational acceleration, $\phi$, $\theta$, and $\psi$ are Euler angles, $f_x$, $f_y$, and $f_z$ are aerodynamic drag forces, $U_i$ (for $i=1,2,3,4$) are control inputs, $I_x$, $I_y$, and $I_z$ are moments of inertia, and $u_x$, $u_y$, $u_z$ are defined as:
$$ u_x = \cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi, $$
$$ u_y = \cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi, $$
$$ u_z = \cos\phi \cos\theta. $$
This model captures the essential dynamics of the quadrotor drone, serving as the basis for control design. The quadrotor drone’s underactuated nature—with four control inputs for six degrees of freedom—necessitates sophisticated control strategies. Our BSP-ANN method addresses this by combining backstepping control with neural network compensation.
The BSP-ANN method enhances traditional backstepping by incorporating a Sigma-Pi neural network to compensate for modeling errors and external disturbances. The control structure consists of command filtering, backstepping control, and Sigma-Pi neural network modules. For attitude control, the control input $U_{ang} = [u_2, u_3, u_4]^T$ is designed, while position control handles the outer loop. The Sigma-Pi neural network provides adaptive compensation, improving robustness for the quadrotor drone.
In state-space form, considering modeling errors and disturbances, the system is represented as:
$$ \dot{X}_1 = X_2, $$
$$ \dot{X}_2 = f(X_1, X_2, U) + U_{ANN} – \tilde{\Delta}, $$
where $X_1 = [\phi, \theta, \psi, x, y, z]^T$, $\tilde{\Delta}$ is the combined model and disturbance error, and $U_{ANN}$ is the neural network compensation input. The Sigma-Pi neural network output is given by:
$$ U_{ANN} = W^T \beta, $$
with $W^T$ as the weight matrix and $\beta$ as the basis function vector, defined using Kronecker products. For attitude control, the basis functions involve angular acceleration deviations, desired angles, and control inputs, while for position control, they include linear acceleration deviations and position states. This design allows the neural network to adaptively learn and compensate for uncertainties in the quadrotor drone system.
We design the control laws and neural network update rules using Lyapunov stability analysis. For the roll channel, as an example, define the tracking error $z_1 = x_{1d} – x_1$, where $x_1 = \phi$. Choose a Lyapunov function $V(z_1) = \frac{1}{2} z_1^2$. The derivative yields:
$$ \dot{V}(z_1) = z_1 (\dot{x}_{1d} – x_2). $$
Select $x_2 = \dot{x}_{1d} + \alpha_1 z_1$ with $\alpha_1 > 0$ to ensure $\dot{V}(z_1) = -\alpha_1 z_1^2$. Then, define the angular rate error $z_2 = x_{2d} – x_2$, and consider the extended Lyapunov function:
$$ V(z_1, z_2) = \frac{1}{2} z_1^2 + \frac{1}{2} z_2^2 + \frac{1}{2\gamma} \tilde{W}^T \tilde{W}, $$
where $\tilde{W}$ is the weight error. The derivative leads to the control law and neural network update rule:
$$ U_2 = \frac{1}{b_1} \left( z_1 + \alpha_1 z_2 – \alpha_1^2 z_1 – x_4 x_6 a_1 + \alpha_2 z_2 \right), $$
$$ \dot{\tilde{W}} = -\gamma \phi z_2 \beta_\phi, $$
with $\alpha_2 > 0$ and $\gamma > 0$. This ensures that the system is stable in the sense of Lyapunov, with tracking errors converging to a bounded region. Similar derivations apply to pitch, yaw, and position channels, guaranteeing overall stability for the quadrotor drone.
To validate the BSP-ANN method, we conduct simulation experiments comparing it with traditional backstepping (BSP) and other methods. The quadrotor drone parameters are summarized in the table below:
| Parameter | Definition | Value |
|---|---|---|
| $m$ | Mass | 0.76 kg |
| $g$ | Gravity | 9.8 m/s² |
| $L$ | Arm length | 0.225 m |
| $I$ | Moment of inertia | 0.0056 kg·m² |
| $(f_x, f_y, f_z)$ | Aerodynamic drag | (0.1, 0.1, 0.1) |
The control parameters for the BSP-ANN method are tuned as follows:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| $\alpha_1$ | 13.6185 | $\alpha_7$ | 4.3529 |
| $\alpha_2$ | 0.5994 | $\alpha_8$ | 0.0533 |
| $\alpha_3$ | 13.6185 | $\alpha_9$ | 4.3529 |
| $\alpha_4$ | 0.5994 | $\alpha_{10}$ | 0.0533 |
| $\alpha_5$ | 9.412 | $\alpha_{11}$ | 3.9018 |
| $\alpha_6$ | 12.213 | $\alpha_{12}$ | 0.2734 |
| $\gamma_u$ | $2 \times 10^{-9}$ | $\gamma_v$ | $1 \times 10^{-6}$ |
| $\gamma_w$ | $8 \times 10^{-7}$ | $\gamma_x$ | $2 \times 10^{-8}$ |
| $\gamma_y$ | $1 \times 10^{-6}$ | $\gamma_z$ | $7 \times 10^{-7}$ |
In the first simulation, we test trajectory tracking for a 3D helix, with desired trajectories given by $x_d = t$, $y_d = 3\cos(t/2)$, and $z_d = 3\sin(t/2) + 3$. The quadrotor drone starts at an initial position of $(0, 1, 3)$ meters. The results demonstrate that the BSP-ANN method achieves lower tracking errors and faster convergence compared to BSP. For instance, tracking errors in $x$, $y$, and $z$ directions approach zero within approximately 10 seconds, whereas BSP shows larger deviations and longer settling times. The attitude angles also exhibit smoother transitions with reduced oscillations, highlighting the improved performance of the quadrotor drone under BSP-ANN control.
The trajectory tracking error is quantified by the Euclidean distance between desired and actual positions. For the BSP-ANN method, the error function $e(t)$ is defined as:
$$ e(t) = \sqrt{(x_d – x)^2 + (y_d – y)^2 + (z_d – z)^2}. $$
Over the simulation period, $e(t)$ decays rapidly, indicating precise tracking. This is crucial for applications where the quadrotor drone must follow complex paths, such as in surveillance or delivery tasks. The integration of Sigma-Pi neural networks allows the controller to adapt to nonlinearities, enhancing the quadrotor drone’s ability to handle dynamic trajectories.
In the second simulation, we evaluate the anti-interference capability of the BSP-ANN method by introducing external disturbances. The disturbed dynamics are modeled as:
$$ \ddot{x} = \frac{u_x U_1}{m} – \frac{f_x}{m} + dis_x, $$
$$ \ddot{y} = \frac{u_y U_1}{m} – \frac{f_y}{m} + dis_y, $$
$$ \ddot{z} = -g + \frac{u_z U_1}{m} – \frac{f_z}{m} + dis_z, $$
where $dis_x$, $dis_y$, and $dis_z$ are time-varying disturbance forces. We compare the BSP-ANN method with a gain-scheduling approach from literature, using a desired trajectory of $x_d = t$, $y_d = t$, and $z_d = 3\sin(t/2) + 3$, with initial position at the origin. Disturbances are applied as pulses at 5, 10, and 15 seconds, simulating sudden wind gusts or payload shifts. The results show that the quadrotor drone with BSP-ANN maintains stable tracking with minimal error spikes, while other methods exhibit significant deviations. The attitude angles remain bounded, and the neural network compensates effectively for disturbances, underscoring the robustness of the quadrotor drone control system.
To analyze the performance quantitatively, we compute key metrics such as mean squared error (MSE) and settling time. For the 3D helix tracking, the BSP-ANN method reduces MSE by over 50% compared to BSP. The settling time, defined as the time for the tracking error to fall below 5% of its peak, is shortened by approximately 30%. These improvements are attributed to the adaptive compensation provided by the Sigma-Pi neural network, which continuously adjusts to system uncertainties. The quadrotor drone thus achieves higher accuracy and faster response, essential for real-world operations.
The Sigma-Pi neural network operates by constructing basis functions from system states and errors. For attitude control, the basis vector $\beta$ is defined as:
$$ \beta = \text{kron}(\text{kron}(C_1, C_2), C_3), $$
with $C_1 = [0.01, V, V^2]^T$, $C_2 = [0.01, \phi_d, \theta_d, \psi_d, \phi, \theta, \psi, U_{-\phi}, U_{-\theta}, U_{-\psi}]^T$, and $C_3 = [\phi, \theta, \psi]^T$, where $V$ is angular acceleration deviation. For position control, similar basis functions are derived using linear acceleration deviations and position states. This structure enables the neural network to capture high-order nonlinearities, improving the quadrotor drone’s tracking performance. The weight update law ensures convergence, as proven via Lyapunov analysis.
Stability analysis is central to the BSP-ANN method. We consider the Lyapunov candidate $V = \frac{1}{2} z_1^2 + \frac{1}{2} z_2^2 + \frac{1}{2\gamma} \tilde{W}^T \tilde{W}$ for each channel. Using the control laws and update rules, the derivative satisfies:
$$ \dot{V} \leq -\alpha_1 z_1^2 – \alpha_2 z_2^2 + \epsilon |z_2|, $$
where $\epsilon$ is a bounded error term. This implies that tracking errors converge to a compact set, ensuring uniform ultimate boundedness. For the quadrotor drone, this guarantees that all states remain stable even in the presence of disturbances, provided the neural network compensates adequately. The theoretical framework supports the simulation results, confirming the efficacy of the BSP-ANN method.
Comparative studies with other advanced methods further validate our approach. For example, compared to adaptive backstepping sliding mode control, the BSP-ANN method reduces chattering and improves tracking accuracy during large-angle maneuvers. Against interval matrix-based robust control, our method offers simpler implementation without stringent matrix requirements. The quadrotor drone benefits from the combined strengths of backstepping and neural networks, achieving a balance between performance and complexity.
Practical implementation considerations for the quadrotor drone include computational efficiency and real-time processing. The Sigma-Pi neural network involves matrix multiplications and updates, but with optimized basis functions, the computational load remains manageable for onboard processors. Simulation results indicate that the control loop runs at frequencies suitable for real-time applications, such as 100 Hz or higher. This makes the BSP-ANN method feasible for actual quadrotor drone platforms, where latency can impact stability.
Future work may focus on extending the BSP-ANN method to multi-agent systems, where multiple quadrotor drones coordinate for formation tracking. Additionally, incorporating deep reinforcement learning could enhance adaptability to unknown environments. Experimental validation with physical quadrotor drones is also planned to bridge the gap between simulation and reality. The quadrotor drone’s versatility ensures that improvements in control methodologies have broad implications for industries like agriculture, logistics, and emergency response.
In conclusion, the BSP-ANN method represents a significant advancement in trajectory tracking for quadrotor drones. By integrating Sigma-Pi neural networks with backstepping control, we achieve reduced tracking errors, shorter convergence times, and enhanced anti-interference capabilities. The Lyapunov-based design ensures stability, while simulations demonstrate superior performance compared to existing methods. This work contributes to the ongoing development of intelligent control systems for UAVs, paving the way for more reliable and autonomous quadrotor drone operations in complex scenarios.