In recent years, the quadrotor drone has garnered significant attention from both industry and academia. Characterized by vertical take-off and landing (VTOL) and relatively low manufacturing cost, this versatile platform is well-suited for a variety of missions. These include tracking and locking onto adversarial targets in aerial combat scenarios, as well as rapidly identifying and homing in on survivors during maritime search and rescue operations. The core enabling technology for the quadrotor drone to execute such tasks is its motion target tracking capability, which has consequently become a focal point of research. Within a tracking system, the control objective for the quadrotor drone is decoupled into two primary components: attitude tracking and position tracking.
Attitude control for a quadrotor drone is challenging due to the system’s inherent nonlinearity and strong coupling between rotational dynamics. Researchers have employed various advanced control techniques to address this, including sliding mode control (SMC), adaptive control, and neural network-based control. Position control, however, presents an even greater challenge. The quadrotor drone is an underactuated system, possessing six degrees of freedom (position and orientation) but only four control inputs (the thrusts of the four rotors). Designing a control law that manages this underactuation while simultaneously maintaining a specific, often fixed, distance from a moving target is a complex problem. Inspired by prior work, this article proposes a novel asymptotic tracking control framework. The primary contributions are threefold. First, we adopt a dual-loop control architecture, segregating the system into an outer position loop and an inner attitude loop. Second, for the position loop, we develop a method that fuses an Artificial Potential Field (APF) with Sliding Mode Control (SMC). This combination enables the quadrotor drone to track a moving target with high precision while maintaining a pre-defined fixed distance. Third, for the robust and accurate attitude loop, we design a controller leveraging a Radial Basis Function (RBF) neural network algorithm. The stability of both subsystems is rigorously proven using Lyapunov methods. Finally, comprehensive simulation results are provided to demonstrate the superiority, robustness, and tracking performance of the proposed control strategy.
Mathematical Model of the Quadrotor Drone
The quadrotor drone is the primary controlled object in the target tracking mission. Its simplified structure, defining the inertial frame {W} and body frame {B}, is considered. Based on the Newton-Euler formulation, the dynamic model of the quadrotor drone is established as follows:
$$ m\ddot{\mathbf{P}} = f\mathbf{R}\mathbf{e}_3 – mg\mathbf{e}_3 + \mathbf{d}_1 $$
$$ \mathbf{J}\ddot{\boldsymbol{\Theta}} = \boldsymbol{\varsigma} – \mathbf{C}\dot{\boldsymbol{\Theta}} + \mathbf{d}_2 $$
Here, \( m \) is the total mass, and \( \mathbf{P} = [x, y, z]^T \) denotes the position of the quadrotor’s center of mass in the inertial frame. \( \boldsymbol{\Theta} = [\phi, \theta, \psi]^T \) represents the Euler angles (roll, pitch, yaw). \( g \) is the gravitational acceleration, and \( \mathbf{e}_3 = [0, 0, 1]^T \) is the unit vector along the vertical direction. The control inputs are the total thrust \( f \) and the torque vector \( \boldsymbol{\varsigma} = [\varsigma_\phi, \varsigma_\theta, \varsigma_\psi]^T \). \( \mathbf{d}_1 \) and \( \mathbf{d}_2 \) represent external disturbance forces (e.g., wind gusts) and moments, respectively. \( \mathbf{R} \) is the rotation matrix transforming coordinates from the body frame to the inertial frame. \( \mathbf{J} \) is the inertia matrix expressed in the inertial frame. The rotation matrix \( \mathbf{R} \) and the inertia matrix \( \mathbf{J} \) are given by:
$$ \mathbf{R} = \begin{bmatrix}
C_\theta C_\psi & S_\phi S_\theta C_\psi – C_\phi S_\psi & C_\phi S_\theta C_\psi + S_\phi S_\psi \\
C_\theta S_\psi & S_\phi S_\theta S_\psi + C_\phi C_\psi & C_\phi S_\theta S_\psi – S_\phi C_\psi \\
-S_\theta & S_\phi C_\theta & C_\phi C_\theta
\end{bmatrix} $$
$$ \mathbf{J} = \begin{bmatrix}
I_{xx} & 0 & -I_{xx}S_\theta\\
0 & I_{yy}C^2_\phi + I_{zz}S^2_\phi & (I_{yy} – I_{zz})S_\phi C_\phi C_\theta\\
-I_{xx}S_\theta & (I_{yy} – I_{zz})S_\phi C_\phi C_\theta & I_{xx}S^2_\theta + I_{yy}S^2_\phi C^2_\theta + I_{zz}C^2_\phi C^2_\theta
\end{bmatrix} $$
where \( C_{(\cdot)} \) and \( S_{(\cdot)} \) denote \( \cos(\cdot) \) and \( \sin(\cdot) \), respectively. The Coriolis and centripetal terms matrix \( \mathbf{C} \) is calculated as:
$$ \mathbf{C} = \dot{\mathbf{J}} – \frac{1}{2} \frac{\partial}{\partial \boldsymbol{\Theta}} (\dot{\boldsymbol{\Theta}}^T \mathbf{J}) $$
Let \( \mathbf{P}_t \) denote the position of the moving target. The relative distance vector from the quadrotor drone to the target is \( \boldsymbol{\rho} = \mathbf{P} – \mathbf{P}_t \). The control objective for positioning is to maintain a desired tracking distance \( \rho_d \), defining the position tracking error as \( \mathbf{e}_p = \boldsymbol{\rho} – \rho_d \frac{\boldsymbol{\rho}}{\|\boldsymbol{\rho}\|} \).
Dual-Loop Controller Design
The overall control system features a cascade structure, as shown in the conceptual diagram. The outer position loop generates virtual control commands, which are then translated into desired attitude angles (\( \phi_d, \theta_d \)) for the inner attitude loop to track. The yaw command \( \psi_d \) is typically given independently.
| Control Loop | Primary Objective | Key Techniques Used | Output |
|---|---|---|---|
| Outer Position Loop | Maintain fixed distance \( \rho_d \) to moving target. | Artificial Potential Field (APF), Sliding Mode Control (SMC), Disturbance Estimation. | Virtual force \( \mathbf{U}_p \), desired roll \( \phi_d \), desired pitch \( \theta_d \). |
| Inner Attitude Loop | Track desired angles \( \phi_d, \theta_d, \psi_d \) robustly and accurately. | RBF Neural Network, Sliding Mode Control (SMC), Adaptive Laws. | Control torques \( \boldsymbol{\varsigma} \). |
Position Controller: APF and SMC Synthesis
The position error dynamics are derived from (1):
$$ \dot{\mathbf{e}}_p = \frac{\mathbf{U}_p + \mathbf{d}_1}{m} – g\mathbf{e}_3 – \ddot{\mathbf{P}}_d $$
where \( \mathbf{U}_p = f\mathbf{R}\mathbf{e}_3 \) is the virtual control input to be designed.
We construct an artificial potential field to govern the relative distance between the quadrotor drone and the target. It consists of a repulsive force for close-range safety and an attractive force to maintain the desired distance. The repulsive potential is modeled as a generalized Morse function:
$$ J_r = \begin{cases}
a b \left( e^{\frac{\|\boldsymbol{\rho}\|}{c}} – e^{\frac{\rho_{\min}}{c}} \right)^2, & \|\boldsymbol{\rho}\| \in D \\
0, & \|\boldsymbol{\rho}\| \notin D
\end{cases} $$
where \( a, b, c \) are positive constants, \( D = [\rho_{\min}, \rho_{\max}] \) defines the active range of the field, \( \rho_{\min} \) is the minimum safe distance, and \( \rho_{\max} \) is the maximum influence distance. The attractive potential is defined to ensure a stable equilibrium at \( \|\boldsymbol{\rho}\| = \rho_d \):
$$ J_a = \begin{cases}
\frac{1}{2} a k \|\boldsymbol{\rho}\|^2, & \|\boldsymbol{\rho}\| \in D \\
0, & \|\boldsymbol{\rho}\| \notin D
\end{cases} $$
where \( k \) is a positive constant. The parameters are chosen such that the net force is zero at the desired distance, satisfying:
$$ -k \rho_d + \frac{b}{c} \left( e^{\frac{\rho_d}{c}} – e^{\frac{\rho_{\min}}{c}} \right)^2 e^{\frac{\rho_d}{c}} = 0 $$
The combined velocity field, derived from the negative gradient of the total potential \( J = J_r + J_a \), guides the quadrotor drone:
$$ \mathbf{v}_f = -\nabla J_r – \nabla J_a = a \left[ -k + \frac{b}{c} \frac{ e^{\frac{\|\boldsymbol{\rho}\|}{c}} }{ \left( e^{\frac{\|\boldsymbol{\rho}\|}{c}} – e^{\frac{\rho_{\min}}{c}} \right)^2 } \right] \frac{\boldsymbol{\rho}}{\|\boldsymbol{\rho}\|} $$
A sliding surface is defined for the position error:
$$ \mathbf{s}_1 = \lambda_1 (\mathbf{v}_f – \dot{\mathbf{e}}_p) + \lambda_2 \mathbf{e}_p, \quad \lambda_1, \lambda_2 > 0 $$
Taking its derivative and substituting the dynamics, we design the virtual control law \( \mathbf{U}_p \) as:
$$ \mathbf{U}_p = -\hat{\mathbf{d}}_1 + m \left[ g\mathbf{e}_3 + \ddot{\mathbf{P}}_d + \dot{\mathbf{v}}_f + \frac{\lambda_2}{\lambda_1} \dot{\mathbf{e}}_p + \frac{\lambda_3}{\lambda_1} \mathbf{s}_1 \right] $$
where \( \lambda_3 > 0 \), and \( \hat{\mathbf{d}}_1 \) is the estimate of the external disturbance \( \mathbf{d}_1 \). The adaptation law for the disturbance estimate is designed as \( \dot{\hat{\mathbf{d}}}_1 = \frac{\lambda_1 \mathbf{s}_1^T}{m} \). Using Lyapunov stability analysis with \( V = \frac{1}{2} \mathbf{s}_1^T \mathbf{s}_1 + \frac{1}{2} \tilde{\mathbf{d}}_1^T \tilde{\mathbf{d}}_1 \) (where \( \tilde{\mathbf{d}}_1 = \mathbf{d}_1 – \hat{\mathbf{d}}_1 \)), we can prove that \( \dot{V} = -\lambda_3 \mathbf{s}_1^T \mathbf{s}_1 \leq 0 \), ensuring global asymptotic stability of the position tracking error.
From the obtained \( \mathbf{U}_p = [U_x, U_y, U_z]^T \), the actual thrust command \( f \) and the desired attitude angles \( \phi_d, \theta_d \) for the inner loop are computed:
$$ f = \frac{U_z}{C_{\phi_d} C_{\theta_d}}, \quad \theta_d = \arctan\left( \frac{U_x C_{\psi_d} + U_y S_{\psi_d}}{U_z} \right), \quad \phi_d = \arctan\left( C_{\theta_d} \frac{U_x S_{\psi_d} – U_y C_{\psi_d}}{U_z} \right) $$
Attitude Controller: RBF Neural Network Enhanced SMC
The inner loop’s objective is to force the actual attitude \( \boldsymbol{\Theta} \) to track the desired angles \( \boldsymbol{\Theta}_d = [\phi_d, \theta_d, \psi_d]^T \) generated by the outer loop. The attitude dynamics (2) can be rewritten in state-space form:
$$ \begin{aligned}
\dot{\boldsymbol{\Theta}}_1 &= \boldsymbol{\Theta}_2 \\
\dot{\boldsymbol{\Theta}}_2 &= \mathbf{f}(\boldsymbol{\Theta}, \dot{\boldsymbol{\Theta}}) + \mathbf{g}(\boldsymbol{\Theta}, \dot{\boldsymbol{\Theta}})\boldsymbol{\varsigma} + \mathbf{d}
\end{aligned} $$
where \( \mathbf{f}(\cdot) = -\mathbf{J}^{-1}\mathbf{C}\dot{\boldsymbol{\Theta}} \), \( \mathbf{g}(\cdot) = \mathbf{J}^{-1} \), and \( \mathbf{d} = \mathbf{J}^{-1}\mathbf{d}_2 \).
Define the attitude tracking error as \( \mathbf{e}_1 = \boldsymbol{\Theta}_d – \boldsymbol{\Theta}_1 \). A sliding surface is defined:
$$ \mathbf{s}_2 = \dot{\mathbf{e}}_1 + \lambda_4 \mathbf{e}_1, \quad \lambda_4 > 0 $$
The dynamics of \( \mathbf{s}_2 \) are:
$$ \dot{\mathbf{s}}_2 = \ddot{\boldsymbol{\Theta}}_d – \mathbf{f}(\cdot) – \mathbf{g}(\cdot)\boldsymbol{\varsigma} – \mathbf{d} + \lambda_4 \dot{\mathbf{e}}_1 $$
The functions \( \mathbf{f}(\cdot) \) and \( \mathbf{g}(\cdot) \) are often uncertain or nonlinear. To address this and mitigate chattering, we employ two RBF neural networks to approximate them:
$$ \mathbf{f}(\cdot) = \mathbf{W}^{*T} \mathbf{h}_f(\mathbf{x}) + \boldsymbol{\varepsilon}_f, \quad \mathbf{g}(\cdot) = \mathbf{V}^{*T} \mathbf{h}_g(\mathbf{x}) + \boldsymbol{\varepsilon}_g $$
where \( \mathbf{x} = [\boldsymbol{\Theta}_1^T, \boldsymbol{\Theta}_2^T]^T \) is the input, \( \mathbf{h}(\cdot) \) is the Gaussian function output, \( \mathbf{W}^*, \mathbf{V}^* \) are ideal weights, and \( \boldsymbol{\varepsilon}_f, \boldsymbol{\varepsilon}_g \) are approximation errors. The actual network outputs are \( \hat{\mathbf{f}}(\mathbf{x}) = \hat{\mathbf{W}}^T \mathbf{h}_f(\mathbf{x}) \) and \( \hat{\mathbf{g}}(\mathbf{x}) = \hat{\mathbf{V}}^T \mathbf{h}_g(\mathbf{x}) \).
The control law for the attitude torques is designed as:
$$ \boldsymbol{\varsigma} = \hat{\mathbf{g}}(\mathbf{x})^{-1} \left[ -\hat{\mathbf{f}}(\mathbf{x}) + \ddot{\boldsymbol{\Theta}}_d + \lambda_4 \dot{\mathbf{e}}_1 + \eta \, \text{sgn}(\mathbf{s}_2) \right], \quad \eta \geq 0 $$
The weight adaptation laws for the RBF networks are designed as:
$$ \dot{\hat{\mathbf{W}}} = -\gamma_1 \mathbf{s}_2^T \mathbf{h}_f(\mathbf{x}), \quad \dot{\hat{\mathbf{V}}} = -\gamma_2 \mathbf{s}_2^T \mathbf{h}_g(\mathbf{x}) \boldsymbol{\varsigma} $$
where \( \gamma_1, \gamma_2 > 0 \) are learning rates. Choosing the Lyapunov function \( L = \frac{1}{2} \mathbf{s}_2^T \mathbf{s}_2 + \frac{1}{2\gamma_1} \text{tr}(\tilde{\mathbf{W}}^T\tilde{\mathbf{W}}) + \frac{1}{2\gamma_2} \text{tr}(\tilde{\mathbf{V}}^T\tilde{\mathbf{V}}) \) (with \( \tilde{\mathbf{W}} = \mathbf{W}^* – \hat{\mathbf{W}} \), \( \tilde{\mathbf{V}} = \mathbf{V}^* – \hat{\mathbf{V}} \)), and substituting the control and adaptation laws, its derivative becomes:
$$ \dot{L} = \mathbf{s}_2^T (-\boldsymbol{\varepsilon}_f – \boldsymbol{\varepsilon}_g\boldsymbol{\varsigma} – \mathbf{d}) – \eta \|\mathbf{s}_2\| $$
If the robust gain \( \eta \) is chosen such that \( \eta \geq \| -\boldsymbol{\varepsilon}_f – \boldsymbol{\varepsilon}_g\boldsymbol{\varsigma} – \mathbf{d} \| \), then \( \dot{L} \leq 0 \), proving the bounded stability of the attitude tracking error.
| Category | Parameter | Symbol | Value |
|---|---|---|---|
| Position Controller (APF+SMC) | Desired Distance | \( \rho_d \) | 2 m |
| Min/Max APF Range | \( \rho_{\min}, \rho_{\max} \) | 1 m, 5 m | |
| APF Constants | \( k, b, c, a \) | 10, 16.83, 2, 1 | |
| SMC Gains | \( \lambda_1, \lambda_2, \lambda_3 \) | 1, 3, 1 | |
| Quadrotor Mass | \( m \) | 1 kg | |
| Inertia Matrix | \( \text{diag}(I_{xx}, I_{yy}, I_{zz}) \) | \( \text{diag}(0.004, 0.004, 0.008) \, \text{kg·m}^2 \) | |
| Attitude Controller (RBF+SMC) | Sliding Surface Gain | \( \lambda_4 \) | 5 |
| Robust Gain | \( \eta \) | 0.5 | |
| Learning Rates | \( \gamma_1, \gamma_2 \) | 10, 1 | |
| RBF Gaussian Width | \( b_j \) | 5 | |
| Target & Initial States | Target Trajectory | \( \mathbf{P}_t(t) \) | \( [0.5\cos(0.5t), \, 0.5\sin(0.5t), \, 2+0.1t] \) m |
| Initial Quadrotor State | \( [\mathbf{P}_0; \boldsymbol{\Theta}_0] \) | \( [0, 0, 1.5] \, \text{m}; \, [0.2, 0.2, \pi/3] \, \text{rad} \) |
Simulation Analysis and Results
To validate the effectiveness and robustness of the proposed control method (hereafter referred to as APF&RBF), extensive simulations were conducted. For comparison, a benchmark controller based on a second-order sliding mode (2-SMC) design was also implemented. The simulation environment included realistic, slow-time-varying disturbances acting on the quadrotor drone, modeling aerodynamic effects:
$$ \mathbf{d}_1 = [-0.125\sin(0.5\pi t), \, -0.125\cos(0.5\pi t), \, 0]^T \, \text{N} $$
$$ \mathbf{d}_2 = [0.3\sin(0.1\pi t)+0.1, \, 0.4\cos(0.1\pi t)+0.1, \, 0.5\sin(0.1\pi t)+0.2]^T \, \text{N·m} $$
The target was set to follow a helical ascent path: \( \mathbf{P}_t(t) = [0.5\cos(0.5t), \, 0.5\sin(0.5t), \, 2+0.1t]^T \) meters.
The three-dimensional trajectory plot clearly demonstrates that the proposed APF&RBF controller enables the quadrotor drone to successfully track the moving target while maintaining the prescribed distance. The convergence to the desired tracking orbit is smoother and faster compared to the 2-SMC benchmark. Analysis of the position tracking errors (\( e_x, e_y, e_z \)) reveals the superior performance of the APF&RBF strategy. The errors converge to a very small neighborhood of zero rapidly and maintain this precision throughout the target’s maneuver, showcasing the effectiveness of the combined APF and disturbance-estimating SMC. In contrast, the 2-SMC exhibits larger initial transient errors and slightly more oscillation during steady-state tracking.
The performance of the inner attitude loop is critical for overall system stability. The attitude tracking errors (\( e_\phi, e_\theta, e_\psi \)) under the RBF-enhanced controller demonstrate excellent convergence. The roll and pitch angles, which are continuously generated by the outer loop, are tracked with a maximum error of less than 0.1 radians after a brief transient phase. The yaw angle, held constant at \( \psi_d = \pi/3 \), is maintained with negligible error. The RBF network’s ability to approximate nonlinear dynamics and adapt is evident in the faster settling time and reduced chattering compared to the standard 2-SMC approach, which shows larger and more oscillatory errors in attitude tracking.
These simulation results comprehensively validate the proposed dual-loop control framework. The outer loop’s APF-SMC synthesis provides precise and robust distance keeping for the quadrotor drone, while the inner loop’s RBF-SMC controller ensures fast and accurate attitude stabilization, even in the presence of model uncertainties and external disturbances. The synergy between these components leads to a high-performance tracking system suitable for real-world applications of quadrotor drones.
Conclusion
This article presented a novel high-precision control algorithm for enabling a quadrotor drone to track a moving target. The control architecture was strategically decomposed into a cascade structure comprising an outer position loop and an inner attitude loop. For position control, a synthesized method combining an Artificial Potential Field (APF) with Sliding Mode Control (SMC) was developed. This approach ensures the quadrotor drone maintains a specific fixed distance from the target, with the APF providing natural guidance and the SMC guaranteeing robustness against disturbances. For the attitude loop, an advanced controller integrating an RBF neural network with SMC was designed. This combination effectively compensates for model nonlinearities, uncertainties, and minimizes chattering, leading to rapid and precise tracking of the attitude commands generated by the position controller. The stability of the entire closed-loop system was rigorously established using Lyapunov theory. Finally, numerical simulations were conducted, comparing the proposed method against a benchmark controller. The results unequivocally demonstrated the superior tracking accuracy, convergence speed, and disturbance rejection capabilities of the proposed APF&RBF strategy. This work provides a reliable and effective control solution that enhances the autonomous tracking capabilities of quadrotor drones, making them more viable for complex real-world missions such as persistent surveillance, search and rescue, and dynamic target engagement.