The operation of a quadrotor drone carrying a suspended load represents a significant challenge in aerial robotics, combining the inherent underactuation and nonlinear dynamics of the aircraft with the complex, oscillatory behavior of a pendulum-like payload. This system is highly relevant for practical applications such as construction, logistics, and search-and-rescue, where transporting cargo via cable is the most efficient method. However, the dynamic coupling between the quadrotor drone and the swinging load can severely degrade flight stability and positioning accuracy. The core of addressing this challenge lies in two fundamental aspects: deriving an efficient yet accurate dynamic model and designing a robust controller capable of rapidly suppressing payload oscillations.
Traditional modeling approaches, primarily the Newton-Euler and Lagrangian methods, present notable drawbacks for this multi-body system. The Newton-Euler method requires the explicit analysis of internal constraint forces (e.g., cable tension), leading to cumbersome derivation steps. The Lagrangian method, while avoiding internal forces, necessitates the construction and differentiation of energy functions, resulting in computationally intensive second-order differential equations. For real-time simulation and control synthesis, a more streamlined modeling technique is desirable.
Furthermore, after a maneuver, the payload continues to swing due to inertia. Without active control, these oscillations decay slowly through natural damping, resulting in unacceptably long stabilization times that hinder operational efficiency. While advanced nonlinear controllers like backstepping or sliding mode control offer solutions, their complexity can be a barrier to implementation. Proportional-Integral-Derivative (PID) control remains widely used for its simplicity but suffers from poor adaptability to the system’s nonlinear and time-varying dynamics. Hybrid strategies like neural network-tuned PID controllers have shown promise by adapting parameters online, yet many rely on standard gradient descent with fixed learning rates, limiting their convergence speed and adaptability in dynamic environments.
This work presents a comprehensive solution to these problems. First, we employ Kane’s method to establish the dynamic model of the quadrotor drone slung-load system. This method offers a middle ground, eliminating the need for internal force analysis and complex energy function differentiation, leading to a more efficient derivation process. Second, we design a novel anti-swing controller. We introduce a swing angle-position compensation strategy and fuse it with an enhanced neural network PID controller. The neural network’s weight update mechanism is improved using the Adaptive Moment Estimation (Adam) algorithm, creating an Adam-NN-PID controller. This controller features an adaptive learning rate, enabling faster and more robust online tuning of PID gains to counteract oscillations induced by maneuvers or external disturbances like wind.
The primary contributions and innovations of this article are summarized as follows:
- We apply Kane’s method for the dynamic modeling of a quadrotor drone with a cable-suspended payload. This approach simplifies the modeling procedure by avoiding the analysis of ideal constraint reactions required in the Newton-Euler formulation and bypassing the computation of kinetic/potential energy functions and their derivatives as in the Lagrangian method.
- We develop an adaptive anti-swing controller by integrating the Adam optimization algorithm with a Backpropagation Neural Network (BPNN) for PID parameter tuning. This Adam-NN-PID controller overcomes the limitation of fixed learning rates in conventional BPNN-PID, improving convergence speed and control performance. A dedicated swing-angle-to-position compensation strategy is designed to work synergistically with this controller for rapid load stabilization.
1. Dynamic Model of the Quadrotor Drone Slung-Load System
1.1 System Description and Assumptions
The system comprises a quadrotor drone, treated as a rigid body \(B\) with mass \(m_b\), and a point-mass payload \(P\) with mass \(m_p\), connected by a massless, inextensible cable of constant length \(l\). The following assumptions are made to formulate a tractable yet representative model:
- The quadrotor drone body is symmetric and rigid.
- The cable is always taut, its mass is negligible, and its length \(l\) is constant.
- The payload is a point mass.
- The cable attachment point coincides with the center of mass (CoM) of the quadrotor drone.
The inertial frame is defined as \(\mathcal{N}: \{O, \mathbf{n}_x, \mathbf{n}_y, \mathbf{n}_z\}\). The body-fixed frame attached to the quadrotor drone is \(\mathcal{B}: \{C, \mathbf{b}_x, \mathbf{b}_y, \mathbf{b}_z\}\), with its origin \(C\) at the CoM. The position of \(C\) in \(\mathcal{N}\) is \((x, y, z)\). The payload position in \(\mathcal{N}\) is \((x_p, y_p, z_p)\). The orientation of the quadrotor drone is given by the Z-Y-X Euler angles \((\phi, \theta, \psi)\) representing roll, pitch, and yaw, respectively. The cable direction is defined by two swing angles: \(\alpha\) (pitch-wise swing) and \(\beta\) (roll-wise swing).
1.2 Kinematic Equations
The system has 8 degrees of freedom, described by the generalized coordinate vector \(\mathbf{q} = [x, y, z, \phi, \theta, \psi, \alpha, \beta]^T\). The corresponding generalized speed vector is \(\mathbf{u} = \dot{\mathbf{q}} = [\dot{x}, \dot{y}, \dot{z}, \omega_x, \omega_y, \omega_z, \dot{\alpha}, \dot{\beta}]^T\), where \((\omega_x, \omega_y, \omega_z)\) are the body-axis angular velocities. The relationship between angular velocities and Euler angle rates is:
$$
\begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z \end{bmatrix} =
\begin{bmatrix}
1 & 0 & -\sin\theta \\
0 & \cos\phi & \cos\theta\sin\phi \\
0 & -\sin\phi & \cos\theta\cos\phi
\end{bmatrix}
\begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix}
$$
The position of the payload is geometrically related to the quadrotor drone‘s position and the swing angles:
$$
\begin{bmatrix} x_p \\ y_p \\ z_p \end{bmatrix} =
\begin{bmatrix} x + l\sin\alpha\cos\beta \\ y – l\sin\beta \\ z + l\cos\alpha\cos\beta \end{bmatrix}
$$
Differentiating yields the velocity relationship, crucial for Kane’s method:
$$
\begin{bmatrix} \dot{x}_p \\ \dot{y}_p \\ \dot{z}_p \end{bmatrix} =
\begin{bmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{bmatrix} +
\begin{bmatrix}
l\cos\beta\cos\alpha & -l\sin\beta\sin\alpha \\
0 & -l\cos\beta \\
-l\cos\beta\sin\alpha & -l\sin\beta\cos\alpha
\end{bmatrix}
\begin{bmatrix} \dot{\alpha} \\ \dot{\beta} \end{bmatrix}
$$
The translational velocity of the quadrotor drone CoM and the payload, and the angular velocity of the quadrotor drone, expressed in the body frame, are:
$$
\begin{aligned}
\mathbf{v} &= \dot{x}\mathbf{b}_x + \dot{y}\mathbf{b}_y + \dot{z}\mathbf{b}_z \\
\mathbf{v}_p &= \dot{x}_p\mathbf{b}_x + \dot{y}_p\mathbf{b}_y + \dot{z}_p\mathbf{b}_z \\
\boldsymbol{\omega} &= \omega_x\mathbf{b}_x + \omega_y\mathbf{b}_y + \omega_z\mathbf{b}_z
\end{aligned}
$$
1.3 Dynamic Modeling Using Kane’s Method
Kane’s method provides a powerful alternative to Newton-Euler and Lagrangian mechanics. It involves calculating generalized active forces and generalized inertia forces corresponding to each independent generalized speed, and setting their sum to zero. The governing equation is:
$$
(\mathbf{F}_r + \mathbf{F}_r^*) = 0, \quad r = 1, 2, …, 8
$$
where \(\mathbf{F}_r\) is the generalized active force and \(\mathbf{F}_r^*\) is the generalized inertia force associated with the \(r\)-th generalized speed.
Generalized Active Forces: The active forces on the system are the total thrust \(\mathbf{U} = U_x\mathbf{b}_x + U_y\mathbf{b}_y + U_z\mathbf{b}_z\) (where \(U_x, U_y, U_z\) are the components of the collective thrust vector in the body frame), the control torques \(\boldsymbol{\tau} = \tau_x\mathbf{b}_x + \tau_y\mathbf{b}_y + \tau_z\mathbf{b}_z\), and gravity acting on both the quadrotor drone and the payload. The total active force is \(\mathbf{F} = \mathbf{U} + (m_b+m_p)g\mathbf{b}_z\). The generalized active forces for the translational and rotational speeds are found by projecting these forces/moments onto the partial velocity vectors:
$$
\begin{aligned}
\tilde{F}_x &= \mathbf{F} \cdot \frac{\partial \mathbf{v}}{\partial \dot{x}} = U_x, \quad \tilde{F}_y = U_y, \quad \tilde{F}_z = U_z + (m_b+m_p)g \\
\tilde{M}_x &= \boldsymbol{\tau} \cdot \frac{\partial \boldsymbol{\omega}}{\partial \omega_x} = \tau_x, \quad \tilde{M}_y = \tau_y, \quad \tilde{M}_z = \tau_z
\end{aligned}
$$
For the swing degrees of freedom, the active force is the gravity on the payload. Their generalized active forces are:
$$
\tilde{F}_{\alpha} = m_p g l \sin\alpha, \quad \tilde{F}_{\beta} = m_p g l \sin\beta \cos\alpha
$$
Generalized Inertia Forces: The inertia force for the quadrotor drone is \(-(m_b \ddot{x}\mathbf{b}_x + m_b \ddot{y}\mathbf{b}_y + m_b \ddot{z}\mathbf{b}_z)\). For the payload, it is \(-m_p (\ddot{x}_p\mathbf{b}_x + \ddot{y}_p\mathbf{b}_y + \ddot{z}_p\mathbf{b}_z)\). The inertia torque for the rigid quadrotor drone, with inertia matrix \(\mathbf{I} = \text{diag}(I_x, I_y, I_z)\), is \(-\left( \mathbf{I} \dot{\boldsymbol{\omega}} + \boldsymbol{\omega} \times (\mathbf{I} \boldsymbol{\omega}) \right)\).
The generalized inertia forces are obtained by projecting these inertia forces/torques onto the corresponding partial velocities. For translation:
$$
\begin{aligned}
\tilde{F}^*_x &= -m_b \ddot{x} – m_p \ddot{x}_p \\
\tilde{F}^*_y &= -m_b \ddot{y} – m_p \ddot{y}_p \\
\tilde{F}^*_z &= -m_b \ddot{z} – m_p \ddot{z}_p
\end{aligned}
$$
For rotation:
$$
\begin{aligned}
\tilde{M}^*_x &= -(I_x \dot{\omega}_x + (I_z – I_y)\omega_y\omega_z) \\
\tilde{M}^*_y &= -(I_y \dot{\omega}_y + (I_x – I_z)\omega_x\omega_z) \\
\tilde{M}^*_z &= -(I_z \dot{\omega}_z + (I_y – I_x)\omega_x\omega_y)
\end{aligned}
$$
For the swing dynamics, the projections involve the relationship between payload acceleration and swing angle accelerations derived from the kinematic equations. The resulting generalized inertia forces \(\tilde{F}^*_{\alpha}\) and \(\tilde{F}^*_{\beta}\) are complex expressions involving \(\ddot{x}, \ddot{y}, \ddot{z}, \ddot{\alpha}, \ddot{\beta}\).
Applying Kane’s Equation: Setting the sum of generalized active and inertia forces to zero for each generalized speed yields the complete dynamic model. The final equations of motion for the quadrotor drone slung-load system are as follows, where \(s_i = \sin(i)\) and \(c_i = \cos(i)\):
Translational dynamics (coupled with swing):
$$
\begin{aligned}
&(m_b + m_p)\ddot{x} + m_p l(\ddot{\alpha}c_\beta c_\alpha – \ddot{\beta}s_\beta s_\alpha – \dot{\alpha}^2 c_\beta s_\alpha – \dot{\beta}^2 c_\beta s_\alpha – 2\dot{\alpha}\dot{\beta}s_\beta c_\alpha) = U_x \\
&(m_b + m_p)\ddot{y} + m_p l(\dot{\beta}^2 s_\beta – \ddot{\beta} c_\beta) = U_y \\
&(m_b + m_p)\ddot{z} + m_p l(2\dot{\alpha}\dot{\beta}s_\beta s_\alpha – \ddot{\alpha}c_\beta s_\alpha – \ddot{\beta}s_\beta c_\alpha – \dot{\alpha}^2 c_\beta c_\alpha – \dot{\beta}^2 c_\beta c_\alpha) = (m_b+m_p)g + U_z
\end{aligned}
$$
Rotational dynamics (standard quadrotor drone dynamics):
$$
\begin{aligned}
I_x \dot{\omega}_x + (I_z – I_y)\omega_y\omega_z &= \tau_x \\
I_y \dot{\omega}_y + (I_x – I_z)\omega_x\omega_z &= \tau_y \\
I_z \dot{\omega}_z + (I_y – I_x)\omega_x\omega_y &= \tau_z
\end{aligned}
$$
Swing dynamics (pendulum motion coupled with drone acceleration):
$$
\begin{aligned}
\ddot{\alpha} &= \frac{\ddot{z}s_\alpha – \ddot{x}c_\alpha + 2l\dot{\alpha}\dot{\beta}s_\beta – g s_\alpha}{l c_\beta} \\
\ddot{\beta} &= \frac{\ddot{y}c_\beta c_\alpha + \ddot{z}s_\beta c_\alpha^2 – l\dot{\alpha}^2 c_\beta c_\alpha s_\beta + \ddot{x}s_\alpha s_\beta c_\alpha + g s_\alpha^2 s_\beta – g s_\beta}{l c_\alpha}
\end{aligned}
$$
The efficiency of Kane’s method is evident when compared to traditional approaches. The table below summarizes the key differences in the modeling process for the quadrotor drone slung-load system.
| Modeling Method | Key Principle | Advantages for This System | Disadvantages for This System |
|---|---|---|---|
| Newton-Euler | Direct application of $\mathbf{F}=m\mathbf{a}$ and $\boldsymbol{\tau}=\mathbf{I}\dot{\boldsymbol{\omega}}+\boldsymbol{\omega}\times\mathbf{I}\boldsymbol{\omega}$ to each body, including internal constraint forces. | Physically intuitive; directly shows force couplings. | Requires explicit solution for cable tension force; derivation becomes cumbersome; equations are often complex. |
| Lagrangian | Based on energy principles. Uses kinetic ($T$) and potential ($V$) energy to form the Lagrangian $L=T-V$, then applies $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) – \frac{\partial L}{\partial q_i} = Q_i$. | Automatically eliminates internal constraint forces; systematic procedure. | Requires formulating $T$ and $V$ for the entire coupled system; involves computing partial derivatives leading to complex second-order equations. |
| Kane’s Method | Uses the concept of generalized active and inertia forces projected onto partial velocity directions: $F_r + F_r^* = 0$. | Eliminates internal forces without constructing energy functions; often leads to simpler first-order differential equations or less complex second-order forms; efficient for computer formulation. | Less physically intuitive than Newton-Euler; requires careful selection of generalized speeds and calculation of partial velocities. |
2. Design of the Anti-Swing Controller
2.1 Swing Angle-Position Compensation Strategy
The fundamental idea for active swing damping is to command the quadrotor drone to move in a way that applies a counteracting acceleration on the payload. The desired swing angles are zero: \(\alpha_{des}=0, \beta_{des}=0\). We design a compensation strategy where a position correction \(\Delta X\) is added to the quadrotor drone‘s current target position \(X_{act}\). This correction is generated based on the swing angle error and is designed to decay to zero as the swing is suppressed.
$$
\begin{aligned}
X_{des} &= X_{act} + \Delta X \\
\Delta X &= K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \dot{e}(t) \\
e(t) &= \xi_{des} – \xi_{act} \quad \text{for} \quad \xi = \alpha, \beta
\end{aligned}
$$
Here, \(K_p, K_i, K_d\) are the PID gains. For horizontal position control (\(x\) and \(y\)), \(\Delta X\) is calculated based on \(\beta\) and \(\alpha\) errors respectively, with appropriate sign conventions. This strategy effectively makes the quadrotor drone “chase” the swinging load to dampen its motion.
2.2 Adaptive PID Controller Based on Adam-Optimized Neural Network (Adam-NN-PID)
The performance of the above strategy critically depends on the PID gains. Fixed gains cannot optimally handle the nonlinear, coupled, and disturbance-prone nature of the system. We propose an adaptive controller where a neural network continuously tunes the PID gains online. To overcome the slow convergence of standard Backpropagation Neural Network PID (BPNN-PID), we integrate the Adam optimization algorithm into the weight update process.
The structure of the Adam-NN-PID controller is as follows: A three-layer BPNN (input, hidden, output) takes the swing angle error \(e(k)\) and its history as input. The output layer produces the three PID parameters \(K_p, K_i, K_d\). These parameters are then used in the compensation strategy (Eq. above). The network is trained online to minimize the swing angle error.
The key innovation lies in the weight update rule. Standard BPNN-PID often uses gradient descent with momentum:
$$
\omega^{(t)} = \omega^{(t-1)} + \eta \delta O^T + \varepsilon (\omega^{(t-1)} – \omega^{(t-2)})
$$
where \(\eta\) is a fixed learning rate and \(\varepsilon\) is the momentum factor. This fixed \(\eta\) can lead to slow or unstable convergence.
The Adam algorithm replaces this with an adaptive, per-parameter update. For each network weight \(\omega\), it computes estimates of the first moment (mean, \(m_t\)) and second moment (uncentered variance, \(v_t\)) of the gradients:
$$
\begin{aligned}
m_t &= \lambda_1 m_{t-1} + (1 – \lambda_1) g_t \\
v_t &= \lambda_2 v_{t-1} + (1 – \lambda_2) g_t^2
\end{aligned}
$$
where \(g_t\) is the current gradient, and \(\lambda_1, \lambda_2 \in [0,1)\) are decay rates. These estimates are biased towards zero initially, so bias-corrected versions are used:
$$
\begin{aligned}
\hat{m}_t &= \frac{m_t}{1 – \lambda_1^t} \\
\hat{v}_t &= \frac{v_t}{1 – \lambda_2^t}
\end{aligned}
$$
The weight update is then performed with an adaptive step size:
$$
\omega^{(t)} = \omega^{(t-1)} – \eta \cdot \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon}
$$
where \(\epsilon\) is a small constant to prevent division by zero. This algorithm automatically adjusts the learning rate for each weight based on the historical gradient information, leading to faster and more robust convergence compared to fixed-learning-rate methods. The complete algorithm flow for the Adam-NN-PID controller is summarized in the table below.
| Step | Action | Mathematical Description / Purpose |
|---|---|---|
| 1 | Initialization | Initialize network weights \(\mathbf{W}_{ih}, \mathbf{W}_{ho}\), Adam moment vectors \(\mathbf{m}=0, \mathbf{v}=0\). Set hyperparameters: learning rate \(\eta\), decay rates \(\lambda_1, \lambda_2\), constant \(\epsilon\). |
| 2 | Forward Pass | For current error \(e(k)\), calculate hidden layer output \(\mathbf{O}_h = f(\mathbf{W}_{ih} \cdot \mathbf{X}_{in})\), then output layer \(\mathbf{K} = [K_p, K_i, K_d]^T = g(\mathbf{W}_{ho} \cdot \mathbf{O}_h)\). |
| 3 | Control Law | Calculate position compensation \(\Delta X(k)\) using Eq. with the neural network outputs \(\mathbf{K}\). |
| 4 | Gradient Calculation | Compute performance index \(E(k) = \frac{1}{2}e(k)^2\). Use backpropagation through the network and the chain rule through the PID controller to find gradients \(\frac{\partial E}{\partial \mathbf{W}_{ho}}\) and \(\frac{\partial E}{\partial \mathbf{W}_{ih}}\). |
| 5 | Adam Weight Update | For each weight matrix (\(\mathbf{W}_{ho}\), \(\mathbf{W}_{ih}\)):
|
| 6 | Iteration | \(k \leftarrow k+1\). Return to Step 2 for the next control cycle. |
The Adam optimizer provides two main advantages over the momentum-based update in standard BPNN-PID: 1) Adaptive learning rates per parameter, allowing for faster progress in flat, consistent gradient directions and smaller steps in noisy, high-curvature regions. 2) More stable and efficient handling of sparse gradients, which is common in real-time control scenarios. This makes the Adam-NN-PID controller particularly suitable for the rapidly changing dynamics of a disturbed quadrotor drone slung-load system.
3. Simulation Analysis and Results
3.1 Simulation Setup and Scenarios
The proposed modeling and control framework is validated in a MATLAB/Simulink environment. The quadrotor drone and payload parameters used in simulation are listed in the table below. The quadrotor drone‘s attitude and position are controlled by cascaded PID loops. The anti-swing controller modifies the position setpoint as described.
| Parameter | Symbol | Value |
|---|---|---|
| Quadrotor Drone Mass | $m_b$ | 1.4 kg |
| Payload Mass | $m_p$ | 0.1 kg |
| Drone Inertia (diagonal) | $[I_x, I_y, I_z]$ | [0.0211, 0.0219, 0.0366] kg·m² |
| Cable Length | $l$ | 1.0 m |
| Gravitational Acceleration | $g$ | 9.81 m/s² |
To comprehensively evaluate performance, five simulation cases are designed, as outlined in the following table. The goal is to isolate and analyze the effects of the payload, the anti-swing strategy, and the different controller types.
| Case | Payload | External Wind Disturbance | Anti-Swing Control Active | Anti-Swing Controller Type | Primary Evaluation Purpose |
|---|---|---|---|---|---|
| Case 1 | No | No | No | – | Baseline performance of the quadrotor drone alone. |
| Case 2 | Yes | No | No | – | Impact of an uncontrolled swinging payload on drone dynamics. |
| Case 3 | Yes | Yes | Yes | Conventional PID | Comparative performance of different anti-swing controllers under disturbances. |
| Case 4 | Yes | Yes | Yes | Standard BPNN-PID | |
| Case 5 | Yes | Yes | Yes | Proposed Adam-NN-PID |
The quadrotor drone is commanded to perform a maneuver: take off to \(z=10\) m, then move to \((x,y)=(6,6)\) m. After the maneuver (post \(t=15\)s), the system is subjected to various wind disturbances to test the anti-swing controller’s robustness: 1) A short-duration step gust, 2) A sinusoidal wind gust, and 3) Continuous random turbulence.
3.2 Control Performance Under Disturbances
The swing angle responses (\(\alpha\) and \(\beta\)) for Cases 3-5 under the different wind disturbances are the key metrics. The results demonstrate the effectiveness of the anti-swing strategy and the superiority of the Adam-NN-PID controller.
1. Step Gust Disturbance: A step wind gust of amplitude 0.3 rad is applied to the swing angles at \(t=40\) s for 1 second. The Adam-NN-PID controller brings the swing back to the \(\pm0.02\) rad band significantly faster than both the conventional PID and the standard BPNN-PID.
2. Sinusoidal Gust Disturbance: A sinusoidal wind disturbance with amplitude 0.3 rad and period \(2\pi\) seconds is applied from \(t=40\) s to \(t=46\) s. The adaptive controllers (BPNN-PID and Adam-NN-PID) handle this periodic perturbation much better than fixed-gain PID. The Adam-NN-PID shows the quickest stabilization once the disturbance ends.
3. Random Turbulence Disturbance: Continuous, random turbulence is applied throughout the flight to simulate a realistic windy environment. The Adam-NN-PID controller maintains the smallest oscillation amplitude and demonstrates superior disturbance rejection throughout the flight, highlighting its adaptability to persistent, unstructured disturbances.
3.3 Quantitative Performance Comparison
The performance is quantified using two metrics: 1) Settling Time (\(T_s\)): Time for the swing angle to enter and remain within the \(\pm0.02\) rad band. 2) Root Mean Square (RMS) of Swing Angle: Calculated over a relevant time interval to measure oscillation magnitude. Lower values for both metrics indicate better performance.
The quantitative results for the swing angle \(\alpha\) under different disturbances are consolidated in the table below. The data clearly shows the progression of improvement from conventional PID to BPNN-PID to the proposed Adam-NN-PID.
| Disturbance Type | Controller | Settling Time \(T_s\) (s) [Post-40s] | RMS(\(\alpha\)) \(\times 10^{-3}\) [40-70s] | Improvement in \(T_s\) vs. PID |
|---|---|---|---|---|
| Step Gust | PID | 11.75 | 51.16 | – |
| BPNN-PID | 2.17 | 43.45 | 81.5% faster | |
| Adam-NN-PID | 2.07 | 43.39 | 82.4% faster | |
| Sinusoidal Gust | PID | 10.97 | 49.84 | – |
| BPNN-PID | 7.17 | 45.18 | 34.6% faster | |
| Adam-NN-PID | 6.07 | 46.29 | 44.7% faster | |
| Random Turbulence | PID | 12.37 | 60.81 | – |
| BPNN-PID | 8.03 | 44.99 | 35.1% faster | |
| Adam-NN-PID | 6.39 | 44.47 | 48.3% faster |
A similar trend is observed for the swing angle \(\beta\). The consistent outperformance of Adam-NN-PID, especially in settling time, validates that the Adam optimizer’s adaptive learning rate mechanism enables faster and more effective online tuning of the neural network, leading to quicker controller adaptation and superior swing damping for the quadrotor drone.
3.4 Hyperparameter Sensitivity Analysis for Adam-NN-PID
The performance of the Adam optimizer depends on its hyperparameters, primarily the decay rates \(\lambda_1\) and \(\lambda_2\). A sensitivity analysis was conducted under turbulent wind conditions. The results indicate that the typical recommended values of \(\lambda_1=0.9\) and \(\lambda_2=0.999\) perform well for this application. Excessively low decay rates cause the moment estimates to be too noisy, leading to oscillatory updates. Excessively high decay rates cause the estimates to rely too heavily on the distant past, making the controller slow to adapt to new conditions. The chosen values provide a good balance, allowing rapid adaptation to changes while filtering out high-frequency noise in the gradient signals, which is crucial for stable real-time control of the quadrotor drone system.
4. Conclusion
This study addressed the dual challenges of efficient modeling and effective swing control for a quadrotor drone transporting a cable-suspended payload. The application of Kane’s method for dynamic modeling proved highly advantageous. It streamlined the derivation process by eliminating the need to solve for internal cable forces (as in Newton-Euler methods) and avoiding the construction and differentiation of complex energy functions (as in Lagrangian methods). The resulting model is both accurate and computationally efficient, providing a solid foundation for controller design and simulation.
To tackle the problem of persistent payload oscillation, a novel active anti-swing control scheme was proposed. A swing angle-to-position compensation strategy was integrated with an adaptive PID controller, where the PID gains are continuously tuned online by a neural network. The key innovation was enhancing this neural network with the Adam optimization algorithm for weight updates. The Adam-NN-PID controller overcomes the limitation of fixed learning rates in conventional BPNN-PID, enabling faster convergence and more robust adaptation to the system’s nonlinear dynamics and external disturbances.
Comprehensive simulations under various wind disturbance scenarios (step, sinusoidal, and random turbulence) demonstrated the superior performance of the proposed framework. Compared to both conventional PID and standard BPNN-PID controllers, the Adam-NN-PID controller achieved significantly faster swing stabilization and maintained smaller oscillation amplitudes. This confirms its effectiveness and robustness for improving the operational safety and efficiency of a quadrotor drone performing slung-load transportation tasks.
Future work will focus on extending the Kane’s method modeling approach to more complex cooperative systems, such as multiple quadrotor drones carrying a shared load. Furthermore, the integration of the adaptive anti-swing controller with more advanced trajectory planning and fault-tolerant control strategies will be explored to enhance the overall autonomy and reliability of aerial transportation systems.