In recent years, the application of quadrotor unmanned aerial vehicles (UAVs) has expanded significantly, particularly in payload transportation tasks. Among various payload-carrying methods, slung-load systems, where a payload is suspended via a cable beneath the quadrotor, offer simplicity and cost-effectiveness. However, the dynamic coupling between the quadrotor, cable, and payload, combined with external disturbances like wind and time-varying payload masses, poses significant challenges for accurate modeling and robust control. Traditional modeling approaches, such as Newton-Euler and Lagrangian methods, often involve complex steps, including explicit analysis of constraint forces or construction of energy functions with second-order derivatives, which can be computationally intensive and less efficient. To address these issues, this paper proposes a dynamics modeling approach based on Kane’s method, which simplifies the process by combining forces and partial velocities, eliminating the need for constraint force analysis or Lagrangian function computations. Additionally, to enhance control performance under composite disturbances—such as time-varying payload oscillations and complex wind fields—an adaptive learning rate model reference adaptive control (MRAC) strategy is introduced. This controller integrates the Adam optimization algorithm to dynamically adjust learning rates and control parameters, improving convergence speed and overshoot suppression compared to conventional methods. The effectiveness of the proposed modeling and control techniques is validated through simulations, demonstrating superior performance in handling nonlinear disturbances and ensuring stable quadrotor operation.
The quadrotor slung-load system consists of a rigid quadrotor body, a massless cable, and a time-varying payload modeled as a point mass. The system is subject to assumptions: the quadrotor is symmetric and rigid, the cable remains taut and length-invariant, the payload is always below the quadrotor, and the quadrotor’s center of mass coincides with the cable attachment point. The dynamic behavior involves 8 degrees of freedom (DOF): 6 DOF for the quadrotor’s position and orientation, and 2 DOF for the payload’s swing angles due to the cable constraint. To model this system, Kane’s method is employed, which utilizes generalized coordinates and speeds to derive equations of motion efficiently. The generalized coordinates are defined as:
$$ q = [\phi, \theta, \psi, x, y, z, \alpha, \beta]^T $$
where $\phi$, $\theta$, and $\psi$ are the roll, pitch, and yaw angles of the quadrotor; $x$, $y$, and $z$ represent the quadrotor’s position in the inertial frame; and $\alpha$ and $\beta$ denote the payload’s swing angles in the longitudinal and lateral directions, respectively. The generalized speeds are selected as:
$$ u = [\omega_x, \omega_y, \omega_z, v_x, v_y, v_z, \dot{\alpha}, \dot{\beta}]^T $$
where $\omega_x$, $\omega_y$, and $\omega_z$ are the angular velocities of the quadrotor, and $v_x$, $v_y$, and $v_z$ are its linear velocities. The relationship between generalized coordinates and speeds is derived through kinematic equations. For instance, the angular velocity of the quadrotor can be expressed as:
$$ \omega_B = \omega_x \mathbf{b}_1 + \omega_y \mathbf{b}_2 + \omega_z \mathbf{b}_3 $$
where $\mathbf{b}_1$, $\mathbf{b}_2$, and $\mathbf{b}_3$ are unit vectors along the body-fixed axes. The position of the payload $P$ relative to the quadrotor’s center of mass is given by:
$$ \begin{bmatrix} x_p \\ y_p \\ z_p \end{bmatrix} = \begin{bmatrix} x + l \sin \alpha \cos \beta \\ y – l \sin \beta \\ z + l \cos \alpha \cos \beta \end{bmatrix} $$
where $l$ is the cable length. The partial velocities, which are crucial in Kane’s method, are computed as the derivatives of the velocities with respect to the generalized speeds. For the quadrotor and payload, these are derived to form the basis for the dynamic equations.
Kane’s equations state that the sum of generalized active forces and generalized inertia forces for each generalized speed is zero:
$$ (F_r + F_r^*) \cdot \mathbf{v}_r = 0, \quad r = 1, 2, \dots, 8 $$
where $F_r$ represents the generalized active forces (e.g., thrust and gravity), and $F_r^*$ represents the generalized inertia forces. The active forces include the thrust generated by the quadrotor’s rotors and gravitational forces. The thrust and moments are defined as:
$$ \mathbf{R} = U_x \mathbf{b}_1 + U_y \mathbf{b}_2 + U_z \mathbf{b}_3 $$
$$ \mathbf{T} = \tau_x \mathbf{b}_1 + \tau_y \mathbf{b}_2 + \tau_z \mathbf{b}_3 $$
where $U_x$, $U_y$, and $U_z$ are the components of the rotor thrust, and $\tau_x$, $\tau_y$, and $\tau_z$ are the moments about the body axes. The gravitational forces on the quadrotor and payload contribute to the generalized active forces. The inertia forces are derived from the accelerations of the quadrotor and payload. For example, the quadrotor’s linear acceleration and the payload’s acceleration are computed using time derivatives of their velocities. The resulting dynamic equations for the system are as follows:
For the quadrotor’s translational motion:
$$ (m_q + m_p) \ddot{x} + m_p l (\ddot{\alpha} \cos \beta \cos \alpha – \ddot{\beta} \sin \beta \sin \alpha – \dot{\alpha}^2 \cos \beta \sin \alpha – \dot{\beta}^2 \cos \beta \sin \alpha – 2 \dot{\alpha} \dot{\beta} \sin \beta \cos \alpha) = U_x + d_x $$
$$ (m_q + m_p) \ddot{y} + m_p l (\ddot{\beta} \sin \beta – \ddot{\beta} \cos \beta) = U_y + d_y $$
$$ (m_q + m_p) \ddot{z} + m_p l (2 \dot{\alpha} \dot{\beta} \sin \beta \sin \alpha – \ddot{\alpha} \cos \beta \sin \alpha – \ddot{\beta} \sin \beta \cos \alpha – \dot{\alpha}^2 \cos \beta \cos \alpha – \dot{\beta}^2 \cos \beta \cos \alpha) = (m_q + m_p)g + U_z + d_z $$
For the quadrotor’s rotational motion:
$$ I_x \dot{\omega}_x + (I_z – I_y) \omega_y \omega_z = \tau_x + d_\phi $$
$$ I_y \dot{\omega}_y + (I_x – I_z) \omega_x \omega_z = \tau_y + d_\theta $$
$$ I_z \dot{\omega}_z + (I_y – I_x) \omega_x \omega_y = \tau_z + d_\psi $$
For the payload’s swing dynamics:
$$ \ddot{\alpha} = \frac{1}{l \cos \beta} (\ddot{z} \sin \alpha – \ddot{x} \cos \alpha + 2 l \dot{\alpha} \dot{\beta} \sin \beta – g \sin \alpha) $$
$$ \ddot{\beta} = \frac{1}{l \cos \alpha} (\ddot{y} \cos \beta \cos \alpha + \ddot{z} \sin \beta \cos^2 \alpha – l \dot{\alpha}^2 \cos \beta \cos \alpha \sin \beta + \ddot{x} \sin \alpha \sin \beta \cos \alpha + g \sin^2 \alpha \sin \beta – g \sin \beta) $$
In these equations, $m_q$ and $m_p$ are the masses of the quadrotor and payload, $I_x$, $I_y$, and $I_z$ are the moments of inertia, $g$ is gravity, and $d_x$, $d_y$, $d_z$, $d_\phi$, $d_\theta$, $d_\psi$ represent external disturbances such as wind. The time-varying nature of the payload mass $m_p$ adds nonlinearity to the system, requiring adaptive control strategies.
The control of a quadrotor with a slung time-varying payload demands robustness against disturbances and parameter variations. Traditional PID controllers, while simple, often lack adaptability to changing conditions and require manual tuning. Model reference adaptive control (MRAC) offers a solution by adjusting controller parameters online to minimize tracking error between the system output and a reference model. However, conventional MRAC uses fixed learning rates, which can lead to slow convergence or oscillations under abrupt changes. To overcome this, an adaptive learning rate MRAC based on the Adam optimization algorithm is proposed. This approach dynamically adjusts both the control gains and the learning rate, enhancing performance in the presence of composite disturbances.
The MRAC framework involves a reference model that defines desired system behavior, and an adaptive law that updates controller parameters. For a quadrotor, the control input for attitude stabilization can be designed as a PID structure with adaptive gains. The tracking error is defined as:
$$ e = y_{\text{ref}} – y_{\text{act}} $$
where $y_{\text{ref}}$ is the reference model output and $y_{\text{act}}$ is the actual system output. The parameter update law in conventional MRAC is:
$$ k_{\text{new}} = \eta e + k_{\text{old}} $$
where $\eta$ is a fixed learning rate and $k_{\text{old}}$ is the previous gain. The control input is computed as $u = k_{\text{new}} e$. While this adapts to errors, the fixed $\eta$ can cause issues like overshoot or slow response. The Adam-MRAC improves this by incorporating momentum and adaptive learning rates from the Adam algorithm. The update rules for Adam are:
$$ m_t = \lambda_1 m_{t-1} + (1 – \lambda_1) g_t $$
$$ v_t = \lambda_2 v_{t-1} + (1 – \lambda_2) g_t^2 $$
$$ \hat{m}_t = \frac{m_t}{1 – \lambda_1^t} $$
$$ \hat{v}_t = \frac{v_t}{1 – \lambda_2^t} $$
$$ \Delta \omega_t = \frac{\eta \hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} $$
where $g_t$ is the gradient of the error, $\lambda_1$ and $\lambda_2$ are decay rates, $\eta$ is the initial learning rate, and $\epsilon$ is a small constant for numerical stability. In Adam-MRAC, the learning rate becomes $\eta / (\sqrt{\hat{v}_t} + \epsilon)$, allowing it to decrease for large error fluctuations to prevent oscillations and increase for small errors to accelerate convergence. This results in a control law that adapts both gains and learning rates, providing better disturbance rejection for the quadrotor system.
To validate the proposed modeling and control approaches, simulations are conducted in MATLAB/Simulink. The quadrotor parameters are listed in Table 1, and the payload mass varies exponentially with time as $m_p = 0.5 e^{-0.05t}$ to simulate time-varying effects. Complex wind disturbances are applied as forces and moments in the x and y directions, representing realistic environmental conditions. Four cases are designed to analyze system behavior and controller performance, as summarized in Table 2.
| Parameter | Value |
|---|---|
| Quadrotor mass, $m_q$ (kg) | 1.4 |
| Payload mass, $m_p$ (kg) | 0.5 (time-varying) |
| Moment of inertia, $I_x$, $I_y$, $I_z$ (kg·m²) | diag(0.0211, 0.0219, 0.0366) |
| Cable length, $l$ (m) | 1.0 |
| Gravity, $g$ (m/s²) | 9.8 |
| Case | Time-Varying Payload | Wind Disturbance | Controller |
|---|---|---|---|
| 1 | No | No | PID |
| 2 | Yes | Yes | PID |
| 3 | Yes | Yes | MRAC |
| 4 | Yes | Yes | Adam-MRAC |
In Case 1, the quadrotor follows a trajectory: taking off along the z-axis at t=0 s, moving along the x-axis at t=5 s, and along the y-axis at t=10 s, before hovering at a target position. This baseline case with no disturbances shows stable performance with PID control. In Case 2, time-varying payload and wind disturbances are introduced, causing oscillations in roll and pitch angles due to the coupled dynamics. For instance, the pitch angle $\theta$ experiences deviations after 10 s, primarily from payload swings, while roll angle $\phi$ is affected after 15 s due to wind moments. This highlights the need for adaptive control.
Cases 3 and 4 compare MRAC and Adam-MRAC controllers under the same disturbances. The MRAC controller adapts gains online without manual tuning, reducing errors compared to PID. However, Adam-MRAC demonstrates faster convergence and lower overshoot, as the adaptive learning rate optimizes parameter updates. For example, in attitude control, the roll and pitch angles stabilize more quickly with Adam-MRAC, and the tracking error in position decreases rapidly. The quadrotor’s ability to reject composite disturbances is enhanced, ensuring robust operation in challenging environments.
The simulation results confirm that Kane’s method provides an efficient modeling framework for the quadrotor slung-load system, avoiding the complexities of traditional approaches. The Adam-MRAC controller outperforms both PID and conventional MRAC in handling time-varying payloads and wind disturbances, offering improved stability and convergence. This combination of simplified modeling and advanced control contributes to the reliability of quadrotor UAVs in practical applications, such as cargo delivery and surveillance, where payload variations and environmental factors are common.
In conclusion, this work addresses the challenges of modeling and controlling quadrotor UAVs with slung time-varying payloads. Kane’s method streamlines the dynamics modeling process by leveraging generalized speeds and partial velocities, reducing computational effort while maintaining accuracy. The proposed Adam-MRAC controller enhances adaptive capabilities through dynamic learning rate adjustment, providing superior disturbance rejection and convergence performance. Future research could explore real-time implementation and extension to multi-quadrotor systems for collaborative payload transport. The methodologies presented here offer a foundation for developing more resilient and efficient quadrotor-based systems in dynamic environments.