In recent years, quadcopter unmanned aerial vehicles (UAVs) have gained significant attention due to their versatility in applications such as search and rescue, surveillance, and payload transportation. However, when a quadcopter carries a suspended load, the system becomes highly underactuated and coupled, leading to challenges in stability and control. The swing of the load introduces additional degrees of freedom, which can cause oscillations and reduce the efficiency of trajectory tracking. This paper addresses these issues by proposing a control strategy based on Linear Active Disturbance Rejection Control (LADRC) combined with an acceleration-based swing suppression algorithm. The LADRC compensates for external disturbances and model uncertainties, while the swing suppression algorithm plans trajectories to minimize load oscillations. Simulation results demonstrate the effectiveness of this approach in achieving precise trajectory tracking and swing reduction for a quadcopter with a suspended load.
The dynamics of a quadcopter with a suspended load are complex, involving eight degrees of freedom and four control inputs. The system is modeled under assumptions such as rigid body dynamics, constant gravity, and ideal cable connections. The position of the load is derived from the quadcopter’s coordinates and the cable length, leading to nonlinear equations of motion. The key variables include the quadcopter’s position (x, y, z), Euler angles (φ, θ, ψ), and the load’s swing angles (α, β). The equations of motion are derived using Lagrangian mechanics, resulting in a set of differential equations that describe the coupled behavior of the quadcopter and load.
The general form of the dynamics can be expressed as:
$$ \ddot{x} = \frac{1}{M + m} \left[ U_{1x} + mL \left( \dot{\beta}^2 \sin \beta – \ddot{\beta} \cos \beta \right) \right] $$
$$ \ddot{y} = \frac{1}{M + m} \left[ U_{1y} + mL \left( \dot{\beta} \sin \alpha \sin \beta – \ddot{\alpha} \cos \beta \cos \alpha + \dot{\beta}^2 \cos \beta \sin \alpha + \dot{\alpha}^2 \sin \alpha \cos \beta + 2 \dot{\alpha} \dot{\beta} \cos \alpha \sin \beta \right) \right] $$
$$ \ddot{z} = \frac{1}{M + m} \left[ U_{1z} – mL \left( \ddot{\beta} \cos \alpha \sin \beta + \ddot{\alpha} \sin \alpha \cos \beta + \dot{\beta}^2 \cos \alpha \cos \beta + \dot{\alpha}^2 \cos \alpha \cos \beta – 2 \dot{\alpha} \dot{\beta} \sin \alpha \sin \beta \right) \right] – g $$
where M is the mass of the quadcopter, m is the load mass, L is the cable length, and U_{1x}, U_{1y}, U_{1z} are the control inputs derived from the total thrust and orientation. The swing angles α and β evolve according to:
$$ \ddot{\alpha} = \frac{1}{M L \cos \beta} \left[ 2 M L \sin \beta \dot{\beta} \dot{\alpha} – \cos \alpha U_{1y} – \sin \alpha U_{1z} \right] $$
$$ \ddot{\beta} = \frac{1}{M L} \left[ -M L \sin \beta \cos \beta \dot{\alpha}^2 – \cos \beta U_{1x} – \sin \beta \sin \alpha U_{1y} – \sin \beta \cos \alpha U_{1z} \right] $$
These equations highlight the underactuated nature of the system, as the swing angles cannot be directly controlled but are influenced by the quadcopter’s motion.
To design the controller, the system parameters are summarized in the following table:
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Quadcopter Mass | M | 1.5 | kg |
| Load Mass | m | 0.5 | kg |
| Gravity | g | 9.8 | m/s² |
| Cable Length | L | 1.0 | m |
| Moment of Inertia (X) | J_x | 1.745e-2 | kg·m² |
| Moment of Inertia (Y) | J_y | 1.745e-2 | kg·m² |
| Moment of Inertia (Z) | J_z | 3.175e-2 | kg·m² |
The LADRC approach is employed for both position and attitude control of the quadcopter. The core idea is to use an Extended State Observer (ESO) to estimate total disturbances, including load swing effects and external forces, and then compensate for them in the control law. For a second-order system, the state-space representation is:
$$ \dot{x} = A x + B u + E f $$
$$ y = C x $$
where x represents the state variables, u is the control input, and f denotes the total disturbance. The ESO is designed as:
$$ \dot{z} = (A – L C) z + B u + L y $$
$$ \hat{y} = C z $$
with L being the observer gain matrix, tuned based on the bandwidth ω₀. For the position control, the virtual control inputs U_{1x}, U_{1y}, U_{1z} are derived from the desired accelerations. The control law for each channel is:
$$ u = \frac{ k_1 (z_1 – y_r) + k_2 z_2 – z_3 }{ b } $$
where z₁, z₂, z₃ are the estimated states from the ESO, y_r is the reference signal, and k₁, k₂ are controller gains determined by the desired closed-loop bandwidth ω_c. Similarly, for attitude control, the roll, pitch, and yaw channels use analogous LADRC structures to ensure stability and disturbance rejection.
To suppress the load swing, an acceleration-based planning algorithm is integrated. The desired accelerations are modified as:
$$ \ddot{x}_d(t) = \ddot{x}_t(t) – k \left( \dot{\beta} \sin \alpha \sin \beta – \dot{\alpha} \cos \alpha \cos \beta \right) $$
$$ \ddot{y}_d(t) = \ddot{y}_t(t) – k \dot{\beta} \cos \beta $$
$$ \ddot{z}_d(t) = \ddot{z}_t(t) – k \left( \dot{\beta} \cos \alpha \sin \beta + \dot{\alpha} \sin \alpha \cos \beta \right) $$
where k is a positive gain, and \ddot{x}_t, \ddot{y}_t, \ddot{z}_t are the target accelerations. This formulation ensures that the Lyapunov function V(t) for the swing energy decreases over time, promoting swing damping. The planned trajectories are obtained by integrating these accelerations:
$$ x_d(t) = x_t(t) + k \int_0^t \sin \alpha \cos \beta \, d\tau $$
$$ y_d(t) = y_t(t) – k \int_0^t \sin \beta \, d\tau $$
$$ z_d(t) = z_t(t) + k \int_0^t \cos \alpha \cos \beta \, d\tau – k t $$
This approach allows the quadcopter to follow a reference path while actively reducing load oscillations.
Simulations are conducted in MATLAB/Simulink to validate the proposed control strategy. Two scenarios are considered: point-to-point motion with disturbance rejection and trajectory tracking under time-varying disturbances. The performance is evaluated based on position tracking accuracy and swing angle reduction. For the first scenario, the quadcopter starts at (0, 0, 0) and rises to (0, 0, 5), then moves to (5, 5, 5). At t = 15 s, disturbances D_x = D_y = D_z = 1 N are applied. The results show that with LADRC alone, the position overshoots are up to 6.3% in x and y directions, and the settling time is around 4.5 s. With the swing suppression algorithm, overshoots reduce to 1.4%, and settling time decreases to 2.8 s, with no steady-state error.
The swing angles α and β are critical indicators of performance. Without suppression, the maximum swing angle reaches 17.5°, with persistent oscillations. With the algorithm, swings are limited to 6° and damped quickly. The root mean square error (RMSE) of the swing angles, defined as RMSE = √(e_α² + e_β²), is significantly lower when swing suppression is active, as shown in the following table:
| Control Method | Max Swing Angle (°) | Settling Time (s) | RMSE |
|---|---|---|---|
| LADRC Only | 17.5 | 4.5 | 0.85 |
| LADRC with Swing Suppression | 6.0 | 2.8 | 0.12 |
In the trajectory tracking scenario, the quadcopter follows a 3D spiral path defined by x = 2 cos(πt/10), y = -2 sin(πt/10) + 2, and z = 0.1t, with time-varying disturbances added. The LADRC with swing suppression achieves smoother tracking and smaller swing oscillations compared to LADRC alone. The swing angles remain bounded within 5°, and the RMSE is reduced by over 70%, demonstrating the robustness of the proposed method.
The effectiveness of the control design is further analyzed through the stability properties. The closed-loop system with LADRC ensures that the estimation error of the ESO converges exponentially, and the swing suppression algorithm guarantees that the Lyapunov function V(t) decreases monotonically. This combined approach enhances the overall performance of the quadcopter in handling suspended loads.
In conclusion, this paper presents a comprehensive control framework for a quadcopter with a suspended load, integrating LADRC for disturbance rejection and an acceleration-based algorithm for swing suppression. The simulations confirm that the method improves trajectory tracking accuracy and reduces load oscillations, even in the presence of external disturbances. Future work will focus on experimental validation and extending the approach to multi-quadcopter systems for cooperative load transportation.
The proposed strategy highlights the potential of advanced control techniques in enhancing the capabilities of quadcopter UAVs in complex applications. By addressing the challenges of underactuation and coupling, this research contributes to the development of more reliable and efficient aerial robotic systems.