Quadrotor unmanned aerial vehicles (UAVs) have gained significant attention in various applications, such as logistics and transportation, due to their ability to perform vertical take-off, landing, and hovering. However, when a quadrotor is used for suspended load transportation, external disturbances like wind can induce oscillations in both the quadrotor and the payload, leading to instability. This paper addresses the challenge of suppressing payload swing in quadrotor-based suspension systems by designing an active anti-swing control system. The system leverages high-precision attitude estimation derived from sensor data, which serves as input to the control algorithm. A dynamic model of the suspension system is established based on kinematic and dynamic principles, and a variable coefficient proportional-integral-derivative (PID) control algorithm is proposed to enhance performance. Experimental results demonstrate the effectiveness of this approach compared to traditional PID control.
The suspension system involves a quadrotor carrying a payload attached via a cable, which can be modeled as a pendulum. The world coordinate system is fixed to the quadrotor, denoted as \( o_e x_e y_e z_e \), while the suspension system coordinate system is \( o_b x_b y_b z_b \). The relationship between these frames is illustrated in the figure below, which shows the geometric configuration of the quadrotor and suspended load.
In this setup, the swing angles of the payload are represented by \( \alpha \) and \( \beta \), where \( \alpha \) is the angle between the projection of the cable onto the \( o_e x_e z_e \) plane and the negative \( z_e \)-axis, and \( \beta \) is the angle between the cable and the \( o_e x_e z_e \) plane. The position of the payload \( (x_L, y_L, z_L) \) is related to the quadrotor’s position \( (x_U, y_U, z_U) \) and the cable length \( l \) by the following equations:
$$ x_L = x_U + l \sin \alpha \cos \beta $$
$$ y_L = y_U + l \sin \beta $$
$$ z_L = z_U – l \cos \alpha \cos \beta $$
The potential energy \( V \) and kinetic energy \( E \) of the system are given by:
$$ V = m_U g z_U + m_L g z_L $$
$$ E = \frac{1}{2} m_U (\dot{x}_U^2 + \dot{y}_U^2 + \dot{z}_U^2) + \frac{1}{2} m_L (\dot{x}_L^2 + \dot{y}_L^2 + \dot{z}_L^2) $$
where \( m_U \) is the mass of the quadrotor, \( m_L \) is the mass of the payload, and \( g \) is the gravitational acceleration. Using the Lagrangian \( L = E – V \), the derivatives of the payload position can be derived, leading to acceleration terms that describe the system’s dynamics. For instance, the velocity components are:
$$ \dot{x}_L = \dot{x}_U + \dot{l} \sin \alpha \cos \beta + l (\dot{\alpha} \cos \alpha \cos \beta – \dot{\beta} \sin \alpha \sin \beta) $$
$$ \dot{y}_L = \dot{y}_U + \dot{l} \sin \beta + l \dot{\beta} \cos \beta $$
$$ \dot{z}_L = \dot{z}_U – \dot{l} \cos \alpha \cos \beta + l (\dot{\alpha} \cos \alpha \cos \beta + \dot{\beta} \cos \alpha \sin \beta) $$
Further differentiation yields the acceleration, which is essential for control design. The overall control system structure integrates a swing observer that monitors the payload’s swing angles, computes the swing period \( T_s \) and phase \( \phi \), and updates the coefficients in the attitude controller. The attitude is estimated using data from gyroscopes, accelerometers, and magnetometers, resulting in a quaternion representation that is converted to Euler angles for control purposes.
The control algorithm employs a variable coefficient PID strategy to adapt to changing swing dynamics. The discrete-time control output \( u(k) \) is expressed as:
$$ u(k) = K_{pa} e_a(k) + K_{ia} \sum_{n=1}^{k} e_a(k) – K_{pg} e_g(k) $$
where \( e_a(k) \) is the angle error, \( e_g(k) \) is the angular velocity error, \( K_{pa} \) is the proportional coefficient for angle control, \( K_{ia} \) is the integral coefficient, and \( K_{pg} \) is the proportional coefficient for angular velocity control. To enhance performance, the coefficients are varied based on the swing period and current phase. Specifically, \( K_{pa} \) is adjusted as a linear function of the swing period \( T_s \), and \( K_{pg} \) is modified according to the phase. The updated control law becomes:
$$ u(k) = P_s T_s e_a(k) + K_{ia} \sum_{n=1}^{k} e_a(k) – p(t) \Phi e_g(k) $$
where \( P_s \) is a tuning coefficient, \( p(t) \) is a correction factor for the angle control, and \( \Phi \) is a correction factor for the angular velocity control. This variable coefficient approach allows the quadrotor to respond more effectively to oscillations compared to fixed-gain PID control.
Attitude estimation is critical for the control system. The quaternion \( q \) representing the orientation is derived from sensor data. The acceleration vector \( a = [a_x, a_y, a_z]^T \) is used to form a quaternion \( q_a \), the angular velocity \( \omega \) from the gyroscope gives the quaternion increment \( \Delta q_\omega \), and the magnetic field vector \( m = [m_x, m_y, m_z]^T \) provides \( q_m \). The quaternion update is performed using:
$$ q_{\text{new}} = q_{\text{old}} \otimes (\Delta q_\omega \otimes q_{\text{old}} + q_a \Delta q_m) $$
where \( \otimes \) denotes quaternion multiplication. The Euler angles \( (\gamma, \phi, \psi) \) are then extracted from the quaternion for control:
$$ \gamma = \text{atan2}[2(q_r q_x + q_y q_z), 1 – 2(q_x^2 + q_y^2)] $$
$$ \phi = \arcsin[2(q_r q_y – q_z q_x)] $$
$$ \psi = \text{atan2}[2(q_r q_z + q_x q_y), 1 – 2(q_y^2 + q_z^2)] $$
To validate the control system, an experimental platform was developed, consisting of a main control module, attitude measurement module, power module, and motor drive module. The hardware was designed with lightweight materials to minimize mass while maintaining rigidity. Tests were conducted in both indoor and outdoor environments under various conditions, such as different cable lengths and wind speeds. The performance of the variable coefficient PID control was compared to traditional PID control in terms of stabilization time.
The following table summarizes the stabilization times for different swing angles and cable lengths, highlighting the improvement offered by the variable coefficient approach:
| Cable Length (m) | Traditional PID Stabilization Time (s) at 10° | Traditional PID Stabilization Time (s) at 30° | Traditional PID Stabilization Time (s) at 45° | Variable Coefficient PID Stabilization Time (s) at 10° | Variable Coefficient PID Stabilization Time (s) at 30° | Variable Coefficient PID Stabilization Time (s) at 45° |
|---|---|---|---|---|---|---|
| 0.5 | 4.2 | 6.2 | 9.3 | 3.6 | 4.8 | 7.8 |
| 1.0 | 5.1 | 7.1 | 11.2 | 4.4 | 6.1 | 9.0 |
| 1.5 | 6.4 | 8.6 | 12.9 | 5.7 | 7.4 | 10.2 |
| 10.0 | 18.6 | 21.2 | 24.4 | 13.3 | 17.6 | 18.9 |
For example, with a cable length of 0.5 m and an initial swing angle of 30°, the variable coefficient PID reduced the stabilization time by 1.4 seconds compared to traditional PID. In outdoor tests with wind speeds of 4 m/s and a cable length of 10.0 m, the active anti-swing system achieved stabilization in approximately 18.9 seconds, whereas without control, it took about 60 seconds. This demonstrates the system’s effectiveness in real-world conditions.
Additionally, drift calibration was performed to improve sensor accuracy. The calibrated attitude \( Y_c \) is obtained by subtracting the drift constant \( ZD_{ri} \) from the raw measurement \( Y_r \):
$$ Y_c = Y_r – ZD_{ri} $$
This calibration reduced angular drift to 0.26° per minute, which is 1/14 of the uncalibrated drift, ensuring more reliable attitude estimates for the quadrotor control.
The variable coefficient PID control algorithm was fine-tuned using a bisection method to optimize parameters. The final control output for a swing period of 4.8 s and an initial angle of 30° is given by:
$$ u(k) = (0.66 T_s – 40) e_a(k) + 4.8 \sum_{n=1}^{k} e_a(k) – 8000 \sin\left(2\pi \frac{t_0}{T_s}\right) e_g(k) $$
where \( t_0 \) is the start time. This formulation allows the quadrotor to adaptively dampen oscillations based on real-time swing characteristics.
In conclusion, the active anti-swing control system for quadrotor UAV suspension systems effectively suppresses payload swing through a combination of dynamic modeling and variable coefficient PID control. The integration of attitude estimation and adaptive control coefficients results in faster stabilization times and improved performance under varying conditions. Future work could focus on automating parameter tuning and extending the system to handle more complex payload dynamics. The proposed approach enhances the reliability and efficiency of quadrotor-based transportation, making it suitable for practical applications in logistics and beyond.