## 1 Introduction  
Bridge cranes play an indispensable role in modern factories as efficient handling equipment. Their flexibility, efficiency, and low impact on the floor and surrounding equipment make them essential handling tools in various types of factories. However, most industrial cranes still rely on manual operation, which often results in low efficiency and poor precision and can even cause the crane to swing due to the influence of the lifting mechanisms. This swinging can cause difficulties for operators in terms of system automation, thereby prolonging task completion times and reducing production efficiency. Therefore, anti-swing control of crane payloads and automatic trolley positioning technology have become key factors in improving the automation level of crane lifting operations. In recent years, with the continuous improvement in industrial automation levels, the intelligent anti-swaying and positioning control of bridge crane systems has garnered significant attention and numerous control methods have been rapidly developed. These methods may differ in their basic objectives, but they can generally be categorized into two main types: closed-loop control [1; 2; 3] and open-loop control [4]. Open-loop control is relatively simple, as it only requires consideration of the input control signals without concerning the position of the crane trolley or the angle of the payload sway. Input shaping [5] is an open-loop technique used in real-time applications and has been widely applied in anti-swaying control for cranes. This technique achieves the reduction in system vibrations by convolving the command input signal with a series of pulses designed based on the system's inherent frequency and damping ratio. Its fundamental principle lies in effectively suppressing vibrations that may occur within the crane system through meticulously designed pulse sequences, thereby enhancing system stability and control performance. The application of this method not only reduces vibration amplitudes but also improves the smoothness and precision of crane operation, thereby enhancing the efficiency and safety of the handling process. To address the impact of payload hoisting and the uncertainty of parameters, [PERSON] and [PERSON] and [PERSON] and [PERSON] (2018) proposed an adaptive input shaping method based on flexible modal frequency variations. This method aims to cope with parameter uncertainties. Although input shaping techniques have achieved certain effectiveness in the anti-swaying control of cranes, their control targets are ideally determined using model parameters. Therefore, their performance may not be ideal when encountering external disturbances. [PERSON] et al. [PERSON] et al. (2018, 2020) employed command smoothing techniques, coupled with dual pendulums and distributed mass payloads, to suppress oscillations in bridge cranes. [PERSON] et al. [PERSON] et al. (2020) proposed an enhanced adaptive unscented Kalman filter. Simulations and practical experiments on a small-scale bridge crane system demonstrated that the proposed method is suitable for estimating payload swaying angles and friction forces. Compared to open-loop control, closed-loop control allows for real-time and precise monitoring of both the payload swing angle and the position of the trolley during crane motion, enabling feedback control based on the acquired information. Therefore, in modern technological developments, the application of closed-loop control in crane sway suppression is becoming increasingly widespread. [PERSON] and [PERSON] and [PERSON] (2020) employed the Linear Quadratic Regulator (LQR) method for the anti-sway control of overhead cranes. [PERSON] et al. [PERSON] et al. (2020) designed a parallel distributed fuzzy LQR controller combined with genetic algorithms for the anti-sway control of double-pendulum overhead cranes. [PERSON] et al. [PERSON] et al. (2020) addressed the sway suppression and positioning issues of a double-pendulum bridge crane system by designing a novel time-varying sliding mode control (NTVSMC).