# Adaptive Hierarchical Sliding Mode Control for Double-Pendulum Gantry Crane Based on Neural Network
## 1 Introduction  
Gantry cranes, with their efficient material handling capabilities and flexible spatial adaptability, play a crucial role in heavy-load logistics operations in fields such as port handling, construction, and industrial manufacturing. The inherent underactuation of these systems contributes to the complexity of anti-swing control [1, 2]. In the context of the rapid development of intelligent control technology, the transition of cranes from automated to intelligent operation has become a major trend [3]. The technology required for path recognition and planning by intelligent cranes necessitates the precise positioning of the payload. Crane anti-swing technology is a prerequisite for achieving this precise positioning. Consequently, the intelligence and adaptability of anti-swing control have become the focus of new research initiatives [4]. The field of crane control has evolved through several key phases. Early research often relied on classic control methods, such as Proportional-Integral-Derivative (PID)control, which provided a foundational approach to the problem. While effective to some extent, these methods struggled with the system's inherent nonlinearities. This led to the development of more advanced robust nonlinear control strategies designed to handle such complexities. In the extant research, scholars have predominantly adopted a simplistic approach to gantry cranes, conceptualizing them as a \"trolley-payload\" single-pendulum model. However, when the mass of the hook is comparable to that of the payload and there is a distance between them, the system exhibits significant double-pendulum effects [5,6]. It is evident that traditional single-pendulum models are inherently limited in their capacity to address strongly coupled systems, a deficiency that arises from an oversight in the consideration of hook-payload coupling effects. Research on anti-swing control for double-pendulum cranes is of theoretical and engineering significance for improving the performance of heavy-load precision lifting operations. Recognizing this, researchers began to establish more accurate, though more complex, nonlinear dynamic models of double-pendulum cranes using frameworks like the Lagrange method to better capture the hook-payload coupling dynamics. In [7], a nonlinear dynamic model of a double-pendulum overhead crane was established, with the study adopting the Lagrange method as its theoretical framework. This model accurately characterized the hook-payload coupling dynamics, thereby establishing a theoretical foundation for high-precision anti-swing control. The advent of anti-swing technology has led to the mainstream adoption of closed-loop control systems. Closed-loop control systems have been developed to correct control inputs through real-time feedback. It is evident that such control methodologies exhibit a measure of robustness, encompassing Proportional-Integral-Derivative (PID) control [8,9], sliding mode control (SMC) [10], and adaptive control [11,12]. In order to achieve intelligent anti-swing for cranes, [PERSON] et al. [13] adopted an adaptive control method and proposed an improved PID control strategy based on an adaptive disturbance observer. This resolved the issue of substandard anti-swing positioning performance due to unidentified disturbances and friction in cranes. However, its adjustment and response speed were comparatively sluggish. Sliding mode control has been extensively applied due to its rapid response and strong robustness to system parameter variations and external disturbances [14]. Reference [15] proposed a control method based on sliding mode control for double-pendulum crane problems, enabling the payload to move along a pre-specified path while suppressing swing. However, conventional sliding mode control is susceptible to chattering. Currently, approaches to solve the chattering problem in sliding mode control have become a research hotspot.