# Adaptive Hierarchical Sliding Mode Control for Double-Pendulum Gantry Crane Based on Neural Network
## 1 Introduction
In addressing the chattering issue in the context of sliding mode control, scholars have previously put forward improved sliding mode methodologies [16]. Nevertheless, these methodologies continue to exhibit deficiencies in the management of the coupled dynamic characteristics inherent in double-pendulum systems [17]. The current state of the field is characterized by the integration of intelligent control techniques to further enhance adaptability and performance. This modern paradigm involves using methods like neural networks and fuzzy logic for online learning and approximation of system uncertainties. Various studies have explored options for integrating neural networks with PID-like structures or as pure controllers [5,8,9]. A primary challenge, however, remains in designing a framework that systematically combines the guaranteed stability of robust control with the adaptive learning capabilities of intelligent methods to solve the complex double-pendulum problem, especially given issues like chattering in standard SMC. To clarify our contribution, it is important to differentiate the proposed method from other advanced control paradigms. While our approach shares similar objectives with synergetic control theory in using a manifold to handle uncertainties, it is fundamentally rooted in the principles of sliding mode control (SMC), relying on its well-established framework for robustness and stability. Furthermore, the selection of the adaptive RBF neural network over deep reinforcement learning (DRL) methods was deliberate. DRL often requires extensive offline training and lacks the formal stability guarantees crucial for physical systems [18,19]. In contrast, our RBFNN-based approach is computationally efficient, adapts online in real-time without prior datasets, and most critically, allows for a rigorous Lyapunov-based proof of stability, which is paramount for this application. A primary challenge in the control of double-pendulum gantry cranes is the accurate modeling of the system's strongly coupled nonlinear characteristics and the design of effective control laws to suppress oscillations and achieve precise positioning. It is often challenging to achieve an optimal control performance in complex systems due to the limited capability of traditional control methods. The present study considers the precise positioning of the trolley and the rapid swing suppression of the hook and payload. This limitation is addressed by leveraging the powerful nonlinear approximation capabilities of neural networks and the robustness of sliding mode control. The present paper puts forward a hypothesis concerning the development of an adaptive hierarchical sliding mode control strategy based on radial basis function (RBF) neural networks. In comparison with conventional hierarchical sliding mode control or standard sliding mode control, the proposed method has the advantage of estimating and compensating for unknown system dynamics online using neural networks, thus reducing the reliance on the system model. Moreover, it employs adaptive laws to adjust parameters, thereby achieving a better control performance and robustness in comparison to conventional control methods. Furthermore, a hierarchical sliding mode structure is adopted in the controller design to facilitate the design process and address different control objectives separately. The following constitute our primary contributions:  
Developing and implementing a robust control strategy for a more accurate and realistic double-pendulum gantry crane model, addressing its inherent strong coupling and nonlinear dynamics. Designing a novel, integrated control framework that synergistically combines a hierarchical sliding mode control (HSMC) structure with an adaptive RBF neural network. This framework is purpose-built to actively compensate for system uncertainties and external disturbances online.