# Adaptive Hierarchical Sliding Mode Control for Double-Pendulum Gantry Crane Based on Neural Network
## 5 Conclusions  
This study has addressed the challenging, strongly coupled nonlinear control problem of a double-pendulum gantry crane system. The core contribution of this work is the proposal and validation of a novel adaptive control strategy that integrates a hierarchical sliding mode control (HSMC) structure with an RBF neural network compensator. By constructing a composite sliding mode surface and leveraging the RBF neural network to perform online estimation and compensation of system uncertainties, the proposed method significantly enhances the control performance and robustness of the system. The simulation analysis results demonstrate that the proposed method successfully achieves precise trolley trajectory tracking, while significantly suppressing the swing of the hook and payload. Specifically, in comparison to conventional sliding mode control (SMC), the RBF-HSMC reduced the maximum residual swing angle of the hook and payload by approximately 84%. Compared to hierarchical sliding mode control (HSMC), the reduction was approximately 80%. This enhanced swing suppression was achieved with greater efficiency and smoothness; the control force amplitude was lowered to 5.23 N, reductions of 37.4% and 30.2% compared to SMC and HSMC, respectively, which also significantly mitigated control signal chattering. The controller also demonstrated excellent robustness by maintaining a high performance under variations in system parameters and in the presence of external disturbances. This research provides an important theoretical basis and practical engineering reference for the intelligent anti-swing control of lifting equipment. It should be noted that the findings of this study are based on a comprehensive set of simulations. This simulation-based approach is a crucial first step, allowing for the development and rigorous validation of the control strategy's performance and robustness under a wide range of conditions. The successful results presented herein provide a strong foundation for future research. Future work will focus on addressing the limitations of the current study by extending the proposed 2D control strategy to more complex 3D crane models. The next logical step will be to implement and validate the proposed RBF-HSMC controller on a real-world gantry crane testbed. This future work will involve integrating the algorithm with a real-time control platform and the necessary sensing hardware, such as encoders and IMUs, to verify its practical effectiveness.  
Formal analysis, L.Y. and Y.C.; investigation, B.L.; data curation, L.Y. and Y.C.; writing--original draft, L.Y., Y.C. and J.Y.; writing--review and editing, L.S.; funding acquisition. All authors have read and agreed to the published version of the manuscript.  
The project is supported by the 2022 Henan Province Industrial Research and Development Joint Fund Major Project (Project No.: 225101610072), 2023 Henan Province Key Research and Development Special Project (Project No.: 231111221700), 2024 Henan Province Industrial Research and Development Joint Fund Major Project (Project No.: 245101610033), 2025 Henan Province Science and Technology Key Project (Project No.: 252102411007), and 2025 National Administration for Market Regulation Science and Technology Program (Project No.: 2024 MK080).  
Not applicable.  
The data presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.