# Adaptive Hierarchical Sliding Mode Control for Double-Pendulum Gantry Crane Based on Neural Network
## Appendix A Derivation of the Equation of Motion  
Let the coordinates of the crane double-pendulum hook be \((x_{1},y_{1})\):  
\[\left\{ \begin{array}{l}x_{1}=x+l_{1}\sin\theta_{1}\\ y_{1}=-l_{1}\cos\theta_{1}\end{array} \right.\]  
The  
* [PERSON] et al. (2024) [PERSON]; [PERSON]; [PERSON] Neural network adaptive PID-like coupling control for double pendulum cranes with time-varying input constraints. _Mech. Syst. Signal Process._**2024**, _208_, 17. [CrossRef]
* [PERSON] et al. (2014) [PERSON]; [PERSON]; [PERSON] Stable neural PID anti-swing control for an overhead crane. _N Intelligent Autom. Soft Comput._**2014**, _20_, 145-158. [CrossRef]
* [PERSON] et al. (2018) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] A novel anti-swing positioning controller for two dimensional bridge crane via dynamic sliding mode variable structure. _Procedia Comput. Sci._**2018**, _131_, 626-632. [CrossRef]
* [PERSON] et al. (2016) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Adaptive tracking control for double-pendulum overhead cranes subject to tracking error limitation, parametric uncertainties and external disturbances. _Mech. Syst. Signal Process._**2016**, _76_, 15-32. [CrossRef]
* [PERSON] et al. (1997) [PERSON]; [PERSON]; [PERSON] Anti-swing Position Control of Crane based on Acceleration Feedback and Predicted Pattern Following Method. _IEEJ Trans. Ind. Appl._**1997**, _117_, 1341-1347. [CrossRef]
* [PERSON] et al. (2024) [PERSON]; [PERSON]; [PERSON] Anti-swing positioning control algorithm for portal slewing crane. _Comput. Integr. Manuf. Syst._**2024**, 1-20. [CrossRef]
* [PERSON] et al. (2025) [PERSON]; [PERSON]; [PERSON]. An integrated solution for 3D overhead cranes: Time-optimal motion planning, obstacle avoidance, and anti-swing. _Eng. Sci. Technol. Int. J._**2025**, _59_, 101852. [CrossRef]
* [PERSON] and [PERSON] (2013) [PERSON]; [PERSON] Sliding mode controls of double-pendulum crane systems. _J. Mech. Sci. Technol._**2013**, _27_, 1863-1873. [CrossRef]
* [PERSON] et al. (2011) [PERSON]; [PERSON]; [PERSON] Control of Overhead Crane Systems by Combining Sliding Mode with Fuzzy Regulator. _IFAC Proc. Vol._**2011**, _44_, 9320-9325. [CrossRef]
* [PERSON] et al. (2025) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Optimal Policy Gradient Based Type-2 Fuzzy Control for Multi-DC Terminal PEC Converter in 5G-Based Commercial Buildings. _IEEE J. Emerg. Sel. Top. Power Electron._**2025**, _in press_. [CrossRef]
* [PERSON] and [PERSON] (2024) [PERSON]; [PERSON] [PERSON] Adaptive Fixed-Time Control Strategy of Generator Excitation and Thyristor-Controlled Series Capacitor in Multi-Machine Energy Systems. _IEEE Access_**2024**, _12_, 12. [CrossRef]
* [PERSON] (2014) [PERSON] Application of intelligent sliding mode control with moving sliding surface for overhead cranes. _Neural Comput. Appl._**2014**, _24_, 1369-1379. [CrossRef]
* [PERSON] et al. (2022) [PERSON]; [PERSON]; [PERSON]; [PERSON] Adaptive tracking of double pendulum crane with payload hoisting/lowering. _Autom. Constr._**2022**, _141_, 104368. [CrossRef]
* [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON] Enhanced-coupling adaptive control for double-pendulum overhead cranes with payload hoisting and lowering. _Automatica_**2019**, _101_, 241-251. [CrossRef]
* [PERSON] et al. (2014) [PERSON]; [PERSON]; [PERSON] Adaptive Fuzzy Hierarchical Sliding-Mode Control for the Trajectory Tracking of Uncertain Underactuated Nonlinear Dynamic Systems. _IEEE Trans. Fuzzy Syst._**2014**, _22_, 286-299. [CrossRef]
* [PERSON] et al. (2012) [PERSON]; [PERSON]; [PERSON]; [PERSON] A motion planning-based adaptive control method for an underactuated crane system. _IEEE Trans. Control Syst. Technol._**2012**, _20_, 241-248. [CrossRef]
* [PERSON] (2005) [PERSON] _Sliding Mode Variable Structure Control MATLAB Simulation_; Tsinghua University Press: Beijing, China, 2005; pp. 348-350. * [PERSON] (2011) Lin, F.K. GB/T14405-2001 General bridge cranes and GB/T14406-2011 General gantry cranes. _Hoisting Conveying Mach._**2011**, _39_, 1.