This paper presents a method for generating an obstacle-avoiding trajectory in optimal time and addresses the control problem of double-pendulum cranes, accounting for external disturbances and system constraints. Firstly, the double pendulum crane system is demonstrated to be a flatness system, with the flat outputs corresponding to the positions of the loads. Subsequently, by analyzing and incorporating various constraints, an optimization problem is formulated to determine the reference trajectory from the initial position to the final position of the load, minimizing travel time while ensuring obstacle avoidance. To mitigate the effects of disturbances and sensor limitations, a fixed-time extended-state observer is employed to estimate the output signal velocities and the total disturbance. These estimated signals and the derivative from the Lyapunov function of the proposed Nonsingular Hierarchical Fixed-Time Sliding Mode Control are utilized to design a Lyapunov-based Model Predictive Controller, enabling the load to track the reference trajectory while minimizing the swing angle and adhering to system constraints. Furthermore, the stability of the Lyapunov-based Model Predictive Controller is established based on the stability properties of the Nonsingular Hierarchical Fixed-Time Sliding Mode Control. Finally, simulation results and comparisons demonstrate the effectiveness of the proposed methods under various conditions.

This paper presents a method for generating an obstacle-avoiding trajectory in optimal time and addresses the control problem of double-pendulum cranes, accounting for external disturbances and system constraints. Firstly, the double pendulum crane system is demonstrated to be a flatness system, with the flat outputs corresponding to the positions of the loads. Subsequently, by analyzing and incorporating various constraints, an optimization problem is formulated to determine the reference trajectory from the initial position to the final position of the load, minimizing travel time while ensuring obstacle avoidance. To mitigate the effects of disturbances and sensor limitations, a fixed-time extended-state observer is employed to estimate the output signal velocities and the total disturbance. These estimated signals and the derivative from the Lyapunov function of the proposed Nonsingular Hierarchical Fixed-Time Sliding Mode Control are utilized to design a Lyapunov-based Model Predictive Controller, enabling the load to track the reference trajectory while minimizing the swing angle and adhering to system constraints. Furthermore, the stability of the Lyapunov-based Model Predictive Controller is established based on the stability properties of the Nonsingular Hierarchical Fixed-Time Sliding Mode Control. Finally, simulation results and comparisons demonstrate the effectiveness of the proposed methods under various conditions.