Input-limited optimal control for overhead cranes with payload hoisting/lowering and double pendulum effects

As an underactuated system, the overhead crane is widely used to transport cargoes in industry according to its flexibility and lifting capacity. Many control strategies are designed based on single pendulum crane models, which ignore the hook's mass and treat the payload as a mass point. However, the payload will swing around the hook when the payload is too large or the hook mass cannot be directly ignored, which is called double pendulum effects, making the dynamic more complex. In practical applications, input saturation and energy consumption should also be considered. To this end, we design an input-limited optimal controller for double pendulum cranes. Specifically, based on the defined performance index function and by solving the Hamilton-Jacobi-Bellman (HJB) equation, we can get an optimal controller satisfying the saturation constraint. In addition, the neural network is utilized to estimate the optimal performance index function. Furthermore, the convergence of state variables is proved theoretically, i.e. the trolley/rope length can converge to corresponding desired positions. Meanwhile, the hook's swing and payload's swing can also be eliminated. Finally, the simulation results are utilized to illustrate the performance of the designed optimal controller.