Pseudospectral method based time optimal trajectory planning for double pendulum cranes

In practice, overhead crane systems are widely used while the common control methods are always designed based on the assumption that the payload can be regarded as a mass point, with the hook mass being ignored. In this situation, the crane system is seen as a single pendulum. When this assumption is not satisfied, the crane may behave like a double pendulum, which has more unactuated states and is more difficult to be controlled properly. Motivated by this observation, we present a time optimal trajectory planning scheme for double pendulum crane systems, which can yield a global optimal swing-free trajectory. Specifically, we first implement some basic transformations on the system dynamics. Then, various constraints, including upper and lower bounds of the two pendulum angles and the allowable trolley velocity and acceleration, are taken into consideration to obtain an optimization problem. After that, the Gauss-pseudospectral method (GPM) is utilized to convert the constrained optimization problem into a nonlinear programming problem, which can be solved more conveniently, while the trajectory constraints are also considered during the transformation. During the entire planning process, no linearization is required, which is different from most existing methods. Finally, numerical simulation results are given to illustrate the satisfactory performance of the proposed method.