Model Inversion for Tip Control of Underactuated Non-minimum Phase Gantry Cranes with Small Inertia Ratio

Motion planning in non-minimum phase underactuated multibody systems is challenging mainly due to the non-collocation of the control force with respect to desired output and due to the system flexibility. A motion planning method for non-minimum phase underactuated multibody systems based on the inverse dynamics is proposed in this paper. The approach is structured as follows. First, the system is partitioned into actuated and unactuated coordinates. Then, the desired output is described as a non-linear separable function of these coordinates. Since the system is non-minimum phase, output redefinition is exploited to virtually redefine the output displacement in order to stabilize the system internal dynamics. This enables the stable integration of the ODEs describing the evolution of the unactuated coordinates. Finally, the trajectory of the actuated coordinates is obtained through non-linear kinematic inversion. The method is then applied to the case of a spatial overhead crane moving a suspended load. The effectiveness of the method in tracking a spatial trajectory is demonstrated through numerical simulations.