A graphical phase plane approach for controlling cargo transfer on quay-side container cranes with hoisting

Input-shaping is a practical open-loop strategy for the control of transient and residual oscillations on cranes, especially those having fixed cable length and predefined payload transfer paths. In this paper we extend the double-step input-shaping control approach to include maneuvers that involve hoisting. The approach is based on using the graphical representation of the phase plane of the payload oscillations. The phase plane is used to derive mathematical constraints to compute the switching times of a double-step acceleration command profile that will result in minimal transient and residual oscillations. The controller design is based on an accurate two-dimensional four-bar-mechanism model of a container crane. For the purpose of controller design, the model is reduced to a constrained double pendulum with variable length link and a kinematic angular constraint. The generated commands were based on both a. linear and a nonlinear frequency approximation of the payload oscillation period. Results demonstrated that in contrast with the single-step input shaping controllers, which are very sensitive to frequency approximations, the proposed double-step controller is less sensitive to small variations in the frequency even for large commanded accelerations. Numerical simulations demonstrated that using this approach, oscillations during and at the end of transfer maneuvers can be reduced to less than 2 cm.