This paper deals with modelling and control of an overhead crane with a flexible cable. The developed dynamical model includes both transverse vibrations of the flexible cable and large swing angles of cable while the trolley is moving horizontally. To carry out the modelling, Rayleigh-Ritz discretization method is used to achieve an ordinary differential equation (ODE) model for transverse deflection of the cable with finite generalized degrees of freedom. Using the Euler-Lagrange formulation, a nonlinear dynamic model of the crane system with a flexible cable is obtained. The control objective is to move the payload position to the desired point and at the same time, to reduce the payload swing and to suppress the cable's transverse vibration. To this end, a static state feedback control combined with integral error feedback is proposed. Simulations are performed using the developed nonlinear model under various conditions to illustrate the effectiveness of the proposed control system.

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This paper aims to develop an advanced control system for an overhead crane with transverse vibrations of flexible cable while considering large angle of the cable swing. The control objective is to move the payload to a desired position and at the same time, to reduce the payload swing and to suppress the cable transverse vibrations only by applying a directional (horizontal) driving force to the trolley. The crane system with cable vibrations and large swing angle is categorized as a multi-degree under-actuated system whose characteristics impose serious challenges when applying control methods. One reasonable way to overcome the challenges is to separate the degrees of under actuation. To do so, singular perturbation approach is used to divide the dynamical system into a fast subsystem for cable vibrations (with flexible un-actuated coordinates) and a slow subsystem for trolley motion and cable swing (with one degree of under actuation). Based on two-time scale control method, a composite control system is designed. Accordingly, the control system consists of a linear controller for fast dynamics (for vibration suppression) and a nonlinear controller for slow dynamics (for reducing cable swing and controlling the payload motion). Simulations are performed to demonstrate the effectiveness of the control system for an illustrative example of the crane systems with flexible cable moving a lightweight payload.

A flexible-cable overhead crane system having large swing is studied as a multi-degree underactuated system. To resolve the system dynamics complexities, a second order singular perturbation (SP) formulation is developed to divide the crane dynamics into two one-degree underactuated fast and slow subsystems. Then, a control system is designed based on the two-time scale control (TTSC) method to: (a) transfer the payload to a desired location and decrease the payload swing, by a nonlinear controller for slow dynamics; and (b) suppress transverse vibrations of the cable, by a linear controller for fast dynamics. The nonlinear controller is designed based on an energy shaping technique according to the controlled Lagrangian method. To demonstrate the control system effectiveness, an example of the flexible cable crane systems with a lightweight payload is considered to perform simulations. In addition to the proposed control system, two other controllers; namely, a linear controller based on the linear–quadratic regulator method and a TTSC based on the approximate SP model and partial feedback linearization, are applied to the system for comparison. Also, by applying a disturbance force to the trolley and considering 10% uncertainty in crane parameters, the control performance against disturbances and parameter uncertainties is investigated.

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