^{1}

^{1}

^{2}

^{1}

^{2}

The paper presents a control approach to a flexible gantry crane system. From Hamilton’s extended principle the equations of motion that characterized coupled transverse-transverse motions with varying rope length of the gantry is obtained. The equations of motion consist of a system of ordinary and partial differential equations. Lyapunov’s direct method is used to derive the control located at the trolley end that can precisely position the gantry payload and minimize vibrations. The designed control is verified through extensive numerical simulations.

Gantry cranes are essential for load handling operation in various fields. Due to the flexibility in the gantry system structure, there exists a trade-off between the requirement of fast operations and minimizing payload vibration. Currently, solving the aforementioned constraint is an open and promising research area.

Renowned for its simplicity in practical applications, input command shaping is one of the earliest attempts to crane control [

The above-mentioned works treat crane motion as pendulum-like motion where the crane cable is considered as a rigid body. Hence, the crane equations of motion consist of ordinary differential equations. In practice, the crane cable is flexible; this property leads to a requirement of a set of partial differential equations describing the crane motion. Joshi et al. [

In the paper, a position control and vibration suppression of a 3D gantry crane with varying rope length are considered. The contribution of the research is the formulation of of 3D flexible gantry with coupling mechanism in transverse-transverse and transverse-longitudinal directions and the control designed for stabilizing the system. The system equations of motion are derived via Hamilton’s extended principle. Subsequently, control forces are constructed based on Lyapunov’s stability. Extensive simulations are presented to demonstrate the effectiveness of the proposed control.

Before deriving the gantry system mathematical model, the following is assumed:

Cable axial deformation is very small in comparison with the cable length.

The payload is considered as a point-mass.

Friction in trolley motion is ignored.

Environmental disturbances are neglected.

Moving mass in

The deflection angle of the cable from vertical axis is very small.

Diameter to length ratio of the cable is very large.

Assumption

The crane system coordinate is depicted in Figure

System coordinate.

For the sake of clarity, from this point onward

The control objective is to simultaneously drive the payload to desired position while keeping swing angle as small as possible. In order to archive the control objective, consider the following Lyapunov candidate function

Numerical simulations for illustrating the ability of positioning trolley and suppressing payload swing angle of the proposed control are given in this section. Initially, the payload is located at (0,0,0); the system is designed to move the payload to the desired position at (15,20,5). The system simulation parameters are

Trolley and payload responses in X direction without vibration control.

Control force

Trolley and payload responses in Y direction without vibration control.

Control force

Payload response in Z direction without vibration control.

Control force

Secondly, the proposed control for positioning and vibration suppressing is activated. The simulation results in Figures

Trolley and payload responses in X direction with vibration control.

Control force

Trolley and payload responses in Y direction with vibration control.

Control force

Payload response in Z direction with vibration control.

Control force

In addition, because payload vibration is fed back to the trolley, control effort is lower in controlled cases.

In the paper, position and vibration suppression problems in the gantry crane system with flexible cable are considered. Based on system energy analysis equations of motion that govern the gantry dynamics deforming in two transverse directions with varying cable length is derived according to Hamilton’s extended principle. The process results in a system of ordinary and partial differential equations including cable dynamics and boundary conditions at trolley and payload ends. Control forces that solve the control problem are designed based on Lyapunov’s direct method. The control forces are placed at trolley end of the gantry. Simulations are given to illustrate the effectiveness of the proposed control. System friction, input, and output constraints are the main target in the future research.

This publication is supported by multiple datasets, which are available at locations cited in the reference section.

The authors declare that they have no conflicts of interest.