1. Introduction

Nowadays, cargo transportation plays an important role in many industrial fields. For carrying the cargo in short distance or small area, such as in automotive factories and shipyards, the overhead cranes are naturally applied. To increase productivity, the overhead cranes today are required in high-speed operation. However, the fast motion of overhead cranes usually leads to the large swings of cargo and non-precise movements of trolley and bridge. The faster the cargo transport is, the larger the cargo swings. This makes dangerous and unsafe situation during the operating process. The crane itself and the concerning equipment in the factory can be damaged without proper control strategies.

In recent decades, the control problems of overhead cranes in both theory and practice have attracted many researchers. Various kinds of crane control techniques have been applied from classical methods such as linear control [1], nonlinear control [2, 5, 6], optimal approach [7], adaptive algorithms [8, 9] to modern techniques such as fuzzy logic [3, 4, 10], neural network [11], command shaping [12], and so on.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

The abovementioned researches deal with crane motion modeled as pendulum or multisection pendulum systems. As a result, their dynamics are described as an ordinary differential equation or a system of ordinary differential equations. In practice, the crane rope exhibits a certain degree of flexibility; hence, the equation of motions of the gantry crane with flexible rope is represented by a set of partial differential and ordinary differential equations. In [13– 15], the authors successfully design a controller that can stabilize the system with the rope flexibility. Flexible rope also is considered in [16, 17] where coupled longitudinal-transverse motion and 3D model are investigated.

This chapter accesses the modeling and control of overhead cranes according to the other research direction. We construct a distributed model of overhead cranes in which the mass and the flexibility of payload suspending cable are fully taken into account. We utilize the analytical mechanics including Hamilton principle for constructing such the mathematical model. With the received model, we analyze and design two nonlinear control algorithms based on two versions of Lyapunov stability: one is the so-called traditional Lyapunov function and the other is the socalled barrier Lyapunov. Dissimilar to the preceding study [18, 19] whereas the problem of actuated payload positioning system is considered, the proposed controllers track the trolley to destination precisely while keeping the payload swing small during the transport process and absolutely suppressed at the payload destination with control forces exerted at the trolley end of the system. The quality of control system is investigated by numerical simulation. Since the system dynamics is characterized by a distributed mass model, the finite difference method is applied to simulate the system responses in MATLAB® environment.

The chapter content is structured as follows. Section 2 constructs a distributed mass model of overhead cranes. Section 3 analyzes and designs two nonlinear controllers based on Lyapunov direct theory. The analysis of system stability is included. Section 4 numerically simulates the system responses and analyzes the received results. Finally, the remarks and conclusions are shown in Section 5.
