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INTRODUCTION
Currently, the gantry crane is widely used in the fields of industry, transportation, construction, and more. The most important problem facing the operation of this crane is the swing of the payload while it is being moved from one point to another as a result of inertia forces and external loads [1]. If this swing exceeds the acceptable limits, it will cause direct damage to the operating environment in the event of a load collision, either to workers or to nearby equipment [1]. This swing also causes stresses in the payload rope, crane cart, and crane steel structure. Long-term swing may lead to crane structure stresses as a result of fatigue that may cause permanent structural deformations and thus fatigue failure, which may lead to accidents, casualties, and property losses [2]. Therefore, the worker on the crane is often forced to stop the cart until the swing angle drops to a limit close to zero, in order to resume moving the cart again, which leads to the loss of a lot of operating time and effort. Some of the skilled workers maneuver the cart back and forth in order to minimize the stopping time, i.e. reach a small swing angle as quickly as possible, however the maneuvering process does not usually lead to a significant reduction in operating time [3].
Several papers have established the dynamic model of gantry crane movement and created a control unit designed only to reduce the swing angle to a minimum, i.e. to reduce the risk of accidents (safety), without paying enough attention to reduce the load transfer time as the swing of the payload, particularly in the start-up period, depends on the acceleration of the cart. However, the mathematical models in some of these papers are very complex models and do not explain how these models were derived, especially those that studied the swing of the payload in 3-dimensions by means of nonlinear equations [4].
The author of this paper believes that he developed a realistic model using Newton's Mechanics and Lagrange's Mechanics, with a detailed explanation of how this model is inferred, which was simplified by excluding nonlinear factors, , which are degrees of freedom that have no significant effect, which means reducing the number of control inputs [5] that have been implemented in the Linear Quadratic Regulator (LQR).Therefore, it is necessary to find the dynamic model using state-space, for ease of approaching it using modern methods of control like (LQR). This type of modeling is enough to analyze and design complex control systems that contain (M.I.M.O.), unlike the modeling of Laplace transforms that is used for (S.I.S.O.) systems [6].This has achieved excellent results in terms of minimizing the payload swing angle and most importantly the time to reach a steady state of the payload while it is moving, and the distance traveled at this time. The author does not claim to have closed most of the doors on the problem of effective control of pendulum cranes, as it is an open problem and needs more and more further research, as the design of mechanisms and structures has become more precise and less tolerant of transient vibrations [5] [6], but he presented a simple model in terms of its ease of derivation, realism of application, and treatment of the most influential factors in the issue of payload swing.
Researchers investigated different control methods to control the gantry crane. In [7], LQR and Fuzzy-LQR based were compared, both had oscillatory behavior in the beginning and a big overshoot, it took the angle 3 and a half seconds to settle down in the Fuzzy LQR and 4 seconds in the LQR. It took the cart 3 and a half seconds to reach 1m in the Fuzzy-LQR based and it took 2 and a half seconds to reach 1 m in the LQR.
In [8] LQR was investigated, the results show that it took a long time for the angle to settle down (12s) and there is a big overshoot.
Our results proved faster and smoother convergence of the swing angle to a very small value close to zero starting from any initial conditions while the cart is moving in a reasonable speed.
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