# Mathematical Matlab Simulink Software Computer Science Essay

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Gantry cranes are common industrial structure which can be found in factories, oil rig platform, shipyard and especially seaports. Gantry crane is the most important logistic facilities to support a seaport operation. Gantry crane normally used to hoist and lowering shipping containers between freight vessel ships and wharf, which imported and exported goods and commodities were delivered thru these gateways.

Millions of trading transactions among domestic and international companies depends on the robustness of gantry crane to move their shipping containers within shortest time. Therefore, seaports must provide adequate facilities in terms of speed and robustness of the crane, to handle thousands of shipping containers to be loaded and unloaded from the vessel.

The performance of the seaport depends heavily on robustness of the crane handling system, in order to maximise the quantity of daily loading and unloading shipping containers capacity. However, these cranes are manually operated by an operator inside control cabin, 45 meters above sea level. The operators use the joystick and acceleration pedal to control the movement and direction of the cranes. The control cabin will move along the gantry crane directly above the shipping container to transfer containers between ships or on the harbour pavements.

Mathematical software MATLAB-SIMULINK will be used to simulate the perturbation parameters due to versatility in setting the elements of Robust Control Design of a Crane Handling System.

## Problem Statements

The robustness of a crane handling system is the key factor for a seaport organization performance. Maximizing the speed of loading and unloading shipping containers between a vessel and wharf at pin point accuracy from 45m above the sea level is a tremendous undertaking which require high precision and robust control system design.

Experienced crane operators have to consider too many parameters such as wind speed, travelling speed, containers weight, and swinging factor while judging the accurate lifting and lowering position. These parameters puts a lot of pressures on the crane operators which significantly reducing their performance over hours of work shift. By designing a robust control system, some of these judging burdens will be lifted from them so they can concentrate more on their works.

Jerking and swinging crane may cause damage to the contents of the container due to improper torque distribution and uncompensated acceleration and deceleration (Tahour A. et al. 2008). Therefore, a robust control design will improve the crane safety features in terms of avoiding overshooting and protecting the content of the containers from unwanted impact during hoisting or lowering.

Although gantry crane move in 2 axis only, the length of the cable and the weight of the shipping containers caused the shipping containers to sway besides axis offset position due to different drive motors and motor characteristics (Bernard et al.2002). Control system design must consider the effects of bad weather such as storm, heavy rains, sea waves and the pendulum effects on the containers itself.

## 1.3 Significance of the Research

According to Hanafi (2003) the main purpose of designing a robust control handling system for gantry cranes, is to transfer loads from point to point as fast as possible and, at the same time, keep the load swing small during the transfer process and completely eliminate it at the load destination.

Some goods are fragile and hazardous, thus, excessive impact can either cracked or spilled the content of the containers which can cause malfunction or contaminations. Therefore, robust control system assures a safer load transfer by compensating the effect of acceleration and deceleration to avoid damage upon impact between contents and the shipping container.

Positioning the shipping containers at pin point accuracy, 45 meters above the sea level is a tremendous challenge. The crane operator must precisely position the load for each and every attempt to avoid overshoot or missed the exact position. If the load missed the exact position, the crane operator will waste a lot time to make correction and reposition the load. By implementing a robust control design, errors can be reduce or eliminate, therefore enhance efficiencies and productivities.

## 1.4 Research Question

i. What are the current crane handling control systems?

ii. What is the control design option for crane handling system?

iii. How to evaluate the implementation of proposed control design for a crane handling system?

## 1.5 Research Objectives

The objectives of this project are:

To analyze the current crane handling control system.

To design a robust control for a crane handling system.

To implement the proposed robust control crane handling system design using simulation of MATLAB Simulink software.

## 1.6 Scope of Work

The scopes of this project are:

Analyze, design and simulate gantry crane robust control.

Number of uncertainty or perturbation limited to two parameters only which are position of the trolley and sway angle of the mass.

Simulated responses of the position of the trolley and sway angle of the mass are presented using MATLAB.

## 1.7 Summary

The next chapter of this proposal briefly describes several types of crane movement controlling method and highlights the problems that usually occurred in crane dynamics. Comparison between controllers in terms of advantages and disadvantages, facilitate the controller selection process for the most suitable one. Chapter III then discusses the proposed development techniques for crane simulation using MATLAB Simulink software to produce plots of the system response. Expected result from the simulations, is achieving satisfactory vibration reduction of a crane system using the proposed method

## 2.1 Introduction

Anti-swing control is a well-known term in gantry crane control. It is designed to move the payload of gantry crane as fast as possible while the payload swing angle should be kept as small as possible at the final position. In this chapter, several method of controlling a crane was studied to obtain general information about the controller's nature.

A number of studies have proposed anti-swing control using the well-known proportional, integral, derivative (PID) control method. However, PID controllers cannot always effectively control systems with changing parameters. Some studies have also proposed intelligent-based control including fuzzy control.

However, the designers often have to face the problem of tuning many parameters during the design to obtain optimum performance. Thus, a lot of effort has to be taken in the design stage. The objective of this project is to design a practical anti-swing control which is simple in the design and also robust. A complex dynamic analysis of the system is not needed.

## 2.2 Input Shaping

When a human operator attempts to maneuver payloads using an overhead gantry crane like the one sketched in Figure 2.1, the oscillations induced into the payload by the motion of the trolley can be significant. The crane oscillations make it difficult to manipulate the payload quickly and with good positioning accuracy. Furthermore, when the workspace is cluttered with obstacles, the oscillations can create significant safety issues, especially when the payload or obstacles are of a hazardous or fragile nature. This problem serves as an excellent case study for a gantry crane dynamics and controls.

Figure 2.1 Sketch of a gantry crane. (Singhose et al. 2000)

The major components of the gantry crane are the bridge, the trolley, and the control pendant. The bridge is an I-beam that moves forwards and backwards. The trolley moves left and right on the bridge. The hoisting mechanism, hoisting cable, and payload hang from the trolley.

Input shaping has been shown to be a simple way of reducing swing in gantry cranes. This is particularly important if the crane must operate in a cluttered workspace. Singhose W. et al (2000) has developed a real-time MATLAB simulation of a gantry crane in a cluttered environment. It was used in an undergraduate controls course to teach both dynamics and input shaping. Data from these simulations show that students drove the crane through more efficient paths when input shaping was used.

Input shaping is a simple, effective strategy for reducing crane sway. Input shaping is easier to derive and implement than time-optimal control schemes and does not require the feedback mechanisms of closed-loop and adaptive controllers. Input shaping is implemented in real time by convolving the command signal with an impulse sequence (an input shaper). This process is illustrated in Figure 2.2 with a step input and an input shaper containing two positive impulses. The benefits of input shaping increases the likelihood that an operator will make a more efficient route through obstacles as opposed to taking a long route around an obstacle field.

Figure 2.2 Input Shaping a Pulse Input. (Singhose et al. 2000)

## 2.3 Input Shaping Crane Simulation

A MATLAB simulation of a gantry crane created as existing in a seaport workspace. The operator's viewpoint is directly above the workspace as shown in figure 2.3. The workspace contains a starting zone (hoist payload), goal region (lowering payload), obstacles (ship's haul and positioned payload), and the crane trolley and payload. The crane's trolley was controlled by using the numeric keypad on a computer's keyboard. The objective was to get from the starting zone to the goal region in the quickest and safest way possible. The level of safety pertaining to any particular path was determined from the number and severity of collisions between payload or trolley and the obstacles in the workspace.

In real life, a crane operator must be very careful to avoid collisions between the payload and obstacles. These could result in an injury or a damaged payload. On the other hand, a crane operator must work with a high level of speed and efficiency. Therefore, a real crane operator must make tough decisions about how fast to go at what heights to position the payload in ships haul. In this computer simulation, there is no loss of a job or personal injury.

Figure 2.3 Overhead View of a Crane Workspace. Figure 2.4 Path Examples.

The simulation of the crane based on with and without input shaping. This allowed highlights the benefits of a vibration control scheme like input shaping. In addition, the different path choices taken crane operator, with and without input shaping revealed some important information about how this vibration control technique effects operator decisions. In particular, it showed that operators tend to be more aggressive and thus take more efficient routes when input shaping is available. An example of an unshaped control of the crane and a shaped control is shown in Figure 2.4.

Figure 2.5 Crane Responses without Input Shaping. (Singhose et al. 2000)

Figures 2.5 and 2.6 compare the shaped and unshaped responses of the crane. The data was taken by fixing a marker to the end of the crane and record its motion. This file was imported into a computer and a MATLAB image-processing program calculated the markers position over time. Figure 2.5 shows that the payload oscillates while the crane is commanded to move and after it is commanded to stop. The hatched box represents approximately when the forward button was pressed on the control pendant. Figure 2.6 shows that input shaping eliminates these oscillations.

Notice that without input shaping, once the payload reaches its final destination it swings for several seconds. Figure 2.6 shows that input shaping eliminates these problems. Current research is underway to find input shapers that optimize move time and robustness to modeling errors

Figure 2.6 Crane Responses with Input Shaping. (Singhose et al. 2000)

## 2.4 Convolution (Impulse Shaping)

The most popular technique for input shaping is to convolve a sequence of impulses and various methods for shaping impulse function and examining their robustness has been reported and applied to flexible spacecraft, robots and to the control of swing of suspended objects transported by cranes (Cho,1995). Impulse shaping is a feedforward control technique for reducing vibrations in computer controlled machines. The method works by creating a command signal that cancels its own vibration. That is, vibration caused by the first part of the command signal is canceled by vibration caused by the second part of the command.

Input shaping is implemented by convolving a sequence of impulses, an input shaper, with any desired command. The shaped command that results from the convolution is then used to drive the system. See Figure 2.7. If the impulses in the shaper are chosen correctly, then the system will respond without vibration to any unshaped command.

Figure 2.7 An example of impulse shaping technique. (Cho, 1995)

The amplitudes and time locations of the impulses are obtained from the system's natural frequencies and damping ratios. Shaping can be made very robust to errors in the system parameters. The impulse sequence is chosen such that in the absence of control input, it itself would not cause residual vibration. Recall that convolution in the time domain is equivalent to multiplication in the Laplace domain. In order to increase the rise time, when using impulse shaping, the impulses are allowed to take negative value (Singhose, W.E. et al, 1997) and multihump shaping of the impulses can be used to increase the system robustness (Singhose, W.E. et al, 1997).

A step shaper is designed by generating acceleration/deceleration pulse profile using step inputs with alternative sign. The main difference between impulse shaper and step shaper is whether it uses the convolution concept or not.

## 2.5 Bang-bang Control

Other methods proposed by Meckl and Seering consist of using a multiswitch bang-bang forcing function, which gives time-optimal performance (Meckl, 1995) or adding up harmonics of ramped sinusoid functions in order to approximate as close as possible a bang-bang function, but minimizing the energy introduced at system resonance frequencies.

## 2.6 Zero Vibration

A zero vibration (ZV) shaper is designed by requiring that it produce commands that cause zero residual vibration when the model is perfect. For a single-mode system, this constraint leads to a shaper containing two. However, the ZV shaper is very sensitive to modeling errors.

## 2.7 Zero Vibration and Derivative

A zero vibration and derivative (ZVD) shaper is obtained by requiring the derivative with respect to the frequency of the residual vibration be equal to zero. That is, the sensitivity curve must have zero slopes at the modeling frequency. It is shown that ZVD shaper provides higher robustness than the ZV shaper. Moreover, higher robustness can further be achieved by higher derivatives.

## 2.8 Time-Optimal Rigid-Body (TORB) Command and Time-Optimal Flexible-Body (TOFB)

The efficiency of cargo handling work at a port depends largely on the operation of container cranes. When a ship is unloaded, containers are first transferred from the ship to a waiting truck by a container crane. The truck then carries the container to an open storage area, where another crane stacks the container to a pre-assigned place. The bottleneck of this cycle lies in the transfer of the containers from the ship to the truck.

Therefore, minimizing this transfer time will bring about a large cost saving. When a ship is loaded, the same problem is encountered. Since a large swing of the container load during the transfer is dangerous, the problem is to transfer a container to the desired place as quickly as possible while minimizing the swing of the container during transfer as well as the swing at the end of transfer.

If the oscillation of the container load is ignored, TORB commands can be easily calculated. Unfortunately, TORB commands will usually result in large amplitude oscillations. Experienced crane operators attempt to eliminate vibration by causing a deceleration oscillation which cancels the oscillation induced during acceleration, or they may brush the payload against obstacles to damp out the vibration.

When the swing is considered, TOFB commands that result in zero residual vibration can be generated. Hoisting of the load during the motion increases the difficulty of generating the control because the system is nonlinear. If the system model is linearized, then the associated frequency is time-varying. Optimal controls based on a nonlinear model can be difficult to generate. One method for developing optimal controls divides the motion into fundamental parts. The control for each part is then derived and pieced together. Even when optimal commands can be generated, implementation is usually impractical because the boundary conditions at the end of the maneuver (move distance) must be known at the start of the move. When feedback is available, both robust controllers and combination of open- and closed-loop controls are possible.

## 2.9 Time-Delayed Control

This method uses the direct and time-delayed signal to cancel the poles of a system with the intention of attenuating the residual vibration. Robust Time-Delayed Control is the method referred to as the proportional plus multiple delay control, involves the use of multiple time delays in conjunction with a proportional part to cancel the dynamics of the system in a robust fashion.

All the above methods start with a parametric input function, which usually involves magnitude and time delay. The parameter values are calculated in order to reduce the residual vibrations at the final position. The speed of the system is determined mainly by the system dynamics and little control can be exercised on the speed of the response. In all cases, the achievement of robustness or the control of vibration leads to an increase in system delays.

## 3.1 Introduction

System inversion based method reverse the process by specifying the system output function and deriving the input. In this way, the designer can choose the speed and shape of the motion within the limitations of the drive system. In the present study, in order to control the motion after the end-point, it is necessary to use output function with only one parameter and this leads to a much simpler input function. The single parameter of this output function determines the motion speed and is limited only by the drive system constraints.

## 3.2 Modelling of the Gantry Crane System

A gantry crane system is a crane carrying the trolley or trolley with a movable or fixed hoisting mechanism, that the bridge is rigidly supported on two or more legs running on fixed rails or other runway. The fundamental motions of a gantry crane consist of traversing, load hosting and load lowering. Gantry cranes are widely used as an efficient means of traversing heavy object in factories, warehouse and shipping yards. Like other crane types, gantry cranes met with some dissatisfactory due to its natural characteristics.

As mentioned, the fundamental motions of a gantry crane consist of traversing, load hosting and load lowering. These significant characteristic is that all motions are performed simultaneous at relatively high speed. Crane traversing motions, particularly when starting or stopping; induce undesirable swinging of the suspended load. This creates another problem that the swing could cause the hosting rope to leave its groove which could lead to over wrapping and damage.

One of the characteristics of these cranes is the flexible hoisting ropes used as a part of the structure for the reduction of system mass, which result in favorable features of high payload ratio, high motion speed and low power consumption. However, the flexible hoisting create serious problems, that is the crane acceleration which required for motion will generate undesirable load swing, which is frequently aggravated by load hoisting. Therefore, such load swing should be suppressed as rapidly as possible to maximize the operations.

The operation of the gantry crane systems in many industrial settings is achieved by relying on the skill of experienced crane operators. Unfortunately, precise payload positioning, meaning that the operator using only visual feedback to position the payload, is difficult due to the reality that the payload is free to swing in a pendulum like motion. In addition, the payload swing can result in several performances and safety concerns that include damage to the payload like spillage or breakage, damage to the surrounding environment of personnel, and large internal forces that can result in reduced payload carrying capacity or premature failure of stressed part.

This chapter will emphasize on the modeling of a gantry crane and the application of the inverse dynamic analysis into the system. The system model will then be represented in time domain and in state-space form.

## 3.3 Model Description

The model of a gantry crane is shown in Figure 4.1. Generally, the configuration of the gantry crane model is specified by the horizontal position of the trolley, x, the length of the hosting rope, l, and the swing angle of the rope, Î¸. The payload, which is suspended from the point of suspension, S, is assumed to be a rigid body symmetric about its axis with mass m and centre point , G of mass m. Before the derivation of the equations of motion, some assumptions are made for simplicity. Firstly, friction force that may exist in the trolley is to be ignored. The trolley and the payload can be considered as point masses. Besides, the tension force that may cause the hoisting rope elongate is also neglected. The trolley and the payload are assumed to move in x-y plane, which means a study of two dimensional.

Figure 3.1 Model of a Gantry Crane

MATLAB is a technical computing environment for high performance numeric computation and visualization. MATLAB integrates numerical analysis, matrix computation, signal processing and graphics in an easy-to-use environment where problems and solutions are expressed just as they written mathematically without traditional programming.

MATLAB can extend functionally and versatility with addition of optional application -specific toolboxes. Toolboxes are comprised of suites of MATLAB function (M-files) written by world-class authorities on each of the respective topic. MATLAB is an open environment for which many specialized toolboxes have been developed: Control System, Signal Processing, Optimization, Robust Control, Micro-analysis and Synthesis (micro-tools), System Identification, MMLE State Space Identification and Neural Network.

Simulink is a graphical environment for the modeling and simulating block diagram and general nonlinear systems. It is a companion program to MATLAB and an interactive system for simulating nonlinear dynamic system. It is a graphical program that allows to model a system by drawing a block diagram on the screen and manipulating it dynamically. It can work with linear, nonlinear, continuous, discrete-time, multivariable and multi rate systems.

Block sets are add-ins to Simulink that provide additional libraries of blocks for specialized applications like communications, signal processing, and power systems. Real-time workshop is a program that allows to generate C code from block diagrams and to run it on a variety of real-time system.

## 3.5 The Simulation Process Flow Chart

The propose simulation procedures can be summarized in Figure 3.2 below.

Start

Derive the dynamic 2D gantry crane model

Construct crane dynamic controller model

Establish connection between blocks (components)

Trolley position

Mass sway angle

Simulated response from disturbances

Utilized MATLAB to plot the response

Re run the simulation with an input shaper

Analyze the response

End

Figure 3.2: Proposed simulation procedure flow chart.

## 11

Problem definition

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Background research

Writing proposal

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Designing Robust Control System & Model Definition

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Run Simulation & Data analysis

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Report Writing & Presentation

Final report

Indicator:

Planning

Actual

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Figure 3.3 Master project progress Gantt Chart.