COMMAND SHAPING CONTROL OF A CRANE SYSTEM KING SHYANG SIEN

Transcription

1 COMMAND SHAPING CONTROL OF A CRANE SYSTEM KING SHYANG SIEN A thesis submitted in fulfilment of the requirements for the award of the degree of Master of Engineering (Electrical - Mechatronics and Automation Control) Faculty of Electrical Engineering Universiti Teknologi Malaysia NOVEMBER 006

2 Dedicated to my beloved family iii

3 iv ACKNOWLEDGEMENT Firstly, I would like to take this opportunity to express my sincere gratitude to my supervisor, Dr. Zaharuddin Mohamed, for his valuable guidance and advice throughout the development of this project. I also wish to thank to my friends and all the personal, whose have directly or indirectly played a part in the completion of this project. Last but not least, I would like to thank my beloved family who gave me the moral support and encouragement.

4 v ABSTRACT Cranes are widely used for transportation of heavy material in factories, warehouse, shipping yards, building construction and nuclear facilities. There are 3 major types of crane system: gantry (overhead) crane, rotary (tower) crane and boom crane. This project will concentrate in controlling of gantry crane. First of all, the equations of motion of the gantry crane system are derived by using Lagrangian approach. Then, the model of the gantry crane system is developed to represent the dynamic equations of motion in state space. SIMULINK is used to simulate the dynamic behaviours of the gantry crane. From the simulation, we noticed that the motion of the payload and trolley are unstable with occurrence of the oscillation. The system became undamped system when the input force is taken off. The system will swing on its varying frequencies in this condition. The challenge of this project is to develop a control algorithm for gantry crane system to reduce the oscillation or vibration of the payload and hook. Input shaping command controller is introduced in this project to control the crane system. Input shaping command controller is a feed-forward controller. This project had studied the performance of this designed controller in the crane system. With this controller the gantry crane system is able to transfer the load from point to point as fast as possible and, at the same time, the load swing is kept small during the transfer process and completely vanishes at the load destination.

5 vi ABSTRAK Kren digunakan secara meluasnya dalam pengakutan and pemindahan barang-barang berat dalam kilang, gudang, pembinaan dan juga kemudahan nuklear. Terdapat 3 jenis sistem kren iaitu: kren gantry, kren rotary dan kren boom. Projek ini akan menumpukan perbinacngan and kajian dalam pengawalan kren gantry. Persamaan pergerakan sistem kren gantry telah diterbitkan dengan menggunakan kaedah Lagrangian. Kemudian, model kren gantry dibentuk untuk mewakili persamaan dinamik pergerakan dalam state space. SIMNULINK telah dipilih dalam simulasi kren gantry untuk mengkaji sifat-sifat dinamik sistem kren gantry. Simulasi ini telah menunjukkan bahawa pergerakan beban dan troli akan jadi tidak stabil dengan kewujudan ayunan. Sistem kren akan menjadi sistem undamped apabila daya yang dikenakan ke atas kren diberhantikan. Sistem kren akan mengayun dengan pelbagai frekuensinya dalan situasi ini. Cabaran projek ini adalah untuk membentuk satu sistem kawalan untuk sistem kren gantry yang dapat mengurangkan ayunan atau getaran beban and tali. Sistem kwalan input shaping command telah diperkenalkan dalam projek ini untuk tujuan ini. Sistem kawalan input shaping command ialah teknik kawalan feed-forward. Projek ini telah mengkaji prestasi sistem kawalan yang diperkenalkan ini. Dengan kehadiran sistem kawalan ini, sistem kren gantry akan berupaya untuk menghantar beban ke destinasinya dengan pantas dan pada masa yang sama ayunan beban adalah yang paling minima dalam proses pergerakan ini.

6 vii TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS LIST OF ABBREVIATIONS ii iii iv v vi vii x xi xiii xv 1 INTRODUCTION Introduction 1 1. Project Objective 1.3 Project Scope 1.4 Methodology Thesis Outline 3 LITERATURE REVIEW 5.1 Type Of Crane Gantry Crane 6.1. Tower Crane Boom Crane 7. Details About Gantry Crane 8.3 Literature Review Of the Crane Controller Open Loop Techniques 10

7 viii.3. Closed Loop Techniques 11.4 Command Shaping Control 1 3 SYSTEM MODELING Gantry Crane Model Description Dynamic Equation Derivation Linearization Of The System State Space Representation Of The System 0 4 DYNAMIC BEHAVIORS SIMULATION AND DISCUSSION Differential Equation Editor (DEE) 3 4. Dynamic Behaviours Simulation of Gantry Crane Dynamic Behaviours Simulation Results Sway Motion of Gantry Crane Trolley Position Sway Angle Natural Frequency System Analysis and Discussion Input Force, F Length of The Hoisting Rope,l Trolley Mass, M Payload Mass, m Summary 40 5 INPUT SHAPING CONTROLLER DESIGN AND SIMULATION Input Shaping Technique Zero Residual Vibration Constraints Robustness constraints Unity Magnitude Summation Constraints Time Optimality Constraints Input Shapers Simulation Zero-Vibration Shaper (ZV) Zero-Vibration-Derivative Shaper (ZVD) Zero-Vibration-Derivative-Derivative Shaper (ZVDD) Simulation Results Analysis and Discussion 56

8 ix ZV Shaper ZVD Shaper ZVDD Shaper Summary 61 6 NON LINEAR SYSTEM Non-Linear System s Equation Simulation of Non-Linear System Simulation Result Analysis and Discussion Summary 68 7 CONCLUSION AND FUTURE WORK Conclusion Future Work 71 REFERENCES 7

9 x LIST OF TABLES TABLE NO. TITLE PAGE 4.1 Comparison of Trolley Position and Sway Angle With different System Parameter Setting Amplitudes and Time Locations of ZV Shaper Amplitudes and Time Locations of ZVD Shaper Amplitudes and Time Locations of ZVDD Shaper Performance Comparison Among Input Shapers 6

10 xi LIST OF FIGURES FIGURE NO. TITLE PAGE.1 Gantry Crane 6. Rotary or Tower Crane 7.3 Boom Crane 8.4 Illustration of A Gantry Crane 9.5 Block Diagram of Input Shaping Command Technique Model of Gantry Crane Block Diagram of the Dynamic of Gantry Crane System 4 4. Bang-bang Input Force by Step Response Bang-bang Input Force by Pulse Generator DEE for Gantry Crane System Sway Motion when Input Force is Positive Sway Motion when Input Force is Negative Bang-bang Input Force, 1N Trolley Position (m) Sway Angle (rad) Bang-bang Input Force at 1N, 0.5N and 5N Trolley Position and Sway Angle With Different Input Force, F Trolley Position and Sway Angle With Different Length of Hoisting Rope,l Trolley Position and Sway Angle With Different Trolley Mass, M Trolley Position and Sway Angle With Different Payload Mass, m Generic Input Shaping Process ZV Shaper Convolution With ZV Shaper 50

11 xii 5.4 Block Diagram of ZV shaper ZVD Shaper Convolution With ZVD Shaper Block Diagram of ZVD shaper ZVDD Shaper Convolution With ZVDD Shaper Block Diagram of ZVDD shaper Block Diagram for Gantry Crane Simulation Response of Trolley Position with ZV Shaper Response of Sway Angle with ZV Shaper Response of Trolley Position with ZVD Shaper Response of Sway Angle with ZVD Shaper Response of Trolley Position with ZVDD Shaper Response of Sway Angle with ZVDD Shaper Trolley Position Response Comparison Sway Angle Response Comparison DEE for Non-Linear Gantry Crane System Block Diagram of ZVDD shaper Block Diagram for Non-Linear Gantry Crane Simulation Simulation Result of Non-Linear System With ZVDD Shaper 68

12 xiii LIST OF SYMBOLS Fx - Input force (N) l - Length of hoisting rope (m) M - Trolley mass (kg) m - Payload mass (kg) G - Centre point S - Point of suspension g - Gravitational acceleration (9.81ms-) rm - Position vector of centre point, G ro - Position vector of point of suspension, S xm - Horizontal position of the payload ym - Vertical position of the payload x - Trolley position (m) x& - Trolley velocity (m/s) & x& - Trolley acceleration (m/s ) - Sway angle (rad) & - Sway angle velocity (rad/s) & - Sway angle acceleration (rad/ s ) T - Kinetic energy P - Potential energy ω n - Natural frequency ζ - Damping ratio A - Impulse s amplitude t o - Time of impulse t N - Time of last impulse

13 xiv LIST OF ABBREVIATIONS DEE - Differential Equation Editor ZV - Zero-Vibration ZVD - Zero-Vibration-Derivative ZVDD - Zero-Vibration-Derivative-Derivative PD - Propotional-Derivative

14 CHAPTER 1 INTRODUCTION 1.1 Introduction Cranes are widely used for transportation of heavy material in factories, warehouse, shipping yards, building construction and nuclear facilities. In order to lift heavy payloads in factories, in building construction, on ships and etc, cranes usually have very strong structures. Crane system is tends to be highly flexible in nature, generally responding to commanded motion with oscillations of the payload and hook. The response of this system to external disturbances such as wind is also oscillatory in nature. The swaying phenomenon introduce not only reduce the efficiency of the crane, but also cause safety problem in the complicated working environment. Previously, all the cranes were manually operated. But manual operation became difficult when cranes became larger, faster and higher. Due to this, efficient controllers are applied into the cranes system to guarantee fast turn over time and to meet safety requirement.

15 1. Project Objective There are two main objectives for this project. The first objective is to investigate the dynamic behaviour of the gantry crane system. The mathematical model of a gantry crane is obtained by Lagrangian technique. Then, MATLAB and SIMULINK are used to simulate the sway angle and trolley position of the crane system. By observing the motions of the gantry crane system, the dynamic behaviours are studied. The second objective of this project is to develop an effective control algorithm for the crane system to reduce the oscillation or vibration of the payload and hook. Command shaping control technique is the target controller that will be studied in this project. This designed controller should be able to transfer the load from point to point as fast as possible and, at the same time, the load swing is kept small during the transfer process and completely vanishes at the load destination. 1.3 Project Scope The scopes of this project are: The project topic understanding. Search for the relevant material. Study and analysis of the mathematical model of the gantry crane system. Simulate and investigate the dynamic performance of the gantry crane system. Study and understand the concept of an effective control algorithm for the gantry crane system. Design and develop the proposed controller mathematically.

16 3 Implement the designed controller by MATLAB. Simulate the dynamic performance of the gantry crane system after the designed controller was implemented. Performance analysis. Project report write-up. 1.4 Methodology Gantry crane system literature review. Understand the mathematical model developed. Study the performance of the response of the gantry crane system. Develop effective control algorithms. Investigate the performance of the controllers. Improve the performance of the developed controllers. 1.5 Thesis Outline This report can be divided into seven chapters. Chapter 1 is the introduction of the project objective, scope, methodology and the report outline. Next some information about crane system and also an overview of the literature that has been published in relation to crane control will be covered in Chapter. In Chapter 3 the mathematical model of a gantry crane system will be derived.

17 4 The system model will be represented in state space for analysis. Following chapter will describe how the MATLAB and SIMULINK are used to simulate the dynamic behaviors of the gantry crane. The discussion on the simulation results will be described in this chapter as well. The development and design of the input shaping controller to reduce the vibration of the gantry crane will be included in Chapter 5. This chapter will include the discussion on the performance of the designed controller based on SIMULINK simulation result. Non-linear gantry crane system will be studied in this project. Chapter 6 will describe the non-linear gantry crane system and its simulation results. The last chapter, we will conclude the works done in this project and will include the suggestion for improvement.

18 CHAPTER LITERATURE REVIEW This chapter consists of some information about crane system and also an overview of the literature that has been published in relation to crane control..1 Type Of Crane A crane consists of a hoisting mechanism such as hook and a support mechanism such as trolley girder. The hoisting mechanism has two main functions. It deposits the payload at the target destination and avoids the obstacle in the path by lifts and lowers the payload. The function of the support mechanism is moves the suspension point around the crane workspace. Crane can be classified based on the degree of freedom the support mechanism offer the suspension point. There are 3 major types of crane system: (a) Gantry (overhead) crane (b) Rotary (tower) crane (c) Boom crane.

19 6.1.1 Gantry Crane Gantry crane is composed of a trolley moving in a girder along a single axis. In some gantry crane, the girder is mounted on the second set of orthogonal railings, adding another degree of freedom of the horizontal plane. Gantry crane is commonly used in factories, Figure.1. Figure.1: Gantry Crane.1. Tower Crane Tower crane is commonly used in construction, Figure.. In this crane, the girder rotates in the horizontal plan about a fixed vertical axis. The trolley that holds the load can move in radial position over the girder. The load is attached to the trolley using a set of cables.

20 7 Figure.: Rotary or Tower Crane.1.3 Boom Crane For the boom crane, a boom is attached to a rotating base. The rotational movement of the base along with the elevation movement of the boom places the boom tip over any point in the horizontal plane. The load hangs from the tip of the boom by a set of cables and pulleys. The radial and vertical positions of the load can be changed by changing the elevation angle of the boom. Boom cranes are very common on ships and in the harbours, Figure.3.

21 8 Figure.3: Boom Crane. Details About Gantry Crane There are three main components in a gantry crane which are trolley, bridge and gantry. Figure.4 shows a typical gantry crane. Trolley with a movable or fixed hoisting mechanism is the load lifting component. It moves on and parallel to a bridge which is rigidly affixed to a supporting structure called gantry. The gantry extends downward from the bridge to the ground where it can be mobilized on wheels or set of tracks. The motion of the gantry on the ground, the trolley on the bridge and the hoisting of the payload provide the 3 degrees of freedom of the payload.

22 9 Figure.4: Illustration of A Gantry Crane This type of system tends to be highly flexible in nature, generally responding to commanded motion with oscillations of the payload and hook. The response of these systems to external disturbances, such as wind, is also oscillatory in nature. The swaying phenomenon introduce not only reduce the efficiency of the crane, but also cause safety problem in the complicated working environment. This project will concentrate in controlling of gantry crane to reduce the vibration of the crane system.

23 10.3 Literature Review Of the Crane Controller This section gives an overview of the literature that reviewed the different approaches of the gantry crane control system. These controllers extend from simple open loop techniques to more complex closed loop control techniques..3.1 Open Loop Techniques The most advanced and practical crane controllers today are controllers based on an open loop approach designed to automated and shorten the cycle time for the gantry cranes operating along a pre-defined path. The open loop approach is intended not to excite the swing motion. It is simple to implement but cannot suppress swing motion caused by external disturbance. The most widely used of the open loop control techniques is input-shaping (Eihab, Ali and Ziyad, 001). Alsop et al. (1965) were the first to propose a strategy to control payload pendulations. The controller accelerates the trolley in steps of constant acceleration then kills the acceleration when the payload reaches zero-pendulation angle and lets the trolley coast at a constant travel speed along the path. The same procedure is replicated in the deceleration stage. An iterative procedure is used to calculate the acceleration profile of the trolley (Eihab, Ali and Ziyad, 001). Jones and Petterson (1988) extended the work of Alsop et al. (1965) using non-linear approximation of the cable-payload period to generate an analytical expression for the duration of the coasting stage as a function of the amplitude and duration of the constant acceleration steps. This analytical expression is then used to generate a two step acceleration profile. However it was not able to dampen out

24 11 initial disturbances of the payload and could even amplify them (Eihab, Ali and Ziyad, 001). Karnopp et al. (199) proposed an input-shaping strategy based on cam design techniques. Given the distance of the desired trolley travel and the pendulation natural frequency, it produces a prescribed input position of the trolley to deliver the payload residual-pendulation free at the target position (Eihab, Ali and Ziyad, 001)..3. Closed Loop Techniques The closed loop approach of swing motion provides the ability to eliminate existing swing motion and rejects disturbance at the cost of extra sensors. There are different types of closed loop approach that developed to control crane system such as linear control, adaptive control, fuzzy logic control and non-linear control. Hazlerigg (197) was the first to propose a feedback control strategy. It employed a second-order lead compensator to dampen the payload pendulations. Experimental verifications of the strategy showed that, while it dampened the payload pendulations at the natural frequency of the cable-payload assembly, it introduced pendulations at higher frequencies (Eihab, Ali and Ziyad, 001). Ridout (1987, 1989) proposed a feedback controller using negative feedback of the trolley position and velocity and positive feedback of the pendulation angle to eliminate residual payload pendulations at a constant cable length (Eihab, Ali and Ziyad, 001).

25 1 Yu et al. (1995) used a perturbation technique, the method of averaging to separate the slow and fast dynamics of the gantry crane model. Two independent PD controllers are then applied. The first is a slow-input controller applied to the trolley to maintain tacking of the pre-defined motion profile. The second is a fast input controller to suppress payload pendulations (Eihab, Ali and Ziyad, 001)..4 Command Shaping Control Command shaping control mechanism will be introduced to reduce the oscillation of the payload in this project. Command shaping does not require the feedback mechanism of closed-loop controllers. Vibration suppression is accomplished with a reference signal that anticipates the error before it occurs, rather than with a correcting signal that attempts to restore the system back to a desired state. This means we do not need any sensing of the payload sway when command shaping technique is used. So, command shaping technique which is a feed-forward controller is an easier solution in vibration reduction compared to feedback controller. Conventional command shaping technique involves signal filtering such as Hamming and Butterworth filters. However the conventional technique has been marginally effective at reducing residual oscillation in mechanical system and it will increase the system s settling time as well. Input shaping command is another form of command shaping that effectively improves the transient and steady state response of mechanical system. It involves modifying a reference command in such a way that resonant modes of a system

26 13 combine destructively, resulting in low residual oscillation. In this project, input shaping command is the control technique that we select to control the gantry crane. Figure.5 shows the block diagram of input shaping command technique controller configuration. We will discuss more detail about input shaping command in Chapter 5. Bang- bang Input Input Shaper Shaped Input Gantry Crane System Output Responses Figure.5: Block Diagram of Input Shaping Command Technique

27 CHAPTER 3 SYSTEM MODELING This chapter will describe the derivation of the mathematical model of a gantry crane. The system model will be represented in state space for analysis. 3.1 Gantry Crane Model Description Figure 3.1 shows the model of a gantry crane, where, x = Horizontal position of the trolley (m) l = Length of the hoisting rope (m) = Swing angle of the rope (rad) M = Mass of the trolley (kg) m = Mass of the payload (kg)

28 15 Figure 3.1: Model of Gantry Crane Below are the assumptions made to simplify the process system modelling. (a) Ignored the trolley friction force. (b) The trolley and the payload can be considered as point masses. (c) Ignored the tension force that may cause the hoisting rope elongate. (d) The trolley and the payload are assumed to move in two dimensional only, x-y plane.

29 16 There are four main variables needed to control the gantry crane system: (a) angular deviation ( ) (b) angular velocity ( & ) (c) cart position ( x ) (d) cart velocity ( x& ) 3. Dynamic Equation Derivation Lagrangian approach is used to derive the equation of motion. Based on Figure 3.1, the load and trolley position vectors are given by r r m o = { x l sin, l cos } { x,0 } = (3.1) The kinetic and potential energy of the whole system are given by T = Ttrolley Tpayload 1 1 = Mr& o mr& m 1 1 = Mx& m x (& l& l & xl & & sin xl & & cos ) (3.) P = mgy m = mglcos (3.3)

30 17 Using the Lagrangian approach, the following equation is derived as L = T P = Mx& 1 m x (& l& l & xl & & sin xl & & cos ) mgl cos 1 (3.4) Let the generalized forces corresponding to the generalized displacements q = { x, } be {,0 } F =. Constructing the Lagrangian L = T P and using F x Lagrangian s equations d dt L q& j L q j = F j j =1, (3.5) We will obtain the equation of motion for the gantry crane system. Firstly, the equation of motion associate with the generalized coordinate q = x can be derived as below: d L L dt x& x = Fx L = 0 x ( x& l& sin l & cos ) L 1 = Mx& m x& = Mx& mx& ml& sin ml & cos d dt L = x& = = Mx && mx && m( && l sin l& & cos ) m[ l& & cos l && cos l & ( sin )& ] ( M m) && x m( && l sin l& & cos ) m( l& & cos l && cos l & sin ) ( M m) && x ml( && cos & sin ) ml& & cos ml && sin

31 18 Thus, d L 0 = dt x& Fx F x = ( M m) & x ml( && cos & sin ) ml& & cos ml & sin (3.6) Secondly, the equation of motion associate with the generalized coordinate q = is as below: d dt L L = 0 & L 1 = m 1 = m L = m & d dt [ xl && cos xl & & ( sin )] mgl( sin ) ( xl & & cos xl & & sin ) mgl sin ( l & xl & cos ) 1 L = & = 1 1 m m [ l && l& & ( l) && xl cos xl & & cos xl & & ( sin )] ( l && 4ll& & && xl cos xl && cos xl & & sin ) Thus,

32 19 d dt L L = 0 & 1 ( l && 4ll& & && xl cos xl & & cos xl & & sin ) m( xl & & cos xl & & sin ) 1 m mgl sin = 0 l && 4ll& & && xl cos xl && cos xl & & sin xl && cos xl & & sin gl sin = 0 l && 4l& & && x cos g sin = 0 l & l& & & x cos g sin = 0 (3.7) The equations of motion of the gantry crane model associated with the generalized coordinates q = { x, } can be summarized, respectively as: x: = ( M m) & x ml( && cos & sin ) ml& & cos ml & sin F x : l & l& & & x cos g sin = Linearization Of The System The above derived model is a nonlinear model. The nonlinear model has to be linearized to simplify the progress of the modelling. For safe operation, two assumptions had been made. First we assume that the swing angle should be kept small. 0 & 0 In this study, we assume that changing the rope length is needed only to avoid

33 0 obstacles in the path of the load. This change can be considered small too. l & & l 0 Using these two assumptions, the simplified equation of motion for the gantry crane system can be obtained. x: F M m & x ml & x = ( ) (3.8) : l & & & x g = 0 (3.9) 3.4 State Space Representation Of The System After getting the linearized equation, equation (3.8) and (3.9) can be written in state space representation. x & = Ax Bu Where, x x = &x &

34 1 x& & x & = && x && From equation (3.9), & x = l & g (3.10) Substitute equation (3.10) into (3.8) will get the following equation, F M m l & g ml & x = ( )( ) & M m Fx = g Ml Ml (3.11) Substitute equation (3.11) into equation (3.8), F x = ( M M m Fx m)&& x ml g Ml Ml Fx m & x = g (3.1) M M

35 Equation (3.11) and (3.1) can be arranged into the matrix form as below: Bu Ax x = & F x Ml M x x Ml g M M m mg x x = ) ( & & && && & & (3.13) The output equation is, Du Cx y = = & &x x x (3.14)

36 CHAPTER 4 DYNAMIC BEHAVIORS SIMULATION AND DISCUSSION This chapter will describe how the MATLAB and SIMULINK is used to simulate the dynamic behaviors of the gantry crane. The discussion on the simulation results will be described in this chapter as well. 4.1 Differential Equation Editor (DEE) MATLAB is a high level technical computing environment suitable for solving scientific and engineering problems. MATLAB can be used to analyze and design control systems problems. Differential Equation Editor is one of the tools in MATLAB. DEE allows us to directly input the differential equations. The differential equations should be re-written in the standard form: dx = f ( x, u, p) ; y = g( x, u, p) dt

37 4 In other words, the LHS of the state equation should only contain derivatives and all other terms should be taken over to the right hand side of the equation. 4. Dynamic Behaviours Simulation of Gantry Crane Figure 4.1 below shows the block diagram of gantry crane system that we used to simulate the dynamic behaviour of the gantry system in this project. As mentioned above, the gantry crane system is represented by the DEE tool. Figure 4.1: Block Diagram of the Dynamic of Gantry Crane System The input to the gantry crane system is a bang-bang torque. Figure 4. and Figure 4.3 show the block diagram of the bang-bang input force. We can generate the bang-bang input force by either method shown in figure below. This bang-bang input force is applied to the trolley of the gantry crane system. Three scopes are used to capture the torque, trolley position and sway angle response.

38 5 Figure 4.: Bang-bang Input Force by Step Response Figure 4.3: Bang-bang Input Force by Pulse Generator Figure 4.4 shows the DEE box where we entered the state space equation of the gantry crane system.

39 6 Figure 4.4: DEE for Gantry Crane System 4.3 Dynamic Behaviours Simulation Results This section will describe the response of the gantry crane which based on fixed parameters.

40 Sway Motion of Gantry Crane F F - Figure 4.5: Sway Motion when Input Force is Positive Figure 4.6: Sway Motion when Input Force is Negative Above figures show the sway angle of the gantry crane, when input force, F is applied to the trolley. The initial position has an angle of zero. When a positive input force is applied to the trolley as showed in Figure 4.5, the payload will swing clock wise direction. The sway angle will be a negative value. Vice versa, the payload will swing anticlockwise direction when the input force is negative as showed in Figure 4.6. The sway angle will be a positive value. The initial position point has a maximum kinetic energy, whereas the final position point has maximum potential energy. used. For the simulation in this project, the following gantry crane parameters are Trolley mass, M = 1kg Payload mass, m = 0.8kg Length of hoisting rope, l = 0.305m Gravity, g = 9.81m/s Input force, F = 1N

41 8 Figure 4.7 shows the bang-bang input force that applied to the trolley. Figure 4.7: Bang-bang Input Force, 1N

42 Trolley Position Figure 4.8: Trolley Position (m) When input force is applied to the trolley, the trolley will travel horizontally along its path. Graph above shows that the trolley reached around 0.58m at s. The input force will be taken away after s. The oscillation after s is due to the non-stop oscillation of the payload.

43 Sway Angle Figure 4.9: Sway Angle (rad) Due to the input force applied is positive, the payload will swing at clock direction at the starting of the simulation. The payload will continue oscillate although the force is taken off after s. This is because the gantry crane system that we modelled is without damping. The payload oscillates with a maximum value of 0.16rad which is around 9. degree from its initial position. The sway angle will increase when the length of the hoisting rope become longer. We must avoid the increasing of the sway angle because this will cause safety problems in practical.

44 Natural Frequency The frequency of oscillation of a system if all the damping is removed is defined as natural frequency. Gantry crane that we study in this project is a vibrating system. When an external force is applied to the gantry crane, it will oscillate at the particular frequency of the external force. However if the external force is taken off from the undamped system, the system will continue oscillate but with its own particular frequency which is its natural frequency. Natural frequency of the gantry crane can be calculated using below formula: g m ω = n 1 l M (4.1) Based on the parameters that specified in Section 4.3.1, the natural frequency of the gantry crane system that we analyze is as below: ω n = 1 = 7.5rad / s (4.)

45 3 4.4 System Analysis and Discussion The effect of input force, length of the hoisting rope, trolley mass and payload mass on the dynamic behaviour of the gantry crane are investigated Input Force, F Figure 4.10: Bang-bang Input Force at 1N, 0.5N and 5N.

46 Figure 4.11: Trolley Position and Sway Angle With Different Input Force, F 33

47 34 Figure 4.10 shows the different input force that applied to the system. Figure 4.11 shows the response of the system in term of trolley position and sway angle when different input force applied to the system. We notice that when the input force decreased from 1N to 0.5N the trolley will only be able to reach around 0.3m from its initial position. The magnitude of the payload swing is decreased to approximate 0.08rad. However the frequency of the payload swing remained the same. When the input force is 5N, the trolley is able to travel more distance where it can reaches around.9m from its initial position. However the vibration of the trolley is greater. The magnitude of the payload swing also increased to around 0.8rad. But the sway frequency of the payload remains the same.

48 Length of The Hoisting Rope,l Figure 4.1: Trolley Position and Sway Angle With Different Length of Hoisting Rope,l

49 36 Trolley still can reach at 0.58m at s although the length of the hoisting rope decreased from 0.305m to 0.m. However the fluctuation of the trolley increased with shorter rope. This is due to the payload oscillation increased when the rope is shorter. From Figure 4.1, we can see that the magnitude and frequency of the payload swing is greater. For the length of the hoisting rope increased from 0.305m to 0.5m, the payload swing magnitude is lower after the input force is taken off at s. With the lower or smaller oscillation of the payload, the trolley can reach the same location with more stable.

50 Trolley Mass, M Figure 4.13: Trolley Position and Sway Angle With Different Trolley Mass, M

51 38 When the trolley mass, M decreased from 1kg to 0.5kg or the trolley mass is smaller than the payload mass (M<m), the magnitude and frequency of the payload swing is the greatest in this simulation. Due to the trolley which is lighter than the payload, the trolley position can reach further distance but with it will vibrate together with payload where cause the trolley position fluctuates with more frequency. If the trolley mass is same as the payload mass (M=m=0.8kg), the trolley position can reach around 0.65m with greater oscillation. The payload swing at higher magnitude and frequency. Graph above shows that when M >> m, the payload will not has much effect on the system. The trolley movement distance decreased to 0.45m and the trolley seems to be more stable after the input force had been taken off after s. The magnitude of the payload swing also decreased.

52 Payload Mass, m Figure 4.14: Trolley Position and Sway Angle With Different Payload Mass, m

53 40 Figure 4.14 shows the performance of the system when we increase and decrease the payload mass, m. When payload mass is decreased from 0.8kg to 0.kg, trolley can easily reach farer location where the trolley position is increased to around 0.7m. Based on figure above, the fluctuation of the trolley position is lesser compared to initial system. This is due to the payload swing is greatly decreased and almost stable after the input force is taken off at s. For heavier payload where the payload mass increased from 0.8kg to 1.6kg, trolley will only reached at approximate 0.4m after s. The fluctuation of the trolley is also very high. For the payload swing, the magnitude at the beginning stage is decreased. However after the input force had been taken off, the swing magnitude and frequency of the payload were increased significantly. This simulation shows that when M << m, the payload will have significant effect on the system. 4.5 Summary From the simulation results, we noticed that the motion of the payload and trolley are unstable with the occurrence of the oscillation. The system became undamped system when the input force is taken off. The system will swing on its varying frequencies in this condition. By changing of the system parameters such as payload mass (m), trolley mass (M), length of the hoisting rope (l) and the input force (F), the dynamic performance of the gantry crane will be affected. The swing magnitude and the travelling distance will change when the system parameters changed. Table below shows the summary of the simulation results of the dynamic performance of the gantry crane with different kind of parameter setting.

54 41 So, it is very important to have an effective control mechanism to reduce the oscillation of the payload because great oscillation will causes damage on the environment and also human around. Trolley Position Sway Angle Input Force (F ref = 1N) Length (l ref = 0.305m) Trolley Mass (M ref = 1kg) Payload Mass (m ref = 0.8kg) Decrease (0.5N) Increase (5N) Decrease (0.m) Increase (0.5m) M < m (0.5kg) M = m (0.8kg) M > M ref (1.5kg) Decrease (0.kg) Increase (1.6kg) Magnitude Frequency Distance Magnitude Frequency Lower Same Lower Lower Same Greater Same Greater Greater Same Greater Greater Same Greater Greater Lower Lower Same Lower Lower Greatest Greatest Greatest Greatest Greatest Greater Greater Greater Greater Greater Lower Lower Lower Lower Lower Lower Lower Greater Lower Lower Greater Greater Lower Greater Greater Table 4.1: Comparison of Trolley Position and Sway Angle With different System Parameter Setting

55 CHAPTER 5 INPUT SHAPING CONTROLLER DESIGN AND SIMULATION This chapter is about the development and design of the input shaping controller to reduce the vibration of the gantry crane. This chapter will include the discussion on the performance of the designed controller based on SIMULINK simulation result. 5.1 Input Shaping Technique Input shaping command is another form of command shaping that effectively improves the transient and steady state response of mechanical system. It involves modifying a reference command in such a way that resonant modes of a system combine destructively, resulting in low residual oscillation. In input shaping, an input shaper which is a sequence of impulses will be convoluted with an arbitrary reference signal to produce a shaped command that is used to drive a system. This process is illustrated in Figure 5.1 with an input shaper composed of two impulses. This input shaper will be convolved with the unshaped input to generate the shaped input where this shaped input has the same oscillation

56 43 reducing properties as the original set of impulses. Unshaped Input Input Shaper Shaped Input Figure 5.1: Generic Input Shaping Process In general, an input shaper can contain any number of impulses which have any amplitude and duration. The amplitudes and time location of the impulses are obtained from the system s natural frequencies and damping ratios. The selection of the correct impulse amplitudes and time location of the impulses is a challenge in input shaping design to ensure the system designed is without vibration to any unshaped command. This amplitudes and time location of the impulses are determined by solving a set of constraint equations. The constraint equations are usually categorized as below: (a) Zero Residual Vibration Constraints (b) Robustness Constraints (c) Unity Magnitude Summation Constraints (d) Time Optimality Constraints.

57 Zero Residual Vibration Constraints This constraint is derived based on the concept that the impulse sequence must achieve zero vibration after the last impulse. Firstly, we assume the system can be modelled as a second-order harmonic oscillator, or can be decomposed into a set of second order systems, then the vibration ratio can be determined from the expression for residual vibration amplitude from a single impulse as below. Aωn y( t) = exp( ζω n ( t t0 ))sin( ωn 1 ζ ( t t0 )) 1 ζ (5.1) where A is the amplitude of the impulse, t 0 is the time the impulse is applied, ω is the natural frequency and ζ is the damping ratio. The residual single-mode vibration amplitude of the response is obtained at the time of the last impulse, t N as V = (5.) V 1 V where, V A ω N i n 1 = exp( ζω n ( t N i d i i= 1 1 ζ t ))cos( ω t ) (5.3) V A ω N i n = exp( ζω n ( t N ti d i i= 1 1 ζ ))sin( ω t ) (5.4)

58 45 To achieve zero vibration after the last impulse, it is required the both V 1 and V are independently zero. This constraint only guarantees no residual vibration if the system natural frequency and damping ratio are correct. below. The constraint equations of the zero residual vibration constraint are as N Aiω n V1 = exp( ζω n ( t N ti ))cos( ωdti ) = 0 i= 1 1 ζ (5.5) N Aiω n V = exp( ζω n ( t N ti ))sin( ωdti ) = 0 i= 1 1 ζ (5.6) 5.1. Robustness constraints Due to the system parameters are never precisely known, some robustness to system uncertainty must be incorporated. By simplifying the first derivative of V 1 and V, the constrain equations of the robustness constraints will be obtained as dv1 dω n N = Ait i= 1 i exp( ζω ( t n N t ))sin( ω t i d i ) = 0 (5.7) dv dω n N = Ait i= 1 i exp( ζω ( t n N t i ))cos( ω t ) = 0 d i (5.8)

59 46 The level of robustness can be further increased by increasing the derivative order of V 1 and V. Overall, the i-th level of the robustness constraints can be represented as i d V 1 = 0 dω n (5.9) i d V = 0 dω n (5.10) Unity Magnitude Summation Constraints This constraint is to ensure that the shaped command produces the same rigid body motion as the unshaped command. To satisfy this requirement, the summation of impulse amplitude must be equal to one. N A n i= 1 = 1 (5.11) where n is the number of impulses.

60 Time Optimality Constraints When the shapers convolved with the unshaped input, the shaped input will be delayed by the length of the shaper. To avoid or minimize the delay of te shaped input response, we require time optimality constraint where the duration of the shaper must be made as short as possible. The time optimality constraint is as below. min( t N ) (5.1) where t N is the time of the final impulse. 5. Input Shapers Simulation In this project we will discuss about three types of input shaper which are zero-vibration (ZV) shaper, zero-vibration-derivative (ZVD) shaper and zero-vibration-derivative-derivative (ZVDD) shaper. As mentioned is section above, the amplitudes and time location of the impulses are obtained from the system s natural frequencies and damping ratio. The natural frequency of the gantry crane is obtained from previous section. Based on the calculation the natural frequency is 7.5rad/s (1.Hz). The damping ratio is zero because the gantry crane model that developed in this project is an undamped system. The simulation parameters are as below.

61 48 Trolley mass, M = 1kg Payload mass, m = 0.8kg Length of hoisting rope, l = 0.305m Gravity, g = 9.81m/s Input force, F = 1N Natural frequency, ω n = 7.5rad/s or 1.Hz Damping ratio, ζ = Zero-Vibration Shaper (ZV) Zero Vibration (ZV) shaper consists of two impulses. ZV shaper does not take robustness constraint into account. It only considers zero residual vibration, unity amplitude summation and time optimality constraints. By solving the constraints equations, the amplitudes and time locations of the ZV shaper are obtained as below. Figure 5. shows the ZV shaper signal and Table 5.1 is the ZV shaper s amplitudes and time location based on the gantry crane s natural frequency and damping ratio.. A A 1 1 = 1 K K = 1 K at at t t 1 = 0 Π = ω 1 ζ n (5.13)

62 49 Figure 5.: ZV Shaper Impulse Magnitude Time Location (s) Table 5.1: Amplitudes and Time Locations of ZV Shaper For gantry crane simulation, the ZV shaper will be convoluted with the input force of the system which is the bang-bang input to generate the shaped input for the system. This is illustrated in Figure 5.3. The block diagram of the ZV shaper is illustrated in Figure 5.4.

63 50 Figure 5.3: Convolution With ZV Shaper Figure 5.4: Block Diagram of ZV shaper

64 Zero-Vibration-Derivative Shaper (ZVD) Zero Vibration-Derivative (ZVD) shaper consists of three impulses. The constraints equations considered in designing of ZVD shaper are zero residual vibration, unity amplitude summation, time optimality constraints and first order robustness constraint equation. By solving the constraints equations, the amplitudes and time locations of the ZVD shaper are obtained as below. Figure 5.5 shows the ZVD shaper signal and Table 5. is the ZVD shaper s amplitudes and time location based on the gantry crane s natural frequency and damping ratio ζ ω ζ ω Π = = Π = = = = n n t at K K K A t at K K K A t at K K A (5.14) Figure 5.5: ZVD Shaper

65 5 Impulse Magnitude Time Location (s) Table 5.: Amplitudes and Time Locations of ZVD Shaper For gantry crane simulation, the ZVD shaper will be convoluted with the input force of the system which is the bang-bang input to generate the shaped input for the system. This is illustrated in Figure 5.6. The block diagram of the ZV shaper is illustrated in Figure 5.7. Figure 5.6: Convolution With ZVD Shaper

66 53 Figure 5.7: Block Diagram of ZVD shaper 5..3 Zero-Vibration-Derivative-Derivative Shaper (ZVDD) Zero Vibration-Derivative-Derivative (ZVDD) shaper consists of four impulses. The constraints equations considered in designing of ZVDD shaper are zero residual vibration, unity amplitude summation, time optimality constraints and second order robustness constraint equation. By solving the constraints equations, the amplitudes and time locations of the ZVDD shaper are obtained as below. Figure 5.8 shows the ZVDD shaper signal and Table 5.3 is the ZVDD shaper s amplitudes and time location based on the gantry crane s natural frequency and damping ratio.

67 ζ ω ζ ω ζ ω Π = = Π = = Π = = = = n n n t at K K K K A t at K K K K A t at K K K K A t at K K K A (5.15) Figure 5.8: ZVDD Shaper Impulse Magnitude Time Location (s) Table 5.3: Amplitudes and Time Locations of ZVDD Shaper

68 55 For gantry crane simulation, the ZVDD shaper will be convoluted with the input force of the system which is the bang-bang input to generate the shaped input for the system. This is illustrated in Figure 5.9. The block diagram of the ZVDD shaper is illustrated in Figure Figure 5.9: Convolution With ZVDD Shaper Figure 5.10: Block Diagram of ZVDD shaper

69 Simulation Results Analysis and Discussion MATLAB and SIMULINK are used to simulate the gantry system with the shaped input force. Figure 5.11 below shows the block diagram for this simulation. Figure 5.11: Block Diagram for Gantry Crane Simulation

70 ZV Shaper Figure 5.1: Response of Trolley Position with ZV Shaper Figure 5.13: Response of Sway Angle with ZV Shaper

71 58 Figure 5.1 and Figure 5.13 show the response of trolley position and sway angle of a gantry crane system with ZV shaper. For the trolley position, we noticed that the fluctuation is significantly reduced after the input force is taken off at s. But the response of the system is slower than original system where the trolley needs more time to travel to its destination. The rise time for the system with ZV shaper is around 1.3s. For the payload swing, we noticed that the maximum amplitude of the sway angle is reduced from 0.16rad to 0.06rad. The oscillation of the payload is significantly reduced ZVD Shaper Figure 5.14: Response of Trolley Position with ZVD Shaper

72 59 Figure 5.15: Response of Sway Angle with ZVD Shaper The response of the trolley position is shown in Figure The fluctuation or oscillation of the trolley is significantly reduced after the input force is taken off at s. But the response of the system is slower than system with ZV shaper where the rise time is around 1.4s. Based on Figure 5.15, the oscillation of the payload is reduced to zero when the input force is taken off. This means the payload is in a static position after the input force is taken off. The maximum sway amplitude is almost similar to ZV shaper which is 0.06rad.

73 ZVDD Shaper Figure 5.16: Response of Trolley Position with ZVDD Shaper Figure 5.17: Response of Sway Angle with ZVDD Shaper

74 61 Figure 5.16 shows that the fluctuation of the trolley is significantly reduced after the input force is taken off at s. But the response of the system is slowest among the three types of input shaper. The rise time for the system with ZVDD shaper is around 1.5s. For the payload swing, we noticed that the maximum amplitude of the sway angle is reduced from 0.16rad to 0.05rad which is the best among all the input shapers. The oscillation of the payload is suppressed effectively to zero when the input force is taken off. This means the payload is in a static position after the input force is taken off. 5.4 Summary The simulation results shows that the vibration of the gantry crane system can be reduced effectively by input shaper technique. The original system with unshaped bang-bang input force will oscillate at its own natural frequency when the input force is taken off. But this oscillation can be effectively reduce or even removed from the system if the bang-bang input force is shaped by the input shaper technique before applied or input to the gantry crane system. From the results above, the payload of the system with ZV shaper will still oscillate when the input force is taken off. System with ZVD and ZVDD shaper show that the payload oscillation is zero radian after the input force is taken off which payload will be static at the final destination. This means the input shaper with more number of impulses has better performance in reducing the oscillation of the payload.

75 6 However the disadvantage of the input shaper technique is that the response of the system will become slower. This means the trolley needs more time to reach its destination. Above simulation results show that ZV shaper has fastest response compared to ZVD and ZVDD shaper. The response of the ZVDD shaper is the slowest among these three shapers. This indicates that the increasing of the number of impulses for the input shaper will slower down the response of the system. Table 5.4 below shows the summary of the simulation results for gantry crane system with different type of input shaper applied. Input Shaper System Response Payload Oscillation Max Sway Amplitude Original Fastest Highest Highest ZV Shaper Slow Lower Lower ZVD Shaper Slower Zero Lower ZVDD Shaper Slowest Zero Lowest Table 5.4: Performance Comparison Among Input Shapers Based on the comparison in the table above, ZVDD shaper is the best command shaping controller for gantry crane system. Although its response is the slowest, the payload oscillation is zero and the max sway amplitude is the smallest among all the controllers. Its performance meets the objective of this project which is to reduce the oscillation or vibration of the payload. Figure 5.18 and Figure 5.19 are the system response comparison between ZV, ZVD and ZVDD shaper.

76 63 Figure 5.18: Trolley Position Response Comparison Figure 5.19: Sway Angle Response Comparison

77 CHAPTER 6 NON LINEAR SYSTEM This chapter will describe the non-linear gantry crane system and its simulation results. 6.1 Non-Linear System s Equation From Section 3., the equations of motion of the gantry crane model associated with the generalized coordinates q = { x, } can be summarized, respectively as: x: = ( M m) & x ml( && cos & sin ) ml& & cos ml & sin (6.1) F x : l & l& & & x cos g sin = 0 (6.) Equation 6. can be rewrite as below: l& & & && x cos g sin = (6.3) l

78 65 Substituting equation 6.3 into equation 6.1 gives, (6.4) sin sin sin cos ) sin (1 sin sin sin cos cos sin sin sin cos sin cos sin sin cos cos cos ) ( sin cos sin ) sin cos ( cos ) ( m M ml gm F x m m M ml ml gm F m m M ml ml gm F x ml ml ml gm xm ml l x m M ml ml ml l g x l ml x m M F x x x x = = = = = & && && & && & && && & & & && & & && && & & & && & & && Replace equation 6.4 into equation 6.3 will get following equation. (6.5) ) sin ( )) cos ( ( sin cos ) sin ( sin sin sin cos sin sin cos cos sin ) sin ( cos sin sin cos cos sin cos m M l m m l gm F m M l gm gm ml gm F l g m M l ml gm F l g x l x x x = = = = & && & & && & & && The non-linear gantry crane system can be represented with below equations. sin sin sin cos m M ml gm F x x = & & & ) sin ( )) cos ( ( sin cos m M l m m l gm F x = & & &

79 66 6. Simulation of Non-Linear System For simulation of the non-linear system, we used DEE tool in SIMULINK and MATLAB. The gantry crane system is represented by the DEE tool as shown in Figure 6.1. Figure 6.1: DEE for Non-Linear Gantry Crane System ZVDD shaper will be used in non-linear gantry crane system simulation. Same as linear system simulation, the ZVDD shaper will be convoluted with the input force of the system which is the bang-bang input to generate the shaped input for the system. This is illustrated in Figure 6.. The block diagram of the ZV shaper is illustrated in Figure 6.3.