Command shaping for vibration reduction of container cranes

2012 12th International Conference on Control, Automation and Systems Oct. 17-21, 2012 in ICC, Jeju Island, Korea1. INTRODUCTIONContainer cranes (or quay cranes) are used for vessel-to-truck and truck-to-vessel loading and unloading of containers at container terminals. Container cranes consist of a supporting structure called the gantry that can traverse the length of a quay or yard, a trolley, and a moving gripper called a spreader. The trolley runs along rails, which are located on the top or sides of the boom, transferring containers to or from the ship. The spreader can be lowered on top of a container, and locks on to the container's four locking points by means of twist-lock mechanisms. The container is then lifted and transferred onto a truck, which takes the container to a storage yard. The crane also unloads containers from the truck and transfers them to the ship. Cranes normally transport single containers; however, some new cranes are capable of loading/unloading up to four 20-foot containers at once.Since the movement of the trolley sways the container during transport, the main issue in container crane control is quick suppression of the vibrations caused by trolley motions at the trolley’s goal position. Additionally, a residual sway, due to crane dynamics and disturbances such as winds, occurs at the end of the trolley’s movement. Thus, researchers working in the area of container crane control always have been obliged to deal with sway suppression.Container oscillation and the obligation to suppress it, in fact, constitute a transportation bottleneck headache for container terminal officials. Although the "container" crane is a special case of many types of cranes, early crane control results are still relevant to the present concern. Feedback control methods and algorithms have been applied to crane systems [1-8]. These algorithms need to measure the sway angle. Normally, an industry camera detects markers on top of the spreader and calculates the sway angle. Sometimes, the sway information is not available because of the weather conditions. Therefore, feedback control cannot be implemented if the system is lack of sensors. Feed-forward control has been also applied to control the sway motion of the payload. When the rope length is constant, it is known that the input (or command) shaping control is very effective [9-14]. In their studies, the reference signal was modified and implemented in real time, where modification depends on the natural frequency of the system. Time optimal control is another frequently used control method suitable for crane control [15-16]. Time optimal control, similarly to input shaping control, requires exact system information. In short, these methods require highly accurate system parameter values to achieve satisfactory system responses.In this study, an input shaping control is applied by utilizing the characteristics of an industrial servo motor, which has a driver consisting of a few control modes. In case of velocity control mode, the motor will track a given velocity command by generating torque automatically. First of all, the dynamic model of container cranes is analyzed. The model is simplified and used to design the input shaping control. The control law for linear model is used to control the full order nonlinear container cranes. Finally, the robustness of the method is also evaluated.The paper is organized as follows. In Section 2, the system dynamics of a container crane are derived. In Section 3, the input shaping control law for linear system is proposed. In Section 4, simulation results of the control system with nonlinear model are discussed. Finally, in Section 5, conclusions are drawn.2. DYNAMIC MODEL OF CRANE Consider the container crane illustrated in Fig. 1. The container (payload) is picked up by the spreader, both being suspended from the trolley by a rope of length l.Command Shaping for Vibration Reduction of Container CranesQuang Hieu Ngo1, Yanghai Nan2, and Keum-Shik Hong3*1College of Technology, Can Tho University, Vietnam(Tel : +84-7103-834267; E-mail: nqhieu@.vn)2School of Mechanical Engineering, Pusan National University, Busan 609-735, Korea(Tel: +82-51-510-1481; E-mail: yhnan@pusan.ac.kr)3Department of Cogno-Mechatronics Engineering, Pusan National University, Busan 609-735, Korea(Tel: +82-51-510-2454; Email: kshong@pusan.ac.kr) * Corresponding authorAbstract: A combination of command shaping and bang/off-bang control scheme for container cranes is investigated in this paper. A linear model of the system is used to design the input shaper, where the trolley acceleration is given as an input, for suppressing the payload vibration. Bang/off-bang control guarantees that the trolley reaches the goal position By utilizing an industrial servo motor and its driver, the acceleration command from input shaper is integrated to become the velocity command for the motor. The control algorithm is simulated with the nonlinear system. The robustness of the control system is also evaluated with the natural frequency has ± 10% uncertainty (rope length variation).Keywords: Container crane; Command shaping; Residue sway; Input shaping.The masses of the trolley and the payload are m t and m p , respectively. A control force f x is applied to the trolley. In an actual crane, four ropes are used to hoist the spreader (including the payload). However, for simplicity, only one rope is assumed in the present paper. It is also assumed that the motions of both the spreader and the rope occur in the vertical plane, that is, the X -Y plane (see Fig. 1). Let x be the trolley position along the X -axis, θ be the sway angle, and g be the gravitational acceleration. Considering the motions of the trolley and the payload in the two-dimensional plane, the kinetic energy K and potential energy P of the entire system aregiven by()()(),cos sin 212121cos sin 21sin cos 2121p p 22p 2p 2p t 22p 2p 2t θθθθθθθθθθ x l m x l m l m l m x m m l l xm l l m x m K +++++=+++−+= (1) ()θcos 1p −=gl m P . (2)Taking ()θ,x q = as the generalized coordinatescorresponding to the generalized forces ()0,x f f =, and using the Lagrange’s equation ,2,1,==∂∂+∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂i f q Pq K q K dt d i ii i (3) the equations of motion can be obtained as(),sin cos 2cos sin 2pp p p p t θθθθθθθ l m l m l m l m x m m f x −++++= (4) θθθθsin 2cos 0p p 2p p gl m l l m l m x l m +++= . (5) The above derived model is a nonlinear model. The nonlinear model has to be linearized to simplify the progress of the modeling for designing the feed-forwardcontrol law. Two assumptions had been made tolinearize the system. Small sway angle is the firstassumption, since .1cos ,sin ≈≈θθθ The second assumption is the small rope length variation, .0=≈l l Based on these two assumptions, the simplifiedequations of motion for the gantry crane system can beobtained,(),θ l m x m m f p p t x ++= (6) θθg l x++= 0. (7) 3. INPUT SHAPING CONTROLIn this section, the input shaper will be designed sothat the trolley can reach the desired position without the sway motion since the system is lack of the sensors. The input shaper is designed based on the systemcharacteristic due to (7). Given xu −= as a controlinput, the Laplace transform of (7) yields,θ Fig. 1 Container crane model. ()()l g s s U s +=21θ. (8) The linear system (8) has two poles, ,ωj s ±= where .,1l g j =−=ω Giving the system an impulse will cause it to vibrate. This vibration can be suppressed if the control input U (s ) can generate two zeros to cancelthe poles at .ωj ± The time delay control or input shaping is the method can generate two zeros. It will be used to suppress the vibration of the sway motion in thispaper. Zero-Vibration (ZV) and Zero-Vibration Derivative (ZVD) are used to evaluate the feed-forwardcontrol method. ZV and ZVD impulse sequences are applied in this paper with the given parameters as follows,⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡215.05.0t t t A i i ,(9) ⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡221225.05.025.0t t t t A i i , (10) where A i is the amplitude of the i-th impulse and t i is the time of their application (time location - in seconds), π=2t , and ω is the natural frequency.and maximum speed of the trolley, the system willgenerate the reference control command .r xInput shaper modifies r x by using ZV or ZVD andintegrated the modified signal to become the referencevelocity. The system is driven by industrial servo motorto track the velocity command in case of the speedcontrol mode is active. The motor driver will generatethe force automatically by built-in controller. The diagram is illustrated in Fig. 2.The reference control command r xare obtained in Fig. 3, where max max up a v t =and ,max d down x t =v max and a max are the maximum speed and the maximum acceleration of the trolley. This called as bang/off-bang control. It guarantees that the trolley is able to move to the goal position, x d , in the shortest time, but the sway is not took into account in this case.Fig. 4 shows the reference velocity command with ZV impulses, which the parameters given as follows.,05.05.05.05.05.05.05.05.08765432⎥⎦⎤⎢⎣⎡−−−−=⎥⎦⎤⎢⎣⎡t t t t t t t t A i i where ,up 3t t =,2up 4t t t +=,down 5t t =,2down 6t t t +=,down up 7t t t += and 2down up 8t t t t ++=. 4. SIMULATION RESULTSIn this section, the comparison of ZV and ZVD is performed with nonlinear model (4)-(5). The robustness of the system is also simulated to evaluate the control method. The simulation parameters are given in Table 1. Fig. 5 shows the control command with ZV and ZVD modification. These commands are integrated to obtain the velocity commands as shown in Fig. 6. By applying these velocity commands to the industrial servo motor, the trolley always reaches the goal position but the sway motions of the payload are different. With unshaped command, the sway motion cannot be suppressed, see Fig. 7. In other cases, the vibration of the payload is eliminated. The ZVD command can suppress the sway motion more thoroughly than ZV command.Fig. 2 Control diagram.a maxv max-a maxtx dFig. 3 Unshaped control input.a max-a max(a)(b)Fig. 4 Control input with ZV .Table 1 Simulation parametersParameters Value Trolley mass 20 kgPayload 15 kg Nominal rope length 2 m Maximum speed 1 m/s Maximum acceleration 0.2 m/s 2 Goal position 10 mUnshapedA c c e l e r a t i o n [m /s 2]ZVTime [s]ZVDFig. 5 Modified control command.Time [s]T r o l l e y v e l o c i t y r e f e r e n c e [m /s ]Fig. 6 Reference velocity command, v r .To test the robustness of the control method, the rope length of the nonlinear system (4)-(5) has been changed between 1.5 m and 2.5 m. By using ZV command, the residues sway motion of the payload more sensitive for rope length variation. On the other hand, the residues sway motion is not larger when the control system using ZVD. These results can be seen in Fig. 8.Time [s]T r o l l e y p o s i t i o n [m ](a)Time [s]S w a y a n g l e [r a d ](b)Fig. 7 System response.Time [s]S w a y a n g l e [r a d ](a)Time [s]S w a y a n g l e [r a d ](b)Fig. 8 System robustness.5. CONCLUSIONSIn this paper, to achieve fast traveling and minimal chattering of the trolley in case of the system is lack of sensor, a command shaping control algorithm for container cranes is investigated. The trolley must reach the goal position without residue sway motion of the payload. Therefore, the combination of command shaping and bang/off-bang control was proposed. The robustness of the involving ± 10 % uncertainty (natural frequency), was demonstrated.ACKNOWLEDGEMENTThis research was supported by the World Class University program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology, KoreaREFERENCES[1] K. S. Hong, B. J. Park, and M. H. 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