Port Automation: Modeling and Control of Container Cranes

Transcription

1 Port Automation: Modeling and Control of Container Cranes Keum-Shik Hong* and Quang Hieu Ngo** * Dept. of Cogno-Mechatronics Engineering and School of Mechanical Engineering, Pusan National University ** School of Mechanical Engineering, Pusan National University 30 Jangjeon-dong, Geumjeong-gu, Busan, , Korea. s: {kshong, }@pusan.ac.kr Abstract The container crane is a key machine in ports. Over the past 15 years, modeling and control of quay cranes for automating ports has been the main research issue. In this paper, crane models and control methodologies appeared in the literature are reviewed. A generalized formulation of the most widely used crane model is introduced. Various control techniques are briefly discussed. In conclusion, an amenable control strategy as well as a plausible model is recommended. Keywords: crane, dynamics, control, pendulum motion, port automation. 1. Introduction Since containers were introduced to the world-trade industry, an increasing number of goods are being put into these containers and loaded onto vessels to be carried to their respective destinations over the world. When a vessel arrives at a port, containers destined for this port must be unloaded and new containers which are bound for other ports must be loaded before the vessel can resume its trip. Demands on container ports to perform the loading and unloading process with maximum efficiency will become greater as transport companies continue to increase both the size of their fleets and also the capacity of the vessels. The problem with this increase is that port authorities are predicting that they will run out of space to expand their operational areas and the option of developing on surrounding land is often hampered from local residents. Therefore, the only remaining solution will be the reduction of the amount of time that the vessel needs to remain in dock. The way to do this is to ensure that the container unload/load process is done as rapidly as possible, and this can be achieved by ensuring that the equipments in the port such as quay cranes (QCs), automated guide vehicles (AGVs), automated lift vehicles (ALVs) and so on, must be operated at their maximum efficiencies (Thurston et al., 00). The unloading/loading container process from/to the vessel to/from yard is performed by QCs, trucks or AGVs. One to six QCs are assigned to pick up containers depending on the vessel size. Each crane will be responsible for several rows of container storage slots along the ship. For each row, the QC must first unload all the containers destined for this port, and then load all the containers scheduled for leaving this port. Containers will be carried between the QCs and the yard by AGVs or trucks. An increasing number of trucks or AGVs will catch of the unloading/loading container demand. Because of the space limitation, the QCs cannot be assigned more to serve the vessel. Thus, the unloading/loading efficiency depends on how fast the container is unloaded/loaded by QCs. However, the fast trolley movement of the QC sways the container during its movement, the main issue in a crane system is the quick suppression of vibrations caused by the trolley motion. In addition, a residual sway occurs at the end of the trolley movement due to crane dynamics and disturbances like winds. Thus, researchers working in the area of crane control have always targeted sway suppression. More generally, payload oscillations and the need to suppress them have been identified as a bottleneck in the operations of the transportation and construction industries even where relatively simple gantry crane are concerned. The control methods developed for the conventional cranes can be applied to container cranes as well. The paper is organized as follows: The modeling of crane systems is addressed in Section. The control strategies for cranes are described in Section 3. In Section 4, the development of the container crane in the future is introduced. Conclusions are given in Section 5.. Modeling crane systems In modeling crane dynamics, two approaches are used. The first approach is the lumped-mass approach, in which the Fig. 1. Loading/unloading process (Hamburg Port Consulting GmbH) 009 ICA, ISBN

2 hoisting rope is modeled as a mass-less rigid rod and the payload is modeled as a lumped point mass. The sway motion is modeled as a pendulum motion in two or three dimensional space. Therefore, an ordinary differential equation (ODE) model is used in deriving the control laws. Another approach is the distributed parameter system approach, in which the hoisting rope is considered as a string (distributed-mass). A typical assumption in the second approach is that the rope is perfectly flexible and inextensible. In this case, the dynamics of the crane system is expressed as a coupled ODE and partial differential equation (PDE): the trolley motion is given in the ODE form and the rope dynamics is given in the PDE form. By using the PDE form, the payload as well as the rope motion is described exactly, especially when the rope length is very large and the payload is very heavy. In addition, the vibration of the rope that affects the payload motion during the trolley movement is also more accurately suppressed..1. Lumped-mass models The most popular model of gantry cranes augments the planar version of the pendulum model with the motion of gantry in the horizontal direction as shown in Fig.. The (simplest) linear equations of motion of the gantry crane are as follows. M & x& mgθ = F, (1) & x + lθ& + gθ = 0, () where M and m are the masses of the trolley and the load, respectively, l is the rope length, x is the trolley position and θ is the sway angle. It is noted that the linear natural frequency of the pendulum is dependent on the trolley and the payload masses. ( ) M + m g ω =. (3) Ml Omar et al. (005) further augmented this model with a friction force affecting the trolley motion. The friction was estimated and cancelled by applying an opposite control action. Liu et al. (005) derived the equations of the motion of a two-dimensional overhead crane. System linearization transformed the two-dimensional system to two independent systems: X-direction transport system and Y-direction transport system. Both the two systems were the same dynamic models; each of them was described by the simple pendulum model. Hong et al. (000) introduced a simple model with an equation describing the dynamics of the cable hoisting. The container was lifted using the hoist motor, while it was lowered by the total weight of the spreader and the container. By winding the trolley rope around trolley drum, the trolley was pulled to a desire spot. During the trolley movement, the container was swayed and it was suppressed by controlling the trolley motion. Takagi et al. (003) dealt with modeling and control of a crane mounted on a tower-like flexible structure. A fast transfer of the load can sway of the load and vibration of the flexible structure. Controlling both the sway and the vibration by the inherent capability of the tower crane was objectives of the paper. Their paper described the design of a centralized control system considering coupling of between the up-and-down direction and the rotational direction. The comparison of experiments and analysis showed that the decentralized control system had almost the same performance and stability as the centralized one... Distributed-mass models The model available in this category is the planar model for a gantry crane linearized around the cable s equilibrium position as shown in Fig.3. The governing equation and boundary conditions are following. ( y, t) ( y, t) w w ρ (, ) = 0 P y t, 0 < y < l, (4) t y y ( 0, t) ( 0 t) w w, M P( 0, t) t y = f g () t, (5) Fig.. Schematic diagram of a gantry crane with the lumped-mass model. Fig. 3. Schematic diagram of a gantry crane with the distributed-mass model. 009 ICA, ISBN

3 (, t) (, t) w l w l m + P( l, t) = 0, (6) t y where M and m are the masses of the trolley and the load, respectively, f g is the input force applied to the trolley, w(y,t) is the transverse motion of the cable around its equilibrium position, y is a curvilinear coordinate representing the arc-length along the cable, ρ is the mass per unit length of the cable, P(y,t) is the tension in the cable, P ( y t) mg + ρg( l y),. (7) d'andrea-novel et al. (1994, 000, 00) used this model for overhead cranes and provided the asymptotic stability of a closed-loop system under the assumption that the rope length was constant. Baicu et al. (1998) extended this model and applied to a gantry robot for flexible link. The electrical subsystem dynamics for a permanent magnet brushed DC motor coupled with the link dynamics to form a hybrid system of partial and ordinary differential equations. Rahn et al. (1999) also used the simple PDE s model and applied control law to stabilize the gantry crane system. Kim et al. (009) augmented the simple model with an axially moving system concept. Because the load was hoisted up and down during the trolley motion, the crane was modeled as an axially moving string system. The dynamics of the moving string were derived using Hamilton s principle for systems with changing mass. 3. Controlling crane systems Researchers in the crane control area have developed many methods to control the sway motion of the load. Two popular categories in crane control are open loop control such as input shaping control, optimal control and so on, and close loop control such as state feedback, nonlinear control, fuzzy control, adaptive control and so on Input shaping and optimal control The command shaping is a reference signal modification technique that is implementable in real time. However, command shaping does not have closed-loop mechanisms of feedback control. Therefore, it must be used in conjunction with a feedback control if it is used for disturbance rejection. Hong et al. (003) introduced a path planning for the purpose of facilitating understanding of the semiautomatics modes. In actual semi-automatic operation mode, the four paths were continuous (AB hoisting up, manual mode, BC hoisting up and travelling of the trolley, auto mode, CD traveling of the trolley, auto mode, DE hoisting down, manual mode). A modified input shaping control methodology had been presented to restrict the swing angle of the pay load within a specified value during the transfer to minimize the residual vibration at the end point. The conventional method was enhanced by adding one more constraint to limit the transient sway angle within a specified value using the sway angle based on a linear time invariant system. Sorensen et al. (007) developed a combined feedback and input shaping controller enabling precise positioning and sway reduction in bridge and gantry cranes. The first feedback module detected and compensated for positioning error. The second module detected and rejected disturbances. Input shaping was used in a third module to mitigate motion induced oscillation. Al-Garni et al. (1995) developed a nonlinear dynamic model of the overhead crane. The optimal control scheme was applied to control the overhead crane to satisfy the minimize sway and final time. Hong et al. (000) proposed a two-stage control of container cranes. The first stage control was a modified time-optimal control with feedback for the purpose of fast trolley traveling. The second stage control was a nonlinear control for the quick suppression of the residual sway while lowering the container at the target trolley position. The secondary control combined the partial feedback linearization to account for the unknown nonlinearities as much as possible and the variable structure control to account for the un-modeled dynamics and disturbances. Klosinski (005) developed a mathematical model included a crane model and a control system. A desired input signal consisting of three phase which were able to change depend on the rope length was applied to control system. Terashima et al. (007) presented an open-loop control strategy for sway-free, point-to-point motion of a load mass in the three-dimensional motion of a rotary crane. In order to suppress the sway of the load during transfer and the residual sway after transfer, an optimal control method was applied. The minimum time-control problem was considered in case of the rope length was varied. The minimum time control was also compared to the pre-shaping control, which effectively controls vibration. Singhose et al. (000) presented the dynamic behavior of a planar gantry crane with hoisting of the load. The command generation method of input shaping was proposed for reduction of the residual vibration. Several versions of input shaping were evaluated and compared with time-optimal rigid-body commands over a wide range of parameters. Input shaping provided significant reduction in both the residual and transient oscillations, even when the hoisting distance was a large percentage of the cable length. Borsc et al. (008) introduced a control strategy based on the changing structure of a control system, ie, on controlled jump changes of the control system parameters, ensures more rapid stabilization than a control system with constant parameters. An algorithm of the time-optimal stabilization was designed in accordance with the Pontriagin s maximum principle in such a way that the resulting behaviour was non-periodic. The designed control algorithm divided the state space into segments in which the parameters took their limit values. The control problem 009 ICA, ISBN

4 was resolved for third-order and higher linear systems. This control strategy was applied to gantry crane to demonstrate the system robustness. Optimal control and input shaping techniques are limited by the fact that they are extremely sensitive to variations on the parameter values about the nominal value and changes in the initial conditions and external disturbances. Therefore, they require highly accurate values of the system parameters to achieve satisfactory system response. Moreover, a couple of the feedback control and input shaping or optimal control should be used to improve the control performance of sensitive systems. 3.. Linear and nonlinear control Kim et al. (004) designed a state feedback controller with an integrator to control a real container crane. The inclinometer was used instead of a vision system, while providing almost the same performance. A number of observers to estimate the angular velocity of the load and the trolley velocity were presented. Park et al. (007) developed a nonlinear anti-sway controller crane with hoisting. The container crane involved a planar motion in conjunction with the hoisting motion. A novel feedback linearization control law provided a simultaneous trolley position regulation, sway suppression and load hoisting control. The advantage of the proposed control law lay in the full incorporation of the nonlinear dynamics by partial feedback linearization. Neyfeh et al. (008) developed a nonlinear model of the crane system by modeling the crane-hoist-payload assembly as a double pendulum. They derived a linear approximation specific to this model and a cubic model of the dynamics for nonlinear analysis. Using linear analysis, the gain and time delay factors for stabilizing controllers were determined. Also, they showed that the controller underwent a Hopf bifurcation at the linear stability boundary. Using the method of multiple scales on the cubic model, the normal form of the Hopf bifurcation was determined. The controller underwent a supercritical bifurcation that helped explain the robustness of the controller. Schaub (008) proposed a method used two active ship motion compensation strategies to stabilize the payload during perator-controlled at sea cargo transfer scenarios using ship-based cranes. A new ship motion sensing strategy was developed using only inertial measurement unit (IMU) information to reduce the cost and complexity of the ship motion sensors, while improving the overall crane performance. A new rate-based control strategy was developed which directly computed required crane joint rates to isolate the payload from the ship motion. Ghigliazza et al. (00) considered the dynamics of a tower crane: a point mass suspended by a light cable (itself assumed massless) from a horizontally moving support. They derived general equations of motion, and analyzed two cases in detail: the linearly accelerating support, and the support describing a circle at constant speed. They also found steadily rotating solutions, discuss their stability and bifurcations, and provide a partial characterization of global orbit structures. Moustafa et al. (1988) derived a nonlinear dynamical model for an overhead crane. The model took into account simultaneous travel and transverse motions of the crane. The aim was to transport an object along a specified transport route in such a way that the swing angles were suppressed as quickly as possible. An anti-swing control system which adopts a feedback control to specify the crane speed at every moment was adopted. The gain matrix was chosen such that a desired rate of decay of the swing angles was obtained. Messineo et al. (008) proposed a novel feedback controller for cranes employed in heavy-lift offshore marine operations. The control objective was to reduce the hydrodynamic slamming load acting on a payload at water-entry of moonpool operations; at the same time the values of the wire tension must be kept within acceptable bounds. Hicar et al. proposed robust control by Ackermann to ensure crane robustness against the burden weight and rope length variation, which provides sharp positioning and forbidden swinging of the burden in the final position. The real crane will be connected to a distributed control system (DSR). An analysis of the working place was performed with accent on the measurable units of burden swinging, and the results of robust controlling were described. Yoshida et al. (199) designed a saturating control law which satisfied a constrained input condition and gave an upper bound of a given quadratic performance index, using a unique guaranteed cost control method. This controller was applied to a crane system to show the practicability of the control law. Wahyudi et al. (007) developed a sensorless automatic gantry crane control strategy using reference modifier. A reference modifier was introduced to produce antiswing cart motion Boundary control Rahn et al. (1999) developed a proportional, derivative, and coupling amplification control law that applied to the system boundary (gantry motion) to stabilize the flexible cable gantry crane. A root locus analysis based on Galerkin s method for tuning control gains was another interesting feature of their work. Baicu et al. (1998) used backstepping boundary control to drive a flexible-link gantry robot as a gantry crane by using a brushed DC motor. The electrical subsystem dynamics for a permanent magnet brushed DC motor coupled with the link dynamics to form a hybrid system of partial and ordinary differential equations. A boundary voltage control law was developed based on Lyapunov theory for distributed parameter systems. Through an embedded 009 ICA, ISBN

5 desired-current control law, the integrator back-stepping controller generated the desired control force on the mechanical subsystem. A velocity observer estimated the gantry velocity, eliminating one feedback sensor. Modal analysis and Galerkin s method generated the close loop modal dynamics. d'andrea-novel et al. (1994) established the uniform exponential stability of the closed-loop system of their original problem by adding an angular-velocity feedback of the rope in addition to the displacement feedback, the velocity of the trolley and the angular-displacement of the rope. Moreover, d Andrea-Novel and Coron (000) proposed a torque control to stabilize an overhead crane with a variable-length flexible cable. Kim et al. (009) applied the Lyapunov function method to derive a boundary control law, where the Lyapunov function candidate took the form of the total mechanical energy of the system. The boundary control law utilized the hoisting speed as well as the sway angle of the rope at the gantry side Fuzzy control Benhidieb et al. (1995) described the comparison of a fuzzy logic control system with the Linear Quadratic Gaussian control (LQG) of an overhead crane, considering the applicability of the control algorithms in real time and assuming that the model was representative of the real system. A number of possible perturbations were examined in a study of the robustness of the control algorithms. Chen et al. (009) considered a practical overhead crane control problem of designing a fuzzy control to ensure that the trolley arrives precisely at its destination; the load swing angle approaches to zero, and the energy of input and states is minimized. Furthermore, the constraints for the overhead crane operation were also considered and satisfied. Those constraints were that when the crane is moving, the swing angle has to be small; the swing speed has to be slow, and the moving force is limited. The fuzzy descriptor system was chosen to represent the dynamics of the overhead crane, and a guaranteed cost fuzzy control design was proposed for the fuzzy descriptor system. Moreover, this study also illustrated why the fuzzy descriptor system instead of traditional T S fuzzy system was selected here as the model of the overhead crane. Chang et al. (008) presented a novel method that accelerated transportation and minimized the payload swing of the overhead cranes. Based on the inertia theorem, this method did not need a complex dynamic model for a crane system, but rather used trolley position and swing angle data to design the proposed fuzzy projection controller. An enhanced fuzzy algorithm was also utilized to eliminate the dead-zone problem. The feasibility and effectiveness of the proposed scheme were also compared with those of the conventional PD and nonlinear coupling methods. Chang et al. (009) proposed a new accelerated method for rapid and smooth transportation of a nonlinear 3-D crane. Based on the projection vectors of swing angles on -D plane and the remaining distance to the destination, this approach was able to be easily incorporated into an intelligent fuzzy controller. Wahyudi et al. (007) introduced a practical and intelligent control method for automatic gantry crane. The design of proposed method was based on a simple open-loop experiment and without the need either to model crane or perform system identification. The effectiveness of the proposed method as well as the robustness to parameter variations were evaluated experimentally in a lab-scale gantry crane system. Its performance was also compared with that of classical PID and fuzzy logic controllers. Yi et al. (003) proposed a new fuzzy controller for anti-swing and position control of an overhead traveling crane based on the Single Input Rule Modules (SIRMs) dynamically connected fuzzy inference model. The trolley position and velocity, the rope swing angle and angular velocity were selected as the input items, and the trolley acceleration as the output item. Each input item was given with a SIRM and a dynamic importance degree. The control system was proved to be asymptotically stable to the destination. The controller was robust to different rope lengths and had generalization ability for different initial positions. Ahmad (009) presented the use of anti-sway angle control approaches for a two-dimensional overhead gantry crane with disturbances effect in the dynamic system. Delayed Feedback Signal (DFS) and proportional-derivative (PD)-type fuzzy logic controller were the techniques used in this investigation to actively control the sway angle of the rope of gantry crane system. A nonlinear overhead gantry crane system was considered and the dynamic model of the system was derived using the Euler-Lagrange formulation. Performances of both controllers are examined in terms of sway suppression, disturbances cancellation, time response specifications and input force. Cho et al. proposed a new fuzzy anti-swing control scheme for a three-dimensional overhead crane. The proposed control consisted of a position servo control and a fuzzy-logic control. The position servo control was used to control crane position and rope length, and the fuzzy-logic control was used to suppress load swing. The proposed control guaranteed not only prompt suppression of load swing but also accurate control of crane position and rope length for simultaneous travel, traverse, and hoisting motions of the crane. Furthermore, the proposed control provided practical gain tuning criteria for easy application Adaptive control Corriga et al. (1998) considered a linear parameter-varying model of the crane, where the time-varying parameter was the length of the suspending rope. The set of models given by frozen values of the rope length was considered to show how all these models have been reduced to a single time-invariant model using suitable time scaling. The time 009 ICA, ISBN

6 scaling relation has been used to derive a control law for the time-varying system that implemented an implicit gain scheduling. Using a Lyapunov-like theorem, it was also possible to find relative upper bounds for the rate of change of the varying parameter that ensured the stability of the time varying system. Hua et al. (007) proposed a nonlinear control scheme incorporating parameter adaptive mechanism to ensure the overall closed-loop system stability. By applying the designed controller, the position error has been driven to zero while the sway angle was rapidly damped to achieve swing stabilization. Stability proof of the overall system was given in terms of Lyapunov concept. Chang et al. (007) provided an effective all-purpose adaptive fuzzy controller for the crane. This method did not need the complex dynamic model of the crane system, but it used trolley position and swing angle information instead to design the fuzzy controller. An adaptive algorithm was provided to tune the free parameters in the crane control system. External disturbance, such as the wind and the hit, which always deteriorates the control performance, was also discussed in the paper to verify the robustness of the proposed adaptive fuzzy algorithm. Cheng et al. (1996) developed a robust controller which combined a feedback linearization approach and a time delay control scheme. The time delay control was applied to complete the feedback linearization for a nonlinear system under the influence of uncertainty. Johansen et al. (003) developed a new strategy for active control in heavy-lift offshore crane operations by introducing a new concept referred to as wave synchronization. Wave synchronization reduced the hydrodynamic forces by minimizing variations in the relative vertical velocity between payload and water using a wave-amplitude measurement. Wave synchronization was combined with conventional active heave compensation to obtain accurate control. Messineo et al. (009) designed an adaptive controller for cranes employed in heavy-lift offshore marine operations. The control objective was to reduce the hydrodynamic slamming load acting on a payload at water entry of moonpool operations, while letting the payload tracked a given velocity profile. The adopted solution relied upon the use of an adaptive observer and two adaptive external models of the disturbance, employed to recover the unavailable information about the error to be regulated. As a result, the closed-loop system was rendered adaptive with respect to both the plant parameters and the frequencies of the harmonic disturbances affecting the system. A certainty-equivalence controller which made use of the estimated parameters and the reconstructed tracking error was proposed, and the performance of the overall scheme was verified experimentally on a scale-model Sliding mode control Liu et al. (005) proposed a sliding mode fuzzy control for both X-direction and Y-direction transport. According to the influences on system dynamic performance, both the slope of sliding mode surface and the coordination between the two subsystems was automatically tuned by real time fuzzy inference respectively. The effectiveness of the proposed control approach was demonstrated. Lee et al. (006) proposed a sliding mode anti-swing for overhead cranes. A sliding-mode anti-swing trajectory control scheme was designed based on the Lyapunov stability theorem, where a sliding surface, coupling the trolley motion with load swing, was adopted for a direct damping control of load swing. The proposed control guaranteed asymptotic stability while keeping all internal signals bounded. In association with a new anti-swing motion planning scheme, the proposed control realized a typical anti-swing trajectory control in practice, allowing high-speed load-hoisting motion and sufficient damping of load swing. The proposed control was simple for a real-time implementation with high-frequency sampling. Bartolini et al. (00) proposed a simple control scheme, based on second-order sliding modes. It guaranteed a fast and precise load transfer and the swing suppression during the load movement, despite of model uncertainties and unmodeled dynamic actuators. 4. Crane control in the future In the future, the size of containers ship becomes bigger and they may not be accessible to the port because of shallow water. Containers will be transferred from/to larger ships to/from smaller ships offshore. A typical arrangement is to place two ships side by side. A small ship has cranes. Along one side of the crane ship is the large container ship. The container crane in the smaller ship will pick up containers and transfer between the two ships. The smaller ship will move containers to the port and the crane in the small ship can unload the containers to the ground, by removing the possibility of constructing an expensive port facility. The sea state is a very important factor in this operation. The sea-excited motion of the crane ship can excite large pendulations of the containers while they are suspended by the cables of the cranes. The large motions can be due to either large excitation amplitudes or small excitation amplitudes near resonant conditions. Therefore, the container cranes have to compensate for the motions induced on the suspended load by the sea wave, wind and other external disturbances as the majority of crane systems used offshore. 5. Conclusions In the literature, nonlinear control models including the feedback linearization occupied the majority of the works done on crane control. The simplified model was used to develop many control strategies, from opened loop control to feedback control. However, this model seems not sufficient when cranes become bigger: the rope length is 009 ICA, ISBN

7 very large; the payload is too heavy, and so on. Then, the distributed-mass model can be used to design a control algorithm because it describes exactly the crane system. The main difficulty is how to design control laws in a systematic way. Many significant research efforts have been attemped to the development of control strategies to improve the efficiency and safety of the cranes. The input shaping technique has shown an ability to control crane systems but it is not robust enough to be used for many cranes. They are not robust enough to reject external disturbance and to stabilize the payload under uncertain crane parameters. The combination between linear control and input shaping techniques are able to be a control strategy candidate. However, they are also not robust enough to allow for variation in the hoisting cable length, payload mass and high trolley speed. Fuzzy logic and adaptive control are plausible candidates for crane control. A combination of all these methods is also a good strategy when it allows variations of the rope length, payload and trolley masses, etc. However, it is remarked that the design of a hybrid controller to produce robust and efficiency control strategy is not a trivial issue. Acknowledgment This work was supported by the Mobile Harbor Project of the Korea Advanced Institute of Science and Technology funded by the Ministry of Education, Science and Technology, Korea. References Ahmad, M. A. (009). Active sway suppression techniques of a gantry crane system. European Journal of Sciences Research, 7(3), Al-Garni, A. Z., Moustafa, K. A. F., & Javeed Nizami, S. S. A. K. (1995). Optimal control of ovehead cranes. Control Engineering Practice, 3(9), Baicu, C. F., Rahn, C. D., & D.M. Dawson. (1998). Backstepping boundary control of flexible-link electrically driven gantry robots. IEEE/ASME Transactions on Mechatronics, 3(1), Bartolini, G., A. Pisano & Usai, E. (00). Second-order sliding-mode control of container cranes. Automatica, 38(10), Benhidjeb, A., & Gissinger, G. L. (1995). Fuzzy control of an overhead crane performance comparison with classical control. Control Engineering Practice, 3(1), Borsc, M., Vitko, A., & Thursky, B. (008). Optimal stabilization of modal control with variable structure. Transactions of the Institute of Measurement and Control, 30(1), Chang, C. Y. (007). Adaptive fuzzy controller of the overhead cranes with nonlinear disturbance. IEEE Transactions on Industrial Informatics, 3(), Chang, C. Y., & Chiang, K. H. (008). Fuzzy projection control law and its application to the overhead crane. Mechatronics, 18(10), Chang, C. Y., & Chiang, K. H. (009). Intelligent fuzzy accelerated method for the nonlinear 3-D crane control. Expert Systems with Applications, 36(3), Chen, Y. J., Wang, W. J., & Chang, C. L. (009). Guaranteed cost control for an overhead crane with practical constraints: fuzzy descriptor system approach. Engineering Application of Artificial Intelligence, (4-5), Cheng, C. C., & Chen, C. Y. (1996). Controller design for an overhead crane system with uncertainty. Control Engineering Practice, 4(5), Cho, S.K., & Lee, H. H. (00). A fuzzy-logic antiswing controller for three-dimensional overhead cranes. ISA Transactions, 41(), Corriga, G., Giua, A., & Usai, G. (1998). An implicit gain scheduling controller for cranes. IEEE Transactions on Control Systems Technology, 6(1), d Andréa-Novel, B., Boustany, F., Conrad, F., & Rao, B. P. (1994). Feedback stabilization of a hybrid PDE-ODE system: application to an overhead crane. Mathematics of Control, Signals, and Systems, 7(1), 1-. d Andrea-Novel, B., & J. M. Coron. (000). Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach. Automatica, 36(4), d Andréa-Novel, B., & Coron, J.M. (00). Stabilization of an overhead crane with a variable length flexible cable. Computational and Applied Mathematics, 1(1), Ghigliazza, R. M., & Holmes, P. (00). On the dynamics of cranes, or spherical pendula with moving supports. International Journal of Non-Linear Mechanics, 37(7), Hicar, M., & Ritok, J. (005). Robust control of real experimental bridge crane. Journal of Electrical Engineering, 56 (3-4), Hong, K. T., Huh, C. D., & Hong, K. S. (003). Command shaping control for limiting the transient sway angle of crane systems. International Journal of Control, Automation, and Systems, 1(1), Hong, K. S., Park, B. J., & Lee, M. H. (000). Two-stage control for container cranes. JSME International Journal, Series C, 43(), Hua, Y. J., & Shine, Y. K. (007). Adaptive coupling control for overhead crane systems. Mechatronics, 009 ICA, ISBN

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