Nonlinear Input-Shaping Controller for Quay-Side Container Cranes

Input-shaping is one of the most practical open-loop control strategies for gantry cranes, especially those having predefined paths and operating at constant cable lengths. However, when applied to quay-side container cranes, its performance is far from satisfactory. A major source of the poor performance can be linked to the significant difference between the gantry crane model and the quay-side container crane model. Gantry cranes are traditionally modeled as a simple pendulum. However, a quay-side container crane has a multi-cable hoisting mechanism.In this paper, a two-dimensional four-bar-mechanism model of a container crane is developed. For the purpose of controller design, the crane model is reduced to a double pendulum with two fixed-length links and a kinematic constraint. The method of multiple scales is used to develop a nonlinear approximation of the oscillation frequency of the simplified model. The resulting frequency approximation is used to determine the switching times for a bang-off-bang input-shaping controller. The performance of the controller is numerically simulated on the full model of the container crane, and is compared to the performance of similar controllers based on a nonlinear frequency approximation of a simple pendulum and a linear frequency approximation of a constraint double pendulum. Results demonstrate a superior performance of the controller based on the nonlinear frequency approximation of the constraint double pendulum.The effect of the oscillation frequency on the controller performance is investigated by varying the model's frequency around the design value. Simulations revealed that the performance of the controller suffers serious degradation due to small changes in the model frequency. To alleviate the shortcomings of the input-shaping controller, a delayed-position feedback controller is successfully applied at the end of each transfer maneuver to eliminate residual oscillations without affecting the commands of the input-shaping controller.     

Effect of hoisting cable elasticity on anti-sway controllers of quay-side container cranes

Oscillation frequency of crane payloads is the main and most important factor in crane anti-sway control systems design. In the summer of 2005, a Smart Sway Control system (SSC) was installed on a 65-ton quay-side container crane at Jeddah Port. During the calibration phase of the installation, it was observed that heavy payloads combined with the dynamic stretch of the hoist cables had a significant impact on the configuration of the hoisting mechanism and the pattern of oscillation. This introduced considerable change in the oscillation frequency of the payload, which resulted in a significant impact on the performance of the anti-sway control system. Empirical formulas had to be used to compensate for the change in the frequency approximation used in the controller algorithm. In this work, an analytic approximation of the oscillation frequency of the hoisting mechanism of a quay-side container crane is developed, which takes into consideration the elasticity of the hoisting cables. A parametric study is performed to investigate the extent of the effect of the hoisting cables stretch on the system behavior for a typical range of payload masses and cable lengths. The performance of the delayed feedback control system used in the SSC controller is simulated on an elastic cables model using both the elastic and rigid cable frequency approximations.     

Nonlinear analysis of time-delay position feedback control of container cranes

Time-delay feedback control of container cranes is robustly stable and insensitive to initial conditions for most of the linearly stable region. To better understand this robustness and any limitations of the technique, we undertake a nonlinear analysis of the system. To this end, we develop a nonlinear model of the crane system by modeling the crane-hoist-payload assembly as a double pendulum. Then, we derive a linear approximation specific to this model. Finally, we derive a cubic model of the dynamics for nonlinear analysis. Using linear analysis, we determine the gain and time delay factors for stabilizing controllers. Also, we show that the controller undergoes a Hopf bifurcation at the linear stability boundary. Using the method of multiple scales on the cubic model, we determine the normal form of the Hopf bifurcation. We then show that for practical operating ranges, the controller undergoes a supercritical bifurcation that helps explain the robustness of the controller.     

Mathematical modelling of the container cranes under seismic loading and proving by shake table

This paper is concerned with modelling of the behaviour of container cranes under seismic loadings. For this purpose, physical and mathematical models are prepared. A six degrees-of-freedom non-linear mathematical model is developed in order to understand the dynamic behaviour of cranes under the seismic loadings. In order to determine the seismic behaviour of the container cranes against earthquakes, a 1/20 scaled crane model was designed and constructed. For the comparison of the models, the real earthquake records were used. The results are used to observe the destructive effects and compared with the period values of the most critical sections on the crane structure. When time and frequency domains are compared, it is seen that mathematical modelling of the container crane structure shows reasonable results under dynamic loadings. It will be available to take precautions and to increase seismic performance of cranes with the help of the developed dynamic model. Also, the developed mathematical model will be able to be used as a crane model in active vibration control studies in order to decrease the structural vibrations on container cranes.     

A partially saturated adaptive learning controller for overhead cranes with payload hoisting/lowering and unknown parameters

For underactuated overhead cranes with payload hoisting/lowering, a partially saturated adaptive controller subject to unknown or uncertain system parameters is presented in this paper. To decrease the convergence time in the case of the overhead crane parameters already experienced by the system, the learning component is added to the proposed partially saturated adaptive controller. By introducing hyperbolic tangent functions into the control methods, the proposed controllers can guarantee soft trolley start even in the case of high initial velocities of trolley and cable. The convergence and stability performance of the closed-loop system is proven by Lyapunov techniques and LaSalle’s invariance theorem. Simulation results are listed to verify the adaptive performance with reduced actuating forces and strong robustness with respect to different external disturbances of the proposed controllers.     

Dynamic response of tower crane induced by the pendulum motion of the payload

Dynamic response of tower cranes coupled with the pendulum motions of the payload is studied in this paper. A simple perturbation scheme and the assumption of small pendulum angle are applied to simplify the governing equation. The tower crane is modeled by the finite element method, while the pendulum motion is represented as rigid-body kinetics. Integrated governing equations for the coupled dynamics problem are derived based on Lagrange’s equations including the dissipation function. Dynamics of a real luffing crane model with the spherical and planar pendulum motions is analyzed using the proposed formulations and computational method. It is found that the dynamic responses of the tower crane are dominated by both the first few natural frequencies of crane structure and the pendulum motion of the payload. The dynamic amplification factors generally increase with the increase of the initial pendulum angle and the changes are just slightly nonlinear for the planar pendulum motion.     

Cargo Pendulation Reduction of Ship-Mounted Cranes

Ship-mounted cranes are used to transfer cargo from large container ships to smaller lighters when deep-water ports are not available. The wave-induced motion of the crane ship can produce large pendulations of the cargo being hoisted and cause operations to be suspended. In this work, we show that it is possible to reduce these pendulations significantly by controlling the slew and luff angles of the boom. Such a control can be achieved with the heavy equipment that is already part of the crane so that retrofitting existing cranes would require a small effort. Moreover, the control is superimposed on the commands of the operator transparently. The successful control strategy is based on delayed feedback of the angles of the cargo-hoisting cable in and out of the plane of the boom and crane tower. Its effectiveness is demonstrated in a fully nonlinear three-dimensional computer simulation and in an experiment with a 1/24th-scale model of a T-ACS (The Auxiliary Crane Ship) crane mounted on a platform moving with three degrees of freedom. The results demonstrate that the pendulations can be significantly reduced, and therefore the range of sea conditions in which cargo-transfer operations can take place can be greatly expanded.     

Pendulation Reduction in Boom Cranes Using Cable Length Manipulation

A technique is proposed to reduce payload pendulations using the reelingand unreeling of the hoisting cable. The payload is modeled as a pointmass, the cable is modeled as a rigid link, and the assembly, aspherical pendulum, is attached to the boom tip. An excitation isapplied to the assembly at the boom tip. The motion of the payload isdescribed using two-dimensional and three-dimensional models. Ourresults demonstrate that cable-length manipulation can be used to reducepayload pendulations due to near-resonance excitations. Significantreductions can be obtained via an appropriate choice of thereeling/unreeling speed. We also demonstrate the limitations inherent intwo-dimensional modelings of a crane.     

Reduced normal forms for nonlinear control of underactuated hoisting systems

Nearly every cargo that is transported by ship is boxed in standardized containers. The land to ship and ship to land handling of containers is performed by container cranes. These cranes dominate the throughput of the containers in ports. The operators need to be well-trained and skilled because of the requirements of the task to load or unload the ships. During container handling, the crane-load system can be compared to a pendulum of rope length L and deflection f{\phi} . It is referred to as an underactuated system. Whenever the fixed point of the pendulum is displaced, the load starts oscillating. In the nonlinear dynamics literature, one finds the field of resonant coupling to transfer energy from actuated controllable modes to underactuated uncontrollable modes. This work presents a novel normal form approach in order to identify the mode coupling. The proposed method decomposes a nonlinear control system into controllable and uncontrollable subsystem. Only the normal form terms in the controllable subsystem are utilized to design a general controller. Simulation results are given to highlight the load oscillation suppression.     

On the Need for Control of Nonlinear Oscillations in Machine Systems

Oscillations in machines are invariably nonlinear. This is either because of inertial coupling effects between different motions of the moving components, material and constitutive phenomena giving rise to stiffness modifications, nonlinear dissipation mechanisms, large deflections, or, as is most likely, some sort of combination of all of these. The net effect of nonlinear vibrations is that at best the machine may well behave a little differently from the way the designer intended, or at worst, in a manner which renders it completely unsuitable for the job. The extent of such problems depends on the nature and the scale of the nonlinearities that are present but it is safe to say that nonlinear oscillations can rarely be completely overlooked in precision machinery analysis and design. The unifying theme in this paper is pendulum motion, firstly in the case of a mobile gantry crane for container stacking where we wish to minimise such motion and converge on a target, and then secondly in the case of a vibration absorber in which we choose to initiate pendulum motion within a special absorber, for the purposes of vibration minimisation. The third example involves the potential for pendulum motion at a very much larger scale and summarises the main control problem that is likely to be encountered in a fully deployed momentum exchange propulsion tether operating in space. The paper discusses the general mathematical issues that pertain to pendulum motion in each of the three cases. This motion is investigated initially in the context of the mobile gantry crane, in the form of a basic three dimensional dynamical model. A feedback linearised controller is shown to offer some advantages for the control of such a system and then a simulation based on data from a practical implementation of this within a real-time control system on a 1/10 laboratory scale model is discussed. It is recalled that the real-time effectiveness of the controller can be compromised by relatively slow sensing and data logging hardware but that despite this some useful performance gains can still be obtainable using this sort of control strategy. The second example comprises an autoparametric vibration absorber and here it is shown how even a simple hunting controller can exploit the mode-locking and wide detuning region effects inherent in autoparametric systems. Further experimental results are discussed in the case of a hunting controller for detuning a vertically oriented parametrically excited pendulum in order to exploit and enhance the powerful and persistent absorption available during autoparametric interaction. The paper concludes with a summary review of the third problem in which the theoretical attitude dynamics of a motorised momentum exchange space propulsion tether are summarised and it is shown that they need to be controlled for reliable and optimal payload velocity boost from both circular parking orbits and elliptical transfer orbits about the Earth.     

Partially saturated nonlinear control for gantry cranes with hardware experiments

Gantry cranes are typically underactuated nonlinear dynamic systems with highly coupled system states. We propose in this paper a partially saturated nonlinear controller for gantry crane systems by converting the crane model into an objective (i.e., desired closed-loop) system. The presented scheme guarantees “soft” cart start by introducing a smooth saturated function into the controller. In particular, we first establish an objective system with desired signal convergence and stability performance. Then, on the basis of the objective dynamics’ structure, we derive a partially saturated control law straightforwardly by solving one partial differential equation, without necessity of performing partial feedback linearization operations to the original crane model. The convergence and stability performance of the objective system is assured with Lyapunov-based methods. In order to verify the practical control performance of the proposed method, we implement both numerical simulation and hardware experiments to illustrate that the new method achieves increased performance with respect to existing methods, with lessened control efforts.     

基于贝叶斯信息融合的岸桥有限元模型修正研究

为建立精确的岸桥有限元模型,研究了基于贝叶斯信息融合的模型修正方法．通过方差分析,确定待修正参数,利用中心复合试验设计获取样本点,根据有限元计算结果与实测的结果残差为目标函数获得响应样本．拟合样本点和响应样本值构建二阶多项式响应面模型,并检验响应面模型的精度．基于贝叶斯理论更新融合系数来优化响应面参数,从而获得修正模型．以宁波大榭3号岸桥为工程背景,对比修正后的模态频率和实测频率,最大频率相对误差不超过5%,进而验证了基于贝叶斯信息融合的动力学有限元模型修正方法的有效性．修正后的有限元模型可进一步应用于岸桥的健康监测和安全评估．     

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简要介绍利用牵引绳与起升绳配合作业、增加工作幅度的有牵引绳的 缆绳式桅杆起重机的结构和工作原理，对其主要构件进行受力分析，建立主要构件受力情况 的计算公式. 通过计算比较，验证在设计许可范围内, 起升绳斜拉作业是可行、安全的. 提 出该类起重机适宜的有关参数和设计时应注意的事项.     

Global Bifurcation Analysis of a Controlled Underactuated Mechanical System

In this paper, bifurcation theory is employed to classify different dynamical behaviors arising in an underactuated mechanical system subject to bounded controls. The methodology is applied to an inertia wheel pendulum consisting of a simple pendulum with a rotating disk at the end. Restricting the magnitude of the control action places an important obstacle to the design of a continuous controller capable of swinging-up and stabilize the pendulum at the inverted position: the arm only can reach that position by means of oscillations of increasing amplitude. The controller is derived from a simple nonlinear state-feedback law, followed by a saturating device that limits the maximum amplitude of the control action applied to the system. This bound gives birth to a rich dynamical behavior, including pitchfork and Hopf bifurcations of equilibria, saddle-node bifurcations of periodic orbits, homoclinic and heteroclinic bifurcations. The global dynamics is analyzed in terms of certain control gains and a two-parameter bifurcation diagram is derived. It is shown that the dynamics on this bifurcation diagram is organized in a pair of codimension-two rotationally symmetric bifurcation points. Finally, it is found out that when the control gains lie on a certain region in the parameter space simultaneous stabilization of the upright position together with a large basin of attraction is obtained. Simulation results show that almost global stabilization of the system can be achieved.     

Stabilization problems for a system with output feedback (review)

Algorithms for solving the problem of design of static output feedback controllers for stationary linear systems with continuous and discrete time are reviewed. The inverse problem is considered. The algorithms of synthesis of output feedback controllers are generalized to the case of a periodic discrete-time system. To solve such problems, it might be more natural to use an approach based on multi-criterion optimization. It is also shown that these algorithms can be used for the optimal stabilization of unstable systems with delay. In this connection, the parameters of a controller with given structure for a controlled unstable scalar system with delay are optimized. To this end, the system is first approximated by a system without delay, with the exponent approximated by a fractionally rational function. Since the structure of the controller is given, the quality of approximation is estimated as the difference (in the space of controller coefficients) between the stability domains of the original and approximating systems. At the next stage, the gain coefficients of the controller for the reduced system are optimized. The efficiency of the thus synthesized controller is assessed through mathematical modeling of a system with delay whose feedback loop is defined by the gain coefficients found. The approach is illustrated by stabilizing an inverted simple pendulum with a proportional–derivative controller with delay. The problem of synthesis of a robust controller for this example is considered. Some examples of designing a robust controller, including for a third-order system in which the delay rather than some parameter is uncertain are presented     

Nonlinear Dynamics of Floating Cranes

The nonlinear dynamic responses of moored crane vessels to regular wavesare investigated experimentally and theoretically. The main subject ofinterest are nonlinear phenomena like bifurcations and the existence ofmultiple attractors. In the experimental part of the work, a mooredmodel of a crane vessel has been excited by regular waves in a wavetank. A special mechanism has been developed to model the nonlinearbehavior of real mooring systems. The theoretical part of the workconcerns the mathematical modeling of the floating cranes. Twomathematical models of different levels of complexity are presented. Twodifferent tools are used to systematically determine the responses ofthe systems to periodic forcing of waves. Firstly, the path-followingtechniques in combination with numerical integration of equations ofmotion applied to a full nonlinear model give insight into the dynamicsin time domain. Secondly, the multiple scales method allows for ananalytical investigation of simplified nonlinear models in frequencydomain. Many results of computations for two crane vessels, barge andship, are presented. Special attention is paid to oscillations near thefrequencies of primary resonances and to subharmonic motions. Anexcellent agreement is found between the results of time-domain andfrequency-domain analysis. The computational examples chosen correspondto the models used not only in the present experiments but in theexperiments of others as well. The results presented in the work allow usto draw several important conclusions concerning the dynamic behavior offloating cranes during offshore operations. Both the developed modelsand the analytical tools can be used to identify the limits of theoperating range of floating cranes.     

Energy coupled–dissipation control for 3-dimensional overhead cranes

Handling loads with small swings is difficult for a 3-dimensional overhead crane due to its hard nonlinearity. Moreover, the nonlinear dynamics increases the complexity of the required feedback, thus making the closed-loop system sensitive to a variation in the cable length that negatively influences the damping feature. To address these problems, a significant storage function characterized by the desired damping is constructed based on passivity. Consequently, a nonlinear controller is delivered by enforcing the coupled–dissipation inequality, thus drastically increasing the damping of the closed-loop system. In particular, new coupled–dissipation signals are fabricated to augment the coupling between the trolley movement and the payload sway. Due to its very simple structure that excludes the cable length, the proposed controller is robust to unknown cable lengths and easy to implement. In the frame of the Lyapunov theory, LaSalle’s invariance principle is applied to illustrate the corresponding stability. The effectiveness of the proposed control on improving the system performance is verified through simulation results.     

Modeling of system dynamics of a slewing flexible beam with moving payload pendulum

When a tower crane is handling payload via rotation and moving the carriage simultaneously the jib structure and the payload can be modeled as a system consisting of a slewing flexible clamed-free beam with the spherical payload pendulum that moves along the beam. The present work completes the dynamic modeling of the system mentioned above. The clamed-free beam attached to a rotating hub is modeled by Euler–Bernoulli beam theory. The payload is modeled as a sphere pendulum of point mass attached to via massless inextensible cable the carriage moving on the rotating beam. Non-linear coupled equations of motion of the in- and out-of-plane of the beam and the payload pendulum are derived by means of the Hamilton principle. Some remarks are made on the equations of motion.     

Motion stability dynamics for spacecraft coupled with partially filled liquid container

This paper presents a canonical Hamiltonian model of liquid sloshing for the container coupled with spacecraft. Elliptical shape of rigid body is considered as spacecraft structure. Hamiltonian system is an important form of mechanical system. It mostly used to stabilize the potential shaping of dynamical system. Free surface movement of liquid inside the container is called sloshing. If there is uncontrolled resonance between the motion of tank and liquid-frequency inside the tank then such sloshing can be a reason of attitude disturbance or structural damage of spacecraft. Equivalent mechanical model of simple pendulum or mass attached with spring for sloshing is used by many researchers. Mass attached with spring is used as an equivalent model of sloshing to derive the mathematical equations in terms of Hamiltonian model. Analytical method of Lyapunov function with Casimir energy function is used to find the stability for spacecraft dynamics. Vertical axial rotation is taken as the major axial steady rotation for the moving rigid body.     

On stabilization of systems with delay

The optimization of the parameters of a controller of given structure for a controlled unstable scalar system with delay is studied. First, the original system with delay is approximated by a system without delay. To this end, the exponent is approximated by a fractional rational function. Since the structure of the controller is fixed, the quality of the approximation is assessed by comparing the stability domains of the original and approximating systems (in the space of controller coefficients). Next, the coefficients of the controller for the reduced system are optimized. The performance of the controller thus synthesized can be assessed by mathematically modeling the original system (with delay) whose feedback is determined by the controller coefficients found. The approach is exemplified by stabilizing an inverted mathematical pendulum with a PD controller. This example is used to examine the issue of synthesis of a robust controller __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 86–100, October 2008.