EP2272785B1 - Method for controlling a drive of a crane - Google Patents

Abstract

The method involves providing the reference-movement of a cantilever arm head (5) as input parameter. A control parameter is calculated for controlling the drive on the basis of input parameter. The vibration dynamics of the system is considered for drive and the crane structure during calculating the control variable, in order to reduce the natural vibration. An independent claim is also included for a crane or crane controller with a control unit.

Description

The present invention relates to methods for controlling drives of a crane. In particular, the present invention relates to a method for controlling a drive of a crane, in particular a slewing gear and / or a luffing gear, wherein a target movement of the boom tip serves as an input, on the basis of which a control variable for driving the drive is calculated. Furthermore, the present invention relates to a method for controlling a hoist of a crane, in which a desired stroke movement of the load serves as input, on the basis of which a control variable for driving the drive is calculated. The drive of the crane according to the invention may in particular be a hydraulic drive. The use of an electric drive is also possible. In this case, the luffing z. B. be realized via a hydraulic cylinder or a retractable.

In known methods for controlling drives of a crane while an operator is by means of hand levers, the target movement of the boom tip and Thus, the target movement of the load in the horizontal direction before, from which due to the kinematics of slewing and luffing a control variable for controlling these drives is calculated. Furthermore, the operator by means of hand levers, the target stroke movement of the load before, from which a control variable for controlling the lifting mechanism is calculated.

Furthermore, methods for load oscillation damping are known for example from DE102004052616 in which, instead of the movement of the cantilever tip, a target movement of the load serves as an input to calculate a control variable for driving the drives. This z. B. a physical model of the movement of the load rope hanging load in response to the movement of the drives are used to avoid by a corresponding control of the drives spherical pendulum vibrations of the load.

However, the known methods for controlling cranes can lead to significant loads on the crane structure.

The object of the present invention is therefore to provide a method for controlling a drive of a crane which reduces such loads on the crane structure.

According to the invention, this object is achieved by a method according to claim 1. In the method according to the invention for controlling a drive of a crane, in particular a slewing gear and / or a luffing gear, a desired movement of the cantilever tip serves as an input quantity, on the basis of which a control variable for controlling the drive is calculated. According to the invention, it is provided that in the calculation of the control variable, the internal vibration dynamics of the system of drive and the crane structure is taken into account in order to dampen natural oscillations. The drive may be a hydraulic drive. The use of an electric drive is also possible.

In this case, the inventors of the present invention have found that the natural vibrations can heavily load the crane structure and the drives. By taking into account the internal vibration dynamics of the drive and the crane structure in the calculation of the control variable, however, natural oscillations can be damped and advantageously largely avoided. On the one hand, this has the advantage that the cantilever tip follows the predetermined target movement without oscillation exactly. On the other hand, the crane structure and the drives are not burdened by the natural vibration. The damping of the natural vibrations according to the invention therefore has a positive effect on the service life and the maintenance costs.

The method according to the invention is advantageously used in cranes, in which a boom is articulated around a horizontal rocking axis so that it can be wiped on a tower. The boom can be up and tipped off by a arranged between the tower and the boom boom cylinder in the rocker. It is also possible to use as a luffing mechanism a retractor, which moves the boom via a stranding in the rocker plane. The tower is in turn rotatable about a slewing gear, in particular in the form of a hydraulic motor about a vertical axis. The tower can be arranged on an undercarriage, which can be moved by a chassis.

The inventive method can be used in any cranes, for example in port cranes and in particular mobile harbor cranes.

Advantageously, according to the invention, the drive is controlled on the basis of a physical model which describes the movement of the crane tip as a function of the control variable. The use of a physical model makes it possible to quickly adapt the control method to different cranes. The vibration behavior does not have to be laboriously determined by measurements, but can be described on the basis of the physical model. In addition, the physical model allows a realistic description of the vibration dynamics of the crane structure, so that all relevant natural oscillations can be damped. The physical model describes not only the kinetics of the drives and the crane structure, but also the vibration dynamics of the drive and the crane structure.

Advantageously, the calculation of the control variable is based on an inversion of the physical model, which describes the movement of the crane tip as a function of the control variable. The inversion thus gives the control variable as a function of the desired movement of the cantilever tip.

Advantageously, the model describing the movement of the crane tip as a function of the control variable is non-linear. This results in a greater accuracy of the control result, since the decisive effects, which lead to natural oscillations of the crane structure, are non-linear.

If a hydraulic drive is used, the model advantageously takes into account the vibration dynamics of the drive due to the compressibility of the hydraulic fluid. This compressibility leads to vibrations of the crane structure, which can significantly burden them. By considering the compressibility of the hydraulic fluid, these vibrations can be damped.

Advantageously, the inventive method is used to control the rocking cylinder used as luffing mechanism, the kinematics of the articulation of the cylinder and the mass and inertia of the boom of the crane are included in the calculation of the control variable. As a result, natural vibrations of the boom can be damped in the rocker plane.

As an alternative to the hydraulic cylinder, a retraction mechanism can be used as a luffing mechanism, wherein advantageously the kinematics and / or dynamics of the pull-in stranding as well as the mass and the inertia of the boom of the crane are included in the calculation of the control variable.

Alternatively or additionally, the inventive method is used to control the slewing gear, wherein the moment of inertia of the boom of the crane enters the model. As a result, natural oscillations of the crane structure can be damped about the vertical axis of rotation.

Advantageously, the vibration damping takes place by means of feedforward control. As a result, costly sensors can be saved, which would otherwise have to be used. In addition, the feedforward control allows effective reduction of the natural oscillations without being limited to a certain frequency range by the response speed of the drives, as in closed-loop control.

Advantageously, the position, the speed, the acceleration and / or the jerk of the cantilever tip serve as reference values of the pilot control. In particular, at least two of these values advantageously serve as nominal values. Further advantageously, in addition to the position, at least one of the further variables is used as the setpoint. Further advantageously, all these variables are used as desired values of the precontrol.

Further advantageously, a desired trajectory of the cantilever tip is generated as inputs of the controller from inputs of an operator and / or an automation system. Thus, from the inputs entered by an operator by means of hand levers and / or the signals of an automation system, a desired trajectory of the cantilever tip is generated. The control method according to the invention now ensures that the drives of the crane are controlled so that the jib tip follows this desired trajectory and natural oscillations of the crane are avoided.

The method according to the invention can be used together with a load oscillation damping, or else completely without a load oscillation damping. Known methods for load oscillation damping focus solely on the avoidance of pendulum vibrations of the load, which sometimes even to a Increasing the natural vibration of the crane structure and thus could lead to a greater load than a control without load oscillation damping. By contrast, the present invention dampens the natural vibrations of the crane structure and thus protects the crane structure.

It can be provided that possible spherical pendulum vibrations of the load are not received as a measure in the control. Therefore, can be dispensed with expensive measuring equipment for measuring the rope angle.

Furthermore, possible spherical pendulum vibrations of the load can be disregarded when controlling the drive. As a result, the method according to the invention can also be used for simpler crane controls without load oscillation damping in order to protect the crane structure.

However, the method according to the invention can also be used in crane controls with load oscillation damping. The method is then implemented in such a way that initially the load movement serves as a set value, from which a desired movement of the cantilever tip is generated. This desired movement of the cantilever tip then serves as input of the method according to the invention. By this two-stage approach, a damping of the natural vibrations of the crane structure can be achieved even with methods with load oscillation damping. Known methods for load-swing damping, on the other hand, are designed solely to avoid vibrations of the load and can thereby even increase the natural vibrations of the crane structure.

The method described so far was preferably used to control a slewing gear and / or a luffing gear of a crane. However, it can also be used to control the hoist of a crane. In particular, the vibration dynamics of the hoist can be taken into account due to the compressibility of the hydraulic fluid.

In the control of the hoist but advantageously the target stroke of the load is used as input, on the basis of which a control variable for driving the drive is calculated.

Object of the present invention is therefore also to allow a structural protection in the control of the hoist of a crane.

This object is achieved by a method according to claim 10. In this case, a method for controlling a hoist of a crane is provided, in which a desired stroke movement of the load serves as input, on the basis of which a control variable for controlling the drive is calculated. According to the invention it is provided that in the calculation of the control variable, the vibration dynamics of the system of hoist, rope and load in the cable direction is taken into account to dampen natural oscillations. The inventors of the present invention have recognized that the vibration dynamics of the system of hoist, rope and load can lead to vibrations of the load or the crane structure, which can significantly burden both the load rope and the boom. Therefore, this vibration dynamics is now considered according to the invention to avoid natural oscillations of the load and / or the hoist. The hoist can be driven hydraulically and / or electrically.

Also, this method is advantageously used in cranes, in which a boom is pivoted about a horizontal rocking axis wippen on a tower. The load rope is advantageously guided by a winch on the tower base via one or more pulleys on the spire to one or more pulleys on the boom tip.

Advantageously, the vibration dynamics of the lifting system is taken into account according to the inventive method in a vibration-reduction operation, while any movements of the support area on which the crane structure is supported, not taken into account in the control of the hoist become. The control thus starts in the vibration reduction operation of a stationary support area. The control according to the invention must therefore take into account only vibrations which arise through the hoisting rope and / or the hoisting gear and / or the crane structure. Movements of the support area, as z. B. arise in a floating crane by wave motion, remain disregarded in the vibration reduction operation, however. The crane control can be made considerably simpler.

The method according to the invention can be used in a crane, which is actually supported on a stationary support area during the lift with the crane structure, in particular on the ground. However, the crane control according to the invention can also be used in a floating crane, but does not take into account the movements of the floating body in the vibration reduction mode. If the crane control system has an operating mode with active coasting sequence, the vibration reduction operation accordingly takes place without simultaneous active coasting sequence operation.

Further advantageously, the method according to the invention in transportable and / or mobile cranes is used. The crane advantageously has support means via which it can be supported at different lifting locations. Further advantageously, the method is used in port cranes, especially in mobile harbor cranes, crawler cranes, vehicle cranes, etc. used.

Advantageously, the oscillation dynamics of the lifting system due to the extensibility of the hoisting rope is taken into account in the calculation of the control variable. The extensibility of the hoisting cable leads to an expansion of the rope in the cable direction, which is attenuated according to the invention by a corresponding control of the hoist. Advantageously, the vibration dynamics of the rope is taken into account with the load hanging freely in the air.

The hoist of the crane according to the invention can be hydraulically driven. Alternatively, a drive via an electric motor is possible.

If a hydraulically driven hoist is used, the oscillation dynamics of the hoisting gear due to the compressibility of the hydraulic fluid are also advantageously taken into account in the calculation of the control variable. Thus, those natural oscillations are taken into account, which arise due to the compressibility of the hydraulic fluid, which is applied to the drive of the hoist.

Advantageously, the variable cable length of the hoist rope is included in the calculation of the control variable. The method according to the invention for controlling the lifting mechanism thus takes into account vibrations of the load suspended on the hoist rope, which are caused by the extensibility of the hoisting rope depending on the rope length of the hoist rope. Advantageously, continue to material constants of the hoisting rope, which affect its extensibility, in the calculation. Advantageously, the rope length is determined by the position of the hoist.

Further advantageously, the weight of the load hanging on the load rope is included in the calculation of the control variable. Advantageously, this weight of the load is measured and enters as a measured value in the control process.

Advantageously, the control of the hoist is based on a physical model of the crane, which describes the stroke movement of the load as a function of the control variable of the hoist. As already stated, such a physical model allows a quick adaptation to new crane types. In addition, this allows a more accurate and better vibration damping. In addition to the kinematics, the model also describes the vibration dynamics due to the extensibility of the hoisting rope and / or due to the compressibility of the hydraulic fluid. The model advantageously starts from a stationary support area of the crane.

Advantageously, the control of the hoist is based on the inversion of the physical model. This inversion allows precise control of the drive. The physical model first describes the movement of the load as a function of the control variable. The inversion therefore gives the control variable as a function of the desired stroke movement of the load.

As already shown with respect to the control of the rocker and the slewing gear, the control of the hoist according to the present invention can be combined with a load oscillation damping, which dampens spherical oscillations of the load. The present method can also be used without a load swing damping to dampen natural oscillations of the system of hoist winch, rope and load, which run in the cable direction, and in particular vibrations of the load in the stroke direction.

The present invention further includes a crane controller for performing a method as set forth above. The crane control advantageously has a control program via which a method, as has been described above, is implemented.

The present invention further comprises a crane having a control unit having a control program via which a method as set forth above is implemented. The crane control or the crane obviously gives rise to the same advantages as already described above with regard to the methods.

The crane advantageously has a slewing gear, a luffing gear and / or a hoist. Advantageously, the crane has a boom, which is pivoted about a horizontal rocking axis on the crane crane and is moved over a luffing cylinder. Alternatively, a retraction mechanism can be used as a luffing mechanism. Furthermore, the crane advantageously has a tower which is rotatable about a vertical axis of rotation. Advantageously, the boom is doing on Tower hinged. Further advantageously, the hoist cable runs from the hoist via one or more pulleys to the load. Furthermore, advantageously, the crane has an undercarriage with a chassis.

Figur 1:

ein Ausführungsbeispiel eines erfindungsgemäßen Krans,

Figur 2:

eine Prinzipzeichnung der Kinematik der Anlenkung des Auslegers eines erfindungsgemäßen Krans,

Figur 3:

eine Prinzipzeichnung der Hydraulik des Wippzylinders eines erfin- dungsgemäßen Krans,

Figur 4:

eine Prinzipzeichnung der Hydraulik des Drehwerks und des Hubwerks eines erfindungsgemäßen Krans, und

Figur 5:

eine Prinzipdarstellung des physikalischen Modells, welches zur Be- schreibung der Dynamik des Lastseils herangezogen wird.

The present invention will now be described in more detail with reference to an embodiment and drawings. Showing:

FIG. 1:

an embodiment of a crane according to the invention,

FIG. 2:

a principle drawing of the kinematics of the articulation of the boom of a crane according to the invention,

FIG. 3:

a schematic drawing of the hydraulics of the luffing cylinder of a crane according to the invention,

FIG. 4:

a schematic drawing of the hydraulics of the slewing gear and the hoist of a crane according to the invention, and

FIG. 5:

a schematic representation of the physical model, which is used to describe the dynamics of the load rope.

In FIG. 1 an embodiment of the crane according to the invention is shown in which an embodiment of a control method according to the invention is implemented. The crane has a boom 1, which is pivoted about a horizontal rocking axis on the tower 2. In the embodiment, a hydraulic cylinder 10 is provided for rocking up and down the boom 1 in the rocker plane, which is articulated between the boom 1 and the tower 2.

The kinematics of the articulation of the boom 1 on the tower 2 is closer in FIG. 2 shown. The boom 1 is pivoted at a pivot point 13 on the tower 2 about a horizontal rocking axis. The hydraulic cylinder 10 is via a Anlenkpunkt 11 on the tower 2 and a pivot point 12 on the boom 1 between them. By a change in length of the hydraulic cylinder 10 so the boom 1 in the rocker plane can be up and tilts. The relevant angles and lengths are in FIG. 2 located.

The tower 2 is as in Fig. 1 shown rotatably disposed about a vertical axis of rotation z, wherein the rotational movement is generated by a slewing gear 20. The tower 2 is arranged for this purpose on an upper carriage 7, which can be rotated over the slewing gear relative to a lower carriage 8. In the embodiment, this is a movable crane, for which the undercarriage 8 is equipped with a chassis 9. At the hub of the crane can then be supported by support members 71.

The lifting of the load takes place via a hoist rope 3, on which a load-receiving element 4, in this case a gripper, is arranged. The hoist rope 3 is guided over pulleys on the jib tip 5 and on the tower top 6 to the hoist 30 on the superstructure over which the length of the hoist rope can be changed.

The inventors of the present invention have now recognized that in known methods for controlling the drives of the crane natural oscillations of the crane structure and the drives can arise, which can significantly burden them.

When driving the slewing gear and / or the luffing gear according to the present invention, therefore, a desired movement of the cantilever tip serves as an input variable, based on which a control variable for controlling the drives is calculated. If the drive is a hydraulic drive, the control variable can include, for example, the hydraulic pressure or the hydraulic flow to the hydraulic drive. According to the invention, in the calculation of the control variable, the internal vibration dynamics of the drives or crane structure. As a result, natural vibrations of the crane structure and the drives can be avoided.

When controlling the hoist, however, vibrations of the load due to the elasticity of the load rope form a decisive factor in the natural vibrations of the crane structure. Therefore, the total system of hoist 30 and rope 3 is used here as the drive system for calculating the control of the hoist. The setpoint stroke position of the load serves as an input variable, on the basis of which the control variable for controlling the hoist is calculated. The vibration dynamics of the hoist, rope and load system are taken into account in the calculation of the control variable in order to avoid inherent vibrations of the system. In particular, the extensibility of the hoisting rope is taken into account in the calculation of the control quantity in order to dampen the expansion vibrations of the rope. So here, unlike in known load pendulum damping no spherical pendulum vibrations of the load are taken into account, but the vibration of the load in the cable direction by the expansion or contraction of the hoisting rope. Furthermore, the vibration of the system of hoist 30 and rope 3 due to the compressibility of the hydraulic fluid can also be considered in the hoist 30.

The present invention thus allows a considerable structural protection of the crane, which in turn saves costs in maintenance as well as in the construction. By taking into account the vibration dynamics of the drives of the crane, that is, the slewing gear, the luffing gear and the system of hoist and rope loads of the crane structure are avoided, which can even be strengthened in known methods for spherical pendulum damping of the load on the contrary.

The actuation of the drives takes place on the basis of a physical model, which describes the movement of the crane tip or the load as a function of the control variable, the model takes into account the internal vibration dynamics of the respective drives.

In FIG. 3 Here is a schematic diagram of the hydraulics of the luffing gear shown. It is z. B. a diesel engine 15 is provided, which drives a variable displacement pump 16. This variable displacement pump 16 acts on the two hydraulic chambers of the luffing cylinder 10 with hydraulic fluid. Alternatively, could be used to drive the variable displacement pump 16 and an electric motor.

FIG. 4 shows a schematic diagram of the hydraulics of the slewing gear and the hoist. Here again z. B. a diesel or electric motor 25 is provided, which drives a variable displacement pump 26. This variable displacement pump 26 forms with a hydraulic motor 27 a hydraulic circuit and drives it. Also, the hydraulic motor 27 is designed as an adjusting motor. Alternatively, a constant motor could be used. About the hydraulic motor 27 then the slewing or the hoist winch is driven.

In FIG. 5 Now, the physical model through which the dynamics of the load cable 3 and the load is described, shown in more detail. The system of load rope and load is considered as a damped spring pendulum, with a spring constant C and a damping constant D. The spring constant C is the length of the hoist rope L, which is determined either on the basis of measured values or calculated on the basis of the control of the hoist winch , Furthermore, the mass M of the load, which is measured via a load mass sensor, enters into the control.

An exemplary embodiment of a method for controlling the respective works will now be described in more detail below:

1 Introduction

At the in FIG. 1 illustrated embodiment is a mobile harbor crane. Here, the boom, the tower and the hoist winch are set in motion by appropriate drives. The boom, the tower and the Hoist winches of the crane in motion offset hydraulic drives generate natural vibrations due to the inherent dynamics of the hydraulic systems. The resulting force oscillations affect the long-term fatigue of the cylinder and the cables and thus reduce the life of the entire crane structure, resulting in increased maintenance. According to the invention, therefore, a tax law is provided which suppresses caused by rocking, turning and lifting movements of the crane natural oscillations and thereby reduces the stress cycles within the Wöhlerdiagramms. A reduction in the stress cycles logically increases the service life of the crane structure.

In the derivation of the tax law returns are to be avoided because they require sensor signals, which must meet certain safety requirements within industrial applications and thereby lead to higher costs.

Therefore, the design of a pure pilot control without feedback is necessary. Within this essay, a flatness-based feedforward control, which inverts the system dynamics, is derived for the rocker, turntable and hoist.

2 luffing mechanism

The boom of the crane is set in motion by a hydraulic luffing cylinder, as in FIG. 1 is shown. The dynamic model and the control law for the luffing cylinder are derived in the following section.

2.1 Dynamic Model

und für die Zylindergeschwindigkeit $${\dot{z}}_c\left({\varphi }_a,{\dot{\varphi }}_a\right)=\frac{\partial z_c\left({\varphi }_a\right)}{\partial {\varphi }_a}\frac{\partial {\varphi }_a}{\partial t}=-d_a{d}_b\sin \left(\frac{\pi }{2}+{\alpha }_2-{\alpha }_1-{\varphi }_a\right)\frac{{\dot{\varphi }}_a}{z_c\left({\varphi }_a\right)}$$

A dynamic model of the hydraulically driven boom of the crane is derived below. The boom is schematically in together with the hydraulic cylinder FIG. 2 shown. The movement of the boom is described by the rocking angle φ _a and the angular velocity φ̇ _a . The movement of the hydraulic cylinder is determined by the cylinder position Z _c , which is considered the distance between the cylinder connection with the tower and the cylinder connection is defined with the boom, and the cylinder speed ż _c described. The geometric dependencies between the movement of the boom and the cylinder are given by the geometric constants d _a , d _b , α ₁ and α ₂ and the Cosinussatz. For the cylinder position: $$z_c\left({\phi }_a\right)=\sqrt{d_a^2+d_b^2-2d_a{d}_b\cos \left(\frac{\pi }{2}+{\alpha }_2-{\alpha }_1-{\phi }_a\right)}$$

and for the cylinder speed $${\dot{z}}_c\left({\phi }_a,{\dot{\phi }}_a\right)=\frac{\partial z_c\left({\phi }_a\right)}{\partial {\phi }_a}\frac{\partial {\phi }_a}{\partial t}=-d_a{d}_{\mathrm{<figure-callout\; id="2"\; label="b\; sin\; \pi "\; filenames="imgf0001.png"\; state="\{\{state\}\}">b\; </figure-callout>}}\sin \left(\frac{\pi }{}\right)\left(\frac{}{2}+{\alpha }_2-{\alpha }_1-{\phi }_a\right)\frac{{\dot{\phi }}_a}{z_c\left({\phi }_a\right)}$$

wobei J_b und m_b das Trägheitsmoment bzw. die Masse des Auslegers bezeichnen, S_b der Abstand zwischen der Auslegerverbindung mit dem Turm und dem Massenschwerpunkt des Auslegers ist, g die Gravitationskonstante ist und F_c und d_c die Zylinderkraft bzw. den Dämpfungskoeffizienten des Zylinders bezeichnen. Es wird angenommen, dass am Ende des Auslegers keine Nutzlast angebracht ist. Der Term cos (γ) in (3) ist durch den Sinussatz gegeben: $$\cos \left(\gamma \right)=\sin \left(\frac{\pi }{2}-\gamma \right)=\frac{d_a}{z_c\left({\varphi }_a\right)}\sin \left(\frac{\pi }{2}+{\alpha }_2-{\varphi }_a\right)$$

wobei α ₁ vernachlässigt wird.Since the geometric angle α _{1 is} small, it is neglected in the derivation of the dynamic model. The method of Newton-Euler gives the equation of motion for the cantilever: $$J_b{\ddot{\phi }}_a=\left(F_c+d_c{\dot{z}}_c\left({\phi }_a,{\dot{\phi }}_a\right)\right)d_b\cos \left(\gamma \right)-m_{b}G{\mathit{s}}_b\cos \left({\phi }_a\right).{\phi }_a\left(0\right)={\phi }_{a0}.{\dot{\phi }}_a\left(0\right)=0$$

where J _b and m _b denote the moment of inertia of the boom, S _{b is} the distance between the boom connection to the tower and the center of gravity of the boom, g is the gravitational constant and F _c and d _{c are} the cylinder force and damping coefficient, respectively Designate cylinders. It is assumed that no payload is attached to the end of the boom. The term cos (γ) in (3) is given by the sinus set: $$\cos \left(\gamma \right)=\sin \left(\frac{\pi }{2}-\gamma \right)=\frac{d_a}{z_c\left({\phi }_a\right)}\mathrm{<figure-callout\; id="2"\; label="sin\; \pi "\; filenames="imgf0001.png"\; state="\{\{state\}\}">sin\; </figure-callout>}\left(\frac{\pi }{}\right)\left(\frac{}{2}+{\alpha }_2-{\phi }_a\right)$$

where α _{1 is} neglected.

wobei A ₁ und A ₂ die wirksamen Flächen in jeder Kammer bezeichnen. Die Drücke p ₁ und p ₂ werden durch die Druckaufbaugleichung unter der Annahme, dass keine innere oder äußere Leckage auftritt, beschrieben. Somit gilt: $${\dot{p}}_1=\frac{1}{\beta V_1\left(z_c\right)}\left(q_l-A_1{\dot{z}}_c\right),p_1\left(0\right)=p_{10}$$

$${\dot{p}}_2=\frac{1}{\beta V_2\left(z_c\right)}\left(-q_l+A_2{\dot{z}}_c\right),p_2\left(0\right)=p_{20}$$

wobei β die Kompressibilität des Öls ist und die Kammervolumina durch $$V_1\left(z_c\right)=V_{\min }+A_1\left(z_c\left({\varphi }_a\right)-z_{c,\min }\right)$$

$$V_2\left(z_c\right)=V_{\min }+V_{2,\max }-A_2\left(z_c\left({\varphi }_a\right)-z_{c,\min }\right)$$

gegeben sind, wobei V_min das Mindestvolumen in jeder Kammer bezeichnet und V_2,max und z _c,min das Höchstvolumen in der zweiten Kammer bzw. die Mindestzylinderposition, die erreicht wird, wenn ϕ _a = ϕ _a,max , ist, sind. Der Öldurchsatz q_l wird durch den Pumpenwinkel vorgegeben und ist gegeben durch: $$q_l=K_l{u}_l$$

wobei u_l und K_l der Ansteuerstrom für den Pumpenwinkel und der Proportionalitätsfaktor sind.The hydraulic circuit of the luffing cylinder consists in principle of a variable displacement pump and the hydraulic cylinder itself, as in FIG. 3 is shown. For the cylinder force follows: $$F_c={\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0001.png"\; state="\{\{state\}\}">p</figure-callout>}}_2{A}_2-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="imgf0001.png"\; state="\{\{state\}\}">p</figure-callout>}}_1{A}_1$$

where A ₁ and A ₂ denote the effective areas in each chamber. The pressures p ₁ and p ₂ are described by the pressure buildup assumption that no internal or external leakage occurs. Thus: $${\dot{p}}_1=\frac{1}{\beta V_1\left(z_c\right)}\left(q_l-{\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="imgf0001.png"\; state="\{\{state\}\}">A</figure-callout>}}_1{\dot{z}}_c\right).{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="imgf0001.png"\; state="\{\{state\}\}">p</figure-callout>}}_1\left(0\right)=p_{10}$$

$${\dot{p}}_2=\frac{1}{\beta V_2\left(z_c\right)}\left(-q_l+A_2{\dot{z}}_c\right).{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="imgf0001.png"\; state="\{\{state\}\}">p</figure-callout>}}_2\left(0\right)={\mathrm{<figure-callout\; id="20"\; label="p"\; filenames="imgf0001.png"\; state="\{\{state\}\}">p</figure-callout>}}_{20}$$

where β is the compressibility of the oil and the chamber volumes through $$V_1\left(z_c\right)=V_{\min }+A_1\left(z_c\left({\phi }_a\right)-z_{c.\min }\right)$$

$$V_2\left(z_c\right)=V_{\min }+{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="imgf0001.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2.\mathrm{Max}}-{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="imgf0001.png"\; state="\{\{state\}\}">A</figure-callout>}}_2\left(z_c\left({\phi }_a\right)-z_{c.\min }\right)$$

where V _min denotes the minimum volume in each chamber and V _{2, max} and z _{c , min are} the maximum volume in the second chamber and the minimum cylinder position achieved when φ _a = φ _{a, max} . The oil flow q _l is given by the pump angle and is given by: $$q_l=K_l{u}_l$$

where u _l and K _{l are} the drive current for the pump angle and the proportionality factor.

2.2 Tax Law

wobei $$\mathbf{f}\left(\mathbf{x}\right)=\left[\begin{array}{l}{x}_2\\ \frac{\left(x_3+d_c{\dot{z}}_c\right)d_b\cos \left(\gamma \right)-m_{b}g{s}_b\cos \left(x_1\right)}{J_b}\\ \left(\frac{A_2^2}{\beta V_2\left(z_c\right)}+\frac{A_1^2}{\beta V_1\left(z_c\right)}\right){\dot{z}}_c\end{array}\right]$$

$$\mathbf{g}\left(\mathbf{x}\right)=\left[\begin{array}{l}0\\ 0\\ -\frac{K_l{A}_2}{\beta V_2\left(z_c\right)}-\frac{K_l{A}_1}{\beta V_1\left(z_c\right)}\end{array}\right]$$

$$h\left(\mathbf{x}\right)=x_1$$

und z_c = z _c (x₁), ż_c = ż_c (x₁,x₂), γ = γ (x_l) und u = u_l. The flatness-based feedforward control according to the invention uses the differential flatness of the system to invert the system dynamics. To derive such a tax law, the dynamic model derived in Section 2.1 must be transformed into the state space. By introducing the state vector x = [φ _a , φ̇ _a , F _c ] ^T , the dynamic model (3), (5), (6) and (7) can be written as a system of first order differential equations given by: $$\dot{\mathbf{x}}=\mathbf{f}\left(\mathbf{x}\right)+\mathbf{G}\left(\mathbf{x}\right)u.y=H\left(\mathbf{x}\right).\mathbf{x}\left(0\right)={\mathbf{x}}_0.t\ge 0$$

in which $$\mathbf{f}\left(\mathbf{x}\right)=\left[\begin{array}{l}{x}_2\\ \frac{\left(x_3+d_c{\dot{z}}_c\right)d_b\cos \left(\gamma \right)-m_{b}G{s}_b\cos \left(x_1\right)}{{\mathrm{<figure-callout\; id="2"\; label="J\; b\; A"\; filenames="imgf0001.png"\; state="\{\{state\}\}">J\; </figure-callout>}}_b}& \\ \left(\frac{A_{}^{}}{}\right)\left(\frac{{}_2^2}{\beta V_2\left(z_c\right)}+\frac{{\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="imgf0001.png"\; state="\{\{state\}\}">A</figure-callout>}}_1^2}{\beta V_1\left(z_c\right)}\right){\dot{z}}_c\end{array}\right]$$

$$\mathbf{G}\left(\mathbf{x}\right)=\left[\begin{array}{l}0\\ 0\\ -\frac{K_l{A}_2}{\beta V_2\left(z_c\right)}-\frac{K_l{A}_1}{\beta V_1\left(z_c\right)}\end{array}\right]$$

$$H\left(\mathbf{x}\right)=x_1$$

and z _c = z _c (x ₁ ) , ż _c = ż _c (x ₁ , x ₂ ), γ = γ (x _l ) and u = u _l .

For the design of a flatness-based feedforward, the relative degree r with respect to the system output must be equal to the order n of the system. Therefore, the relative degree of the considered system (11) is examined below. The relative degree of system output is determined by the following conditions: $$\begin{array}{l}{L}_{\mathrm{G}}{L}_{\mathrm{f}}^{i}H\left(\mathbf{x}\right)=0\forall i=0.....r-2\\ L_{\mathrm{G}}{L}_{\mathrm{f}}^{r-1}H\left(\mathbf{x}\right)\ne 0\forall \mathrm{x}\in {\mathrm{R}}^n\end{array}$$

The operators L _f and Lg represent the Lie derivatives along the vector fields f and g, respectively. Using (15) gives r = n = 3 , so the system (11) is denoted by (12), (13) and ( 14) flat and a flatness-based feedforward control can be designed.

$$\dot{y}=\frac{\partial h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}h\left(\mathbf{x}\right)+\underset{=0}{\underbrace{L_{\mathrm{g}}h\left(\mathbf{x}\right)u}}=x_2$$

$$\ddot{y}=\frac{\partial L_{\mathrm{f}}h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_f^{2}h\left(\mathbf{x}\right)+\underset{=0}{\underbrace{L_g{L}_{f}h\left(\mathbf{x}\right)u}}=f_2\left(\mathbf{x}\right)$$

$$\begin{array}{ll}\stackrel{¨\dot{}}{y}& =\frac{\partial L_f^{2}h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_f^{3}h\left(\mathbf{x}\right)+L_g{L}_f^{2}h\left(\mathbf{x}\right)u\\ & =\frac{x_2}{J_b}{m}_{b}g{s}_b\sin \left(x_1\right)-\frac{x_2}{J_b}\left(x_3+d_c{\dot{z}}_c\left(x_1,x_2\right)\right)d_b\sin \left(\gamma \left(x_1\right)\right)\mathit{\gamma ʹ}\left(x_1\right)\\ & +\frac{x_2}{J_b}{d}_c{d}_b\cos \left(\gamma \left(x_1\right)\right)\frac{\partial {\dot{z}}_c\left(x_1,x_2\right)}{\partial x_1}+\frac{f_2\left(\mathbf{x}\right)}{J_b}{d}_c{d}_b\cos \left(\gamma \left(x_1\right)\right)\frac{\partial {\dot{z}}_c\left(x_1,x_2\right)}{\partial x_2}\\ & +\frac{f_3\left(\mathbf{x}\right)+g_3\left(\mathbf{x}\right)u}{J_b}{d}_b\cos \left(\gamma \left(x_1\right)\right)\end{array}$$

wobei f_i( x ) und g_i( x ) die i-te Reihe des Vektorfelds f(x) und g(x) bezeichnen, die durch (12) und (13) gegeben sind. Die Zustände in Abhängigkeit des Systemausgangs und dessen Ableitungen folgen aus (16), (17) und (18) und lassen sich schreiben als: $$x_1=y$$

$$x_2=\dot{y}$$

$$x_3=\frac{J_b\ddot{y}+m_{b}g{s}_b\cos \left(\gamma \left(y\right)\right)}{d_b\cos \left(\gamma \left(y\right)\right)}-d_c{\dot{z}}_c\left(y,\dot{y}\right)$$

The output of the system (14) and its time derivatives are used to invert the system dynamics. The derivatives are formed by the Lie derivatives, thus: $$y=H\left(\mathrm{x}\right)=x_1$$

$$\dot{y}=\frac{\partial H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}H\left(\mathbf{x}\right)+\underset{=0}{\underset{\}}{L_{\mathrm{G}}H\left(\mathbf{x}\right)u}}=x_2$$

$$\ddot{y}=\frac{\partial L_{\mathrm{f}}H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}={\mathrm{<figure-callout\; id="2"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_f^{}H\left(\mathbf{x}\right)+\underset{=0}{\underset{\}}{L_G{L}_{f}H\left(\mathbf{x}\right)u}}=f_2\left(\mathbf{x}\right)$$

$$\begin{array}{ll}\stackrel{¨\dot{}}{y}& =\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_f^{}H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}={\mathrm{<figure-callout\; id="3"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_f^{}H\left(\mathbf{x}\right)+L_G<figure-callout\; id="2"\; label="\; L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">\; </figure-callout>L_f^{}{}_2^{}H\left(\mathbf{x}\right)u\\ & =\frac{x_2}{J_b}{m}_{b}G{s}_b\sin \left(x_1\right)-\frac{x_2}{J_b}\left(x_3+d_c{\dot{z}}_c\left(x_1,x_2\right)\right)d_b\sin \left(\gamma \left(x_1\right)\right)\mathit{\gamma \text{'}}\left(x_1\right)\\ & +\frac{x_2}{J_b}{d}_c{d}_b\cos \left(\gamma \left(x_1\right)\right)\frac{\partial {\dot{z}}_c\left(x_1,x_2\right)}{\partial x_1}+\frac{f_2\left(\mathbf{x}\right)}{J_b}{d}_c{d}_b\cos \left(\gamma \left(x_1\right)\right)\frac{\partial {\dot{z}}_c\left(x_1,x_2\right)}{\partial x_2}\\ & +\frac{f_3\left(\mathbf{x}\right)+G_3\left(\mathbf{x}\right)u}{J_b}{d}_b\cos \left(\gamma \left(x_1\right)\right)\end{array}$$

where f _i ( x ) and g _i ( x ) denote the i-th row of the vector field f ( x ) and g ( x ) given by (12) and (13). The states depending on the system output and its derivatives follow from (16), (17) and (18) and can be written as: $$x_1=y$$

$$x_2=\dot{y}$$

$$x_3=\frac{J_b\ddot{y}+m_{b}G{s}_b\cos \left(\gamma \left(y\right)\right)}{d_b\cos \left(\gamma \left(y\right)\right)}-d_c{\dot{z}}_c\left(y,\dot{y}\right)$$

welche die Systemdynamik invertiert. Die Referenzsignale y und die entsprechenden Ableitungen werden durch eine numerische Trajektoriengenerierung aus dem Handhebelsignal des Kranbedieners oder aus den Steuersignalen eines Automatisierungssystems gewonnen. Da der Ansteuerstrom u_l die Zylindergeschwindigkeit vorgibt (siehe (10)), werden die Trajektorien ursprünglich in Zylinderkoordinaten für z_c , ż_c , z̈_c und z̈ż_c geplant. Anschließend werden die so erhaltenen Trajektorien in ϕ _a -Koordinaten transformiert und der tatsächliche Ansteuerstrom berechnet.The dissolution of (19) after system input u, using (20), (21), and (22), provides the control law for the flatness-based feedforward control for the luffing cylinder $$u_l=f\left(y,\dot{y},\ddot{y},\stackrel{\dot{}¨}{y}\right)$$

which inverts the system dynamics. The reference signals y and the corresponding derivatives are obtained by a numerical trajectory generation from the hand lever signal of the crane operator or from the control signals of an automation system. Since the drive current u _{l gives} the cylinder speed (see (10)), the trajectories are originally planned in cylinder coordinates for z _c , ż _c , z _c and z _c . Subsequently, the trajectories thus obtained are transformed into φ _a coordinates and the actual drive current is calculated.

3 slewing gear

The tower is rotated by a hydraulic rotary motor. The dynamic model and the control law for the slewing gear are derived within the following section.

3.1 Dynamic Model

wobei J_t und J_m das Trägheitsmoment des Turms und des Motors bezeichnen, i_s das Übersetzungsverhältnis des Drehwerks ist, Δp_s die Druckdifferenz zwischen den Druckkammern des Motors ist und D_m die Verdrängung des Hydraulikmotors bezeichnet. Das Trägheitsmoment des Turms J_t umfasst das Trägheitsmoment des Turms selbst, des Auslegers, der angebrachten Nutzlast des Turms um die z-Achse des Turms (siehe Figur 1). Der Hydraulikkreislauf des Drehwerks besteht im Prinzip aus einer Verstellpumpe und dem Hydraulikmotor selbst, wie in Figur 4 dargestellt ist. Die Druckdifferenz zwischen beiden Druckkammern des Motors wird durch die Druckaufbaugleichung unter der Annahme, dass es zu keinen inneren oder äußeren Leckagen kommt, beschrieben. Zudem wird im Folgenden die kleine Volumenänderung aufgrund des Motorwinkels ϕ _m vernachlässigt. Somit wird das Volumen in beiden Druckkammern als konstant angenommen und mit V_m bezeichnet. Mit Hilfe dieser Annahmen lässt sich die Druckaufbaugleichung als $$\mathrm{\Delta }{\dot{p}}_s=\frac{4}{V_m\beta }\left(q_s-D_m{i}_s{\dot{\varphi }}_s\right),\mathrm{\Delta }{p}_s\left(0\right)=\mathrm{\Delta }{p}_{s0}$$

schreiben, wobei β die Kompressibilität des Öls ist. Der Öldurchsatz q_s wird durch den Pumpenwinkel vorgegeben und ist gegeben durch: $$q_s=K_s{u}_s$$

wobei u_s und K_s der Ansteuerstrom des Pumpenwinkels und der Proportionalitätsfaktor sind.The movement of the tower about the z-axis (see FIG. 1 ) is described by the rotation angle φ _s and the angular velocity φ̇ _s . Using the method of Newton-Euler gives the equation of motion for the hydraulically powered tower: $$\left(J_t+i_s^2{J}_m\right){\ddot{\phi }}_s=i_s{D}_m\mathrm{\Delta }{p}_s.{\phi }_s\left(0\right)={\phi }_{s0}.{\dot{\phi }}_s\left(0\right)=0$$

where J _t and J _m denote the moment of inertia of the tower and the motor, i _{s is} the gear ratio of the slewing gear, Δ p _{s is} the pressure difference between the pressure chambers of the engine and D _m denotes the displacement of the hydraulic motor. The moment of inertia of the tower J _t comprises the moment of inertia of the tower itself, the boom, the attached payload of the tower about the z-axis of the tower (see FIG. 1 ). The hydraulic circuit of the slewing gear consists in principle of a variable displacement pump and the hydraulic motor itself, as in FIG. 4 is shown. The pressure difference between both pressure chambers of the engine is described by the pressure buildup assuming that there are no internal or external leakages. In addition, in the following, the small volume change due to the motor angle φ _{m is} neglected. Thus, the volume in both pressure chambers is assumed to be constant and denoted by V _m . With the help of these assumptions, the pressure build-up equation can be described as $$\mathrm{\Delta }{\dot{p}}_s=\frac{4}{V_m\beta }\left(q_s-D_m{i}_s{\dot{\phi }}_s\right).\mathrm{\Delta }{p}_s\left(0\right)=\mathrm{\Delta }{p}_{s0}$$

where β is the compressibility of the oil. The oil flow rate q _s is given by the pump angle and is given by: $$q_s=K_s{u}_s$$

where u _s and K _{s are} the drive current of the pump angle and the proportionality factor.

3.2 Tax Law

$$\mathbf{g}\left(\mathbf{x}\right)=\left[\begin{array}{c}0\\ 0\\ \frac{4K_s}{V_m\beta }\end{array}\right]$$

$$h\left(\mathbf{x}\right)=x_1$$

und u=u_s. In the following, the dynamic model for the slewing gear is transformed into the state space and a flatness-based feedforward control is designed. The state vector for the slewing gear is defined as x = [φ _s , φ _s , Δ p _s ] ^T With the aid of the state vector, the dynamic model consisting of (24), (25) and (26) can be written as a system of first-order differential equations. that is given by (11) with: $$\mathbf{f}\left(\mathbf{x}\right)=\left[\begin{array}{c}{x}_2\\ \frac{i_s{D}_m{x}_3}{J_t+i_s^2{J}_m}\\ \frac{-4D_m{i}_s{x}_2}{V_m\beta }\end{array}\right]$$

$$\mathbf{G}\left(\mathbf{x}\right)=\left[\begin{array}{c}0\\ 0\\ \frac{4K_s}{V_m\beta }\end{array}\right]$$

$$H\left(\mathbf{x}\right)=x_1$$

and u = u _s .

Again, the relative degree r with respect to the system output must be equal to the order n of the system. Using (15) gives r = n = 3, thus the system (11) is flat with (27), (28) and (29) and a flatness-based feedforward control can be formulated.

$$\dot{y}=\frac{\partial h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}h\left(\mathbf{x}\right)+\underset{=0}{\underbrace{L_{g}h\left(\mathbf{x}\right)u}}=x_2$$

$$\ddot{y}=\frac{\partial L_{\mathrm{f}}h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}^{2}h\left(\mathbf{x}\right)+\underset{=0}{\underbrace{L_g{L}_{\mathrm{f}}h\left(\mathbf{x}\right)u}}=\frac{i_s{D}_m{x}_3}{J_t+i_s^2{J}_m}$$

$$\stackrel{\dot{}¨}{y}=\frac{\partial L_{\mathrm{f}}^{2}h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}^{3}h\left(\mathbf{x}\right)+L_g{L}_{\mathrm{f}}^{2}h\left(\mathbf{x}\right)u=\frac{-4D_m{i}_s{x}_2}{V_m\beta }+\frac{4K_s}{V_m\beta }u$$

The system output (29) and its time derivatives are used to invert the system dynamics. The derivatives are given by the Lie derivatives, so $$y=H\left(\mathbf{x}\right)=x_1$$

$$\dot{y}=\frac{\partial H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}H\left(\mathbf{x}\right)+\underset{=0}{\underset{\}}{L_{G}H\left(\mathbf{x}\right)u}}=x_2$$

$$\ddot{y}=\frac{\partial L_{\mathrm{f}}H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}={\mathrm{<figure-callout\; id="2"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)+\underset{=0}{\underset{\}}{L_G{L}_{\mathrm{f}}H\left(\mathbf{x}\right)u}}=\frac{i_s{D}_m{x}_3}{J_t+i_s^2{J}_m}$$

$$\stackrel{\dot{}¨}{y}=\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}={\mathrm{<figure-callout\; id="3"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)+L_G<figure-callout\; id="2"\; label="\; L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">\; </figure-callout>L_{\mathrm{f}}^{}{\mathrm{}}_2^{}H\left(\mathbf{x}\right)u=\frac{-4D_m{i}_s{x}_2}{V_m\beta }+\frac{4K_s}{V_m\beta }u$$

$$x_2=\dot{y}$$

$$x_3=\frac{J_t+i_s^2{J}_m}{i_s{D}_m}\ddot{y}$$

The states depending on the system output and its derivatives follow from (30), (31) and (32) and can be written as: $$x_1=y$$

$$x_2=\dot{y}$$

$$x_3=\frac{J_t+i_s^2{J}_m}{i_s{D}_m}\ddot{y}$$

welche die Systemdynamik invertiert. Das Referenzsignal y und seine Ableitungen werden durch eine numerische Trajektoriengenerierung aus dem Handhebelsignal des Kranbedieners gewonnen.Solving (33) for the system input u , using (34), (35) and (36), gives the control law for the flatness-based feedforward control for the slewing gear $$u_s=f\left(y,\dot{y},\ddot{y},\stackrel{\dot{}¨}{y}\right)$$

which inverts the system dynamics. The reference signal y and its derivatives are obtained by a numerical trajectory generation from the hand lever signal of the crane operator.

4 hoist winches

The hoist winch of the crane is driven by a hydraulically operated rotary motor. The dynamic model and the hoist winch control law are derived in the following section.

4.1 Dynamic Model

wobei l₁ , l₂ und l₃ die Teillängen von der Hubwinde zum Turm, vom Turm zum Ende des Auslegers und vom Ende des Auslegers zum Haken bezeichnen. Das Hubsystem des Krans, das aus der Hubwinde, dem Seil und der Nutzlast besteht, wird im Folgenden als Feder-Masse-Dämpfer-System betrachtet und ist in Figur 5 dargestellt. Das Verwenden des Verfahrens von Newton-Euler ergibt die Bewegungsgleichung für die Nutzlast: $$m_p{\ddot{z}}_p=m_{p}g-\underset{F_s}{\underbrace{\left(c\left(z_p-r_w{\varphi }_w\right)+d\left({\dot{z}}_p-r_w{\dot{\varphi }}_w\right)\right)}},z_p\left(0\right)=z_{p0},{\dot{z}}_p\left(0\right)=0$$

mit der Gravitationskonstante g, der Federkonstante c, der Dämpfungskonstante d, dem Radius der Hubwinde r_w, dem Winkel ϕ _w , der Hubwinde, der Winkelgeschwindigkeit ϕ̇ _w , der Nutzlastposition z_p , der Nutzlastgeschwindigkeit ż_p , und der Nutzlastbeschleunigung z̈_p . Die Seillänge l_r ist gegeben durch $$l_r\left(t\right)=r_w{\varphi }_w\left(t\right)$$

mit $${\varphi }_w\left(0\right)={\varphi }_{w0}=\frac{l_1\left(0\right)+3l_2\left(0\right)+l_3\left(0\right)}{r_w}$$

Since the lifting force is directly influenced by the payload movement, the dynamics of the payload movement must be taken into account. As in FIG. 1 illustrated, the payload with the mass m _{p is} attached to a hook and can be raised or lowered by the crane by means of a rope of length l _r . The rope is deflected by a pulley on the jib tip and on the tower. However, the rope is not deflected directly from the end of the boom to the hoist winch, but from the end of the boom to the tower, from there back to the end of the boom and then over the tower to the hoist winch (see FIG. 1 ). Thus, the entire rope length is given by: $$l_r={\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="imgf0001.png"\; state="\{\{state\}\}">l</figure-callout>}}_1+3{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="imgf0001.png"\; state="\{\{state\}\}">l</figure-callout>}}_2+{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="imgf0001.png"\; state="\{\{state\}\}">l</figure-callout>}}_3$$

where l ₁ , l ₂ and l ₃ denote the partial lengths of the hoist winch to the tower, from the tower to the end of the boom and from the end of the boom to the hook. The lifting system of the crane, which consists of the hoist winch, the rope and the payload, is considered below as a spring-mass-damper system and is in FIG. 5 shown. Using the Newton-Euler method yields the equation of motion for the payload: $$m_p{\ddot{z}}_p=m_{p}G-\underset{F_s}{\underset{\}}{\left(c\left(z_p-r_w{\phi }_w\right)+d\left({\dot{z}}_p-r_w{\dot{\phi }}_w\right)\right)}}.z_p\left(0\right)=z_{p0}.{\dot{z}}_p\left(0\right)=0$$

with the gravitational constant g , the spring constant c , the damping constant d , the radius of the hoisting winch r _w , the angle φ _w , the hoisting winch, the angular velocity φ̇ _w , the payload position z _p , the payload speed ż _p , and the payload acceleration z̈ _p . The rope length l _r is given by $$l_r\left(t\right)=r_w{\phi }_w\left(t\right)$$

With $${\phi }_w\left(0\right)={\phi }_{w0}=\frac{{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="imgf0001.png"\; state="\{\{state\}\}">l</figure-callout>}}_1\left(0\right)+3{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="imgf0001.png"\; state="\{\{state\}\}">l</figure-callout>}}_2\left(0\right)+{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="imgf0001.png"\; state="\{\{state\}\}">l</figure-callout>}}_3\left(0\right)}{r_w}$$

wobei E_r und A_r das Elastizitätsmodul und die Schnittfläche des Seils bezeichnen. Der Kran hat n_r parallele Seile (siehe Figur 1), somit ist die Federkonstante des Hubwerks des Krans gegeben durch: $$c=n_r{c}_r$$

The spring constant c _{r of} a rope of length l _r is given by Hooke's law and can be written as $$c_r=\frac{e_r{A}_r}{l_r}$$

where E _r and A _r denote the modulus of elasticity and the sectional area of the rope. The crane has n _r parallel ropes (see FIG. 1 ), so the spring constant of the crane's hoist is given by: $$c=n_r{c}_r$$

The damping constant d can be specified using Lehr's damping factor D. $$d=2D\sqrt{{\mathit{cm}}_p}$$

wobei J_w und J_m das Trägheitsmoment der Winde bzw. des Motors bezeichnen, i_w das Übersetzungsverhältnis zwischen dem Motor und der Winde ist, Δp_w die Druckdifferenz zwischen Hoch- und Niederdruckkammer des Motors ist, D_m die Verdrängung des Hydraulikmotors ist und F_s die in (39) gegebene Federkraft ist. Die anfängliche Bedingung ϕ _w0 für den Winkel der Hubwinde wird durch (41) gegeben. Der Hydraulikkreislauf für die Hubwinde ist im Grunde der gleiche wie für das Drehwerk und ist in Figur 4 dargestellt. Die Druckdifferenz Δp_w kann somit analog zum Drehwerk (siehe (25)) geschrieben werden als $$\mathrm{\Delta }{\dot{p}}_w=\frac{4}{V_m\beta }\left(q_w-D_m{i}_w{\dot{\varphi }}_w\right),\mathrm{\Delta }{p}_w\left(0\right)=\mathrm{\Delta }{p}_{w0}$$

The differential equation for the rotational movement of the hoist winch is given by the method of Newton-Euler as $$\left(J_w+i_{\mathrm{<figure-callout\; id="2"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}^2{J}_m\right){\ddot{\phi }}_w=i_w{D}_m\mathrm{\Delta }{p}_w+r_w{F}_s.{\phi }_w\left(0\right)={\phi }_{w0}.{\dot{\phi }}_w\left(0\right)=0$$

where J _w and J _m denote the moment of inertia of the winch and the motor, i _{w is} the gear ratio between the engine and the winch, Δ p _{w is} the pressure difference between high and low pressure chambers of the engine, D _{m is} the displacement of the hydraulic motor, and F _{s is} the spring force given in (39). The initial condition φ _w0 for the angle of the _{hoist winch} is given by (41). The hydraulic circuit for the hoist winch is basically the same as for the slewing gear and is in FIG. 4 shown. The pressure difference Δp _w can thus be written analogously to the slewing gear (see (25)) as $$\mathrm{\Delta }{\dot{p}}_w=\frac{4}{V_m\beta }\left(q_w-D_m{i}_w{\dot{\phi }}_w\right).\mathrm{\Delta }{p}_w\left(0\right)=\mathrm{\Delta }{p}_{w0}$$

wobei u_w und K_w der Ansteuerstrom des Pumpenwinkels und der Proportionalitätsfaktor sind.The oil flow rate q _w is controlled by the pump angle and is given by $$q_w=K_w{u}_w$$

where u _w and K _{w are} the drive current of the pump angle and the proportionality factor.

4.2 Tax Law

$$\mathbf{g}\left(\mathbf{x}\right)=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ \frac{4K_w}{V_m\beta }\end{array}\right]$$

$$h\left(\mathbf{x}\right)=x_3$$

und u = u_w. In the following, the dynamic model for the hoist winch is transformed into the state space to design a flatness-based feedforward control. The derivation of the tax law neglects the damping, therefore D = 0 . The state vector of the crane's hoist is defined as x = [φ _w , φ _w , z _p , ¿ _p , Δ p _w ] ^T. Thus, the dynamic model consisting of (39), (40), (43), (45), (46), and (47) can be written as a system of first order differential equations given by (11), with: $$\mathbf{f}\left(\mathbf{x}\right)=\left[\begin{array}{c}{x}_2\\ \frac{1}{J_w+i_{\mathrm{<figure-callout\; id="2"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}^2{J}_m}\left(i_w{D}_m{x}_5+r_w\left(\frac{e_r{A}_r{n}_r}{r_w{x}_1}\left(x_3-r_w{x}_1\right)\right)\right)\\ x_4\\ G-\frac{e_r{A}_r{n}_r}{r_w{x}_1{m}_p}\left(x_3-r_w{x}_1\right)\\ \frac{-4D_m{i}_w{x}_2}{V_m\beta }\end{array}\right]$$

$$\mathbf{G}\left(\mathbf{x}\right)=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ \frac{4K_w}{V_m\beta }\end{array}\right]$$

$$H\left(\mathbf{x}\right)=x_3$$

and u = u _w .

Again, the relative degree r with respect to the system output must be equal to the order n of the system. Using (15) gives r = n = 5 , thus the system (11) is flat with (48), (49) and (50) and a flatness-based feedforward for D = 0 can be designed.

The system output (50) and its time derivatives are used to invert the system dynamics, as was done for the seesaw and slewing gear.

$$\dot{y}=\frac{\partial h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}h\left(\mathbf{x}\right)+\underset{=0}{\underbrace{L_{g}h\left(\mathbf{x}\right)u}}$$

$$\ddot{y}=\frac{\partial L_{\mathrm{f}}h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}^{2}h\left(\mathbf{x}\right)+\underset{=0}{\underbrace{L_g{L}_{\mathrm{f}}h\left(\mathbf{x}\right)u}}$$

$$\stackrel{\dot{}¨}{y}=\frac{\partial L_{\mathrm{f}}^{2}h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}^{3}h\left(\mathbf{x}\right)+\underset{=0}{\underbrace{L_g{L}_{\mathrm{f}}^{2}h\left(\mathbf{x}\right)u}}$$

$$\stackrel{\left(4\right)}{y}=\frac{\partial L_{\mathrm{f}}^{3}h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}^{4}h\left(\mathbf{x}\right)+\underset{=0}{\underbrace{L_g{L}_{\mathrm{f}}^{3}h\left(\mathbf{x}\right)u}}$$

$$\stackrel{\left(5\right)}{y}=\frac{\partial L_{\mathrm{f}}^{4}h\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}^{5}h\left(\mathbf{x}\right)+L_g{L}_{\mathrm{f}}^{4}h\left(\mathbf{x}\right)u$$

The derivatives are given by the Lie derivatives, so $$y=H\left(\mathbf{x}\right)$$

$$\dot{y}=\frac{\partial H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}=L_{\mathrm{f}}H\left(\mathbf{x}\right)+\underset{=0}{\underset{\}}{L_{G}H\left(\mathbf{x}\right)u}}$$

$$\ddot{y}=\frac{\partial L_{\mathrm{f}}H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}={\mathrm{<figure-callout\; id="2"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)+\underset{=0}{\underset{\}}{L_G{L}_{\mathrm{f}}H\left(\mathbf{x}\right)u}}$$

$$\stackrel{\dot{}¨}{y}=\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}={\mathrm{<figure-callout\; id="3"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)+\underset{=0}{\underset{\}}{L_G<figure-callout\; id="2"\; label="\; L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">\; </figure-callout>L_{\mathrm{f}}^{}{\mathrm{}}_2^{}H\left(\mathbf{x}\right)u}}$$

$$\stackrel{\left(4\right)}{\mathrm{<figure-callout\; id="4"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}=\frac{\partial {\mathrm{<figure-callout\; id="3"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}={\mathrm{<figure-callout\; id="4"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)+\underset{=0}{\underset{\}}{L_G<figure-callout\; id="3"\; label="\; L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">\; </figure-callout>L_{\mathrm{f}}^{}{\mathrm{}}_3^{}H\left(\mathbf{x}\right)u}}$$

$$\stackrel{\left(5\right)}{\mathrm{<figure-callout\; id="5"\; label="y"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">y</figure-callout>}}=\frac{\partial {\mathrm{<figure-callout\; id="4"\; label="L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)}{\partial \mathbf{x}}\frac{\partial \mathbf{x}}{\partial t}={\mathrm{<figure-callout\; id="5"\; label="L\; f"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">L\; </figure-callout>}}_{\mathrm{f}}^{\mathrm{}}H\left(\mathbf{x}\right)+L_G<figure-callout\; id="4"\; label="\; L\; f"\; filenames="imgf0001.png"\; state="\{\{state\}\}">\; </figure-callout>L_{\mathrm{f}}^{}{\mathrm{}}_4^{}H\left(\mathbf{x}\right)u$$

$$x_2=x_2\left(y,\dot{y},\ddot{y},\stackrel{\dot{}¨}{y}\right)$$

$$x_3=y$$

$$x_4=\ddot{y}$$

$$x_5=x_5\left(y,\dot{y},\ddot{y},\stackrel{\dot{}¨}{y},\stackrel{\left(4\right)}{y}\right)$$

The states depending on the system output and its derivatives follow from (51), (52), (53), (54) and (55) and can be written as: $$x_1=\frac{A_r{e}_r{n}_{r}y}{r_w\left(G{m}_p+A_r{e}_r{n}_r-m_p\ddot{y}\right)}$$

$$x_2=x_2\left(y,\dot{y},\ddot{y},\stackrel{\dot{}¨}{y}\right)$$

$$x_3=y$$

$$x_4=\ddot{y}$$

$$x_5=x_5\left(y,\dot{y},\ddot{y},\stackrel{\dot{}¨}{y},\stackrel{\left(4\right)}{y}\right)$$

welche die Systemdynamik invertiert. Das Referenzsignal y und seine Ableitungen werden durch eine numerische Trajektoriengenerierung aus dem Handhebelsignal des Kranbedieners gewonnen.Solving (56) for system input u , using (57), (58), (59), (60), and (61), gives the control law for the flatness-based feedforward control for the hoist $$u_w=f\left(y,\dot{y},\ddot{y},\stackrel{\dot{}¨}{y},\stackrel{\left(4\right)}{y},\stackrel{\left(5\right)}{\mathrm{<figure-callout\; id="5"\; label="y"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">y</figure-callout>}}\right)$$

which inverts the system dynamics. The reference signal y and its derivatives are obtained by a numerical trajectory generation from the hand lever signal of the crane operator.

Claims (15)

1. A method for controlling a drive of a crane, in particular of a slewing gear and/or a luffing gear, an intended movement of the boom tip serving as an input variable on the basis of which a control variable for controlling the drive is calculated, characterised in that, when calculating the control variable, the vibration dynamic of the system formed of the drive and the crane structure is taken into account so as to reduce natural vibrations.
2. The method according to claim 1, wherein the drive is controlled on the basis of a physical model which describes the movement of the crane tip as a function of the control variable, and wherein the model is advantageously non-linear.
3. The method according to claim 2, wherein the drive is controlled on the basis of an inversion of the model.
4. The method according to any one of the preceding claims, wherein the drive is a hydraulic drive and the model takes into account the vibration dynamic of the drive based on the compressibility of the hydraulic fluid.
5. The method according to any one of the preceding claims for controlling the luffing cylinder used as a luffing drive, wherein the kinematics of the pivotal connection of the cylinder as well as the mass and moment of inertia of the boom of the crane are included in the calculation of the control variable.
6. The method according to any one of the preceding claims for controlling the slewing gear, wherein the moment of inertia of the boom of the crane is included in the
   model.
7. The method according to any one of the preceding claims, wherein the vibration is dampened by way of the pilot control, wherein the position, speed, acceleration and/or jolt of the boom tip are advantageously used as setpoint variables of the pilot control.
8. The method according to any one of the preceding claims, wherein an intended trajectory of the boom tip is generated as an input variable of the control from inputs by an operator and/or an automation system.
9. The method according to any one of the preceding claims, wherein possible spherical sway oscillations of the load are not included as measurement variables in the control and/or wherein possible spherical sway oscillations of the load are not taken into account in the control of the drive.
10. A method for controlling a lifting gear of a crane, an intended lifting movement of the load serving as an input variable on the basis of which a control variable for controlling the drive is calculated, characterised in that, when calculating the control variable, the vibration dynamic of the system formed of a lifting gear, cable and load in the cable direction is taken into account so as to reduce natural vibrations.
11. The method according to claim 10, wherein the vibration dynamic based on the stretchability of the lifting cable is taken into account when calculating the control variable.
12. The method according to either claim 10 or claim 11, wherein the lifting gear is driven hydraulically and the vibration dynamic based on the compressibility of the hydraulic fluid is taken into account when calculating the control variable.
13. The method according to any one of claims 10 to 12, wherein the variable cable length of the lifting cable and/or the weight of the load suspended at the carrying cable is/are included in the calculation of the control variable.
14. The method according to any one of claims 10 to 13, wherein the control of the lifting gear is based on a physical model of the crane which describes the lifting movement of the load as a function of the control variable of the lifting gear, the control of the lifting gear advantageously being based on the inversion of the physical model.
15. A crane or crane control having a control unit which comprises a control program via which a method according to any one of claims 1 to 14 is implemented.

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