EP1326798B1 - Crane or digger for swinging a load hanging on a support cable with damping of load oscillations - Google Patents

Abstract

The invention concerns a crane or excavator for traversing a load hanging from a load cable, which is movable in three spatial directions. The crane or excavator has a computer-controlled regulation for the damping of load swings, which contains a path planning module, a centripetal force compensation unit and at least one shaft regulator for the rotating gear, one shaft regulator for the luffing gear and one shaft regulator for the lifting gear.

Description

The invention relates to a crane or excavator for handling a load suspended on a load rope, which has a computer-controlled control for damping the load oscillation.

In particular, the invention is concerned with the load swing damping in cranes or excavators which allows movement of the load suspended on a rope in at least three degrees of freedom. Such cranes or excavators have a slewing gear, which can be mounted on a chassis, which serves for rotating the crane or excavator. Furthermore, a luffing mechanism for erecting or tilting a boom is available. Finally, the crane or excavator includes a hoist for lifting and lowering the load suspended on the rope. Such cranes or excavators are used in various designs. Exemplary here are mobile harbor cranes, ship cranes, offshore cranes, crawler cranes and crawler excavators.

When handling a hanging on a rope load by means of such a crane or excavator oscillations arise, on the one hand on the movement of the crane or excavator itself or even on external disturbances, such as wind are due. Efforts have already been made in the past to suppress pendulum vibrations in load cranes.

Thus, DE 127 80 79 describes an arrangement for the automatic suppression of oscillations of a hanging by means of a rope at a level movable rope suspension point load on movement of Seilaufhängepunktes in at least one horizontal coordinate, in which the speed of Seilaufhängepunktes in the horizontal plane by a control loop is influenced in dependence on a derived from the deflection angle of the load cable against the Endlot size.

DE 20 22 745 shows an arrangement for the suppression of pendulum vibrations of a load suspended by means of a rope on the cat of a crane, the drive of which is equipped with a speeding device and a travel control device, with a control arrangement which takes the cat into consideration during the oscillation period of a first part of the path traveled by the cat so accelerated and delayed during a last part of this way so that the movement of the cat and the vibration of the load at the destination are equal to zero.

From DE 321 04 50 a device is known to hoists for the automatic control of the movement of the load carrier with reassurance occurring during acceleration or deceleration of the load hanging on it pendulum of the load during an acceleration or Abbremszeitintervalles. The basic idea is based on the simple mathematical pendulum. The cat and load mass is not included in the calculation of the movement. Coulombic and velocity-proportional friction of the cat or bridge drives are not considered.

In order to transport a load body as quickly as possible from the site to the destination, DE 322 83 02 proposes to control the speed of the drive motor of the trolley by means of a computer so that the trolley and the load carrier are moved during the steady drive at the same speed and the pendulum damping be achieved in no time. The computer known from DE 322 83 02 works according to a computer program for solving the differential equations applicable to the undamped two-mass vibration system formed from trolley and load body, whereby the coulomb and speed-proportional friction of the cat or bridge drives are not taken into account.

In the method known from DE 37 10 492 the speed between the destinations on the way to be chosen such that after covering half the total distance between the starting point and destination of the pendulum deflection is always equal to zero.

The known from DE 39 33 527 method for damping of load oscillations oscillations comprises a normal speed position control.

DE 691 19 913 deals with a method for controlling the adjustment of a swinging load, in which the deviation between the theoretical and the actual position of the load is formed in a first control loop. This is derived, multiplied by a correction factor and added to the theoretical position of the mobile carrier. In a second control loop, the theoretical position of the mobile carrier is compared with the actual position, multiplied by a constant and added to the theoretical speed of the mobile carrier.

DE 44 02 563 deals with a method for the regulation of electric traction drives of hoists with a load suspended from a rope, the equations describing the dynamic behavior of the setpoint course of the speed the trolley generated and gives to a speed and current controller. Furthermore, the computing device can be extended by a position controller for the load.

The well-known from DE 127 80 79, DE 393 35 27 and DE 691 19 913 control method need for load swing damping a cable angle sensor. In the extended version according to DE 44 02 563, this sensor is also required. Since this rope angle sensor causes considerable costs, it is advantageous if the load oscillation can be compensated without this sensor.

The method of DE 44 02 563 in the basic version also requires at least the crane paw speed. Also in DE 20 22 745 several sensors are required for the load oscillation damping. Thus, in DE 20 22 745 at least one speed and position measurement of the trolley must be made.

DE 37 10 492 requires at least the cat or bridge position as an additional sensor.

As an alternative to this method, another approach, which has become known, for example, from DE 32 10 450 and DE 322 83 02, proposes to solve the differential equations on which the system is based and based on beer. To determine control strategy for the system to suppress a load oscillation, which in the case of DE 32 10 450 the rope length and in the case of DE 322 83 02 the rope length and load mass is measured. In these systems, however, the frictional effects of static friction and velocity-proportional friction, which are not negligible in the crane system, are not taken into account. Also DE 44 02 563 does not take into account friction and damping terms.

The document "Modeling and Control of a Rotary Crane for Swing-Free Transport of Payloads" (by R. Souissi & al, Control Applications, 1992; First IEEE Conference Dayton, USA, 13-16 Sept. 1992, pp. 782-787; shows a regulation.

The object of the present invention is to develop a crane or excavator for transferring from a load suspended on a load rope, which can move the load at least over three degrees of freedom of movement, such that during the movement actively occurring pendulum movement of the load can be damped and the load can be performed as accurately on a given path.

According to the invention this object is achieved by a crane or excavator with the features of claim 1. Accordingly, the crane or excavator on a computer-controlled control for damping the load oscillation, which has a Bahnplanungsmodul, a Zentripetalkraftkompensationseinrichtung and at least one axis controller for the slewing, an axle controller for the luffing and an axis controller for the hoist.

The path control with active damping of the pendulum motion is based on the basic idea of initially modeling the dynamic behavior of the mechanical and hydraulic system of the crane or excavator in a dynamic model based on differential equations. Based on this dynamic model, a feedforward control can be designed which, under these idealized conceptions of the dynamic model, suppresses oscillations when the load is moved by the slewing gear, luffing gear and hoist and guides the load exactly in the given path.

Prerequisite for the precontrol is first the generation of the web in the work space, which is made by the path planning module. The path planning module generates the path, which is given in the form of time functions for the load position, speed, acceleration, jerk and possibly the derivative of the jerk to the feedforward, from the specification of the desired speed proportional to the deflection of the hand lever in the case of semi-automatic operation or from set points in the case of fully automatic operation.

The particular problem with a crane or excavator of the type mentioned is the coupling between the turning and rocking, which results in particular in the formation of the centripetal effect in the rotational movement. This creates vibrations of the load, which no longer after the rotation can be compensated. According to the present invention, these effects are taken into account in a centripetal force compensation device provided in the control.

Further details and advantages of the invention will become apparent from the subsequent claims to the main claim.

For example, if vibrations or deviations from the desired trajectory occur despite the existing control, the system of feedforward and trajectory planning module may be assisted by a state controller in the event of large deviations from the idealized dynamic model (e.g., due to disturbances such as wind effects, etc.). This then performs at least one of the measured variables: pendulum angle in the radial and tangential direction, Aufrichtwinkel, rotation angle, boom bending in the horizontal and vertical direction and their derivative and the load mass back.

It may be advantageous if a decentralized control concept is based on a spatially decoupled dynamic model in which an independent control algorithm is assigned to each individual movement direction.

The present invention provides a particularly efficient and easy to maintain control for a crane or excavator of the type mentioned.

Further details and advantages of the invention will be explained with reference to an embodiment shown in the drawing. As a typical representative of a crane or excavator of the type mentioned, the invention is described here with reference to a mobile harbor crane.

Fig. 1:

Prinzipielle mechanische Struktur eines Hafenmobilkranes

Fig. 2:

Zusammenwirken von hydraulischer Steuerung und Bahnsteuerung

Fig. 3:

Gesamtstruktur der Bahnsteuerung

Fig. 4:

Struktur des Bahnplanungsmoduls

Fig. 5.:

Beispielhafte Bahngenerierung mit dem vollautomatischen Bahnplanungsmodul

Fig. 6:

Struktur des halbautomatischen Bahnplanungsmoduls

Fig. 7:

Struktur des Achsregler im Falle des Drehwerks

Fig. 8:

Mechanischer Aufbau des Drehwerks und Definition von Modellvariablen

Fig. 9:

Struktur des Achsreglers im Falle des Wippwerks

Fig. 10:

Mechanischer Aufbau des Wippwerks und Definition von Modellvariablen

Fig. 11:

Aufrichtkinematik des Wippwerks

Fig. 12:

Struktur des Achsregler im Falle des Hubwerks

Fig. 13:

Struktur des Achsregler im Falle des Lastschwenkwerks

Show it:

Fig. 1:

Basic mechanical structure of a mobile harbor crane

Fig. 2:

Interaction of hydraulic control and path control

3:

Overall structure of the railway control

4:

Structure of the railway planning module

Fig. 5 .:

Exemplary web generation with the fully automatic path planning module

Fig. 6:

Structure of semi-automatic path planning module

Fig. 7:

Structure of the axis controller in the case of the slewing gear

Fig. 8:

Mechanical structure of the slewing gear and definition of model variables

Fig. 9:

Structure of the axis controller in the case of the luffing mechanism

Fig. 10:

Mechanical structure of the luffing gear and definition of model variables

Fig. 11:

Rigging kinematics of the luffing mechanism

Fig. 12:

Structure of the axis controller in the case of the hoist

Fig. 13:

Structure of the axis controller in the case of the load swing mechanism

In Fig. 1. The basic mechanical structure of a mobile harbor crane is shown. The mobile harbor crane is usually mounted on a chassis 1. To position the load 3 in the working space of the boom 5 can be tilted with the hydraulic cylinder of the luffing mechanism 7 by the angle φ _A. With the hoist, the rope length l _{S can be} varied. The tower 11 allows the rotation of the boom by the angle φ _D about the vertical axis. With the load pivot 9, the load at the target point can be rotated by the angle φ _red .

Fig. 2 shows the interaction of hydraulic control and path control 31. As a rule, the mobile harbor crane has a hydraulic drive system 21. An internal combustion engine 23 feeds the hydraulic control circuits via a transfer case. The hydraulic control circuits each consist of a variable displacement pump 25, which is controlled via a proportional valve in the pilot circuit, and an engine 27 or cylinder 29 as a working machine. By means of the proportional valve, a delivery flow Q _FD , Q _FA , Q _FL , Q _FR is thus set, independent of the load pressure. The Poportionalventile are controlled by the signals U _{StD, StA U,} U _StL, U _st. The hydraulic control is usually equipped with a subordinate flow control. It is essential that the control voltages u _StD , U _StA , U _StL , U _{StR are implemented} on the proportional valves by the subordinate flow control in this proportional flow rates Q _FD , Q _FA , Q _FL , Q _FR in the corresponding hydraulic circuit.

It is essential that the time functions for the control voltages of the proportional valves are no longer derived directly from the hand levers, for example via ramp functions, but are calculated in the path control 31, that when moving the crane no oscillations of the load occur and the load of the desired path in the Working space follows.

The fully automated operation of the mobile harbor crane also results in pendulum-free operation.

The basis for this is a dynamic model of the crane with the help of this based on the sensor data at least one of the variables w _v , w _h , l _S , φ _A , φ _D , φφ _red , φ _Stm , φ̇ _Srm , and the guide specifications q̇ _target or q _Goal this task is solved.

The overall structure of the web control 31 will be explained with reference to FIG. The operator 33 inputs the target speeds or the target points either via the hand levers 35 at the control stations or via a set point matrix 37 which was stored in the computer in a previous trip of the crane. The fully automatic or semi-automatic path planning module 39 or 41 calculates, taking into account the kinematic limitations (maximum speed, acceleration and jerk) of the crane, the time functions of the target load position with respect to the luffing, luffing, lifting and load swinging mechanism and their derivatives, which are in the vectors φ _Dref , φ _Aref , l _ref , φ _{Rref are} summarized. The setpoint position vectors are applied to the axis controllers 43,45,47 and 49, which are then evaluated by evaluating at least one of the sensor values φ _A , φ _D , W _v , W _h , l _s , φ̇ _red , φ̇ _Stn , φ̇ _Srm , (see 2) calculate the drive functions u _StD , u _StA , u _StL , u _StR for the proportional valves 25 of the hydraulic drive system 21. In the case of rotary motion, a compensation trajectory for the luffing gear is generated from the guidance specification for the slewing gear in the module for Zentripetalkraftkompensation 150, so that the caused by the Zentripetalbeschleunigung Auswandem of the load is compensated. In order to ensure a constant lifting height in this case, the compensation movement of the luffing gear is synchronized with the hoist movement. At the same time, a permissible cable deflection φ _SrZul due to the rotational movement is calculated for the luffing mechanism _controller .

In the following, the individual components of the path control will now be described in detail.

• ϕ _Dref : Soll - Winkelposition Lastmittelpunkt in Drehrichtung
• ϕ̇ _Dref : Soll - Winkelgeschwindigkeit Lastmittelpunkt in Drehrichtung
• ϕ̈ _Dref : Soll - Winkelbeschleunigung Lastmittelpunkt in Drehrichtung
• 
  Soll - Ruck Lastmittelpunkt in Drehrichtung
• ${\varphi }_{\mathrm{Dref}}^{\left(IV\right)}:$
  
  Ableitung Soll - Ruck Lastimittelpunkt in Drehrichtung

Fig. 4, the interface shows the path planning module 39 or 41. In the case of the fully automatic path planning module 39, the target position vector for the load center in terms of the coordinate q _objective = [φ _Dzieł, r _LAZiel, l _target, φ _RZiel] ^T given. φ _DZiel is the target _{turning angle} , r _LAziel is the radial target position for the load and l _target is the target position for the hoist or lifting height. φ _RZiel is the setpoint for the load _{swivel angle} . In the case of semi-automatic path planning module 41, the input variable is the target velocity vector q = _target [.phi _Dzieł R _LAZiel, L _Target, .phi _RZiel] ^T. The components of the target speed vector are analogous to the target position _vector , the target speed in the direction of the slewing φ̇ _DZiel , following the target speed of the load in the radial direction ṙ _LAZiel , the target speed for the hoist i _target and the target rotational speed in the direction of the load pivot _{φ̇ RZiel} . In the path planning module 39 or 41, the time function vectors for the load position with respect to the rotation angle coordinate and its derivatives φ _{Dref ,} for the load position in the radial direction and their derivatives r _LAref and for the lifting height of the load and their derivatives l _{ref are} calculated from these predetermined variables. Each vector comprises a maximum of 5 components up to the 4th derivative. In the case of the slewing gear, the individual components are:

• φ _Dref : Target - angular position of the load center in the direction of rotation
• φ̇ _Dref : Target angular velocity of the load center in the direction of rotation
• φ̈ _Dref : Target angular acceleration of the load center in the direction of rotation
• 
  Target - jerk load center in direction of rotation
• ${\phi }_{\mathrm{Dref}}^{\left(IV\right)}:$
  
  Derivative reference - jerk load center in direction of rotation

The vectors for the other directions of movement are constructed analogously.

,

aus dem vollautomatischen Bahnplanungsmodul für eine Bewegung mit dem Dreh- und Wippwerk vom Startpunkt ϕ _Dstart =0°, r _LAstart =10m zum Zielpunkt ϕ _DZiel =90°, r _LAZiel =20m. Die Zeitfunktionen werden dabei so berechnet, daß keine der vorgegebenen kinematischen Beschränkungen, wie die maximalen Geschwindigkeiten ϕ̇ _{D max} , ṙ _{LA max} , die maximalen Beschleunigungen ϕ̈ _{D max} ,r̈ _{LA max} oder der maximale Ruck

,

überschritten wird. Hierzu wird die Bewegung in drei Phasen eingeteilt. Eine Beschleunigungsphase I, eine Phase konstanter Geschwindigkeit II, die auch entfallen kann, und eine Abbremsphase III. Für die Phasen I und III wird als Zeitfunktion für den Ruck ein Polynom 3. Ordnung angenommen. Als Zeitfunktion für die Phase II wird eine konstante Geschwindigkeit angenommen. Durch Integration der Ruckfunktion werden die fehlenden Zeitfunktionen für die Beschleunigung, Geschwindigkeit und Position errechnet. Die noch freien Koeffizienten in den Zeitfunktionen werden durch die Randbedingungen beim Start der Bewegung, an den Übergangsstellen zur nächsten bzw. vorangegangenen Bewegungsphase bzw. am Zielpunkt sowie die kinematischen Beschränkungen festgelegt, wobei bezüglich jeder Achse alle kinematischen Bedingungen überprüft werden müssen. Im Falle des Beispieles aus Fig. 5 ist in der Phase I und III die kinematische Beschränkung der maximalen Beschleunigung ϕ̈ _Dmaxund der Ruck

für die Drehachse limitierend wirksam, in Phase II die maximale Geschwindigkeit des Wippwerks Drehachse ṙ _LAmax. Die anderen Achsen werden zu der die Bewegung hinsichtlich der Fahrzeit begrenzenden Achse synchronisiert. Die Zeitoptimierung der Bewegung wird dadurch erreicht, daß in einem Optimierungslauf die minimale Gesamtfahrzeit über die Variierung des Anteils der Beschleunigungs- und Abbremsphase an der Gesamtbewegung bestimmt wird.FIG. 5 shows, by way of example, the generated time functions for the desired angular position φ _Dref, the radial desired position r _LAref , target speeds φ̇ _Dref , ṙ _LAref , desired accelerations φ̈ _Dref , r̈ _Lref and set jerk

.

from the fully automatic path planning module for a movement with the luffing and luffing gear from the starting point φ _Dstart = 0 °, r _LAstart = 10m to the target point φ _DZiel = 90 °, r _LAZiel = 20m. The time functions are calculated so that none of the predetermined kinematic restrictions, such as the maximum speeds φ̇ _{D max} , ṙ _{LA max} , the maximum accelerations φ̈ _{D max} , r̈ _{LA max} or the maximum jerk

.

is exceeded. For this purpose, the movement is divided into three phases. An acceleration phase I, a phase of constant speed II, which may also be omitted, and a braking phase III. For phases I and III, a third-order polynomial is assumed as the time function for the jerk. As a time function for the phase II a constant speed is assumed. By integrating the jerk function, the missing time functions for the acceleration, speed and position are calculated. The remaining free coefficients in the time functions are determined by the boundary conditions at the start of the movement, at the transition points to the next or previous movement phase or at the target point and the kinematic restrictions, with respect to each axis, all kinematic conditions must be checked. In the case of the example of FIG. 5, in phase I and III, the kinematic limitation of the maximum acceleration φ̈ _{D max} and the jerk

for the axis of rotation limiting effective, in Phase II the maximum speed of the luffing mechanism Rotary axis ṙ _{LA max} . The other axes are synchronized to the movement time limiting axis. The time optimization of the movement is achieved by determining the minimum total travel time by varying the proportion of the acceleration and deceleration phase in the overall movement in an optimization run.

The semi-automatic path planner consists of steepness limiters, which are assigned to the individual directions of movement.

bei der derzeitigen Soll-Beschleunigung ϕ̈ _Dref nur die maximale Geschwindigkeitsänderung $$\Delta {\dot{\varphi }}_{D\max }=\frac{{\ddot{\varphi }}_{\mathit{Dref}}\left|{\ddot{\varphi }}_{\mathit{Dref}}\right|}{2{\stackrel{⃛}{\varphi }}_{D\max }}$$

erreichbar ist, die im Block 611 berechnet wird. Deshalb wird dieser Wert auf die aktuelle Soll-Geschwindigkeit ϕ̇ _Dref addiert, wodurch die Dynamik des Gesamtsystems verbessert wird. Hinter dem Begrenzungsglied 69 liegt dann die Zielbeschleunigung ϕ̈ _DZiel vor. Mit der gegenwärtigen Sollbeschleunigung ϕ̈ _Dref wird wiederum eine Soll-Ist-Wert-Differenz gebildet Im Kennlinienblock 615 wird daraus der Soll-Ruck

gemäß $${\stackrel{⃛}{\varphi }}_{\mathit{Dref}}=\{\begin{array}{ll}+{\stackrel{⃛}{\varphi }}_{D\max }\hfill & \mathrm{für}\hspace{0.17em}{\ddot{\varphi }}_{\mathit{DZiel}}-{\ddot{\varphi }}_{\mathit{Dref}}>0\hfill \\ 0\hfill & \mathrm{für}\hspace{0.17em}{\ddot{\varphi }}_{\mathit{DZiel}}-{\ddot{\varphi }}_{\mathit{Dref}}=0\hfill \\ -{\stackrel{⃛}{\varphi }}_{D\max }\hfill & \mathrm{für}\hspace{0.17em}{\ddot{\varphi }}_{\mathit{DZiel}}-{\ddot{\varphi }}_{\mathit{Dref}}<0\hfill \end{array}$$

gebildet. Durch Filterung wird der blockförmige Verlauf dieser Funktion abgeschwächt. Aus der nun berechneten Sollruckfunktion

werden durch Integration im Block 65 die Soll-Beschleunigung ϕ̈ _Dref , die Soll-Geschwindigkeit ϕ̇ _Dref und die Soll-Position ϕ _Dref bestimmt. Die Ableitung des Soll-Ruckes wird durch Differentiation im Block 65 und gleichzeitige Filterung aus dem Soll-Ruck

bestimmt. Im Normalbetrieb werden die kinematischen Beschränkungen ϕ̈ _Dmax und

sowie die Proportionalverstärkung K _S1 so vorgegeben, daß sich für Kranfahrer ein subjektiv angenehmes und sanftes dynamisches Verhalten ergibt. Dies bedeutet, daß maximaler Ruck und Beschleunigung etwas niedriger angesetzt werden, als es das mechanische System erlauben würde. Jedoch ist insbesondere bei hohen Verfahrgeschwindigkeiten der Nachlauf des Systems hoch. D.h. gibt der Bediener aus voller Geschwindigkeit die Zielgeschwindigkeit 0 vor, so benötigt die Last einige Sekunden bis sie zum Stillstand kommt. Da derartige Vorgaben insbesondere in Notsituation mit drohender Kollision gemacht werden, wird deshalb ein zweiter Betriebsmodus eingeführt, der einen Schnellstop des Krans vorsieht. Hierzu wird dem Steilheitsbegrenzerblock für den Nomialbetrieb 61 ein zweiter Steilheitsbegrenzerblock 63 parallelgeschaltet, der strukturell einen identischen Aufbau hat. Jedoch werden die Parameter, die den Nachlauf bestimmen, bis zur mechanischen Belastbarkeitsgrenze des Krans erhöht. Deshalb ist dieser Block mit der maximalen Schnellstopbeschleunigung ϕ̈ _Dmax2 und dem maximalen Schnellstopruck

sowie die Schnellstop-Proportionalverstärkung K _S2 parametrisiert. Zwischen den beiden Steilheitsbegrenzem wird über eine Umschaltlogik 67 hin- und hergeschaltet, die aus dem Handhebelsignal, den Notstop identifiziert. Ausgang des Schnellstop-Steilheitsbegrenzer 63 ist wie beim Steilheitsbegrenzer für den Nor malbeitreb der Soll-Ruck

. Die Berechnung der anderen Zeitfunktionen erfolgt auf gleiche Art und Weise wie beim Normalbetrieb im Block 65.Fig. 6 shows the transconductance limiter 60 for the rotational movement. The target speed of the load 3 by the hand lever of the operating state of .phi _Dzieł is the input signal. This is first normalized to the value range of the maximum achievable speed φ̇ _{D max} . The slope limiter itself consists of two limiter boundary blocks with different parameterization, one for the normal operation 61 and one for the quick stop 63, between which can be switched back and forth via the switching logic 67. The time functions at the output are formed by integration 65. The signal flow in the slope limiter will now be explained with reference to FIG.
In the slope limiter _block for normal operation 61, a setpoint-actual value difference between the target speed φ̇ _DZiel and the current setpoint speed φ̇ _{Dref is first} formed. The difference is amplified by the constant K _S1 (block 613) and gives the target acceleration _φ̈ D _target . A downstream limiting element 69 limits the value to the maximum acceleration ± φ̈ _{D max} . In order to improve the dynamic behavior, it is taken into account in the formation of the desired-actual value difference between the target speed and the current setpoint speed that the jerk limitation ±

at the current target acceleration φ̈ _Dref only the maximum speed change $$\Delta {\dot{\phi }}_{D\mathrm{Max}}=\frac{{\ddot{\phi }}_{\mathit{Dref}}\left|{\ddot{\phi }}_{\mathit{Dref}}\right|}{2{\stackrel{⃛}{\phi }}_{D\mathrm{Max}}}$$

reached, which is calculated in block 611. Therefore, this value is added to the current target speed φ̇ _Dref , thereby improving the dynamics of the entire system. Behind the limiting member 69 is then the target acceleration _φ̈ DZiel _ago . With the current setpoint acceleration φ̈ _Dref , in turn, a setpoint-actual-value difference is formed. In the characteristic _block 615, this results in the setpoint jerk

according to $${\stackrel{⃛}{\phi }}_{\mathit{Dref}}=\{\begin{array}{ll}+{\stackrel{⃛}{\phi }}_{D\mathrm{Max}}\hfill & \mathrm{For}{\ddot{\phi }}_{\mathit{Dzieł}}-{\ddot{\phi }}_{\mathit{Dref}}>0\hfill \\ 0\hfill & \mathrm{For}{\ddot{\phi }}_{\mathit{Dzieł}}-{\ddot{\phi }}_{\mathit{Dref}}=0\hfill \\ -{\stackrel{⃛}{\phi }}_{D\mathrm{Max}}\hfill & \mathrm{For}{\ddot{\phi }}_{\mathit{Dzieł}}-{\ddot{\phi }}_{\mathit{Dref}}<0\hfill \end{array}$$

educated. By filtering, the block-shaped course of this function is weakened. From the calculated nominal pressure function

By integration in block 65, the target acceleration φ̈ _Dref , the target speed φ̇ _Dref and the target position φ _{Dref are} determined. The derivative of the target jerk is by differentiation in block 65 and simultaneous filtering from the target jerk

certainly. In normal operation, the kinematic constraints _{.phi..sub.D max} and

and the proportional gain K _S1 predetermined so that results in a subjectively pleasant and gentle dynamic behavior for crane operators. This means that maximum jerk and acceleration are set slightly lower than the mechanical system would allow. However, especially at high speeds, the wake of the system is high. This means that the operator sets the target speed 0 at full speed, so the load takes a few seconds to come to a standstill. Since such specifications are made especially in an emergency situation with imminent collision, therefore, a second operating mode is introduced, which provides a quick stop of the crane. For this For example, a second transconductance limiter block 63, structurally identical in construction, is connected in parallel with the transconductance limiter block for the nominal operation 61. However, the parameters that determine the caster are increased up to the mechanical load limit of the crane. Therefore this block with the maximum rapid stop acceleration φ̈ _{D max2} and the maximum quick _{stop pressure is}

and the rapid stop proportional gain K _{S2 are} parameterized. Between the two transconductance limiter is switched back and forth via a switching logic 67, which identifies the emergency stop from the hand lever signal. Output of the quick-stop Steibheitsbegrenzer 63 is like the slope limiter for the Nor malbeitreb the target jerk

, The calculation of the other time functions takes place in the same way as in normal operation in block 65.

Thus, at the exit of the semi-automatic path planner as well as the fully automatic path planner, the time functions for the setpoint position of the load in the direction of rotation and its derivation are available taking into account the kinematic restrictions.

nochmals real differenziert.As an alternative to this steepness limiter, it is also possible to use a structure in which the speed setpoint signal, limited to the maximum speed in the steepness of the rising and falling edge in block (691), is limited to a defined value corresponding to the maximum acceleration (FIG. 6 aa). This signal is then differentiated and filtered. The result is the desired acceleration φ̈ _Dref . To calculate the setpoint speed φ̇ _Dref and setpoint position φ _Dref , this signal is integrated, for the calculation of

again differentiated in real terms.

The slope limiter from the semi-automatic path planner can also be used for the fully automated path planner (Figure 6a). This is advantageous since, in particular during the movement in the radial direction, the kinematic limitations are dependent on the erection angle. Therefore, in a block depending on the position of the boom position on the kinematics of the luffing gear (See also FIG. 11) calculates the kinematic constraints ṙ _{LA max} and r̈ _{LA max} and tracks the limits (block 617). This will shorten the journey time. In addition, an extension can be introduced for fully automatic operation (block 621). New input is the target position instead of the target speed. It is advantageous that in the extension 621 in the target-actual comparison between the target position r _target and the target position r _LAref alternatively also the target-actual comparison between target position r _target and measured actual position r _LA calculated and as input can be used for the slope limiter 60. As a result, position errors can be eliminated in this additional control loop. However, since the movements between the individual directions of movement are no longer synchronized, a synchronization module (623) is introduced (FIG. 6b) which adjusts the maximum speeds via proportionality factors p _D , p _r , p _{L in} such a way that a synchronous linear movement results ,

For this purpose, a position vector is calculated from the start and end point, which indicates the direction for the desired movement. The load will always move on this orbit in the direction of the location vector if and only if the current velocity direction vector always points in the same direction as the position vector. However, the current velocity _{vector is} influenced by the proportionality factors p _D , p _r , p _L ; that is, by targeted modification of these proportionality factors, the synchronization task is solved.

The time functions are given to the axis controllers. First, the structure of the Achsregiers for the slewing with reference to FIG. 7 will be explained.

The output functions of the path planning module in the form of the target position of the load in the direction of rotation and their derivatives (speed, acceleration, jerk, and derivative of the jerk) are given to the pre-control block 71. In the pre-control block, these functions are so amplified that results in a web-accurate driving the load with respect to the rotation angle without vibrations under the idealized conditions of the dynamic model.

The basis for determining the feedforward gains is the dynamic model derived in the following sections for rotary motion. Thus, under these idealized conditions, the pendulum of the load is suppressed and the load follows the generated path.

However, since disturbances such as wind influences on the crane load can attack and the idealized model can only reflect the real existing dynamic conditions in partial aspects, optionally the feedforward control can be supplemented by a conditioner control block 73. In this block, at least one of the measured quantities rotational angle φ _D , rotational angular velocity φ̇ _D , bending of the cantilever in the horizontal direction (direction of rotation) W _h , derivative of the bend ẇ _h , rope angle φ _St or the Seilwinkeigeschwindigkeit φ̇ _St amplified and returned to the control input. The derivatives of the measured quantities φ _D and w _h are formed numerically in the microprocessor control. The cable angle can be detected, for example, via a gyroscopic sensor, an acceleration sensor on the load hook, a Hallmeßrahmen, an image processing system or the strain gauges on the boom. Since each of these measurement methods does not directly determine the rope angle, the measurement signal is processed in a disturbance observer module (block 77). The example of the Meßsignalaufbereitung for the measurement signal of a gyroscope on the load hook this is exemplified. In the interference observer, the relevant part of the dynamic model is stored for this purpose and, by comparing the measured variables with the calculated value from the idealized model, estimation variables for the measured variable and their interference components are formed, so that thereafter a noise-compensated measured variable can be reconstructed.

Since the hydraulic drive units are characterized by non-linear dynamic properties (hysteresis, backlash), the value formed for the _actuating input U _Dref in the hydraulic _compensation 75 block from the _precontrol and optional state _{controller output} is changed so that the resulting linear behavior of the overall system can be assumed. Output of block 75 (hydraulic compensation) is the corrected manipulated variable U _StD . This value is then applied to the proportional valve of the hydraulic circuit for the slewing gear.

For a detailed explanation of the procedure is now the derivation of the dynamic model for the axis of rotation serve, which is the basis for the calculation of the Vorsteuerungsverstärkungen of the state controller and the Störbeobachter.

l_s ist dabei die resultierende Seillänge vom Auslegerkopf bis zum Lastmittelpunkt. ϕ _A ist der aktuelle Aufrichtwinkel des Wippwerks, l _A ist die Länge des Auslegers, ϕ _St ist der aktuelle Seilwinkel in tangentialer Richtung.
Das dynamische System für die Bewegung der Last in Drehrichtung kann durch die folgenden Differentialgleichungen beschrieben werden. $$\begin{array}{l}\left[J_T+\left(J_{AZ}+m_A{s}_A^2+m_L{l}_A^2\right){\cos }^2{\varphi }_A\right]{\ddot{\varphi }}_D+m_L{l}_A{l}_s\cos {\varphi }_A{\ddot{\varphi }}_{st}+b_D{\dot{\varphi }}_D=M_{MD}-M_{RD}\\ m_L{l}_A{l}_s\cos {\varphi }_A{\ddot{\varphi }}_D+m_L{l}_s^2{\ddot{\varphi }}_{st}+m_{L}g{l}_s{\varphi }_{st}=0\end{array}$$

For this purpose, FIG. 8 gives explanations of the definition of the model variables. Essential here is the relationship shown there between the rotational position φ _{D of} the crane tower and the load position φ _LD in the direction of rotation. Furthermore, the boom is assumed to be rigid and thus neglects the bending w _{h of} the boom. However, there are no great requirements to integrate them into the model approach. However, this increases the system order and the derivation becomes more complex. The load rotation angle position corrected by the pendulum angle is then calculated to $${\phi }_{LD}={\phi }_D+\frac{l_S}{l_A\cos {\phi }_A}\sin {\phi }_{St}$$

l _s is the resulting rope length from the boom head to the load center. φ _A is the current upright angle of the luffing gear, l _A is the length of the boom, φ _St is the current rope angle in the tangential direction.
The dynamic system for the movement of the load in the direction of rotation can be described by the following differential equations. $$\begin{array}{l}\left[J_T+\left(J_{AZ}+m_A{s}_A^2+m_L{l}_A^2\right){\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0005.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\phi }_A\right]{\ddot{\phi }}_D+m_L{l}_A{l}_s\cos {\phi }_A{\ddot{\phi }}_{st}+b_D{\dot{\phi }}_D=M_{MD}-M_{RD}\\ m_L{l}_A{l}_s\cos {\phi }_A{\ddot{\phi }}_D+m_L{l}_s^2{\ddot{\phi }}_{st}+m_{L}G{l}_s{\phi }_{st}=0\end{array}$$

m _L

Lastmasse

l _s

Seillänge

m _A

Masse des Auslegers

J _AZ

Massenträgheitsmoment des Auslegers bezüglich Schwerpunkt bei Drehung um Hochachse

l _A

Länge des Auslegers

S _A

Schwerpunktsabstand des Auslegers

J _T

Massenträgheitsmoment des Turmes

b _D

viskose Dämpfung im Antrieb

M _MD

Antriebsmoment

M _RD

Reibmoment

designations:

m _L

load mass

l _s

cable length

m _A

Mass of the jib

J _AZ

Mass moment of inertia of the boom with respect to the center of gravity when rotating about the vertical axis

l _A

Length of the boom

S _A

Center of gravity of the jib

J _T

Mass moment of inertia of the tower

b _D

viscous damping in the drive

M _MD

drive torque

M _RD

friction

The first equation of (4) essentially describes the equation of motion for the cantilever crane tower, taking into account the feedback due to the load oscillation. The second equation of (4) is the equation of motion which describes the load oscillation by the angle φ _St , the excitation of the load oscillation caused by the rotation of the tower over the angular acceleration of the tower or an external disturbance expressed by initial conditions for these differential equations ,

$$\Delta p_D=\frac{1}{V\beta }\left(Q_{FD}-i_D\frac{V}{2\pi }{\dot{\varphi }}_D\right)$$

$$Q_{FD}=K_{PD}{u}_{StD}$$

The hydraulic drive is described by the following equations. $$M_{MD}=i_{\mathrm{<figure-callout\; id="2"\; label="D\; V"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D\; </figure-callout>}}\frac{V}{}\frac{}{2\pi }\Delta p_D$$

$$\Delta p_D=\frac{1}{V\beta }\left(Q_{FD}-{\mathrm{<figure-callout\; id="2"\; label="i\; D\; V"\; filenames="imgf0005.png"\; state="\{\{state\}\}">i\; </figure-callout>}}_D\frac{V}{}\frac{}{2\pi }{\dot{\phi }}_D\right)$$

$$Q_{FD}=K_{PD}{u}_{StD}$$

i _D is the gear ratio between engine speed and tower rotation speed, V is the displacement of the hydraulic motors, Δ p _D is the pressure drop across the hydraulic drive motor, β is the oil compressibility, Q _FD is the flow in the hydraulic circuit for turning and K _PD is the Proportionality constant, the relationship between the flow rate and drive voltage indicates the proportional valve. Dynamic effects of subordinate flow control are neglected.

The equations can now be transformed into state space representation (see also O. Föllinger Regelungstechnik, 7th ed., Hüthig Verlag, Heidelberg, 1992). The result is the following state space representation of the system.

mit:State space representation: $$\begin{array}{l}{\underset{~}{\dot{x}}}_D={\underset{~}{A}}_D{\underset{~}{x}}_D+{\underset{~}{B}}_D{\underset{~}{u}}_D\\ {\underset{~}{y}}_D={\underset{~}{C}}_D{\underset{~}{x}}_D\end{array}$$

With:

State vector: $${\underset{~}{x}}_D=\left[\begin{array}{c}{\phi }_D\\ {\dot{\phi }}_D\\ {\phi }_{St}\\ {\dot{\phi }}_{St}\end{array}\right]$$

Tax Size: $$u_D=u_{StD}$$

Output: $$y_D={\phi }_{LD}$$

$$a=J_T+\left(J_{AZ}+m_A{S}_A^2+m_L{l}_A^2\right)\cos {\left({\varphi }_A\right)}^2$$

$$b=m_L{l}_A{l}_S\cos \left({\varphi }_A\right)$$

$$c=b_D+\frac{1}{4}\frac{{{\mathit{i}}_{\mathit{D}}}^2\mathit{V}}{{\pi }^2\beta }$$

$$d=\frac{1}{2}\frac{i_D{K}_{PD}}{\pi \beta }$$

$$e=m_L{l}_S^2$$

$$f=m_{L}g{l}_S$$

Matrix: $${\underset{~}{A}}_D=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& -\frac{ce}{ae-b^2}& \frac{fb}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}& 0\\ 0& 0& 0& 1\\ 0& \frac{cb}{ae-b^2}& -\frac{af}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}& 0\end{array}\right]$$

$$a=J_T+\left(J_{AZ}+m_A{S}_A^2+m_L{l}_A^2\right)\cos {\left({\phi }_A\right)}^2$$

$$b=m_L{l}_A{l}_S\cos \left({\phi }_A\right)$$

$$c=b_D+\frac{1}{4}\frac{{{\mathit{i}}_{\mathit{D}}}^2\mathit{<figure-callout\; id="2"\; label="V\; \pi "\; filenames="imgf0005.png"\; state="\{\{state\}\}">V\; </figure-callout>}}{{\pi }^{}}$$

$$d=\frac{1}{2}\frac{i_D{K}_{PD}}{\pi \beta }$$

$$e=m_L{l}_S^2$$

$$f=m_{L}G{l}_S$$

Control vector: $${\underset{~}{B}}_D=\left[\begin{array}{c}0\\ \frac{de}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}\\ 0\\ -\frac{bd}{ae-b^2}\end{array}\right]$$

Output vector: $${\underset{~}{C}}_D=\left[\begin{array}{cccc}1& 0& \frac{l_S}{\cos \left({\phi }_A\right)l_A}& 0\end{array}\right]$$

The dynamic model of the slewing gear is conceived as a parameter-variable system with regard to the cable length l _S , the righting angle φ _A , the load mass m _L.

Equations (6) to (12) are the basis for the now described design of the feedforward control 71, the state controller 73 and the disturbance observer 77.

und ggf. die Ableitung des Soll-Rucks ϕ ⁽⁴⁾ _Dref· Der Führungsgrößenvektor W _D ist damit $${\underset{‾}{w}}_D=\left[\begin{array}{c}{\varphi }_{\mathit{Dref}}\\ {\dot{\varphi }}_{\mathit{Dref}}\\ {\ddot{\varphi }}_{\mathit{Dref}}\\ {\stackrel{⃛}{\varphi }}_{\mathit{Dref}}\\ {\varphi }_{\mathit{Dref}}^{\left(IV\right)}\end{array}\right]$$

Input variables of the pilot control _block 71 are the desired angular position φ _Dref , the target angular velocity φ̇ _Dref , the target angular acceleration φ̈ _Dref , the target jerk

and if necessary, the derivative of the desired jerk φ ⁽⁴⁾ _{Dref ·} The _{reference variable vector} W _D is with it $${\underset{~}{w}}_D=\left[\begin{array}{c}{\phi }_{\mathit{Dref}}\\ {\dot{\phi }}_{\mathit{Dref}}\\ {\ddot{\phi }}_{\mathit{Dref}}\\ {\stackrel{⃛}{\phi }}_{\mathit{Dref}}\\ {\phi }_{\mathit{Dref}}^{\left(IV\right)}\end{array}\right]$$

mit der Vorsteuerungsmatrix $${\underset{‾}{S}}_D=\left[\begin{array}{ccccc}{K}_{\mathit{VD}0}& K_{\mathit{VD}1}& K_{\mathit{VD}2}& K_{\mathit{VD}3}& K_{\mathit{VD}4}\end{array}\right]$$

In pre-control _block 71, the components of W _{D are} weighted with _{feedforward gains} K _VD0 to K _VD4 and their sum is given to the set input. In the event that the axis controller for the rotation axis does not include a state controller block 73, then the size U _Dvorst from the _{feedforward block is} equal to the reference _{drive voltage} u _dref , which is given after compensation of the hydraulic non-linearity as drive _voltage U _StD on the proportional valve. The state space representation (6) is thereby expanded $$\begin{array}{l}{\underset{~}{\dot{x}}}_D={\underset{~}{A}}_D{\underset{~}{x}}_D+{\underset{~}{B}}_D{\underset{~}{S}}_D{\underset{~}{w}}_D\\ {\underset{~}{y}}_D={\underset{~}{C}}_D{\underset{~}{x}}_D\end{array}$$

with the feedforward matrix $${\underset{~}{S}}_D=\left[\begin{array}{ccccc}{K}_{\mathit{VD}0}& K_{\mathit{VD}1}& K_{\mathit{VD}2}& K_{\mathit{VD}3}& K_{\mathit{VD}4}\end{array}\right]$$

When the die equation (14) is evaluated, it may be written as an algebraic equation for the _{feedforward block} , where U _{Dvorst is} the uncorrected target drive voltage for the proportional valve based on the idealized model. $$u_{\mathit{Dvorst}}={\mathit{<figure-callout\; id="0"\; label="K\; VD"\; filenames="imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}{\mathit{\phi }}_{\mathit{Dref}}+{\mathit{<figure-callout\; id="1"\; label="K\; VD"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}{\dot{\mathit{\phi }}}_{\mathit{Dref}}+{\mathit{<figure-callout\; id="2"\; label="K\; VD"\; filenames="imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}{\ddot{\mathit{\phi }}}_{\mathit{Dref}}+{\mathit{<figure-callout\; id="3"\; label="K\; VD"\; filenames="imgf0001.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}{\stackrel{\mathit{⃛}}{\mathit{\phi }}}_{\mathit{Dref}}+{\mathit{K}}_{\mathit{VD}\mathit{4}}{\mathit{\phi }}_{\mathit{Dref}}^{\mathrm{\left(}\mathit{I}\mathit{V}\mathrm{\right)}}$$

The K _VD0 to K _VD4 are the _{Vorsteuerungsverstärkungen which} are calculated as a function of the current Aufrichtwinkels φ _A , the rope length l _s and the load mass m _L , so that the load without vibrations track exactly the desired trajectory.

angeben. Nun muß der Vorsteuerungsblock bei der Übertragungsfunktion berücksichtigt werden. Dadurch wird aus (17): $$G_{VD}\left(s\right)=\frac{{\varphi }_{LD}}{{\varphi }_{\mathrm{Dref}}}=G\left(s\right)\cdot \left(K_{\mathit{VD}0}+K_{\mathit{VD}1}s+K_{\mathit{VD}2}{s}^2+K_{\mathit{VD}3}{s}^3+K_{\mathit{VD}4}{s}^4\right)$$

The _{feedforward gains} K _VD0 to K _VD4 are calculated as follows. With regard to the controlled variable angular position of the load φ _LD can be the transfer function without pre-control block as follows from the equations of state (6) to (12) according to the context $$G\left(s\right)=\frac{{\phi }_{LD}\left(s\right)}{u_{\mathit{Dvorst}}\left(s\right)}={\underset{~}{C}}_D{\left(s\underset{~}{I}-{\underset{~}{A}}_D\right)}^{-1}{\underset{~}{B}}_D$$

specify. Now the feedforward block must be considered in the transfer function. This turns (17): $$G_{VD}\left(s\right)=\frac{{\phi }_{LD}}{{\phi }_{\mathrm{Dref}}}=G\left(s\right)\cdot \left({\mathrm{<figure-callout\; id="0"\; label="K\; VD"\; filenames="imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}+{\mathrm{<figure-callout\; id="1s"\; label="K\; VD"\; filenames="imgf0012.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}s+K_{\mathit{VD}2}{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+K_{\mathit{VD}3}{\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="imgf0001.png"\; state="\{\{state\}\}">s</figure-callout>}}^3+K_{\mathit{VD}4}{s}^4\right)$$

This expression has the following structure after being multiplied: $$\frac{{\phi }_{LD}}{{\phi }_{\mathit{Dref}}}=\frac{...b_2\left(K_{\mathit{VD}\mathit{i}}\right)\cdot {\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+b_1\left(K_{\mathit{VD}\mathit{i}}\right)\cdot s+b_0\left(K_{\mathit{VD}\mathit{i}}\right)}{...a_2\cdot {\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+a_1\cdot s+a_0},$$

For the calculation of the gains K _VDi (K _VD0 to K _VD4 ) , only the coefficients b ₄ to b ₀ and a ₄ to a _{0 are} of interest. Ideal system behavior with regard to position, speed, acceleration, jerk and possibly the derivative of the jerk results exactly if the transfer function of the overall system consisting of feedforward control and transfer function of the slewing gear according to Eq. 19 or 20 in their coefficients b _i and a _i satisfies the following conditions: $$\begin{array}{l}\frac{b_0}{a_0}=1\\ \frac{b_1}{a_1}=1\\ \frac{b_2}{a_2}=1\\ \frac{b_3}{a_3}=1\\ \frac{b_4}{a_4}=1\end{array}$$

This linear system of equations can be solved in analytical form for the desired pilot _gains K _VD0 to K _VD4 .

$$K_{\mathit{VD}3}=-\frac{-l_{S}cb}{\cos \left({\varphi }_A\right)l_{A}df}$$

$$K_{\mathit{VD}4}=-\frac{l_{S}ab-\cos \left({\varphi }_A\right)l_A{b}^2}{d\cos \left({\varphi }_A\right)l_{A}f}$$

By way of example, this is the case for the model according to Eq. 6 to 12 shown. The evaluation of Eq. 20 according to the conditions of Eq. 21 gives for the pilot control _gains K _VD0 to K _VD4 : $$\begin{array}{l}{\mathrm{<figure-callout\; id="0"\; label="K\; VD"\; filenames="imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}=0\\ {\mathrm{<figure-callout\; id="1"\; label="K\; VD"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}=\frac{c}{d}\\ {\mathrm{<figure-callout\; id="2"\; label="K\; VD"\; filenames="imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}=\frac{-a}{\mathrm{<figure-callout\; id="3"\; label="d\; K\; VD"\; filenames="imgf0001.png"\; state="\{\{state\}\}">d\; </figure-callout>}}& \\ \end{array}$$

$$K_{\mathit{VD}}$$$${\mathit{}3}_{}=-\frac{-l_{S}cb}{\cos \left({\phi }_A\right)l_{A}df}$$

$$K_{\mathit{VD}4}=-\frac{l_{S}ab-\cos \left({\phi }_A\right)l_A{b}^2}{d\cos \left({\phi }_A\right)l_{A}f}$$

This has the advantage that these Vorsteuerungsverstärkungen now be present depending on the model parameters. In the case of model according to Eq. (6) to (12) are the model parameters K _PD i _D, V, _A φ, β, _T J, J _AZ, m _A, s _A, m _L, l _A, l _s, b _D.

The change of model parameters such as the pitch angle φ _A , the load mass m _L and the rope length l _s can be taken into account immediately in the change of the pilot gains. Thus, these can always be tracked depending on the measured values of φ _A , m _L and l _S. That is, the rope length is changed with the hoist, then automatically change the Vorsteuerungsver reinforcements of the slewing, so that as a result, always keep the pendulum damping Ver pre-control in the process of the load is maintained.

Furthermore, when transferred to another crane type with other specifications, the pilot gains can be adjusted very quickly.

The parameters K _PD , i _D , V, β, J _T , J _AZ , m _A , s _A and l _A are available from the data sheet of the technical data. Basically, as variable system parameters, the parameters l _s , φ _A and m _{L are} determined from sensor data. The parameters J _T , J _AZ are known from FEM investigations. The attenuation parameter b _D is determined from frequency response measurements.

With the pilot block, it is now possible to control the axis of rotation of the crane so that under the idealized conditions of the dynamic model according to Eq. (6) to (12) no pendulum movements of the load occur in the slewing process and the load of the web generated by the path planning module accurately follows the web. The function quality of the pilot control depends on the derivation of the desired functions. Optimized system behavior is obtained when switching to the degree of system order, in the case of Eq. 6 to 12 this is the degree 4. A gradual improvement is obtained with each additional setpoint function applied starting with the grade 1 over the case where the system is designed only for stationarity with respect to the position. This applies in principle and is to be transmitted in an analogous manner for the luffing mechanism.

However, the dynamic model is only an abstract representation of the real dynamic conditions. In addition, external disturbances (such as strong wind attack or the like) act.

Therefore, the feedforward block 71 is supported by a state controller 73. In the state controller at least one of the measured variables φ _{St, St} .phi, φ _{D, .phi..sub.D} is weighted with a controller gain and fed back to the control input. (In the case of modeling the boom deflection, one of the measured quantities w _h or ẇ _h could also be returned to compensate for the boom oscillation ). There, the difference between the output value of the feedforward block 71 and the output value of the state energizing block 73 is formed. If the state controller block is present, this must be taken into account in the calculation of the feed forward gains.

Due to the feedback, Eq. (14) too $$\begin{array}{l}{\underset{~}{\dot{x}}}_D=\left({\underset{~}{A}}_D-{\underset{~}{B}}_D{\underset{~}{K}}_D\right){\underset{~}{x}}_D+{\underset{~}{B}}_D{\underset{~}{S}}_D{\underset{~}{w}}_D\\ {\underset{~}{y}}_D={\underset{~}{C}}_D{\underset{~}{x}}_D\end{array}$$

K _D is the matrix of the controller gains of the state controller with the entries k _1D , k _2D , k _3D , k _4D . Accordingly, the descriptive transfer function, which is the basis for calculating the feedforward gains, also changes according to (17) $$G_{DR}\left(s\right)=\frac{{\phi }_{LD}\left(s\right)}{u_{\mathit{Dvorst}}\left(s\right)}={\underset{~}{C}}_D{\left(s\underset{~}{I}-{\underset{~}{A}}_D+{\underset{~}{B}}_D{\underset{~}{K}}_D\right)}^{-1}{\underset{~}{B}}_D$$

To calculate the pilot control _gains K _VDi (K _VD0 to K _VD4 ) , in turn, first (25) is extended analogously to (18) by _{adding the reference} variables. $$G_{\mathit{VDR}}\left(s\right)=\frac{{\phi }_{LD}}{{\phi }_{\mathit{Dref}}}=G_{DR}\left(s\right)\cdot \left({\mathrm{<figure-callout\; id="0"\; label="K\; VD"\; filenames="imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}+{\mathrm{<figure-callout\; id="1s"\; label="K\; VD"\; filenames="imgf0012.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}s+K_{\mathit{VD}2}{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+K_{\mathit{VD}3}{\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="imgf0001.png"\; state="\{\{state\}\}">s</figure-callout>}}^3+K_{\mathit{VD}4}{s}^4\right)$$

In the case of feedback, however, the transfer function is also dependent on the control gains k _1D , k _2D , k _3D , k _4D . This results in the structure $$\frac{{\phi }_{LD}}{{\phi }_{\mathit{Dref}}}=\frac{...b_2\left(K_{\mathit{VD}i}.k_{Di}\right)\cdot {\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+b_1\left(K_{\mathit{VD}i}.k_{Di}\right)\cdot s+b_0\left(K_{\mathit{VD}i}.k_{Di}\right)}{...a_2\cdot {\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+a_1\cdot s+a_0},$$

This expression has the same structure with respect to K _VDi (K _VD0 to K _VD4 ) as Eq. (20). Ideal system behavior in terms of position, speed, acceleration, jerk and possibly the derivative of the jerk arises when and if the transfer function of the entire system from pre-control and transfer function of the axis of rotation of the crane according to Eq. 26 in their coefficients b _i and a _i satisfies the condition (21).

This leads again to a linear system of equations, which can be solved in analytical form for the desired pilot control _gains K _VD0 to K _VD4 . However, the coefficients b _i and a _{i are} now also dependent on the known control gains k _1D , k _2D , k _3D , k _{4D of} the state controller, the derivation of which is explained in the following part of the description of the invention , _in addition to the desired _{feed forward gains} K _VD0 to K _VD4 .

$$K_{\mathit{VD}1}=\frac{c+d{k}_2}{d}$$

$$K_{\mathit{VD}2}=\frac{-\cos \left({\varphi }_A\right)l_{A}fa+\cos \left({\varphi }_A\right)l_{A}bd{k}_3-d{l}_{S}b{k}_1}{\cos \left({\varphi }_A\right)l_{A}df}\cdot \left(-1\right)$$

$$K_{\mathit{VD}3}=-\frac{\left(\cos \left({\varphi }_A\right)l_{A}d{k}_4-l_{S}c-l_{S}d{k}_2\right)b}{\cos \left({\varphi }_A\right)l_{A}df}$$

$$K_{\mathit{VD}4}=\frac{\left(\left(e\cos {\left({\varphi }_A\right)}^2{l}_A^{2}d{k}_3-e\cos \left({\varphi }_A\right)l_{A}d{l}_S{k}_1+l_S\cos \left({\varphi }_A\right)l_{A}fa-\cos {\left({\varphi }_A\right)}^2{l}_A^{2}bf+l_S^{2}bd{k}_1-l_{S}b\cos \left({\varphi }_A\right)l_{A}d{k}_3\right)b\right)}{\left(d\cos {\left({\varphi }_A\right)}^2{l}_A^2{f}^2\right)}$$

For the pilot control _gains K _VD0 to K _{VD4 of} the pilot _block 71, taking into account the state _{regulator block} 73, one obtains: $${\mathrm{<figure-callout\; id="0"\; label="K\; VD"\; filenames="imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}=k_1$$

$${\mathrm{<figure-callout\; id="1"\; label="K\; VD"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}=\frac{c+d{k}_2}{\mathrm{<figure-callout\; id="2"\; label="d\; K\; VD"\; filenames="imgf0005.png"\; state="\{\{state\}\}">d\; </figure-callout>}}$$

$$K_{\mathit{VD}}$$$${\mathit{}2}_{}=\frac{-\cos \left({\phi }_A\right)l_{A}fa+\cos \left({\phi }_A\right)l_{A}bd{k}_3-d{l}_{\mathrm{<figure-callout\; id="1"\; label="S\; b\; k"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">S\; </figure-callout>}}b{k}_{}{}_1}{\cos \left({\phi }_A\right)l_{A}df}\cdot \left(-1\right)$$

$${\mathrm{<figure-callout\; id="3"\; label="K\; VD"\; filenames="imgf0001.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VD}}=-\frac{\left(\cos \left({\phi }_A\right)l_{A}d{k}_4-l_{S}c-l_{S}d{k}_2\right)b}{\cos \left({\phi }_A\right)l_{A}df}$$

$$K_{\mathit{VD}4}=\frac{\left(\left(e\cos {\left({\phi }_A\right)}^2{l}_A^{2}d{k}_3-e\cos \left({\phi }_A\right)l_{A}d{\mathrm{<figure-callout\; id="1"\; label="l\; S\; k"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">l\; </figure-callout>}}_S{k}_{}{}_1+l_S\cos \left({\phi }_A\right)l_{A}fa-\cos {\left({\phi }_A\right)}^2{l}_A^{2}bf+l_S^{2}bd{k}_1-l_{S}b\cos \left({\phi }_A\right)l_{A}d{k}_3\right)b\right)}{\left(d\cos {\left({\phi }_A\right)}^2{l}_A^2{f}^2\right)}$$

Thus, with Eq. (28) analogous to Eq. (23) the pre-control reinforcements are known, which guarantee a vibration-free and web-accurate method of load in the direction of rotation based on the idealized model. Now the state controller _gains k _1D , k _2D , k _3D , k _{4D are} to be determined. This will be explained below.

The regulator feedback 73 is designed as a complete state controller. A complete state controller is characterized in that each state variable, that is, each component of the state vector x _{D is} weighted with a control _gain k _iD and _fed back to the set input of the path. The control _gains k _iD are combined to the control vector K _D.

l ist die Einheitsmatrix. Die Auswertung von (29) führt im Falle des gewählten Zustandsraummodells nach Gl. 6-12 auf ein Polynom 4-ter Ordnung der Form: $$p\left(s\right)=s^4+p_3{s}^3+p_2{s}^2+p_{1}s+p_0$$

According to "Unbehauen, Regelungstechnik 2, loc. Cit.", The dynamic behavior of the system is determined by the position of the eigenvalues of the system matrix A _D , which are also poles of the transfer function in the frequency domain. The eigenvalues of the matrix can be determined by calculating the zeros with respect to the variable s of the characteristic polynomial p (s) from the determinants as follows. $$\begin{array}{l}\det \left(s\underset{~}{I}-{\underset{~}{A}}_D\right)\equiv 0\\ \mathrm{in\; which}p\left(s\right)=\det \left(s\underset{~}{I}-{\underset{~}{A}}_D\right)\end{array}$$

l is the unit matrix. The evaluation of (29) leads in the case of the chosen state space model according to Eq. 6-12 to a polynomial of the 4th order of the form: $$p\left(s\right)=s^4+p_3{\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="imgf0001.png"\; state="\{\{state\}\}">s</figure-callout>}}^3+p_2{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+{\mathrm{<figure-callout\; id="1s"\; label="p"\; filenames="imgf0012.png"\; state="\{\{state\}\}">p</figure-callout>}}_{1}s+{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0005.png"\; state="\{\{state\}\}">p</figure-callout>}}_0$$

By returning the state variables via the controller matrix K _D to the control input, these eigenvalues can be selectively shifted, since the position of the eigenvalues is now determined by the evaluation of the following determinant: $$p\left(s\right)=\det \left(s\underset{~}{I}-{\underset{~}{A}}_D+{\underset{~}{B}}_D\cdot {\underset{~}{K}}_D\right)$$

The evaluation of (31) again leads to a polynomial of the 4th order, which now however depends on the controller _gains k _iD ( i = 1..4). In the case of the model according to Eq. 6-12 becomes (30) too $$p\left(s\right)=s^4+\frac{\left(ce-bd{k}_{4D}+de{k}_{2D}\right)s^3}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}+\frac{\left(af-bd{k}_{3D}+de{k}_{1D}\right)s^2}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}+\frac{\left(d{k}_{2D}f+cf\right)s}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}+\frac{d{k}_{1D}f}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}$$

wobei n die Systemordnung ist, die mit der Dimension des Zustandsvektors gleichzusetzen ist. Im Falle des Modells nach Gl. 6-12 ist n=4 und damit p(s): $$p\left(s\right)=\left(s-r_1\right)\left(s-r_2\right)\left(s-r_3\right)\left(s-r_4\right)=s^4+p_3{s}^3+p_2{s}^2+p_{1}s+p_0$$

It is now _required that the controller _gains k _iD, the Eq. 31 or 32 occupies certain zeros, thereby specifically affecting the dynamics of the system, which is reflected in the zeros of this polynomial. This results in a specification for this polynomial according to: $$p\left(s\right)={\displaystyle \underset{i=1}{\overset{n}{\Pi }}\left(s-r_i\right)}$$

where n is the system order to be equated with the dimension of the state vector. In the case of the model according to Eq. 6-12, n = 4 and thus p (s): $$p\left(s\right)=\left(s-r_1\right)\left(s-r_2\right)\left(s-r_3\right)\left(s-r_4\right)=s^4+p_3{\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="imgf0001.png"\; state="\{\{state\}\}">s</figure-callout>}}^3+p_2{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+{\mathrm{<figure-callout\; id="1s"\; label="p"\; filenames="imgf0012.png"\; state="\{\{state\}\}">p</figure-callout>}}_{1}s+{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="imgf0005.png"\; state="\{\{state\}\}">p</figure-callout>}}_0$$

The poles r _i are to be chosen so that the system is stable, the control operates sufficiently fast with good damping and the manipulated variable limitation is not achieved for typical occurring control deviations. The r _i can be determined before commissioning in simulations according to these criteria.

The control gains can now be calculated by comparing the coefficients of the polynomials Eq. 31 and 33 are determined. $$\det \left(s\underset{~}{I}-{\underset{~}{A}}_D+{\underset{~}{B}}_D\cdot {\underset{~}{K}}_D\right)\equiv {\displaystyle \underset{i=1}{\overset{n}{\Pi }}\left(s-r_i\right)}$$

$$k_{2D}=-\frac{\left(cf+r_1{r}_2{r}_{4}ae-r_1{r}_2{r}_4{b}^2+r_1{r}_2{r}_{3}ae-r_1{r}_2{r}_3{b}^2+r_2{r}_3{r}_{4}ae-r_2{r}_3{r}_4{b}^2+r_1{r}_3{r}_{4}ae-r_1{r}_3{r}_4{b}^2\right)}{df}$$

$$k_{3D}=\frac{\left(a{f}^2+e^2{r}_1{r}_2{r}_3{r}_{4}a-e{r}_1{r}_2{r}_3{r}_4{b}^2-r_1{r}_{2}aef+r_1{r}_2{b}^{2}f-r_2{r}_{4}aef+r_2{r}_4{b}^{2}f-r_3{r}_{4}aef+r_3{r}_4{b}^{2}f\right)}{bdf}$$

$$k_4=\frac{\left(r_1{r}_2{r}_{4}a{e}^2-e{r}_1{r}_2{r}_4{b}^2+r_1{r}_2{r}_{3}a{e}^2-e{r}_1{r}_2{r}_3{b}^2+r_2{r}_3{r}_{4}a{e}^2-e{r}_2{r}_3{r}_4{b}^2+r_1{r}_3{r}_{4}a{e}^2-e{r}_1{r}_3{r}_4{b}^2-r_{2}aef+r_2{b}^{2}f-r_{1}aef+r_1{b}^{2}f-r_{4}aef+r_4{b}^{2}f-r_{3}aef+r_3{b}^{2}f\right)}{bdf}$$

In the case of the model according to Eq. 6-12 results in a linear equation system as a function of the control _gains k _iD . The evaluation of the equation system leads to analytical mathematical expressions for the controller gains as a function of the desired poles r _i and the system parameters. $$k_{1D}=\frac{r_1{r}_2{r}_3{r}_4\left(ae-b^2\right)}{df}$$

$$k_{2D}=-\frac{\left(cf+r_1{r}_2{r}_{4}ae-r_1{r}_2{r}_4{b}^2+r_1{r}_2{r}_{3}ae-r_1{r}_2{r}_3{b}^2+r_2{r}_3{r}_{4}ae-r_2{r}_3{r}_4{b}^2+r_1{r}_3{r}_{4}ae-r_1{r}_3{r}_4{b}^2\right)}{df}$$

$$k_{3D}=\frac{\left(a{f}^2+e^2{r}_1{r}_2{r}_3{r}_{4}a-e{r}_1{r}_2{r}_3{r}_4{b}^2-r_1{r}_{2}aef+r_1{r}_2{b}^{2}f-r_2{r}_{4}aef+r_2{r}_4{b}^{2}f-r_3{r}_{4}aef+r_3{r}_4{b}^{2}f\right)}{bdf}$$

$$k_4=\frac{\left(r_1{r}_2{r}_{4}a{e}^2-e{r}_1{r}_2{r}_4{b}^2+r_1{r}_2{r}_{3}a{e}^2-e{r}_1{r}_2{r}_3{b}^2+r_2{r}_3{r}_{4}a{e}^2-e{r}_2{r}_3{r}_4{b}^2+r_1{r}_3{r}_{4}a{e}^2-e{r}_1{r}_3{r}_4{b}^2-r_{2}aef+r_2{b}^{2}f-r_{1}aef+r_1{b}^{2}f-r_{4}aef+r_4{b}^{2}f-r_{3}aef+r_3{b}^{2}f\right)}{bdf}$$

In the case of model according to Eq. 6-12 are the model parameters K _PD i _D, V, _A φ, β, _T J, J _AZ, m _A, s _A, m _L, l _A, l _{S, D} b. An advantage of this controller design is that now parameter changes of the system, such as the cable length l _S , the righting angle φ _A or the load mass m _L can be considered immediately in changed controller gains. This is crucial for optimized control behavior.

By doing this, the controller gains from the analytic expressions of Eq. 36 are calculated, also possible for individual poles during operation _i r as a function of measured values such as load mass m _L, rope length l _s or of elevation φ _A. to be changed. This results in a very advantageous dynamic behavior.

Alternatively, a numerical design according to the design method of Riccati (see also O. Föllinger: control technology, 7th ed., Hüthig Verlag, Heidelberg, 1992) are performed and the controller gains in look-up tables depending on load mass, Aufrichtwinkel and Rope length are stored.

über die Berechnung nach Gl. 31 numerisch überprüft werden. Da dies nur numerisch erfolgen kann, muß der gesamte durch die veränderlichen Systemparameter aufgespannte Raum erfaßt werden. In diesem Falle wären dies die veränderlichen Systemparameter m _L , l _S und ϕ _A . Diese Parameter schwanken im Intervall [m _Lmin , m _Lmax ],[l _Smin , l _Smax ] bzw. [ϕ _Amin , ϕ _Amax ]. D.h. in diesen Intervallen müssen mehrere Stützstellen m _Lk , l _i bzw. ϕ _Aj gewählt werden und für alle möglichen Kombinationen dieser veränderlichen Systemparameter die Systemmatrix A _ljk (m _Lk. , l _i , ϕ _Aj ) berechnet und in Gl. 31 eingesetzt und mit K _D aus Gl. 37 ausgewertet werden: $$\det \left(s\underset{‾}{I}-{\underset{‾}{A}}_{ijk}+\underset{‾}{B}\cdot {\underset{‾}{K}}_D\right)\equiv 0\hspace{1em}\mathrm{für}\hspace{0.17em}\mathrm{alle}\hspace{0.17em}i,j,k$$

Since a complete state controller requires the knowledge of all state variables, it is advantageous to execute the regulation as output feedback instead of a state observer. This means that not all state variables are fed back via the controller, but only those which are detected by measurements. There are therefore individual k _iD to zero. In the case of the model according to Eq. 6 to 12, for example, the measurement of the rope angle could be omitted. This will make k _3D = 0 . Nevertheless, the calculation of k _1D , k _2D and k _4D can be done analogously to Eq. (36). In addition, it may be useful, due to the considerable computational effort for an individual Operating point to calculate the controller parameters. However, it must then the actual eigenvalue of the system with the controller matrix $${\underset{~}{K}}_D=\left[\begin{array}{cccc}{k}_{1D}& k_{2D}& 0& k_{4D}\end{array}\right]$$

on the calculation according to Eq. 31 are checked numerically. Since this can only be done numerically, the entire space spanned by the variable system parameters must be detected. In this case, these would be the variable system parameters m _L , l _S and φ _A. These parameters vary in the interval [ m _Lmin , m _Lmax ] , [ l _Smin , l _Smax ] and [ φ _Amin , φ _Amax ] . That is, in these intervals, several interpolation points m _Lk , l _i or φ _{Aj must be} selected and calculated for all possible combinations of these variable system parameters, the system matrix A _ljk ( m _Lk. , L _i , φ _Aj ) and in Eq. 31 used and with K _D from Eq. 37 are evaluated: $$\det \left(s\underset{~}{I}-{\underset{~}{A}}_{ijk}+\underset{~}{B}\cdot {\underset{~}{K}}_D\right)\equiv 0\mathrm{For}\mathrm{all}i.j.k$$

If all zeros of (38) are always less than zero, the stability of the system is preserved and the originally selected poles r _i can be retained. If this is not the case, a correction of the poles r _i according to Eq. (33) become necessary.

If a state variable is not measurable, it can be reconstructed from other measurands in an observer. In this case, caused by the measurement principle disturbances can be eliminated. In Fig. 7, this module is referred to as Störbeobachter 77. Depending on which sensor system is used for the rope angle measurement, the interference observer is suitable to configure. If, for example, an acceleration sensor is used, then the observer must estimate the pendulum angle from the pendulum dynamics and the acceleration signal of the load. In an image processing system, it is necessary that the oscillations of the cantilever be compensated by the observer so that a usable signal can be detected. When measuring the deflection of the boom with strain gauges, the signal from the retro-active bending of the cantilever to extract the observer. In the following, the reconstruction of the cable angle and the cable angular velocity will be shown on the basis of the measurement with a gyroscope sensor on the load hook.

1. 1.) Korrektur des meßprinzipbedingten Offsets auf dem Meßsignal
2. 2.) offsetkompensierte Integration des gemessenen Winkelgeschwindigkeitssignals zum Winkelsignal
3. 3.) Eliminierung der Oberschwingungen auf dem Meßsignal, die durch Oberschwingungen des Seiles verursacht werden.

The gyroscope sensor measures the angular velocity in the corresponding sensitivity direction. By a suitable choice of the installation location on the load hook, the sensitivity direction corresponds to the direction of the tangential angle φ _St. The observer now has the following tasks:

1. 1.) Correction of the measurement principle-related offset on the measurement signal
2. 2.) Offset compensated integration of the measured angular velocity signal to the angle signal
3. 3.) Elimination of the harmonics on the measuring signal, which are caused by harmonics of the rope.

The disturbances are first to be modeled as differential equations. First, the _offset error φ̇ _{Offset, D is} introduced as a disturbance variable. The disturbance is assumed to be constant as sections. The fault model is accordingly $${\ddot{\phi }}_{\mathrm{offset}.D}=0$$

bestimmen, wobei µ _Seil die Masse des Seiles bezogen auf die Längeneinheit ist. Die korrespondierende linearisierte Schwingungsdifferentialgleichung für die Oberschwingung ist $${\ddot{\varphi }}_{\mathrm{Ober},D}=-{\varpi }_1^2{\varphi }_{\mathrm{Ober},D}$$

Furthermore, the measurement signal of the angular velocity of the simple pendulum motion of harmonics of the rope is superimposed. The resonant frequency with respect to the harmonics tensioned ropes (see also Beitz W. Küttner K.-H .: Dubbel paperback for mechanical engineering, 17th ed., Springer Verlag, Heidelberg, 1990) can be in the 2-wire suspension on the context $${\pi }_{}$$$${}_1=\frac{\pi }{l_S}\sqrt{\frac{m_{\mathrm{<figure-callout\; id="2"\; label="L\; G"\; filenames="imgf0005.png"\; state="\{\{state\}\}">L\; </figure-callout>}}G}{2{\mu }_{\mathrm{rope}}}}$$

determine, where μ _{rope is} the mass of the rope relative to the unit length. The corresponding linearized vibration differential equation for the harmonic is $${\ddot{\phi }}_{\mathrm{upper}.D}=-{\mathrm{<figure-callout\; id="1"\; label="\pi "\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">\pi </figure-callout>}}_1^2{\phi }_{\mathrm{upper}.D}$$

wobei in Ergänzung zu Gl. 6-12 die folgenden Matrizen und Vektoren eingeführt werden.The state space representation of the submodel for the slewing gear according to Eq. 6-12 is extended by the fault model. In the present case, a complete observer is derived. The observer equation for the modified state space model is therefore: $${\underset{~}{\dot{\widehat{x}}}}_{Dz}=\left({\underset{~}{A}}_{Dz}-{\underset{~}{H}}_{Dx}{\underset{~}{C}}_{mDz}\right)\cdot {\underset{~}{x}}_{Dz}+{\underset{~}{B}}_{Dz}\cdot {\underset{~}{u}}_D+{\underset{~}{H}}_{Dz}{\underset{~}{y}}_{Dm}$$

in addition to Eq. 6-12 the following matrices and vectors are introduced.

Eingangsmatrix: $${\underset{‾}{B}}_{Dz}=\left[\begin{array}{c}0\\ \frac{de}{ae-b^2}\\ 0\\ \frac{-bd}{ae-b^2}\\ 0\\ 0\\ 0\end{array}\right]$$

State vector: $${\underset{~}{x}}_{Dz}=\left[\begin{array}{c}{\phi }_D\\ {\dot{\phi }}_D\\ {\phi }_{St}\\ {\dot{\phi }}_{St}\\ {\dot{\phi }}_{\mathrm{offset}.D}\\ {\phi }_{\mathrm{upper}.D}\\ {\dot{\phi }}_{\mathrm{upper}.D}\end{array}\right]$$

Input matrix: $${\underset{~}{B}}_{Dz}=\left[\begin{array}{c}0\\ \frac{de}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}\\ 0\\ \frac{-bd}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}\\ 0\\ 0\\ 0\end{array}\right]$$

Matrix: $${\underset{~}{A}}_{Dz}=\left[\begin{array}{ccccccc}0& 1& 0& 0& 0& 0& 0\\ 0& -\frac{ce}{ae-b^2}& \frac{fb}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& \frac{cb}{ae-b^2}& -\frac{af}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& -{\mathrm{<figure-callout\; id="1"\; label="w"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">w</figure-callout>}}_1^2& 0\end{array}\right]$$

Störbeobachtermatrix: $${\underset{~}{H}}_{Dz}=\left[\begin{array}{ccc}{H}_{11D}& H_{12D}& H_{13D}\\ H_{21D}& H_{22D}& H_{23D}\\ H_{31D}& H_{32D}& H_{33D}\\ H_{41D}& H_{42D}& H_{43D}\\ H_{51D}& H_{52D}& H_{53D}\\ H_{61D}& H_{62D}& H_{63D}\\ H_{71D}& H_{72D}& H_{73D}\end{array}\right]$$

Observers output matrix: $${\underset{~}{C}}_{mDz}=\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 1& 0& 1\end{array}\right]$$

Output vector of the measured quantities: $${\underset{~}{y}}_{mD}=\left[\begin{array}{c}{\phi }_D\\ {\dot{\phi }}_D\\ {\dot{\phi }}_{Stm}\end{array}\right]$$

The determination of the observer _gains h _ljD is carried out either by transformation into observation _{normal form} or by the Riccati design _method . It is essential that in the observer also variable rope length, righting angle and load mass are taken into account by adapting the observer differential equation and the observer reinforcements.

$$H_{\mathit{dZred}}=\left[\begin{array}{ccccc}{h}_{1\mathit{red}}& h_{2\mathit{red}}& h_{3\mathit{red}}& h_{4\mathit{red}}& h_{5\mathit{red}}\end{array}\right]\hspace{1em}{\underset{‾}{x}}_{\mathit{DZred}}=\left[\begin{array}{c}{\widehat{\varphi }}_{St}\\ {\widehat{\dot{\varphi }}}_{St}\\ {\widehat{\dot{\varphi }}}_{\mathit{Offset},D}\\ {\widehat{\varphi }}_{\mathit{Ober},D}\\ {\widehat{\dot{\varphi }}}_{\mathit{Ober},D}\end{array}\right]$$

$$C_{\mathit{mDZred}}=\left[\begin{array}{ccccc}0& 1& 1& 0& 1\end{array}\right]\hspace{1em}{y}_{\mathit{mDred}}={\dot{\varphi }}_{Stm}$$

$${\underset{‾}{u}}_{\mathit{DZred}}={\ddot{\varphi }}_D$$

The estimation may advantageously also be based on a reduced model. For this, only the second equation from the model approach according to equation 4, which describes the rope oscillation, is considered. The input of the observer is defined as φ̈ _D , which can be calculated either from the measured quantity or U _Dref (see equation 40). The reduced observer state space model taking into account the disturbance variables is then: $${\underset{~}{A}}_{\mathit{DZred}}=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ -\frac{af}{ae-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}& 0& 1& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& -{\mathrm{<figure-callout\; id="1"\; label="w"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">w</figure-callout>}}_1^2& 0\end{array}\right]{\underset{~}{B}}_{\mathit{dZred}}=\left[\begin{array}{c}0\\ \frac{m_L\cdot l_A\cdot \cos {\phi }_A}{m_L{l}_S}\\ 0\\ 0\\ 0\end{array}\right]$$

$$H_{\mathit{dZred}}=\left[\begin{array}{ccccc}{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}}_{1\mathit{<figure-callout\; id="2"\; label="red\; H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">red\; </figure-callout>}}& H_{}{}_{2\mathit{<figure-callout\; id="3"\; label="red\; H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">red\; </figure-callout>}}& H_{}{}_{3\mathit{red}}& H_{4\mathit{<figure-callout\; id="5"\; label="red\; H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">red\; </figure-callout>}}& H_{}{}_{5\mathit{red}}\end{array}\right]{\underset{~}{x}}_{\mathit{DZred}}=\left[\begin{array}{c}{\widehat{\phi }}_{St}\\ {\widehat{\dot{\phi }}}_{St}\\ {\widehat{\dot{\phi }}}_{\mathit{offset}.D}\\ {\widehat{\phi }}_{\mathit{upper}.D}\\ {\widehat{\dot{\phi }}}_{\mathit{upper}.D}\end{array}\right]$$

$$C_{\mathit{mDZred}}=\left[\begin{array}{ccccc}0& 1& 1& 0& 1\end{array}\right]y_{\mathit{mDred}}={\dot{\phi }}_{Stm}$$

$${\underset{~}{u}}_{\mathit{DZred}}={\ddot{\phi }}_D$$

, aus dem reduzierten Störbeobachter 771 (Fig. 7a) können entweder direkt auf den Zustandsregler gegeben werden oder, da das Signal ϕ̂ _St aus dem Beobachter 771 immer noch mit einem geringen Offset überlagert ist, in einem zweiten Offsetbeobachter 773, der nun einen Offset

bezüglich des Winkelsignales ϕ̂ _St annimmt, weiterverarbeitet werden. Hierzu wird als Strömodell

= 0 angenommen.The estimates φ _St ,

, from the reduced observer 771 (Figure 7a) can either be given directly to the state controller or, since the signal φ _St from the observer 771 is still superimposed with a low offset, in a second offset observer 773, which now has an offset

with respect to the angle signal φ _St , be further processed. This is called Strömodell

= 0 accepted.

$$y_{\mathit{mOff}}={\widehat{\varphi }}_{St},\hspace{1em}{u}_{\mathit{DOff}}={\ddot{\varphi }}_D\hspace{1em}{\underset{‾}{B}}_{\mathit{DOff}}=\left[\begin{array}{c}0\\ -\frac{l_A\cos {\varphi }_A}{l_S}\\ 0\end{array}\right]$$

The underlying model based on the second equation of (4) is then $${\underset{~}{A}}_{\mathit{doff}}=\left[\begin{array}{ccc}\mathit{0}& \mathit{1}& \mathit{1}\\ \mathit{-}\frac{\mathit{<figure-callout\; id="0"\; label="G\; l\; S"\; filenames="imgf0005.png"\; state="\{\{state\}\}">G\; </figure-callout>}}{{\mathit{l}}_{\mathit{S}}}& \mathit{0}& \mathit{0}\\ \mathit{0}& \mathit{0}& \mathit{0}\end{array}\right]{\underset{~}{x}}_{\mathit{doff}}=\left[\begin{array}{c}{\widehat{\widehat{\phi }}}_{St}\\ {\widehat{\widehat{\dot{\phi }}}}_{St}\\ {\widehat{\widehat{\phi }}}_{\mathrm{off}}\end{array}\right]$$

$$y_{\mathit{Moff}}={\widehat{\phi }}_{St}.u_{\mathit{doff}}={\ddot{\phi }}_D{\underset{~}{B}}_{\mathit{doff}}=\left[\begin{array}{c}0\\ -\frac{l_A\cos {\phi }_A}{l_S}\\ 0\end{array}\right]$$

The observer gains are determined via pole specification as in the controller design (Eq. 29 ff.). The resulting structure for the two-stage reduced observer is shown in Figure 7a. This variant guarantees an even better compensation of the offset on the measured value and a better estimation for φ _St and φ̇ _St.

bzw.

werden auf den Zustandregler zurückgeführt. Damit erhält man am Ausgang des Zustandsreglerblocks 73 bei Rückführung von ϕ _D ,ϕ̇ _D ,ϕ̂ _St ,

dann $$u_{\mathit{Drück}}=k_{1D}{\varphi }_D+k_{2D}{\dot{\varphi }}_D+k_{3D}{\widehat{\varphi }}_{St}+k_{4D}{\widehat{\dot{\varphi }}}_{St}$$

The estimated values φ _St ,

respectively.

are returned to the state controller. This results in the output of the state regulator block 73 when φ _D , φ̇ _D , φ _St , is returned.

then $$u_{\mathit{suppressive}}=k_{1D}{\phi }_D+k_{2D}{\dot{\phi }}_D+k_{3D}{\widehat{\phi }}_{St}+k_{4D}{\widehat{\dot{\phi }}}_{St}$$

The Sollansteuerspannung the proportional valve for the slewing is taking into account the feedforward 71 then $$u_{\mathit{Dref}}=u_{\mathit{Dvorst}}-u_{\mathit{suppressive}}$$

Since in the state space model according to Eq. If only linear system components can be taken into account, optionally static non-linearities of the hydraulics in block 75 of the hydraulic compensation can be taken into account so as to result in a linear system behavior with respect to the system input. The most significant nonlinear effects of hydraulics are the backlash of the proportional valve around the zero point and hysteresis effects of the subordinate flow control. For this purpose, the static characteristic between the drive _voltage u _{StD of} the proportional valve and the resulting flow Q _{FD is} recorded experimentally. The characteristic can be described by a mathematical function. $$Q_{FD}=f\left(u_{StD}\right)$$

Regarding the system input linearity is required. That is, the proportional valve and the block of hydraulic compensation should, according to Eq. (5) summarized have the following transmission behavior. $$Q_{FD}=K_{PD}{u}_{StD}$$

so ist Bedingung (42) genau dann erfüllt, wenn als statische Kompensationskennlinie $$h\left(u_{\mathit{Dref}}\right)=f^{-1}\left(K_{PD}{u}_{\mathit{Dref}}\right)$$

gewählt wird.Compensation block 75 has the static characteristic $$u_{StD}=H\left(u_{\mathit{Dref}}\right).$$

Thus condition (42) is fulfilled if and only if as a static compensation characteristic $$H\left(u_{\mathit{Dref}}\right)=f^{-1}\left(K_{PD}{u}_{\mathit{Dref}}\right)$$

is selected.

This explains the individual components of the axis controller for the slewing gear. As a result, the combination of path planning module and axis controller slewing meets the requirement of a vibration-free and precise web movement of the load.

Based on these results, the axis controller for the luffing mechanism 7 will now be explained. Fig. 9 shows the basic structure of the axis controller for the luffing mechanism.

The output functions of the path planning module in the form of the Sollastposition, in the radial direction, and their derivatives (speed, acceleration, jerk, and derivative of the jerk) are given to the feedforward block 91 (block 71 in the slewing gear). In the pre-control block, these functions are so amplified that results in a web-accurate driving the load without vibrations under the idealized conditions of the dynamic model. The basis for determining the feedforward gains is the dynamic model derived in the following sections for the luffing gear. Thus, under these idealized conditions, the swinging of the load is suppressed and the load follows the generated lane.

As with the slewing gear, the feedforward can optionally be supplemented by a state control block 93 (see slewing gear 73) for compensating for disturbances (eg wind influences) and compensating for model errors. In this block at least one of the measured values Aufrichtwinkel φ _A , Aufrichtwinkelgeschwindigkeit φ̇ _A , bending of the boom in the vertical direction w _v , the derivation of the vertical bend ẇ _v , the radial cable angle φ _Sr or the radial cable angular velocity φ̇ _Sr amplified and back to the control input recycled. The derivative of the measured quantities φ _A , φ _Sr and w _v is formed numerically in the microprocessor control.

Due to the dominant static non-linearity of the hydraulic drive units (hysteresis, backlash), the now _Vorsteuerung U _Avorst and optional State _{controller output} U _Arück formed value for the control input U _Aref in block hydraulic _compensation 95 (analogous to block 75) changed so that can be assumed to result linear behavior of the overall system. Output of the block 95 (hydraulic compensation) is the corrected control variable u _StA. This value is then applied to the proportional valve of the hydraulic circuit for the cylinder of the luffing gear.

For a detailed explanation of the procedure, the derivation of the dynamic model for the luffing gear is now to serve, which is the basis for the calculation of the pilot control gains, the state controller and the observer.

For this purpose, FIG. 10 gives explanations of the definition of the model variables. Essential here is the relationship shown there between the Aufrichtwinkelposition φ _{A of} the boom and the load position in the radial direction r _LA $$r_{LA}=l_A\cos {\phi }_A+l_S\sin {\phi }_{Sr}$$

However, the small-signal behavior is decisive for the control behavior. Therefore, Eq. (45) linearized and selected an operating point φ _A0 . The radial deviation is then defined as a controlled variable. $$\Delta r_{LA}=-l_A{\phi }_A\mathrm{<figure-callout\; id="0"\; label="sin\; \phi \; A"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sin\; </figure-callout>}{\phi }_A{0}_{}+l_S\sin {\phi }_{Sr}$$

$$-m_L{l}_A{l}_s\sin {\varphi }_{A0}{\ddot{\varphi }}_A+m_L{l}_S^2{\ddot{\varphi }}_{sr}+m_L{l}_{s}g{\varphi }_{sr}=m_L{l}_s{\dot{\varphi }}_D^2\left(l_S{\varphi }_{sr}+l_A\cos {\varphi }_{A0}\right)$$

The dynamic system can be described by the following differential equations. $$\begin{array}{l}\left(J_{AY}+m_A{s}_A^2+m_L{l}_A^2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\phi }_{A0}\right){\ddot{\phi }}_A-m_L{l}_A{l}_s\mathrm{<figure-callout\; id="0"\; label="sin\; \phi \; A"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sin\; </figure-callout>}{\phi }_A{0}_{}{\ddot{\phi }}_{sr}+\\ b_A{\dot{\phi }}_A-m_A{s}_A\mathrm{G}\mathrm{<figure-callout\; id="0"\; label="sin\; \phi \; A"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sin\; </figure-callout>}{\phi }_A{0}_{}\cdot {\phi }_A=\\ M_{MA}-M_{RA}-m_A{s}_A\mathrm{G}\cos {\phi }_{A0}\end{array}$$

$$-m_L{l}_A{l}_s\mathrm{<figure-callout\; id="0"\; label="sin\; \phi \; A"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sin\; </figure-callout>}{\phi }_A{0}_{}{\ddot{\phi }}_A+m_L{l}_S^2{\ddot{\phi }}_{sr}+m_L{l}_{s}G{\phi }_{sr}=m_L{l}_s{\dot{\phi }}_D^2\left(l_S{\phi }_{sr}+l_A\cos {\phi }_{A0}\right)$$

m _L

Lastmasse

l _s

Seillänge

m _A

Masse des Auslegers

J _AY

Massenträgheitsmoment bezüglich Schwerpunkt bei Drehung um horizontale Achse inkl. Antriebsstrang

l _A

Länge des Auslegers

S _A

Schwerpunktsabstand des Auslegers

b _A

viskose Dämpfung

M _MA

Antriebsmoment

M _RA

Reibmoment

designations:

m _L

load mass

l _s

cable length

m _A

Mass of the jib

J _AY

Mass moment of inertia with respect to center of gravity when turning about horizontal axis incl. Drive train

l _A

Length of the boom

S _A

Center of gravity of the jib

b _A

viscous damping

M _MA

drive torque

M _RA

friction

The first equation of (4) essentially describes the equation of motion of the boom with the driving hydraulic cylinder, taking into account the retroactivity of the pendulum of the load. In this case, the proportion acting through the gravity of the boom and the viscous friction in the drive is taken into account. The second equation of (4) is the equation of motion describing the load swing φ _Sr , where the excitation of the vibration is caused by the canting of the cantilever over the angular acceleration of the cantilever or an external perturbation expressed by initial conditions for these differential equations , The term on the right side of the differential equation describes the influence of the centripetal force on the load as the load rotates with the slewing gear. As a result, a typical for a turntable problem is described, as it is a coupling between slewing and luffing. Clearly, this problem can be described by the fact that a slewing movement with a quadratic rotational speed dependency also causes an angular deflection in the radial direction. If exact web Driving the load to be achieved, this problem must be taken into account. First, this effect is set to 0. Once the components of the axis controller have been explained, the point of coupling between lathes and luffing gear is taken up again and possible solutions are shown.

The hydraulic drive is described by the following equations. $$\begin{array}{l}{M}_{MA}=F_{\mathrm{Zyl}}{d}_b\cos {\phi }_p\left({\phi }_A\right)\\ F_{Zyl}=p_{\mathit{Zyl}}{A}_{\mathit{Zyl}}\\ {\dot{p}}_{\mathit{Zyl}}=\frac{2}{\beta V_{\mathrm{Zyl}}}\left(Q_{FA}-A_{\mathit{Zyl}}{\dot{z}}_{\mathit{Zyl}}\left({\phi }_A.{\dot{\phi }}_A\right)\right)\\ Q_{FA}=K_{PA}{u}_{StA}\end{array}$$

F _Zyl is the force of the hydraulic cylinder on the piston rod, p _Zyl is the pressure in the cylinder (depending on the direction of movement piston or ring side), A _Zyl is the cross-sectional area of the cylinder (depending on the direction of movement piston or ring side), β is the oil compressibility, V _Zyl is the cylinder volume, Q _FA is the flow rate in the hydraulic circuit for the luffing gear and K _PA is the proportionality constant, which indicates the relationship between the flow rate and the control voltage of the proportional valve. Dynamic effects of subordinate flow control are neglected. In the oil compression in the cylinder, the relevant cylinder volume is assumed to be half the total volume of the hydraulic cylinder. z _Zyl , ż _Zyl are the position or speed of the cylinder rod. These, like the geometric parameters d _b and φ _{p, are} dependent on the erecting kinematics.

FIG. 11 shows the erecting kinematics of the luffing mechanism. By way of example, the hydraulic cylinder is anchored to the lower end of the crane tower. From design data, the distance d _a between this point and the pivot point of the boom can be taken. The piston rod of the hydraulic cylinder is attached to the boom at a distance d _b . φ ₀ is also known from design data. from that can derive the following relationship between Aufrichtwinkel φ _A and hydraulic cylinder z _{zyl position} . $$z_{\mathit{Zyl}}=\sqrt{d_a^2+d_b^2-2d_b{d}_a\cos \left({\phi }_A+{\phi }_0\right)}$$

$${\dot{\varphi }}_A=\frac{\partial {\varphi }_A}{\partial z_{\mathit{Zyl}}}{\dot{z}}_{\mathit{Zyl}}=\frac{\sqrt{d_a^2+d_b^2-2d_b{d}_a\cos \left({\varphi }_A+{\varphi }_0\right)}}{d_b{d}_a\sin \left({\varphi }_A+{\varphi }_0\right)}{\dot{z}}_{\mathit{Zyl}}$$

Since only the Aufrichtwinkel φ _{A is} measured, the inverse relationship of (48) and the relationship between piston rod speed ż _Zyl and Aufrichtgeschwindigkeit φ̇ _{A is} also of interest. $${\phi }_A=\arccos \left(\frac{d_a^2+d_b^2-{\mathrm{<figure-callout\; id="2"\; label="z\; Zyl"\; filenames="imgf0005.png"\; state="\{\{state\}\}">z\; </figure-callout>}}_{\mathrm{Zyl}}^{}}{2d_a{d}_b}\right)-{\mathrm{<figure-callout\; id="0"\; label="\phi "\; filenames="imgf0005.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_0$$

$${\dot{\phi }}_A=\frac{\partial {\phi }_A}{\partial z_{\mathit{Zyl}}}{\dot{z}}_{\mathit{Zyl}}=\frac{\sqrt{d_a^2+d_b^2-2d_b{d}_a\cos \left({\phi }_A+{\phi }_0\right)}}{d_b{d}_a\sin \left({\phi }_A+{\phi }_0\right)}{\dot{z}}_{\mathit{Zyl}}$$

For the calculation of the effective torque on the boom, the calculation of the projection angle φ _{p is also} required. $$\cos {\phi }_p=\frac{d_a\sin \left({\phi }_A+{\phi }_0\right)}{\sqrt{d_a^2+d_b^2-2d_b{d}_a\cos \left({\phi }_A+{\phi }_0\right)}}=\frac{H_1}{{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}}_2}$$

For a compact notation, in Eq. 51, the auxiliary variables h ₁ and h _{2 is} inserted. This can be done in the Eqs. 46-51 described dynamic model of the luffing gear now in the state space representation (see also O. Föllinger: control technology, 7th ed., Hüthig Verlag, Heidelberg, 1992) to be transformed. Since linearity is assumed, first the centripetal force coupling term with the slewing gear due to the rotational speed φ̇ _{D is} neglected. In addition, the components of Equation 46, which are due to gravitation, are set to zero. The result is the following state space representation of the system.

mit:State space representation: $$\begin{array}{l}{\dot{\underset{~}{x}}}_A={\underset{~}{A}}_A{\underset{~}{x}}_A+{\underset{~}{B}}_A{\underset{~}{u}}_A\\ {\underset{~}{y}}_A={\underset{~}{C}}_A{\underset{~}{x}}_A\end{array}$$

With:

State vector: $${\underset{~}{x}}_A=\left[\begin{array}{c}{\phi }_A\\ {\dot{\phi }}_A\\ {\phi }_{Sr}\\ {\dot{\phi }}_{Sr}\end{array}\right]$$

Tax Size: $$u_A=u_{StA}$$

Output: $$y_A=r_{LA}$$

$$a=J_{AY}+m_A{S}_A^2+m_L{l}_A^2\sin {\left({\varphi }_A\right)}^2$$

$$b=m_L{l}_A\sin \left({\varphi }_A\right)l_S$$

$$c=b_A+\frac{A_{\mathit{zyl}}^2{d}_b^2{h}_1^2}{\beta V_{\mathit{zyl}}{h}_2^2}$$

$$e=\frac{K_{PA}{A}_{\mathit{zyl}}{d}_b{h}_1}{\beta V_{\mathit{zyl}}{h}_2}$$

$$f=m_L{l}_S^2$$

$$g_g=m_{L}g{l}_S$$

wobei: $$\begin{array}{l}{h}_1=\sqrt{d_a^2+d_b^2-2d_b{d}_a\cos \left({\varphi }_A+{\varphi }_0\right)}\\ h_2=d_a\sin \left({\varphi }_A+{\varphi }_0\right)\end{array}$$

Matrix: $${\underset{~}{A}}_A=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& -\frac{fc}{af-b^2}& \frac{b{G}_G}{af-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}& 0\\ 0& 0& 0& 1\\ 0& \frac{bc}{af-b^2}& -\frac{a{G}_G}{af-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}& 0\end{array}\right]$$

$$a=J_{AY}+m_A{S}_A^2+m_L{l}_A^2\sin {\left({\phi }_A\right)}^2$$

$$b=m_L{l}_A\sin \left({\phi }_A\right)l_S$$

$$c=b_A+\frac{A_{\mathit{cyl}}^2{d}_b^2{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}}_1^2}{\beta {\mathrm{<figure-callout\; id="2"\; label="V\; cyl\; H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">V\; </figure-callout>}}_{\mathit{cyl}}{H}_{}^{}{}_2^2}$$

$$e=\frac{K_{PA}{A}_{\mathit{cyl}}{\mathrm{<figure-callout\; id="1"\; label="d\; b\; H"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">d\; </figure-callout>}}_b{H}_{}{}_1}{\beta V_{\mathit{cyl}}{H}_2}$$

$$f=m_L{l}_S^2$$

$$G_G=m_{L}G{l}_S$$

in which: $$\begin{array}{l}{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}}_1=\sqrt{d_a^2+d_b^2-2d_b{d}_a\cos \left({\phi }_A+{\phi }_0\right)}\\ {\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}}_2=d_a\sin \left({\phi }_A+{\phi }_0\right)\end{array}$$

Control vector. $${\underset{~}{B}}_A=\left[\begin{array}{c}0\\ \frac{fe}{af-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}\\ 0\\ \frac{be}{af-b^2}\end{array}\right]$$

Output vector: $${\underset{~}{C}}_A=\left[\begin{array}{cccc}-l_A\sin \left({\phi }_{A0}\right)& 0& l_S& 0\end{array}\right]$$

The dynamic model of the luffing gear is considered as a parameter variable system with respect to the rope length l _s and the trigonometric function components of the boom angle φ _A and the load mass m _L The equations (52) to (58) are the basis for the now described design of the pilot control 91, the state controller 93 and the interference observer 97.

und die Ableitung des Soll-Rucks $r_{LA}^{\left(IV\right)}.$

Der Führungsgrößenvektor W _A ist damit analog zu (13) $${\underset{‾}{w}}_A=\left[\begin{array}{c}{r}_{\mathit{LAref}}\\ {\dot{r}}_{\mathit{LAref}}\\ {\ddot{r}}_{\mathit{LAref}}\\ {\stackrel{⃛}{r}}_{\mathit{LAref}}\\ r_{\mathit{LAref}}^{\left(IV\right)}\end{array}\right]$$

Input variables of the pilot control block 91 are the desired position r _LA , the target speed ṙ _LA , the target acceleration r̈ _LA , the target jerk

and the derivative of the target jerk $r_{LA}^{\left(IV\right)},$

The reference variable vector W _A is thus analogous to (13) $${\underset{~}{w}}_A=\left[\begin{array}{c}{r}_{\mathit{LAREF}}\\ {\dot{r}}_{\mathit{LAREF}}\\ {\ddot{r}}_{\mathit{LAREF}}\\ {\stackrel{⃛}{r}}_{\mathit{LAREF}}\\ r_{\mathit{LAREF}}^{\left(IV\right)}\end{array}\right]$$

mit der Vorsteuerungsmatrix $${\underset{‾}{S}}_A=\left[\begin{array}{ccccc}{K}_{VA0}& K_{VA1}& K_{VA2}& K_{VA3}& K_{VA4}\end{array}\right].$$

In the pre-control _block 91, the components of W _{A are} weighted with the _{feedforward gains} K _VA0 to K _VA4 and their sum is given to the set input. In the event that the axis controller for the Aufrichtachse does not include a state controller block 93, then the size U _Avorst from the _{feedforward block} equal to the reference _{drive voltage} U _Aref , which is given after compensation of the hydraulic non-linearity as drive voltage U _StA on the proportional valve. The state space representation (52) thereby expands analogously to (14) $$\begin{array}{l}{\dot{\underset{~}{x}}}_A={\underset{~}{A}}_A{\underset{~}{x}}_A+{\underset{~}{B}}_A{\underset{~}{S}}_A{\underset{~}{w}}_A\\ {\underset{~}{y}}_A={\underset{~}{C}}_A{\underset{~}{x}}_A\end{array}$$

with the feedforward matrix $${\underset{~}{S}}_A=\left[\begin{array}{ccccc}{K}_{VA0}& K_{VA1}& K_{VA2}& K_{VA3}& K_{VA4}\end{array}\right],$$

When the die equation (60) is evaluated, it may be written as an algebraic equation for the _{feedforward block} , where u _{Avor is} the uncorrected target drive voltage for the proportional valve based on the idealized model. $$u_{\mathit{Avorst}}=K_{\mathit{VA}0}{r}_{\mathit{LAREF}}+K_{\mathit{VA}1}{\dot{r}}_{\mathit{LAREF}}+K_{\mathit{VA}2}{\ddot{r}}_{\mathit{LAREF}}+K_{\mathit{VA}3}{\stackrel{⃛}{r}}_{\mathit{LAREF}}+K_{\mathit{VA}4}{r}_{\mathit{LAREF}}^{\left(IV\right)}$$

The K _VA0 to K _VA4 are the _{Vorsteuerungsverstärkungen which} are calculated as a function of the current Aufrichtwinkels φ _A , the load mass m _L and the rope length l _s , so that the load without vibrations track exactly the desired trajectory.

angeben. Damit kann mit Gl. (63) die Übertragungsfunktion zwischen Ausgang Vorsteuerungsblock und Lastposition berechnet werden. Unter Berücksichtigung des Vorsteuerungsblocks 91 in Gl. (63) erhält man eine Beziehung, die nach Ausmultiplizieren die Form $$\frac{r_{\mathit{LA}}}{r_{\mathit{LAref}}}=\frac{\dots b_2\left(K_{\mathit{VA}i}\right)\cdot s^2+b_1\left(K_{\mathit{VA}i}\right)\cdot s+b_0\left(K_{\mathit{VA}i}\right)}{\dots a_2\cdot s^2+a_1\cdot s+a_0}.$$

hat. Zur Berechnung der Verstärkungen K _VAi (K _VA0 bis K _VA4 ) sind lediglich die Koeffizienten b ₄ bis b ₀ und a ₄ bis a ₀ von Interesse. Ideales Systemverhalten bezüglich Position, der Geschwindigkeit, der Beschleunigung, des Ruckes und der Ableitung des Ruckes ergibt sich genau dann, wenn die Übertragungsfunktion des Gesamtsystems aus Vorsteuerung und Übertragungsfunktion des Wippwerks den Bedingungen nach Gl. (21) für die Koeffizienten b _i und a _i genügt.The _{feedforward gains} K _VA0 to K _VA4 are calculated as follows. Regarding the controlled variable of the radial load position r _LA , the transfer function without feedforward block can be calculated as follows from the state equations (52) to (58) according to the relationship $$G\left(s\right)=\frac{r_{\mathit{LA}}\left(s\right)}{u_{\mathit{Avorst}}\left(s\right)}={\underset{~}{C}}_A{\left(s\underset{~}{I}-{\underset{~}{A}}_A\right)}^{-1}{\underset{~}{B}}_A$$

specify. Thus, with Eq. (63) the transfer function between output pilot block and load position are calculated. Considering the pilot block 91 in Eq. (63) one obtains a relationship which, after multiplying out, the form $$\frac{r_{\mathit{LA}}}{r_{\mathit{LAREF}}}=\frac{...b_2\left(K_{\mathit{VA}i}\right)\cdot {\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+b_1\left(K_{\mathit{VA}i}\right)\cdot s+b_0\left(K_{\mathit{VA}i}\right)}{...a_2\cdot {\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="imgf0005.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+a_1\cdot s+a_0},$$

Has. For the calculation of the gains K _VAi (K _VA0 to K _VA4 ) , only the coefficients b ₄ to b ₀ and a ₄ to a _{0 are} of interest. Ideal system behavior with respect to position, velocity, acceleration, jerk and jerk deflection results exactly if the transfer function of the entire system of feedforward control and transfer function of the luffing gear meets the requirements of Eq. (21) for the coefficients b _i and a _i is sufficient.

This in turn results in a linear equation system which can be resolved in analytical form according to the desired pilot control _gains K _VA0 to K _VA4 .

$$K_{\mathit{VA}1}=\frac{-c}{e{l}_A\sin \left({\varphi }_{A0}\right)}$$

$$K_{\mathit{VA}2}=-\frac{a{g}_g\left(af-b^2\right)}{e{l}_A\sin {\varphi }_{A0}\left(fa{g}_g-b^{2}g\right)}$$

$$K_{\mathit{VA}3}=\frac{-b\left(l_S{b}^{2}c-l_{S}afc\right)}{\left(e{l}_A^2\sin {\left({\varphi }_{A0}\right)}^2\left(fa{g}_g-b^{2}g\right)\right)}$$

$$K_{\mathit{VA}4}=\frac{b\left(a^2{f}^2{l}_A\sin \left({\varphi }_A\right)bg-l_S{a}^3{f}^2{g}_g-l_S{b}^{4}a{g}_g+2l_S{b}^2{a}^{2}f{g}_g-2af{b}^3{l}_A\sin \left({\varphi }_A\right)g+b^5{l}_A\sin \left({\varphi }_A\right)g\right)}{e{l}_A^2\sin {\left({\varphi }_A\right)}^2\left(-2fa{g}_g{b}^{2}g+b^4{g}^2+f^2{a}^2{g}_g^2\right)}$$

In the case of the model according to Eq. 52 to 58 results analogously to the calculation method in the slewing gear (equation 18-23) then for the pilot control amplifications $$K_{\mathit{VA}}$$$${\mathit{}0}_{}=0$$

$${\mathrm{<figure-callout\; id="1"\; label="K\; VA"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VA}}=\frac{-c}{e{l}_A\sin \left({\phi }_{A0}\right)}$$

$${\mathrm{<figure-callout\; id="2"\; label="K\; VA"\; filenames="imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VA}}=-\frac{a{G}_G\left(af-b^2\right)}{e{l}_A\sin {\phi }_{A0}\left(fa{G}_G-b^{2}G\right)}$$

$${\mathrm{<figure-callout\; id="3"\; label="K\; VA"\; filenames="imgf0001.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{VA}}=\frac{-b\left(l_S{b}^{2}c-l_{S}afc\right)}{\left(e{l}_A^2\sin {\left({\phi }_{A0}\right)}^2\left(fa{G}_G-b^{2}G\right)\right)}$$

$$K_{\mathit{VA}4}=\frac{b\left(a^2{f}^2{l}_A\sin \left({\phi }_A\right)bG-l_S{a}^3{f}^2{G}_G-l_S{b}^{4}a{G}_G+2l_S{b}^2{a}^{2}f{G}_G-2af{b}^3{l}_A\sin \left({\phi }_A\right)G+b^5{l}_A\sin \left({\phi }_A\right)G\right)}{e{l}_A^2\sin {\left({\phi }_A\right)}^2\left(-2fa{G}_G{b}^{2}G+b^4{\mathrm{<figure-callout\; id="2"\; label="G"\; filenames="imgf0005.png"\; state="\{\{state\}\}">G</figure-callout>}}^2+f^2{a}^2{G}_G^2\right)}$$

As already shown in the slewing, this has the advantage that the Vorsteuerungsver reinforcements are present depending on the model parameters. In the case of model according to Eq. 52 to 58 are system parameters J _AY, m _A, S _A, I _A, m _L, trigonometri specific terms of φ _A, l _s, b _A, K _PA, A _Zyl, V _Zyl β, _b d, d _a ,

Thus, the change of model parameters such as the righting angle φ _A , the load mass m _L and the rope length l _{s can} be taken into account immediately in the change of the pilot gains. So they can always be tracked depending on the measured values. That is, when the hoist another rope length l _s approached, so automatically change the Vorsteuerungsverstärkungen, so that as a result always the pendulum damping behavior of the feedforward control is maintained during the process of the load.

The parameters J _AY , m _A , s _A , I _A , K _PA , A _Zyl , V _Zyl , β, d _b and d _a are available from the data sheet of the technical data. Basically, as variable system parameters, the parameters / _S , m _L and φ _{A are} determined from sensor data. The attenuation parameter b _A is determined from frequency response measurements. With the pilot block, it is now possible to control the luffing mechanism of the crane so that under the idealized conditions of the dynamic model according to Eq. 52 to 58 no vibrations of the load occur when the luffing gear is moving and the load of the path generated by the path planning module follows the track exactly. However, the dynamic model is only an abstract representation of the real dynamic conditions. In addition, external disturbances (eg wind attack, etc.) can act on the crane.

Therefore, the feedforward block 91 is supported by a state controller 93. In the state controller, at least one of the measured variables φ _A , φ̇ _A , φ _Sr , φ̇ _{Sr is} weighted with a controller gain and fed back to the actuating input. There, the difference between the output value of the pre-control block 91 and the output value of the state-control block 93 is formed. If the state controller block is present, it must be taken into account when calculating the feed forward gains.

Due to the feedback, Eq. (60) too $$\begin{array}{l}{\dot{\underset{~}{x}}}_A=\left({\underset{~}{A}}_A-{\underset{~}{B}}_A{\underset{~}{K}}_A\right){\underset{~}{x}}_A+{\underset{~}{B}}_A{\underset{~}{S}}_A{\underset{~}{w}}_A\\ {\underset{~}{y}}_A={\underset{~}{C}}_A{\underset{~}{x}}_A\end{array}$$

K _A is the matrix of the controller gains of the state controller of the luffing gear analogous to the controller matrix K _D in the slewing gear. Analogous to the calculation method in the slewing gear of Gl. 25 to 28, the descriptive transfer function changes too $$G_{\mathit{AR}}\left(s\right)=\frac{r_{\mathit{LA}}\left(s\right)}{u_{\mathit{Avorst}}\left(s\right)}={\underset{~}{C}}_A{\left(s\underset{~}{I}-{\underset{~}{A}}_A+{\underset{~}{B}}_A{\underset{~}{K}}_A\right)}^{-1}{\underset{~}{B}}_A$$

In the case of the righting axis , for example, the quantities φ _A , φ̇ _A , φ _SR , φ̇ _{Sr can be} returned. The corresponding governing gains of K _A are k _1A , k _2A , k _3A , k _4A . After consideration of the pilot control 91 in Eq. 68, the pilot _gains K _VAi (K _VA0 to K _VA4 ) can be _calculated according to the condition of Eq. 21 are calculated.

This leads again to a linear system of equations analogous to Eq. 22, which can be resolved in analytical form according to the sought pilot _gains K _VA0 to K _VA4 . It should be noted, however, that the coefficients b _i and a _{i are} now also dependent on the known control gains k _1A , k _2A , k _3A , k _{4A of} the state controller , _in addition to the desired pilot _gains K _VA0 to K _VA4 .

$$K_{\mathit{VA}1}=\frac{c+e{k}_{2A}}{e{l}_A\sin \left({\varphi }_{A0}\right)}\cdot \left(-1\right)$$

$$K_{\mathit{VA}2}=\frac{-\left(afbe{k}_{3A}{l}_A\sin \left({\varphi }_A\right)-b^{2}a{g}_g{l}_A\sin \left({\varphi }_A\right)-b^{3}e{k}_{3A}{l}_A\sin \left({\varphi }_A\right)+a^{2}f{g}_g{l}_A\sin \left({\varphi }_A\right)-e{l}_S{b}^3{k}_{1A}+e{l}_{S}baf{k}_{1A}\right)}{\left(e{l}_A^2\sin {\left({\varphi }_{A0}\right)}^2\left(fa{g}_g-b^{2}g\right)\right)}$$

$$K_{\mathit{VA}3}=\frac{-b\left(afe{k}_{4A}{l}_A\sin \left({\varphi }_A\right)-b^{2}e{k}_{4A}{l}_A\sin \left({\varphi }_A\right)-l_S{b}^{2}c-l_S{b}^{2}e{k}_{2A}+l_{S}afc+l_{S}afe{k}_{2A}\right)}{\left(e{l}_A^2\sin {\left({\varphi }_{A0}\right)}^2\left(fa{g}_g-b^{2}g\right)\right)}$$

$$K_{\mathit{VA}4}=\frac{\begin{array}{l}-b(l_{S}b{a}^2{f}^{2}e{k}_{3A}{l}_A\sin \left({\varphi }_A\right)e{l}_S{k}_{1A}-f^3{a}^2{l}_A^2\sin {\left({\varphi }_A\right)}^{2}e{k}_{3A}+l_S^{2}b{a}^2{f}^{2}e{k}_{1A}+l_S{a}^3{f}^2{g}_g{l}_A\sin \left({\varphi }_A\right)\\ -2l_S{b}^{3}afe{k}_{3A}{l}_A\sin \left({\varphi }_A\right)+l_S{b}^{4}a{g}_g{l}_A\sin \left({\varphi }_A\right)+l_S{b}^{5}e{k}_{3A}{l}_A\sin \left({\varphi }_A\right)-2l_S{b}^2{a}^{2}f{g}_g{l}_A\sin \left({\varphi }_A\right)-2l_S^2{b}^{3}eafe{k}_{1A}-f{b}^4{l}_A\sin \left({\varphi }_A\right)e{l}_S{k}_{1A}+2f^2{b}^2{l}_A\sin \left({\varphi }_A\right)e{l}_S{k}_{1A}+2f^2{b}^2{l}_A^2\sin {\left({\varphi }_A\right)}^{2}ae{k}_{3A}\\ -f{b}^4{l}_A^2\sin {\left({\varphi }_A\right)}^{2}e{k}_{3A}+2af{b}^3{l}_A^2\sin {\left({\varphi }_A\right)}^{2}g-b^5{l}_A^2\sin {\left({\varphi }_A\right)}^{2}g+l_A^2{b}^{5}e{k}_{1A}-a^2{f}^2{l}_A^2\sin {\left({\varphi }_A\right)}^{2}bg\end{array}}{\left(e{l}_A^3\sin {\left({\varphi }_A\right)}^3\left(-2fa{g}_g{b}^{2}g+b^4{g}^2+f^2{a}^2{g}_g^2\right)\right)}$$

For the pilot control _gains K _vA0 to K _{VA4 of} the pilot control _block 91, taking into account the state _{regulator block} 93, analogous to Eq. 28 at the axis of rotation: $$K_{\mathit{VA}0}=\frac{k_{1A}}{l_A\sin \left({\phi }_{A0}\right)}\cdot \left(-1\right)$$

$$K_{\mathit{VA}1}=\frac{c+e{k}_{2A}}{e{l}_A\sin \left({\phi }_{A0}\right)}\cdot \left(-1\right)$$

$$K_{\mathit{VA}2}=\frac{-\left(afbe{k}_{3A}{l}_A\sin \left({\phi }_A\right)-b^{2}a{G}_G{l}_A\sin \left({\phi }_A\right)-b^{3}e{k}_{3A}{l}_A\sin \left({\phi }_A\right)+a^{2}f{G}_G{l}_A\sin \left({\phi }_A\right)-e{l}_S{b}^3{k}_{1A}+e{l}_{S}baf{k}_{1A}\right)}{\left(e{l}_A^2\sin {\left({\phi }_{A0}\right)}^2\left(fa{G}_G-b^{2}G\right)\right)}$$

$$K_{\mathit{VA}3}=\frac{-b\left(afe{k}_{4A}{l}_A\sin \left({\phi }_A\right)-b^{2}e{k}_{4A}{l}_A\sin \left({\phi }_A\right)-l_S{b}^{2}c-l_S{b}^{2}e{k}_{2A}+l_{S}afc+l_{S}afe{k}_{2A}\right)}{\left(e{l}_A^2\sin {\left({\phi }_{A0}\right)}^2\left(fa{G}_G-b^{2}G\right)\right)}$$

$$K_{\mathit{VA}4}=\frac{\begin{array}{l}-b(l_{S}b{a}^2{f}^{2}e{k}_{3A}{l}_A\sin \left({\phi }_A\right)e{l}_S{k}_{1A}-f^3{a}^2{l}_A^2\sin {\left({\phi }_A\right)}^{2}e{k}_{3A}+l_S^{2}b{a}^2{f}^{2}e{k}_{1A}+l_S{a}^3{f}^2{G}_G{l}_A\sin \left({\phi }_A\right)\\ -2l_S{b}^{3}afe{k}_{3A}{l}_A\sin \left({\phi }_A\right)+l_S{b}^{4}a{G}_G{l}_A\sin \left({\phi }_A\right)+l_S{b}^{5}e{k}_{3A}{l}_A\sin \left({\phi }_A\right)-2l_S{b}^2{a}^{2}f{G}_G{l}_A\sin \left({\phi }_A\right)-2l_S^2{b}^{3}eafe{k}_{1A}-f{b}^4{l}_A\sin \left({\phi }_A\right)e{l}_S{k}_{1A}+2f^2{b}^2{l}_A\sin \left({\phi }_A\right)e{l}_S{k}_{1A}+2f^2{b}^2{l}_A^2\sin {\left({\phi }_A\right)}^{2}ae{k}_{3A}\\ -f{b}^4{l}_A^2\sin {\left({\phi }_A\right)}^{2}e{k}_{3A}+2af{b}^3{l}_A^2\sin {\left({\phi }_A\right)}^{2}G-b^5{l}_A^2\sin {\left({\phi }_A\right)}^{2}G+l_A^2{b}^{5}e{k}_{1A}-a^2{f}^2{l}_A^2\sin {\left({\phi }_A\right)}^{2}bG\end{array}}{\left(e{l}_A^3\sin {\left({\phi }_A\right)}^3\left(-2fa{G}_G{b}^{2}G+b^4{G}^2+f^2{a}^2{G}_G^2\right)\right)}$$

With Eq. 69 now also the feedforward gains are known, which guarantee a vibration-free and accurate web of the load in the direction of rotation based on the idealized model, taking into account the state of control block 93. It should be noted that the centripetal force term in the model approach for Eq. 68 was neglected and thus is not taken into account in the feedforward control. Again, it is true that the dynamic behavior can already be improved by switching on the first derivative of the nominal function and can be progressively improved step by step by connecting the higher derivatives. Now, state governor _gains k _1A , k _2A , k _3A , k _{4A are} to be determined. This will be explained below.

The regulator feedback 93 is designed as a state controller. The controller gains are calculated analogously to the calculation method of Eq. 29 to 39 at the turning.

The components of the state vector x _A are weighted with control gains k _iA the controller matrix K _A and fed back to the control input of the route.

bestimmt. Da das Modell des Wippwerks wie das der Drehachse die Ordnung n=4 hat, ergibt sich für das charakteristische Polynom p(s) des Wippwerks analog zu Gl. 30, 31, 32 beim Drehwerk $$\begin{array}{l}\mathrm{p}\left(s\right)=s^4\\ +\frac{\left(afbe{k}_{4A}-b^{3}e{k}_{4A}+f^{2}ca-fc{b}^2+f^{2}e{k}_{2A}a-fe{k}_{2A}{b}^2\right)s^3}{{\left(af-b^2\right)}^2}\\ +\frac{\left(-fe{k}_{1A}{b}^2+afbe{k}_{3A}-b^{2}a{g}_g+a^{2}f{g}_g-b^{3}e{k}_{3A}+f^{2}e{k}_{1A}a\right)s^2}{{\left(af-b^2\right)}^2}\\ +\frac{\left(fca{g}_g+fe{k}_{2A}a{g}_g-c{b}^{2}g-e{k}_{2A}{b}^{2}g\right)s}{{\left(af-b^2\right)}^2}+\frac{fe{k}_{1A}a{g}_g-b^{2}e{k}_{1A}g}{{\left(af-b^2\right)}^2}\end{array}$$

As with the slewing gear, the controller gains are calculated by comparing the coefficients of the polynomials analogously to Eq. 35 $$\det \left(s\underset{~}{I}-{\underset{~}{A}}_A+{\underset{~}{B}}_A\cdot {\underset{~}{K}}_A\right)\equiv {\displaystyle \underset{i=1}{\overset{n}{\Pi }}\left(s-r_i\right)}$$

certainly. Since the model of the luffing gear has the order n = 4 like that of the axis of rotation, the characteristic polynomial p (s) of the luffing gear is analogous to Eq. 30, 31, 32 in the slewing gear $$\begin{array}{l}\mathrm{p}\left(s\right)=s^4\\ +\frac{\left(afbe{k}_{4A}-b^{3}e{k}_{4A}+f^{2}ca-\mathrm{<figure-callout\; id="2"\; label="f\; c\; b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">f\; </figure-callout>}c{b}^{}{}^2+f^{2}e{k}_{2A}a-fe{k}_{2A}{b}^2\right)s^3}{{\left(af-b^2\right)}^2}\\ +\frac{\left(-fe{k}_{1\mathrm{<figure-callout\; id="2"\; label="A\; b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">A\; </figure-callout>}}{b}^{}{}^2+afbe{k}_{3A}-b^{2}a{G}_G+a^{2}f{G}_G-b^{3}e{k}_{3A}+f^{2}e{k}_{1A}a\right)s^2}{{\left(af-b^2\right)}^2}\\ +\frac{\left(fca{G}_G+fe{k}_{2A}a{G}_G-c{b}^{2}G-e{k}_{2A}{b}^{2}G\right)s}{{\left(af-b^2\right)}^2}+\frac{fe{k}_{1A}a{G}_G-b^{2}e{k}_{1A}G}{{\left(af-b^2\right)}^2}\end{array}$$

The coefficient comparison with the pole specification polynomial according to Eq. 35 leads again to a linear system of equations for the control _gains k _iA .

The poles r _{i of} the Polvorgabepolynoms be chosen so that the system is stable, the control operates sufficiently fast with good damping and the manipulated variable limitation is not reached at typical occurring deviations. The r _i can be determined before commissioning in simulations according to these criteria.

The determination of the controller gains, analogous to Eq. 36, again leads to analytical mathematical expressions for the controller gains as a function of the desired poles r _i and the system parameters. As with turning, it may be favorable to vary the pole position as a function of measured values of load mass, rope length and elevation angle. In the case of model according to Eq. 52 to 58 are the system parameters J _AY m _A , s _A , l _A , m _L , l _s , b _A , K _PA , A _Zyl , V _Zyl , β, d _b , d _a . As with the slewing gear, parameter changes of the system, such as the rope length l _{S of} the load mass m _L or the righting angle φ _{A, can} now be changed in modified controller gains be taken into account. This is crucial for optimized control behavior.

Alternatively, a numerical design according to the design method of Riccati (see also O. Föllinger: control technology, 7th ed., Hüthig Verlag, Heidelberg, 1992) are performed and the controller gains in look-up tables depending on load mass, Aufrichtwinkel and Rope length are stored.

As with the slewing gear, the control can also be executed as output feedback. In doing so, individual k _iA become zero. The calculation then takes place analogously to Eq. 37 to 38 in the slewing gear.

If a state variable is not measurable, it can be reconstructed from other measurands in an observer. In this case, caused by the measurement principle disturbances can be eliminated. In Fig. 9, this module is referred to as interference observer 97. Depending on which sensor system is used for the rope angle measurement, the interference observer is suitable to configure. In the following, the measurement is again carried out with a gyroscope sensor on the load hook and the reconstruction of the cable angle and the cable angular velocity are shown. In this case occurs as an additional problem, the excitation of pitching vibrations of the load hook, which must also be eliminated by the observer or suitable filter techniques.

1. 1.) Korrektur des meßprinzipbedingten Offsets auf dem Meßsignal
2. 2.) offsetkompensierte Integration des gemessenen Winkelgeschwindigkeitssignals zum Winkelsignal
3. 3.) Eliminierung der Oberschwingungen auf dem Meßsignal, die durch Oberschwingungen des Seiles verursacht werden.
4. 4.) Eliminierung der Nickschwingungen durch geeignetes Störmodell

The gyroscope sensor measures the angular velocity in the corresponding sensitivity direction. By suitable choice of the installation location on the load hook, the sensitivity direction of the direction corresponding to the radial angle φ _Sr. The observer has again the following tasks:

1. 1.) Correction of the measurement principle-related offset on the measurement signal
2. 2.) Offset compensated integration of the measured angular velocity signal to the angle signal
3. 3.) Elimination of the harmonics on the measuring signal, which are caused by harmonics of the rope.
4. 4.) Elimination of the pitching vibrations by suitable interference model

The _offset error φ̇ _offset is again assumed to be constant in sections. $${\ddot{\phi }}_{\mathit{offset}\mathit{.}\mathit{w}}=0$$

To eliminate the pitching vibration of the hook, the resonance frequency w _{Nick 'w is} determined experimentally. The corresponding vibration differential equation corresponds to Eq. 39b $${\ddot{\phi }}_{\mathit{Nick}\mathit{.}\mathit{w}}=-w_{\mathit{Nick}\mathit{.}\mathit{<figure-callout\; id="2"\; label="w"\; filenames="imgf0005.png"\; state="\{\{state\}\}">w</figure-callout>}}^2{\phi }_{\mathit{Nick}\mathit{.}\mathit{w}}$$

wobei in Ergänzung zu Gl. 52-58 die folgenden Matrizen und Vektoren eingeführt werden.The state space representation of the submodel for the luffing gear according to Eq. 52-58 is extended by the fault model. In the present case, a complete observer is derived. The observer equation for the modified state space model is therefore: $${\widehat{\dot{\underset{~}{x}}}}_{Az}=\left({\underset{~}{A}}_{Az}-{\underset{~}{H}}_{Az}{\underset{~}{C}}_{mAz}\right)\cdot {\underset{~}{x}}_{Az}+{\underset{~}{B}}_{Az}\cdot {\underset{~}{u}}_A+{\underset{~}{H}}_{Az}{\underset{~}{y}}_{Am}$$

in addition to Eq. 52-58 the following matrices and vectors are introduced.

Eingangsmatrix: $${\underset{‾}{B}}_{Az}=\left[\begin{array}{c}0\\ \frac{fe}{af-b^2}\\ 0\\ \frac{be}{af-b^2}\\ 0\\ 0\\ 0\end{array}\right]$$

State vector: $${\underset{~}{x}}_{Dz}=\left[\begin{array}{c}{\phi }_A\\ {\dot{\phi }}_A\\ {\phi }_{Sr}\\ {\dot{\phi }}_{Sr}\\ {\dot{\phi }}_{\mathit{offset}\mathit{.}\mathit{w}}\\ {\phi }_{\mathit{Nick}\mathit{.}\mathit{w}}\\ {\dot{\phi }}_{\mathit{Nick}\mathit{.}\mathit{w}}\end{array}\right]$$

Input matrix: $${\underset{~}{B}}_{Az}=\left[\begin{array}{c}0\\ \frac{fe}{af-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}\\ 0\\ \frac{be}{af-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}\\ 0\\ 0\\ 0\end{array}\right]$$

Matrix: $${\underset{~}{A}}_{Az}=\left[\begin{array}{ccccccc}0& 1& 0& 0& 0& 0& 0\\ 0& -\frac{fc}{af-b^2}& -\frac{b{G}_G}{af-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& -\frac{bc}{af-b^2}& -\frac{a{G}_G}{af-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& -w_{\mathit{Nick}\mathit{.}\mathit{<figure-callout\; id="2"\; label="w"\; filenames="imgf0005.png"\; state="\{\{state\}\}">w</figure-callout>}}^2& 0\end{array}\right]$$

Störbeobachtermatrix: $${\underset{~}{H}}_{Az}=\left[\begin{array}{lll}{H}_{11A}\hfill & H_{12A}\hfill & H_{13A}\hfill \\ H_{21A}\hfill & H_{22A}\hfill & H_{23A}\hfill \\ H_{31A}\hfill & H_{32A}\hfill & H_{33A}\hfill \\ H_{41A}\hfill & H_{42A}\hfill & H_{43A}\hfill \\ H_{51A}\hfill & H_{52A}\hfill & H_{53A}\hfill \\ H_{61A}\hfill & H_{62A}\hfill & H_{63A}\hfill \\ H_{71A}\hfill & H_{72A}\hfill & H_{73A}\hfill \end{array}\right]$$

Observers output matrix: $${\underset{~}{C}}_{mAz}=\left[\begin{array}{lllllll}1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 1\hfill \end{array}\right]$$

Output vector of the measured quantities: $${\underset{~}{y}}_{mA}=\left[\begin{array}{c}{\phi }_A\\ {\dot{\phi }}_A\\ {\dot{\phi }}_{Srm}\end{array}\right]$$

geschätzt und eliminiert wird und das dann geschätzte Winkelsignal

für die Zustandsregelung verwendet wird.Alternatively, a reduced model approach as in the slewing is possible again. In addition, an improved offset compensation can be achieved in that in a second observer, the remaining offset on the angle signal φ _Sr , by the additional disturbance variable

estimated and eliminated and the then estimated angle signal

is used for the state control.

auf den Zustandregler zurückgeführt. Damit erhält man am Ausgang des Zustandsreglerblocks 93 bei Rückführung von ϕ _A ,ϕ̇ _A ϕ̂ _Sr ,

bzw.

im Falle des zweistufigen Beobachters (siehe auch Fig. 7a) dann $$u_{\mathit{Arück}}=k_{1A}{\varphi }_A+k_{2A}{\dot{\varphi }}_A+k_{3A}{\widehat{\varphi }}_{Sr}+k_{4A}{\dot{\widehat{\varphi }}}_{Sr}$$

The determination of the observer _gains h _ijD is carried out either by transformation into observational _{normal form} or by the Riccati or _{Polvorgabe design method} . It is essential that in the observer also variable rope length, righting angle and load mass are taken into account by adapting the observer differential equation and the observer reinforcements. From the estimated state vector x _Az , the estimated values φ _Sr ,

returned to the state controller. This results in the output of the state regulator block 93 when φ _A , φ̇ _A φ _{Sr is returned} ,

respectively.

in the case of the two-stage observer (see also Fig. 7a) then $$u_{\mathit{Arück}}=k_{1A}{\phi }_A+k_{2A}{\dot{\phi }}_A+k_{3A}{\widehat{\phi }}_{Sr}+k_{4A}{\dot{\widehat{\phi }}}_{Sr}$$

The Sollansteuerspannung the proportional valve for the rocking axis is taking into account the feedforward control 91 analogous to Eq. 40 then $$u_{\mathit{Aref}}=u_{\mathit{Avorst}}-u_{\mathit{Arück}}$$

As with the slewing gear, optional non-linearities of the hydraulics can be compensated in block 95 of the hydraulic compensation, resulting in a linear system behavior with respect to the system input. In the luffing gear, correction factors for the drive voltage of the righting angle φ _A and for the gain factor K _PA and the relevant cylinder diameter A _Zyl can be provided in addition to the valve output and the hysteresis. Thus, a direction-dependent structure changeover of the axis controller can be avoided.

In order to calculate the necessary compensation function, the static characteristic between the drive _voltage U _{StD of} the proportional valve and the resulting flow Q _FD is experimentally recorded. The characteristic can be described by a mathematical function. $$Q_{FA}=f\left(u_{StA}\right)$$

Regarding the system input linearity is required. That is, the proportional valve and the block of hydraulic compensation should, according to Eq. 47 summarized have the following transmission behavior. $$Q_{FA}=K_{PA}{u}_{StA}$$

so ist Bedingung (76) genau dann erfüllt, wenn als statische Kompensationskennlinie $$h\left(u_{\mathit{Aref}}\right)=f^{-1}\left(K_{PA}{u}_{\mathit{Aref}}\right)$$

gewählt wird.Compensation block 95 has the static characteristic $$u_{StA}=H\left(u_{\mathit{Aref}}\right)$$

Thus, condition (76) is fulfilled if and only if as a static compensation characteristic $$H\left(u_{\mathit{Aref}}\right)=f^{-1}\left(K_{PA}{u}_{\mathit{Aref}}\right)$$

is selected.

This explains the individual components of the axis controller for the luffing gear. As a result, the combination of the path planning module and the Wippwerk axis controller fulfills the requirement of a vibration-free and pin-accurate movement of the load when erecting and tilting the boom.

ausgedrückt. Die entstehenden Pendelbewegungen werden durch die Zustandsregler von Drehwerk und Wippwerk gedämpft. Eine Verbesserung der Bahngenauigkeit und Kompensation der Schwingungsneigung bezüglich der radialen Schwingungen beim Drehen kann durch eine geeignete Vorsteuerung in einem Block zur Kompensation der Zentripetalkräfte erreicht werden. Hierzu wird bei einer Drehbewegung das Wippwerk mit einer Ausgleichsbewegung beaufschlagt, die den Zentripetaleffekt kompensiert.Not considered so far was that upon actuation of the slewing by the centripetal forces the load (as in a chain carousel) deflected in the radial direction becomes. At fast deceleration and acceleration, this effect causes spherical pendulum movements of the load. In the differential equations Eq. 4 and 46 this is determined by the terms depending on ${\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2$

expressed. The resulting oscillations are dampened by the state controllers of Drehwerk and Wippwerk. An improvement in the path accuracy and compensation of the tendency to oscillate with respect to the radial vibrations during rotation can be achieved by a suitable feedforward control in a block for compensating the centripetal forces. For this purpose, during a rotary movement, the luffing mechanism is acted upon by a compensation movement which compensates for the centripetal effect.

eine Auslenkung des Pendels um den Winkel ϕ _Sr . Die Gleichgewichtsbedingung für das Kräftegleichgewicht in diesem Fall lautet: $$m_L\cdot \left(r_{LA}+\Delta r_{LA}\right){\dot{\varphi }}_D^2=m_L\cdot g\cdot \tan {\varphi }_{sr}$$

In Fig. 12 this effect is shown. The sole rotation of the load causes the centripetal force $$F_z=m_L\cdot r_{LA}{\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2$$

a deflection of the pendulum at the angle φ _Sr. The equilibrium condition for the equilibrium of forces in this case is: $$m_L\cdot \left(r_{LA}+\Delta r_{LA}\right){\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2=m_L\cdot G\cdot \tan {\phi }_{sr}$$

$$\Delta z=l_s\cdot \left(1-\cos {\varphi }_{sr}\right).$$

The resulting path deviation in the radial direction Δr _LA and in the direction of the hoist movement Δz can then be described as a function of the radial rope angle φ _Sr $$\Delta r_{LA}=l_s\cdot \sin {\phi }_{sr}$$

$$\Delta z=l_s\cdot \left(1-\cos {\phi }_{sr}\right),$$

eingestellt. Obige Gleichungen werden um ϕ _Sr =0 linearisiert. Damit wird tan ϕ _Sr ≈ sin ϕ _Sr ≈ ϕ _Sr . Die sich dann ergebende radiale Abweichung ist $$\frac{R_1{\dot{\varphi }}_{\mathit{Dref}}^2}{g}\cdot l_s=\Delta r_{LA}$$

The module 150 for compensating the centripetal force (FIG. 3) now has the task of achieving this by simultaneously compensating movement of the luffing mechanism and the lifting mechanism Compensate for deviation as a function of the rotational movement. Instead of the actual rotational speed of the tower φ̇ _D , the target rotational speed of the load φ̇ _Dref generated in the path _{planning module is} used. Depending on the input for the reference variable, the desired position to be set in the radial direction or the angular position of the cantilever to be set is then calculated from the equations (78 ac), so that the original radius is traversed by the load position. About the rocking angle φ _A1 , the resulting radius of rotation of the load of $${\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">R</figure-callout>}}_1=\cos {\phi }_{A1}\cdot l_A$$

set. The above equations are linearized by φ _Sr = 0. This will tan φ _Sr ≈ sin φ _Sr ≈ φ _Sr. The resulting radial deviation is then $$\frac{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">R</figure-callout>}}_1{\dot{\phi }}_{\mathit{Dref}}^2}{G}\cdot l_s=\Delta r_{LA}$$

The radius of rotation maintained by the load is then: $$R_{\mathit{ges}}=R_1\left[1+\frac{{\dot{\phi }}_{\mathit{Dref}}^2}{G}\cdot l_s\right]$$

Now the demand is made that a radius r _{LAkomp should be} given, so that, taking into account the centripetal _deviation , r _{LA is} observed. $$r_{\mathit{LAkomp}}=\frac{1}{1+\frac{{\dot{\phi }}_{\mathit{Dref}}^2}{G}\cdot l_s}{r}_{LA}$$

If the angular position is used as the reference variable input for the luffing mechanism, then, because of Eq. 78e $$\cos {\phi }_{\mathit{Akomp}}=\frac{1}{1+\frac{{\phi }_{\mathit{Dref}}^2}{G}\cdot l_s}\cos {\phi }_{\mathit{Aref}}$$

In order to keep the lifting height of the load constant, the lifting of the load due to the centripetal force effect can be optionally compensated by synchronous control of the hoist. With Eq. (78d) is obtained from the equilibrium condition $$\Delta z=l_s\cdot \left(1-\cos \left(\arctan \left(\frac{R{\dot{\phi }}_D^2}{G}\right)\right)\right)$$

The values for compensating the centripetal force that follow from the calculation of (78i) and (78j) are additionally switched to the reference variable inputs of the axis controllers.

zugelassen wird. Damit die beabsichtigte Seilaustenkung von der unterlagerten Regelung nicht ausgeglichen wird, wird diese mit k _3A gewichtet mit auf den Stelleingang gegeben.In addition, a then permissible Seilaustenkung for φ _{Sr must be} introduced. By raising the boom, the load passes over the target radius r _LAref if _and only if the boom is set to a desired radius of r _LArefkomp and at the same time a cable link of $${\phi }_{\mathit{Srzul}}=\frac{{\dot{\phi }}_{\mathit{Dref}}^2\cdot r_{\mathit{LArefkomp}}}{G-l_S{\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2}$$

is allowed. So that the intended Seilaustenkung is not compensated by the subordinate control, this is weighted with k _3A with given to the control input.

The above relationships are based on a stationarity approach that is applicable in the case of low acceleration in turning. If very high accelerations occur during turning, a dynamic model approach is selected for the taxing compensation.

The pendulum motion of the load can be described taking into account the centrifugal force by the following differential equation, the influence on the pendulum motion by φ̈ _A deliberately was not considered here, because one aims exclusively at the sole effect of centrifugal force. $$m_L{l}_s^2\cdot {\ddot{\phi }}_{Srz}=F_Z\cdot l_S\cdot \cos {\phi }_{Srz}-mG\cdot l_S\cdot \sin {\phi }_{Srz}$$

erhält man $${\ddot{\varphi }}_{Srz}=\left(\sin {\varphi }_{Srz}+\frac{l_A}{l_S}\cos {\varphi }_A\right){\dot{\varphi }}_D^2\cdot \cos {\varphi }_{Srz}-\frac{g}{l_S}\sin {\varphi }_{Srz}$$

ϕ _Srz ist der durch die Zentrifugalkraft bedingte Seilwinkel. Nach Linearisierung um ϕ _Srz = 0 und Vernachlässigung des Terms ${\varphi }_{Sr}\cdot {\dot{\varphi }}_D^2$

gegenüber $\frac{l_A}{l_S}\cos {\varphi }_A\cdot {\dot{\varphi }}_D^2$

erhält man $${\ddot{\varphi }}_{Srz}=\frac{l_A}{l_S}\cos {\varphi }_A{\dot{\varphi }}_D^2-\frac{g}{l_S}{\varphi }_{Srz}$$

Gl. 78jd ist eine Differentialgleichung für eine ungedämpfte Schwingung, die durch außen über $\frac{l_A}{l_S}\cos {\varphi }_A\cdot {\dot{\varphi }}_D^2$

angeregt wird. Diese hat die Eigenfrequenz $\sqrt{\frac{g}{l_S}}.$

Für die Radiuskompensation ist man nur an dem Trend der Abweichung interessiert, da die Schwingung von dem unterlagerten Wippwerksregler gedämpft wird. Der Wippwerksregler ist so eingestellt, dass er mit einem Dämpfungskoeffizient d _R in der obigen Differentialgleichung gleichgesetzt werden kann. Dieser wird in Gl. 78jd eingefügt. Ergebnis ist die folgende Übertragungsfunktion im Frequenzbereich: $${\varphi }_{Srz}\left(s\right)\frac{\frac{l_A}{l_S}\cos {\varphi }_A}{s^2+2\sqrt{\frac{g}{l_S}}\cdot d_R\cdot s+\frac{g}{l_S}}{\dot{\varphi }}_D^2\left(s\right)$$

oder $${\ddot{\varphi }}_{Srz}=-\frac{g}{l_S}{\varphi }_{Srz}-2\sqrt{\frac{g}{l_S}}\cdot d_R\cdot {\dot{\varphi }}_{Srz}+\frac{l_A}{l_S}\cos {\varphi }_A\cdot {\dot{\varphi }}_D^2$$

im Zeitbereich. Diese Differentialgleichung kann nun mit der Messgröße ${\dot{\varphi }}_D^2$

oder der Sollgröße ${\dot{\varphi }}_{\mathit{Dref}}^2$

als Eingang während des Kranbetriebs mitsimuliert werden. Sie liefert die zu erwartenden Seilwinkel aufgrund der Zentrifugalkraft, wobei die Messgrößen der Seillänge l _S und Aufrichtwinkel ϕ _A stets nachgeführt werden.With $$F_Z=\left(\sin {\phi }_{Srz}{l}_S+l_A\cos {\phi }_A\right)m_L\cdot {\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2$$

you get $${\ddot{\phi }}_{Srz}=\left(\sin {\phi }_{Srz}+\frac{l_A}{l_S}\cos {\phi }_A\right){\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2\cdot \cos {\phi }_{Srz}-\frac{G}{l_S}\sin {\phi }_{Srz}$$

φ _Srz is the rope angle caused by the centrifugal force. After linearization by φ _Srz = 0 and neglecting the term ${\phi }_{Sr}\cdot {\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2$

across from $\frac{l_A}{l_S}\cos {\phi }_A\cdot {\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2$

you get $${\ddot{\phi }}_{Srz}=\frac{l_A}{l_S}\cos {\phi }_A{\dot{\phi }}_D^2-\frac{G}{l_S}{\phi }_{Srz}$$

Eq. 78jd is a differential equation for an undamped vibration passing through the outside $\frac{l_A}{l_S}\cos {\phi }_A\cdot {\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2$

is stimulated. This has the natural frequency $\sqrt{\frac{G}{l_S}},$

For the radius compensation, one is only interested in the trend of the deviation, since the vibration is damped by the subordinate luffing mechanism controller. The luffing gear controller is set so that it can be equated with a damping coefficient d _R in the above differential equation. This is reflected in Eq. 78jd inserted. The result is the following transfer function in the frequency domain: $${\phi }_{Srz}\left(s\right)\frac{\frac{l_A}{l_S}\cos {\phi }_A}{s^2+2\sqrt{\frac{G}{l_S}}\cdot d_R\cdot s+\frac{G}{l_S}}{\dot{\phi }}_D^2\left(s\right)$$

or $${\ddot{\phi }}_{Srz}=-\frac{G}{l_S}{\phi }_{Srz}-2\sqrt{\frac{G}{l_S}}\cdot d_R\cdot {\dot{\phi }}_{Srz}+\frac{l_A}{l_S}\cos {\phi }_A\cdot {\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2$$

in the time domain. This differential equation can now be used with the measured variable ${\dot{\phi }}_{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">D</figure-callout>}}^2$

or the target size ${\dot{\phi }}_{\mathit{Dref}}^2$

be simulated as input during crane operation. It delivers the expected rope angles due to the centrifugal force, with the parameters of the rope length l _S and Aufrichtwinkel φ _{A are} always tracked.

und damit $$r_{\mathit{LAkomp}}=r_{\mathit{LAref}}-l_S\mathit{sin}{\varphi }_{\mathit{Srz}}.$$

The resulting radius deviation Δr _LA is then $$\Delta r_{LA}=l_S\sin {\phi }_{Srz}$$

and thus $$r_{\mathit{LAkomp}}=r_{\mathit{LAREF}}-l_S\mathit{sin}{\phi }_{\mathit{Srz}},$$

The higher derivatives are formed accordingly. The simulated angle φ _Srz , caused by the centrifugal force, is weighted compensated by k _3A to the control input.

$$a_3{\ddot{\varphi }}_D+a_4{\ddot{\varphi }}_{St}+a_5{\varphi }_{St}=a_6{\dot{\varphi }}_D^2{\varphi }_{St}$$

$$b_0{\ddot{\varphi }}_{\mathit{A}}+b_1{\ddot{\varphi }}_{Sr}+b_2{\dot{\varphi }}_A=M_A$$

$$b_4{\ddot{\varphi }}_A+b_5{\ddot{\varphi }}_{Sr}+b_6{\varphi }_{Sr}=b_7{\dot{\varphi }}_D^2{\varphi }_{Sr}$$

Further, in order to deal with the problem of coupling the differential equations 4 and 46, the method of flatness-based control and regulation based on the non-linear system equations is applicable. The structure of Eq. 4 and 46 can be written as $$a_0{\ddot{\phi }}_D+a_1{\ddot{\phi }}_{St}+a_2{\dot{\phi }}_D=M_D$$

$$a_3{\ddot{\phi }}_D+a_4{\ddot{\phi }}_{St}+a_5{\phi }_{St}=a_6{\dot{\mathrm{<figure-callout\; id="2"\; label="\phi \; ̇\; D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">\phi \; </figure-callout>}}}_D^{}{\phi }_{\mathrm{<figure-callout\; id="0"\; label="S\; t\; b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">S\; </figure-callout>}t}$$

$$b_{}$$$${}_0{\ddot{\phi }}_{\mathit{A}}+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_1{\ddot{\phi }}_{Sr}+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_2{\dot{\phi }}_A=M_A$$

$$b_4{\ddot{\phi }}_A+{\mathrm{<figure-callout\; id="5"\; label="b"\; filenames="imgf0001.png"\; state="\{\{state\}\}">b</figure-callout>}}_5{\ddot{\phi }}_{Sr}+b_6{\phi }_{Sr}={\mathrm{<figure-callout\; id="7"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_7{\dot{\mathrm{<figure-callout\; id="2"\; label="\phi \; ̇\; D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">\phi \; </figure-callout>}}}_D^{}{\phi }_{Sr}$$

$${\ddot{\varphi }}_{Sr}=\frac{1}{b_1}\left(M_A-b_0{\ddot{\varphi }}_A-b_2{\dot{\varphi }}_A\right)$$

Now Eq. 78k and 78m after φ̈ _St and φ̈ _Sr , respectively. This gives you $${\ddot{\phi }}_{St}=\frac{1}{a_1}\left(M_D-a_0{\ddot{\phi }}_D-a_2{\dot{\phi }}_D\right)$$

$${\ddot{\phi }}_{Sr}=\frac{1}{b_1}\left(M_A-{\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_0{\ddot{\phi }}_A-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_2{\dot{\phi }}_A\right)$$

$$M_A=\frac{b_1}{b_5}\left(b_7{\dot{\varphi }}_A^2{\varphi }_{Sr}-b_6{\varphi }_{Sr}-b_4{\ddot{\varphi }}_A\right)+b_0{\ddot{\varphi }}_A+b_2{\dot{\varphi }}_A$$

In Eq. 781 and 78n, respectively, becomes Eq. 78o or 78p used. Then these equations can be transformed according to the moment to be applied. $$M_D=\frac{a_1}{a_4}\left(a_6{\dot{\mathrm{<figure-callout\; id="2"\; label="\phi \; ̇\; D"\; filenames="imgf0005.png"\; state="\{\{state\}\}">\phi \; </figure-callout>}}}_D^{}{\phi }_{St}-a_5{\phi }_{St}-a_3{\ddot{\phi }}_D\right)+a_0{\ddot{\phi }}_D+a_2{\dot{\phi }}_D$$

$$M_A=\frac{b_1}{b_5}\left({\mathrm{<figure-callout\; id="7"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_7{\dot{\phi }}_A^2{\phi }_{Sr}-b_6{\phi }_{Sr}-b_4{\ddot{\phi }}_A\right)+{\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_0{\ddot{\phi }}_A+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_2{\dot{\phi }}_A$$

$$M_A=\frac{b_1}{b_5}\left(b_7{\dot{\varphi }}_{\mathit{Dref}}^2{\varphi }_{Sr}-b_6{\varphi }_{Sr}-b_4{v}_2\right)+b_0{v}_2+b_2{\dot{\varphi }}_{\mathit{Aref}}$$

mit $$\begin{array}{l}{v}_1={\ddot{\varphi }}_D-P_{10}\left({\varphi }_D-{\varphi }_{\mathit{Dref}}\right)-P_{11}\left({\dot{\varphi }}_D-{\dot{\varphi }}_{\mathit{Dref}}\right)\\ v_2={\ddot{\varphi }}_A-P_{20}\left({\varphi }_A-{\varphi }_{\mathit{Aref}}\right)-P_{21}\left({\dot{\varphi }}_A-{\dot{\varphi }}_{\mathit{Aref}}\right)\end{array}$$

With Eq. 78q and 78r are now given correlations for the setpoint moments as a function of the state variables. If instead of the rotation angle or Aufrichtwinkels now the target rotation angle or Sollauftichtwinkel in Eq. 78q and 78r and the measured actual rope angle φ _St and φ _Sr used so a linear slave controller can be defined (see also A.lsidori: Nonlinear Control Systems 2nd Edition, Springer Verlag Berlin, Rothfuß R. et al.: Flatness: Ein new access to control and regulation, automation technology 11/97 p. 517-525). The presentation is too $$M_D=\frac{a_1}{a_4}\left(a_6{\dot{\phi }}_{\mathit{Dref}}^2{\phi }_{St}-a_5{\phi }_{St}-a_3{v}_1\right)+a_0{\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">v</figure-callout>}}_1+a_2{\dot{\phi }}_{\mathit{Dref}}$$

$$M_A=\frac{b_1}{b_5}\left({\mathrm{<figure-callout\; id="7"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_7{\dot{\phi }}_{\mathit{Dref}}^2{\phi }_{Sr}-b_6{\phi }_{Sr}-b_4{v}_2\right)+b_0{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="imgf0005.png"\; state="\{\{state\}\}">v</figure-callout>}}_2+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}_2{\dot{\phi }}_{\mathit{Aref}}$$

With $$\begin{array}{l}{\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="imgf0002.png,imgf0005.png"\; state="\{\{state\}\}">v</figure-callout>}}_1={\ddot{\phi }}_D-P_{10}\left({\phi }_D-{\phi }_{\mathit{Dref}}\right)-P_{11}\left({\dot{\phi }}_D-{\dot{\phi }}_{\mathit{Dref}}\right)\\ {\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="imgf0005.png"\; state="\{\{state\}\}">v</figure-callout>}}_2={\ddot{\phi }}_A-P_{20}\left({\phi }_A-{\phi }_{\mathit{Aref}}\right)-P_{21}\left({\dot{\phi }}_A-{\dot{\phi }}_{\mathit{Aref}}\right)\end{array}$$

The P ₁₀ , P ₁₁ , P ₂₀ , P ₂₁ are to be chosen so that the scheme works with high dynamics with sufficient damping.

Another way to treat the non-linearity in addition to the two methods shown is the method of exact linearization and decoupling of the system. In the present case, however, this succeeds only incompletely, since the system does not have the full order of difference. Nevertheless, a regulator based on this method can be used.

Lastly, the structure of the axis controller for the hoist will now be explained. The structure of the axis controller is shown in FIG. In contrast to the Achsreglem Drehwerk 43 and Wippwerk 45, the axis controller for the hoist 47, since this axis shows little tendency to oscillate, equipped with a conventional cascade control with an outer loop for the position and an inner for the speed.

From the path planning module 39 or 41, only the time functions setpoint position of the hoist l _ref and the setpoint speed i _{ref are} required to control the axis controller. These are weighted in a pre-control block 121 so as to give a fast responding and steady state system behavior with respect to position. Since the setpoint actual comparison between the reference variable l _ref and the measured variable l _S takes place directly behind the pilot control block, stationarity with respect to the position is then met when the pilot control gain for the position is 1. The pre-control gain for the target speed i _ref is to be determined so subjectively gives a quick but well damped response in the hand lever operation. The position control loop controller 123 may be implemented as a proportional controller (P controller). The control gain is to be determined according to the criteria stability and sufficient damping of the closed loop. Output of regulator 123 is the ideal drive voltage of the proportional valve. As with the axis controller slewing gear 43 and luffing gear 45, the nonlinearities of the hydraulics are compensated in a compensation block 125. The calculation is the same as for turning (equations 42-44). The output variable is the corrected drive voltage of the proportional _valve u _StL . Internal control loop for the speed is the subordinate flow control of the hydraulic circuit.

The last direction of movement is the turning of the load on the load hook itself by the load pivot mechanism. A corresponding description of this regulation results from the German patent application DE 100 29 579 from 15.06.2000, on the Content is expressly referred to here. The rotation of the load is made via the arranged between a hanging on the rope bottom block and a load receiving device load swing mechanism. In this case occurring torsional vibrations are suppressed. Thus, in most cases, just not rotationally symmetric load can be accurately recorded, moved by a corresponding bottleneck and discontinued. Of course, this direction of movement is integrated in the path planning module, as shown for example with reference to the overview in Fig. 3. In a particularly advantageous manner, the load can be moved here after picking during transport through the air in the corresponding desired pivot position by means of the load pivoting mechanism, in which case the individual pumps and motors are controlled synchronously. Optionally, a mode for a rotation-independent orientation can be selected.

In summary, in the embodiment shown here results in a mobile harbor crane, the path control allows a web-accurate method of the load with all axes and actively suppresses vibrations and oscillations.

In particular, for the semi-automatic operation of a crane or excavator, it may be sufficient in the context of the present invention, if only turns on the position and speed function in the feedforward. This leads to a subjectively calmer behavior. It is therefore not necessary to map all the values of the dynamic model up to the derivative of the jerk and to generate from these control variables, which are to be connected for active damping of the load pendulum motion.

Claims (12)

1. Crane or excavator (1, 11, 5) for swinging a load (3) hanging on a load cable, having a rotating mechanism (1) for rotating the crane or excavator, a luffing mechanism (7) for elevating or depressing a jib (5), and a lifting mechanism for raising or lowering the load (3) hanging on the cable, having a computer-based control system (31) for damping the oscillations of the load, which has a path planning module (39), a centripetal force compensating device (150) and at least one axis controller (43) for the rotating mechanism, an axis controller (45) for the luffing mechanism and an axis controller (47) for the lifting mechanism, characterized in that the angle of oscillation and the speed of oscillation of the load (ϕ _Sr ,ϕ _St ,ϕ̇ _Sr ,ϕ̇ _St ) is calculated from gyroscope signals from at least one gyroscope.
2. Crane or excavator according to Claim 1, characterized in that, in addition, a load swivelling mechanism is arranged between a bottom block of the load cable and a load-lifting means, and in that the control system additionally has an axis controller for damping the oscillation of the load, which is connected to the path planning module.
3. Crane or excavator according to Claim 1 or Claim 2, characterized in that, first of all, the path of the load in the working space can be generated in the path planning module and can be passed on to the respective axis controller in the form of a time function for the load position, speed, acceleration of the jerk and, if appropriate, the derivative of the jerk.
4. Crane or excavator according to Claim 3, characterized in that each axis controller has a feed-forward control unit in which, on the basis of a dynamic model based on differential equations, the dynamic behaviour of the mechanical and hydraulic system of the crane or excavator can be depicted, so that it is possible to generate control variables which can be used for the active damping of the oscillating movement of the load.
5. Crane or excavator according to Claim 4, characterized in that the control system additionally has a state control unit, in which real deviations from the idealized dynamic model of the feed-forward control system are registered.
6. Crane or excavator according to Claim 5, characterized in that, in the state control unit, at least one of the measured variables comprising the angle of swing in the radial or tangential direction (ϕ _Sr , or ϕ _St ), angle of elevation (ϕ _A ), angle of rotation (ϕ _D ), cable length (l_s), jib deflection in the horizontal and vertical direction, and also their derivatives and the mass of the load can be fed back.
7. Crane or excavator according to Claim 1, characterized in that the disturbances in the measured signal from the gyroscope can be estimated and compensated for in the disturbance observer.
8. Crane or excavator according to one of Claims 2 to 7, characterized in that the axis controller for the lifting mechanism has a cascade control system having an outer control loop for the position and an inner control loop for the speed.
9. Crane or excavator according to one of Claims 1 to 8, characterized in that, in the path planning module, it is possible to generate the path of the load for semi-automatic operation proportional to the deflection of a hand lever and, in fully automatic operation, corresponding target coordinates.
10. Crane or excavator according to Claim 9, characterized in that the path planning module for semi-automatic operation substantially comprises a elope limiter of second order for normal operation and a slope limiter of second order for a rapid stop.
11. Crane or excavator according to one of Claims 4 to 10, characterized in that only the position function and the speed function can be used as control variables for the active damping of the oscillating movement of the load.
12. Crane or excavator according to Claim 11, characterized in that, in addition, the acceleration function and the jerk function can in each case also be used in the feed-forward control.

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