EP1628902B1 - Crane or excavator for handling a cable-suspended load provided with optimised motion guidance - Google Patents

Description

The invention relates to a crane or excavator for handling a load suspended on a load rope according to the preamble of claim 1.

In particular, the invention deals with the generation of reference variables as control functions in cranes or excavators, which allows a movement of the suspended on a rope load 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 and a slewing gear 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 turning over a load suspended from a rope by means of such a crane or excavator, pendulum movements of the load occur, which are due to the movement the crane or excavator itself are due. Efforts have already been made in the past to reduce or suppress pendulum vibrations in load cranes.

The WO 02/32805 A1 describes a crane or excavator for transferring from a load suspended on a load rope with a computerized control for damping the load oscillation, which has a trajectory planning module, a Zentripetalkraftkompensationseinrichtung and at least one axis controller for the slewing, an axle controller for the luffing and an axis controller for the hoist. Only the kinematic limits of the system are taken into account in the path planning module. The dynamic behavior is only considered in the design of the control.

The DE-A-100 64 182 describes a crane for transferring from a load suspended on a load rope according to the preamble of claim 1.

The object of the invention is to further optimize the motion control of the load hanging on the load rope.

To solve this problem, a generic crane or excavator on a control in which the reference variables for the control are generated so that there is an optimized movement with minimized pendulum swings. In this case, the wandered path of the swinging load can be predicted and, based on this, a collision avoidance strategy can be realized.

Advantageous embodiments of the invention will become apparent from the subsequent claims to the main claim.

It is particularly advantageous that model-based optimal control trajectories are calculated and updated online in the path control of the present invention. In this case, the model-based optimal control trajectories can be constructed based on a model linearized around reference trajectories. Alternatively, the model-based optimal control trajectories may be based on a non-linear model approach.

The model-based optimal control jitters can be determined by returning all state variables.

Alternatively, the model-based optimal control trajectories can be determined by returning at least one measured variable and estimating the remaining state variables.

As an alternative, the model-based optimal control trajectories can be determined by returning at least one measured variable and tracking the remaining state variables by model-based feedforward control.

The path control can advantageously be carried out as fully automatic or as semi-automatic.

Thus, in connection with a control for load oscillation damping, an optimized movement behavior with reduced residual oscillation and lower pendulum deflection while driving. Without the control for load-swing damping, the required sensor technology on the crane can be reduced. It can be a fully automatic operation in which start and finish point are established as well as a hand lever operation, which is referred to below as semi-automatic operation.

In the present invention, the desired functions are in contrast to WO 02/32805 A1 now generated in such a way that the dynamic behavior of the crane is taken into account even before switching to the control. Thus, the scheme has only the task of compensating for model deviations and disturbances, resulting in improved driving behavior. In addition, if the positional accuracy and the tolerable residual oscillation allow it, the control can be completely eliminated and the crane can be operated with this optimized control function. However, the behavior will be slightly less favorable than when operating with the control because the model does not match the actual conditions in every detail.

The method provides for two modes of operation. The hand lever operation, where the operator sets a target speed of the load by the hand lever deflection, and the fully automatic operation where the start and end point are specified.

In addition, the optimized control function calculation can be operated alone or in conjunction with a control for load swing damping.

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.

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 mit Modul zur optimierten Bewegungsführung als Steuerfunktion des Kranes

Fig. 3:

Struktur der Bahnsteuerung mit Modul zur optimierten Bewegungsführung mit Regelung zur Lastpendeldämpfung

Fig. 4:

Struktur der Bahnsteuerung mit Modul zur optimierten Bewegungsführung als Steuerfunktion ohne Regelung zur Lastpendeldämpfung (ggf. mit unterlagerten Positionsreglern für die Antriebe)

Fig. 5:

Mechanischer Aufbau des Drehwerks und Definition von Modellvariablen

Fig. 6:

Mechanischer Aufbau des Wippwerks und Definition von Modellvariablen

Fig. 7:

Aufrichtkinematik des Wippwerks

Fig.8:

Ablaufdiagramm für die Berechnung der optimierten Steuergröße im vollautomatischen Betrieb

Fig. 9:

Ablaufdiagramm für die Berechnung der optimierten Steuergröße im halbautomatischen Betrieb

Fig. 10:

Beispielhafte Führungsgrößengenerierung im vollautomatischen Betrieb

Fig. 11:

Beispielhafte Zeitverläufe von Steuergrößen und Regelgrößen im Handhebelbetrieb

Show it:

Fig. 1:

Basic mechanical structure of a mobile harbor crane

Fig. 2:

Interaction of hydraulic control and path control with module for optimized motion control as a control function of the crane

3:

Structure of the path control with module for optimized motion control with control for load-swing damping

4:

Structure of the path control with module for optimized motion control as a control function without control for load oscillation damping (possibly with subordinate position controllers for the drives)

Fig. 5:

Mechanical structure of the slewing gear and definition of model variables

Fig. 6:

Mechanical structure of the luffing gear and definition of model variables

Fig. 7:

Rigging kinematics of the luffing mechanism

Figure 8:

Flowchart for the calculation of the optimized control variable in fully automatic operation

Fig. 9:

Flow chart for the calculation of the optimized control variable in semi-automatic mode

Fig. 10:

Exemplary reference variable generation in fully automatic operation

Fig. 11:

Exemplary time profiles of control variables and controlled variables in hand lever operation

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. The rope length I _{S can be} varied with the hoist. 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 with module for optimized movement guidance. 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 proportional valves are controlled by the u _SLD , u _StA , u _StL , u _StR signals. 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 _{SrR be implemented} at 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.

The structure of the web control is now shown in FIGS. 3 and 4. FIG. 3 shows the path control with the module for optimized motion guidance Control for load oscillation damping and Figure 4 the path control with the module for optimized motion control without control for load oscillation damping. This Lastpendeldämpfung can, for example, according to the script PCT / EP01 / 12080 have been designed. Therefore, the content disclosed therein is fully included in this document.

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 or a path planner, which takes into account the kinematic constraints of the system, but are calculated in the path control 31, that when moving the crane No or little pendulum movements of the load occur and the load follows the desired path in the working space. That When calculating the optimized control variable, not only the kinematic description but also the dynamic description of the system is considered.

Input values of the module 37 is a set point matrix 35 for the position and orientation of the load, which in the simplest case consists of start and end point. The position is usually described on rotating cranes by polar coordinates ( φ _LD , r _LA , l ) . Since this does not completely describe the position of the extended body (eg of a container) in space, another angle size can be added (angle of rotation γ _L about the vertical axis which is parallel to the cable). The target position _variables φ _{LD target} , r _{LAZ target} , _{Ziel target} , γ _{L target} are _summarized in vector q _target .

Input variables of the module 39 are the current positions of the hand lever 34 for controlling the crane. The deflection of the hand lever corresponds to the desired target speed of the load in the respective direction of movement. Accordingly, the target velocities φ̇ LD _target , ṙ _LAZiel , l̇ _target , γ̇ _target to target velocity vector q̇ _{target are} summarized.

In the case of the module for optimized motion control in fully automatic operation 37, the optimal control problem can be solved from this information about the stored model for describing the dynamic behavior and the selected boundary and secondary conditions. Output variables are then the time functions u _{out, D} , u _{out, A} , u _{out, l} , u _{out, R} , the same input variables of the subordinate control for load sway 36 and the subordinate control for position or speed of the crane 41. Also a direct control 41 of the crane without subordinate control is possible with appropriate formulation of the equations in FIG. 37. In this case, in fully automatic operation, the hand lever value can be used to change the secondary condition of the maximum permissible Geschiwndigkeit in optimal control problem. This is particularly advantageous in that even in fully automatic operation, the user has the opportunity to influence the fully automatic process online in speed. The changes made are immediately adopted and taken into account in the next run of the algorithm.

• Drehwerkswinkel ϕ_D,
• Wippwerkswinkel ϕ_A,
• Seillänge l_S, und
• relative Lasthakenposition c
  und die Winkel zur Beschreibung der Lastposition:

• tangentialer Seilwinkel ϕ_St ,
• radialer Seilwinkel ϕ_Sr, und
• absoluter Rotationswinkel der Last γ _L .

In the case of the module for optimized movement guidance in the semi-automatic operation 39, however, the current desired system state in addition to the boundary and secondary conditions is required to inform the currently desired target speed of the load by the hand lever position as further information. Therefore, in semiautomatic operation, the measured quantities of the position of the crane and the load must be continuously fed back to the module 39. In detail these are:

• Slewing angle φ _D ,
• Luffing angle φ _A ,
• Rope length l _S , and
• relative load hook position c
  and the angles for the description of the load position:

• tangential rope angle φ _St ,
• radial cable angle φ _Sr, and
• absolute angle of rotation of the load γ _L.

In particular, the last-mentioned parameters for rope angles and absolute angles of rotation of the load can only be detected metrologically with considerable effort. For the realization of a load oscillation damping, however, these are indispensably necessary to compensate for disturbances. As a result, a very high positioning accuracy with low residual oscillation can be achieved even under the influence of disturbance variables (such as wind). In the case of Fig. 3, these sizes are all available.

However, if the method is used in a system in which no sensors for the rope angle measurement and the absolute rotation angle exist, then these variables must be reconstructed for the module for optimized motion control in semi-automatic operation. Model-based estimation methods 43, such as observer structures, are available here. In this case, the missing state variables are estimated or tracked from the measured variables of the crane position and the drive functions u _{out, D,} u _{out, A,} u _{out, I,} u _{out, R} in a stored dynamic model (see FIG. 4).

The basis for the method of optimized motion control is the method of dynamic optimization. For this, the dynamic behavior of the crane must be mapped in a differential equation model. To derive the model equations either the Lagrangian formalism or the Newton-Euler method can be used.

In the following several possible model approaches are presented. First, the definition of the model variables will be made with reference to FIGS. 5 and 6. For the sake of clarity, FIG. 5 shows the model variables, the model variables associated with the rotational movement, and FIG. 6, the model variables for the radial movement.

I_S ist dabei die resultierende Seillänge vom Auslegerkopf bis zum Lastmittelpunkt. ϕ_A ist der aktuelle Aufrichtwinkel des Wippwerks, I_A ist die Länge des Auslegers, ϕ_St ist der aktuelle Seilwinkel in tangentialer Richtung (da ϕ_St klein ist, kann sinϕ_St≈ϕ_St angenähert werden).
Das dynamische System für die Bewegung der Last in Drehrichtung kann durch die folgenden Differentialgleichungen beschrieben werden. $$\lfloor J_T+\left(J_{AZ}+m_A{s}_A^2+m_L{l}_A^2\right){\cos }^2{\varphi }_A\rfloor {\ddot{\varphi }}_D+m_L{l}_A{l}_s\cos {\varphi }_A{\ddot{\varphi }}_{st}+b_D{\dot{\varphi }}_D=M_{MD}-M_{MD}$$

$$m_L{l}_A{l}_s{\mathrm{cos\varphi }}_A{\ddot{\varphi }}_D+m_L{l}_s^2{\ddot{\varphi }}_{st}+m_{L}g{\mathit{\text{l}}}_s{\varphi }_{st}=0$$

First, Fig. 5 will be explained in detail. 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. The load rotation angle position corrected by the pendulum angle is then calculated to $${\phi }_{LD}={\phi }_D+\arctan \frac{l_s{\phi }_{St}}{l_A\cos {\phi }_A}$$

I _S is the resulting cable length from the boom head to the load center. φ _A is the current upright angle of the luffing gear, I _A is the length of the boom, φ _St is the actual rope angle in the tangential direction (since φ _{St is} small, sin φ _St ≈φ _St can be approximated).
The dynamic system for the movement of the load in the direction of rotation can be described by the following differential equations. $$\lfloor J_T+\left(J_{AZ}+m_A{s}_A^2+m_L{l}_A^2\right){\cos }^2{\phi }_A\rfloor {\ddot{\phi }}_D+m_L{l}_A{l}_s\cos {\phi }_A{\ddot{\phi }}_{st}+b_D{\dot{\phi }}_D=M_{MD}-M_{MD}$$

$$m_L{l}_A{l}_s{\cos }_A{\ddot{\phi }}_D+m_L{l}_s^2{\ddot{\phi }}_{st}+m_{L}G{\mathit{\text{l}}}_s{\phi }_{st}=0$$

designations:

m_L

Lastmasse

l_S

Seillänge

m_A

Masse des Auslegers

J_AZ

Massenträgheitsmoment des Auslegers bezüglich Schwerpunkt bei Drehung um Hochachse

I_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

(2) beschreibt im wesentlichen die Bewegungsgleichung für den Kranturm mit Ausleger, wobei die Rückwirkung durch die Lastpendelung berücksichtigt wird. (3) ist die Bewegungsgleichung, welche die Lastpendelung um den Winkel ϕ _St beschreibt, wobei die Anregung der Lastpendelung durch die Drehung des Turmes über die Winkelbeschleunigung des Turmes oder eine äußere Störung, ausgedrückt durch Anfangsbedingungen für diese Differentialgleichungen, verursacht wird.

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

I _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

(2) essentially describes the equation of motion for the crane tower with boom, taking into account the feedback caused by the load oscillation. (3) is the equation of motion which describes the load oscillation by the angle φ _St , the excitation of the load oscillation being caused by the rotation of the tower via the angular acceleration of the tower or an external disturbance expressed by initial conditions for these differential equations.

i_D ist das Übersetzungsverhältnis zwischen Motordrehzahl und Drehgeschwindigkeit des Turms, V ist das Schluckvolumen der Hydraulikmotoren, Δp_D ist der Druckabfall über dem hydraulischen Antriebsmotor, β ist die Ölkompressibilität, Q _FD ist der Förderstrom im Hydraulikkreis für das Drehen und K_PD ist die Proportionalitätskonstante, die den Zusammenhang zwischen Förderstrom und Ansteuerspannung des Proportionalventils angibt. Dynamische Effekte der unterlagerten Förderstromregelung werden vernachlässigt.The hydraulic drive is described by the following equations. $$\begin{array}{l}{M}_{\mathit{MD}}=i_D\frac{V}{2\pi }\mathrm{\Delta }{p}_D\\ \mathrm{\Delta }{p}_D=\frac{1}{V\mathrm{\beta }}\left(Q_{FD}-i_D\frac{V}{2\pi }{\dot{\mathrm{\phi }}}_D\right)\\ Q_{FD}=K_{P{D}^{u}StD}\end{array}$$

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, which indicates the relationship between the flow rate and the drive voltage of the proportional valve. Dynamic effects of subordinate flow control are neglected.

bzw. im Zeitbereich $${\ddot{\varphi }}_D=-\frac{1}{T_{D\mathit{Antr}}}{\dot{\varphi }}_D+\frac{K_{PD\mathit{Antr}}}{T_{D\mathit{Antr}}}{u}_{StD}$$

Alternatively, the transmission behavior of the drive units can be represented by an approximate relationship as delay element 1st or higher order instead of the equation 4. The approximation with a delay element of the first order is shown below. Then the transfer function results $$s{\Phi }_D\left(s\right)=\frac{K_{\mathit{P}\mathit{DAntr}}}{1+T_{\mathit{DAntr}}}{U}_{StD}\left(s\right)$$

or in the time domain $${\ddot{\phi }}_D=-\frac{1}{T_{D\mathit{Antr}}}{\dot{\phi }}_D+\frac{K_{PD\mathit{Antr}}}{T_{D\mathit{Antr}}}{u}_{StD}$$

Thus, from the equations (6) and (3) also an adequate model description can be built; Equation (2) is not needed.
T _DAntr is the approximate time constant (derived from measurements to describe the deceleration _{behavior of} the drives) K _PDAntr the resulting gain between drive voltage and resulting velocity in the staionary case.
With a negligible time constant with respect to the drive dynamics, a proportionality between the speed and the drive voltage of the proportional valve can be assumed directly. $${\dot{\mathrm{\phi }}}_D=K_{P{\mathit{Ddirekt}}^{u}StD}$$

Here, too, an adequate model description can be constructed from equations (7) and (3).

For the radial movement shown in FIG. 6, the equations of motion can be established analogously to equations (2) and (3). For this purpose, FIG. 6 gives explanations for 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 $$r_{LA}=l_A{\cos }_A+l_s{\phi }_{Sr}$$

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

The dynamic system can then be described by applying the Newton Euler method through the following differential equations. $$\left(J_{AY}+m_A{s}_A^2+m_L{l}_A^2{\sin }^2{\mathrm{\phi }}_A\right){\ddot{\mathrm{\phi }}}_A-m_L{l}_A{l}_s\sin {\mathrm{\phi }}_A{\ddot{\mathrm{\phi }}}_{sr}+b_A{\dot{\mathrm{\phi }}}_A-m_A{s}_{A}G\sin {\mathrm{\phi }}_A\cdot {\mathrm{\phi }}_A=M_{MA}-M_{RA}-m_A{s}_{A}G\cos {\mathrm{\phi }}_A$$

$$-m_L{l}_A{l}_s{\mathrm{sin\phi }}_A+m_L{l}_s^2{\ddot{\mathrm{\phi }}}_{sr}+m_L{l}_{s}G{\mathrm{\phi }}_{sr}=m_L{l}_s{\ddot{\mathrm{\phi }}}_D^2\left(l_S{\mathrm{\phi }}_{sr}+l_A{\cos }_A\right)$$

designations:

m _L

load mass

I _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

I _A

Length of the boom

S _A

Center of gravity of the jib

b _A

viscous damping

_MA

drive torque

M _RA

friction

Equation (9) 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. Equation (10) 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. clear can be described this problem in that a slewing movement with quadratic rotational speed dependence also causes an angular deflection in the radial direction.

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

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. 7 shows the erecting kinematics of the luffing mechanism. By way of example, the hydraulic cylinder is anchored above the pivot point of the boom on 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 . The correction angle φ ₀ takes into account the deviations of the attachment points from the boom or tower axis and is also known from design data. From this it is possible to derive the following relationship between erecting angle φ _A and hydraulic cylinder position Z _cyl . $$z_{\mathit{cyl}}=\sqrt{d_a^2+d_b^2-2d_b{d}_a\sin \left({\phi }_A-{\mathrm{<figure-callout\; id="0"\; label="\phi "\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_0\right)}$$

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

Since only the Aufrichtwinkel φ _{A is} measured, the inverse relation of (12) and the dependence between piston rod speed ż _Zyl and erection speed φ̇ _{A is} also of interest. $${\phi }_A=\arcsin \left(\frac{d_a^2+d_b^2-z_{Zyl}^2}{2d_a{d}_b}\right)+{\mathrm{<figure-callout\; id="0"\; label="\phi "\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_0$$

$${\phi }_A=\frac{\partial {\phi }_A}{\partial z_{\mathit{cyl}}}{\dot{z}}_{\mathit{cyl}}=\frac{\sqrt{d_a^2+d_b^2-2d_b{d}_a\sin \left({\phi }_A-{\phi }_0\right)}}{-d_b{d}_a\cos \left({\phi }_A-{\mathrm{<figure-callout\; id="0"\; label="\phi "\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_0\right)}{\dot{z}}_{\mathit{cyl}}$$

For the calculation of the effective moment on the boom, the calculation of the projection angle φ _{P is also} required. $$\cos {\phi }_p=\frac{d_a\cos \left({\phi }_A-{\phi }_0\right)}{\sqrt{d_a^2+d_b^2-2d_b{d}_a\sin \left({\phi }_A-{\mathrm{<figure-callout\; id="0"\; label="\phi "\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_0\right)}}$$

bzw. im Zeitbereich $${\ddot{z}}_{\mathit{zyl}}=-\frac{1}{T_{\mathit{AAntr}}}{\dot{z}}_{\mathit{Zyl}}+\frac{K_{\mathit{PAAntr}}}{T_{\mathit{AAntr}}}{u}_{StA}$$

Alternatively, instead of the hydraulic equations (11), an approximation for the dynamics of the drives with an approximate relationship as delay element 1 or higher order can again be provided for this purpose. This gives an example $$s{Z}_{\mathit{cyl}}\left(s\right)=\frac{{\mathrm{<figure-callout\; id="1"\; label="K\; PAAntr"\; filenames="imgf0005.png"\; state="\{\{state\}\}">K\; </figure-callout>}}_{\mathit{PAAntr}}}{1+T_{\mathit{AAntr}}s}{U}_{StA}\left(s\right)$$

or in the time domain $${\ddot{z}}_{\mathit{cyl}}=-\frac{1}{T_{\mathit{AAntr}}}{\dot{z}}_{\mathit{Zyl}}+\frac{K_{\mathit{PAAntr}}}{T_{\mathit{AAntr}}}{u}_{StA}$$

Thus, from the equations (17), (14) and (10) also an adequate model description can be constructed; Equation (9) is not needed. T _AAntr is the approximate time constant (derived from measurements to describe the deceleration _{behavior of} the drives) K _PAAntr the resulting gain between the drive _voltage and the resulting velocity in the stationary case.
With a negligible time constant with respect to the drive dynamics, a proportionality between the speed and the drive voltage of the proportional valve can be assumed directly. $${\dot{z}}_{\mathit{Zyl}}=K_{\mathit{PAdirekt}}{u}_{StA}$$

Here, too, an adequate model description can be constructed from the equations (18), (10) and (14).

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 dated 15.06.2000, the contents of which are 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 module for optimized motion control, as shown for example in 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.

This results in the following equation of motion. The Varaiablenbezeichnung correspond to the DE 100 29 579 from 15.06.2000. No linearization was done. $$({\mathrm{\Theta }}_{Lc}+{\mathrm{\Theta }}_{Uc}){\ddot{\mathrm{\gamma }}}_{\mathit{drill}}=-m_{L}G\sin \left(\frac{d_c{\mathrm{\gamma }}_{\mathit{drill}}}{2l_S}\right)\frac{d_c}{2}-{\mathrm{\Theta }}_{Lc}\ddot{c}$$

Also for the load swing unit now differential equations for the description of the drive dynamics to improve the function as in the rotational movement can be additionally taken into account. Here should be dispensed with a detailed presentation.

The dynamics of the hoist is neglected, since the dynamics of the hoist movement is fast compared to the system dynamics of the load oscillation of the crane. However, as with the load swing mechanism, the corresponding dynamic equations describing the hoist dynamics can be added at any time if required.

mit: $$\mathrm{Zustandsvektor}:\underline{x}={\left[{\mathrm{\varphi }}_D{\dot{\mathrm{\varphi }}}_D{\mathrm{\varphi }}_A{\dot{\mathrm{\varphi }}}_A{\mathrm{\varphi }}_{St}{\dot{\mathrm{\varphi }}}_{St}{\mathrm{\varphi }}_{Sr}{\dot{\mathrm{\varphi }}}_{Sr}{p}_{Zyl}\right]}^T$$

$$\mathrm{Steuergröße}:\underline{u}={\left[u_{StD}{u}_{StA}\right]}^T$$

$$\mathrm{Ausgansgröße}:\underline{y}=\left[{\mathrm{\varphi }}_{LD}{r}_{LA}\right]$$

The remaining equations for the description of the system behavior are now to be brought into a non-linear state space representation according to Isidori, Nonlinear Control Systems Springer Verlag 1995. This is exemplified based on equations (2), (3), (9), (10), (14), (15). The axis of rotation of the load around the vertical axis and the hoist axis is not taken into account in this example below. However, it is not difficult to include in the model description. For the present case of application, a crane without automatic load swiveling mechanism is assumed; for safety reasons the hoist is operated manually by the crane operator. Accordingly you get: $$\mathrm{State\; space\; representation}:\begin{array}{l}\underset{}{\dot{\begin{array}{c}x\end{array}}}=\underset{}{a}\left(\underset{}{x}\right)+\underset{}{b}\left(\underset{}{x}\right)\underset{}{u}\\ \underset{}{\begin{array}{c}y\end{array}}=\underset{}{c}\left(\underset{}{x}\right)\end{array}$$

With: $$\mathrm{state\; vector}:\underset{}{x}={\left[{\mathrm{\phi }}_D{\dot{\mathrm{\phi }}}_D{\mathrm{\phi }}_A{\dot{\mathrm{\phi }}}_A{\mathrm{\phi }}_{St}{\dot{\mathrm{\phi }}}_{St}{\mathrm{\phi }}_{Sr}{\dot{\mathrm{\phi }}}_{Sr}{p}_{Zyl}\right]}^T$$

$$\mathrm{control\; variable}:\underset{}{u}={\left[u_{StD}{u}_{StA}\right]}^T$$

$$\mathrm{Ausgansgröße}:\underset{}{y}=\left[{\mathrm{\phi }}_{LD}{r}_{LA}\right]$$

The vectors a ( x ), b ( x ), c ( x ) are obtained by transforming the equations (2) - (4), (8) - (15).

In the operation of the module for optimized motion control without underlying load oscillation damping occurs in semi-automatic operation, the problem that the state x must be completely present as a measurement vector. Since in this case, however, no pendulum angle sensors are installed, in this case described above, the pendulum angle quantities φ _St , φ _ST , φ _Sr , φ̇ _Sr from the control variables u _StD , u _StA and the measured variables φ _D , φ̇ _D , φ _A , φ̇ _A , p _{Zyl be} reconstructed. For this purpose, the nonlinear model is linearized according to equation (20-23) and, for example, a parameter-adaptive state observer (see also FIG. 4, block 43) is designed. In the case of reduced accuracy requirements, a state tracking of the cable angle quantities based on the model equations and the known progressions of the input variables as well as the measurable state variables can also be simplified.

The target curves for the input signals (control quantities) u _StD (t), u _StA (t) are obtained by the solution of an optimal control problem, that is, □ h. a task of dynamic optimization. For this purpose, the desired reduction of load oscillation in a target function is recorded. Boundary conditions and trajectory restrictions of the optimal control problem result from the orbit data, the technical restrictions of the crane system (eg limited drive power, as well as restrictions due to dynamic load torque limitations to prevent tilting of the crane) and extended demands on the movement of the load. For example, with the method described below, it is possible for the first time to predict the path corridor required by the load when connecting the calculated control functions exactly in advance. This provides automation options that were previously not solvable. Such a formulation of the optimum control problem is given below by way of example both for the fully automatic operation of the system with predetermined start and end point of the load path and for the hand lever operation.

In the case of fully automatic operation, the entire movement is considered from the given start to the given target point. In the objective function of the optimal control problem, the load pendulum angles are evaluated quadratically. The minimization of this target function therefore provides a movement with reduced load oscillation. An additional evaluation of the load pendulum angular velocities with a time-variant (increasing towards the end of the optimization horizon) Strafterm results in a calming of the load movement at the end of the optimization horizon. A regularization term with a quadratic evaluation of the amplitudes of the control variables can favorably influence the numerical condition of the task. $$J=\underset{t_0}{\overset{t_f}{\int }}\left({\phi }_{St}^2\left(t\right)+{\phi }_{Sr}^2\left(t\right)+\rho \left(t\right)\left({\dot{\phi }}_{St}^2\left(t\right)+{\dot{\phi }}_{Sr}^2\left(t\right)\right)\right)+{\rho }_u\left(u_{StD}\left(t\right).u_{StA}\left(t\right)\right))dt$$

designations:

t ₀

Predetermined start time

t _f

Predetermined end time

ρ (t)

Time variant penalty coefficient

ρ _u (u _StD, u _StA )

Regularization term (quadratic evaluation of the tax variables)

In hand lever operation, on the other hand, the complete load movement between the given start and end point is not considered, but the optimal control problem is on a time window which is moved together with the dynamic operation [ t _o , t _f ]. The start time of the optimization horizon t _o is the current time, and in the optimal control problem, the dynamics of the crane system in the forecast horizon is up t _f . This time horizon is an essential tuning parameter of the method and is limited downwards by the oscillation period of the load pendulum movement.

In the objective function of the optimal control problem, the deviation of the actual load speed from the setpoint speeds given by the manual lever positions must be taken into account in addition to the intended reduction in load oscillation. $$J=\underset{{\tilde{t}}_0}{\overset{{\tilde{t}}_f}{\int }}\left(\begin{array}{l}{\begin{array}{c}{\rho }_{LD}\left({\dot{\phi }}_{LD}\left(t\right)-{\dot{\phi }}_{LD.\mathit{should}}\right)\end{array}}^2+{\rho }_{LA}{\left({\dot{r}}_{LA}\left(t\right)-{\dot{r}}_{LA.\mathit{should}}\right)}^2+\\ +{\phi }_{St}^2\left(t\right)+{\phi }_{\mathit{Sr}}^2\left(t\right)+\rho \left(t\right)({\dot{\phi }}_{St}^2\left(t\right)+{\dot{\phi }}_{\mathit{Sr}}^2\left(t\right))+{\rho }_u\left(u_{StD}\left(t\right).u_{StA}\left(t\right)\right)\end{array}\right)dt$$

designations:

t _O

Predetermined start time of the optimization horizon

t _f

Predetermined end time of the forecast period

ρ _LD

Weighting coefficient deviation load rotational speed

φ _{LD, should}

By hand lever position predetermined Lastdrehwinkelgeschwindigkeit

ρ _LA

Weighting coefficient deviation radial load speed

ṙ _{LA , shall}

By Handhebeistellung predetermined radial load speed

In fully automatic operation with a given start and end point, the boundary conditions for the optimal control problem arise from their coordinates and the requirements of a rest position in the start and end positions. $$\begin{array}{l}{\mathrm{<figure-callout\; id="0"\; label="\phi \; D\; t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\phi \; </figure-callout>}}_D\left(t_{}\right)\left({}_0\right)=0.{\phi }_{D.0}.{\phi }_D\left(t_f\right)={\phi }_{D.f}\\ {\stackrel{<figure-callout\; id="0"\; label="\; ̇\; D\; t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\dot{}\; </figure-callout>}{\begin{array}{c}\phi \end{array}}}_D\left(t_{}\right)\left({}_0\right)=0.{\dot{\begin{array}{c}\phi \end{array}}}_D\left(t_f\right)=0\\ {\mathrm{<figure-callout\; id="0"\; label="\phi \; A\; t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\phi \; </figure-callout>}}_A\left(t_{}\right)\left({}_0\right)=\arccos \left(\frac{r_{LA.0}}{l_A}\right).{\phi }_A\left(t_f\right)=\arccos \left(\frac{r_{LA.f}}{l_A}\right)\\ {\dot{\begin{array}{c}\phi \end{array}}}_A\left({\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">t</figure-callout>}}_0\right)=0.{\dot{\begin{array}{c}\phi \end{array}}}_A\left(t_f\right)=0\\ {\begin{array}{c}\mathrm{<figure-callout\; id="0"\; label="\phi \; St\; t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\phi \; </figure-callout>}& \\ \end{array}}_{\mathit{St}}\left(t_{}\right)\left({}_0\right)=0.{\begin{array}{c}\phi \end{array}}_{\mathit{St}}\left(t_f\right)=0\\ {\dot{\begin{array}{c}\phi \end{array}}}_{S\mathrm{<figure-callout\; id="0"\; label="t\; t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">t\; </figure-callout>}}\left(t_{}\right)\left({}_0\right)=0.{\dot{\begin{array}{c}\phi \end{array}}}_{St}\left(t_f\right)=0\\ {\begin{array}{c}\phi \end{array}}_{S\mathrm{<figure-callout\; id="0"\; label="r\; t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">r\; </figure-callout>}}\left(t_{}\right)\left({}_0\right)=0.{\begin{array}{c}\phi \end{array}}_{Sr}\left(t_f\right)=0\\ {\dot{\begin{array}{c}\phi \end{array}}}_{S\mathrm{<figure-callout\; id="0"\; label="r\; t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">r\; </figure-callout>}}\left(t_{}\right)\left({}_0\right)=0.{\dot{\begin{array}{c}\phi \end{array}}}_{Sr}\left(t_f\right)=0\end{array}$$

designations:

φ _{D, 0}

Starting point of turning angle

φ _{D, f}

Endpoint of turning angle

r _{LA, 0}

Starting point load position

r _{LA, f}

End point load position

The boundary conditions for the pressure in the cylinder result from the stationary values in the start and end point according to equation (11).

In hand lever operation, however, must be taken into account in the boundary conditions that the movement does not start from a rest position and generally does not end in a rest position. The boundary conditions at the start time of the optimization horizon t ₀ result from the current system state x ( t ₀ ), which is measured or is reconstructed via an accompanying model from the control variables u _{S, tD} , u _StA and the measured quantities φ _D , φ̇'D, φA, φ̇A, P _Zyl via a parameter- adaptive state observer .
The boundary conditions at the end of the optimization horizon t _f are free.

Due to the technical parameters of the crane system, there are a number of restrictions that have to be considered regardless of the operating mode in the optimal control problem. So the drive power is limited. This can be over a maximum flow rate in the hydraulic drives are described and included in the optimal control problem via amplitude limitations for the control variables. $$\begin{array}{c}-{\dot{u}}_{StD.\mathrm{Max}}\le {\dot{u}}_{StD}\left(t\right)\le {\dot{u}}_{StD.\mathrm{Max}}\\ -{\dot{u}}_{StA.\mathrm{Max}}\le {\dot{u}}_{S\mathit{tA}}\left(t\right)\le {\dot{u}}_{StA.\mathrm{Max}}\end{array}$$

To avoid stresses of the system due to abrupt load changes whose consequences are not covered in the simplified dynamic model described above, the rate of change of the control variables is limited. Thereby defined the mechanical stress can be limited. $$\begin{array}{c}-{\dot{u}}_{StD.\mathrm{Max}}\le {\dot{u}}_{StD}\left(t\right)\le {\dot{u}}_{StD.\mathrm{Max}}\\ -{\dot{u}}_{StA.\mathrm{Max}}\le {\dot{u}}_{S\mathit{tA}}\left(t\right)\le {\dot{u}}_{StA.\mathrm{Max}}\end{array}$$

In addition, it may be required that the control variables should be continuous as functions of time and have continuous first derivatives with respect to time.

The elevation angle is limited due to the crane design $${\phi }_{A.\min }\le {\phi }_A\left(t\right)\le {\phi }_{A.\mathrm{Max}}$$

designations:

u _{StD, max}

Maximum value control function slewing gear

u̇ _{StD, max}

Maximum rate of change Control function Slewing

u _{StA, max}

Maximum value control function luffing mechanism

U _{StA, max}

Maximum rate of change Control function Wippwerk

φ _{A, min}

Minimum value of elevation angle

φ _{A, max}

Maximum value of elevation angle

Additional restrictions arise from further requirements for the movement of the load. Thus, in fully automatic operation, in which the entire load movement is considered from the start to the target point, a monotonous change in the rotation angle can be required. $${\dot{\phi }}_D\left(t\right)\left({\phi }_D\left(t_f\right)-{\mathrm{<figure-callout\; id="0"\; label="\phi \; D\; t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\phi \; </figure-callout>}}_D\left(t_{}\right)\left({}_0\right)\right)\ge 0$$

Rail corridors can be included in the calculation of the optimal control both in fully automatic and in manual lever operation via the analytical description of the permissible load positions with the aid of inequality restrictions. $$G_{\min }\le G\left({\phi }_{LD}\left(t\right).r_{LA}\left(t\right)\right)\le G_{\mathrm{Max}}$$

With the aid of these inequality conditions, a trajectory is enforced inside an allowable range, here the trajectory, the limits of this permissible range limit the load movement and thus represent 'virtual walls'.

If the trajectory to be traversed does not only consist of start and finish point, but further points have to be traced in a given sequence, this can be included in the optimal control problem due to internal boundary conditions. $${\phi }_D\left(t_i\right)={\phi }_{D.i}.{\phi }_A\left(t_i\right)=\arccos \left(\frac{r_{LA.i}}{l_A}\right)$$

designations:

t _i

(free) time of reaching the predetermined path point i

φ _{D, i}

Rotation angle coordinate of the predetermined path point i

r _{LA, i}

Radial position of the predetermined path point i

The claim is not tied to a specific method for numerically calculating the optimal controls. The claim also expressly relates to an approximate solution to the above-mentioned optimal control problems, in which only a solution of sufficient (non-maximal) accuracy is determined in view of a reduced computational outlay for on-line use. In addition, for efficiency reasons, a number of the hard constraints (constraints or trajectory inequality limitations) formulated above may be treated numerically as a soft constraint via an assessment of the constraint violation in the target function.
By way of example, however, the numerical solution will be explained here by means of multi-stage control parameterization.

For approximate numerical solution of the optimal control problem, the optimization horizon is discretized. $${\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">t</figure-callout>}}_0={\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">t</figure-callout>}}^0<{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="imgf0005.png"\; state="\{\{state\}\}">t</figure-callout>}}^1<...<t^K=t_f$$

The length of the subintervals [t ^k , t ^{k + 1} ] can be adapted to the dynamics of the problem. A larger number of subintervals usually leads to an improvement of the approximate solution, but also to an increased calculation effort.
On each of these sub-intervals, the time characteristic of the control variables is now approximated by a starting function U ^k with a fixed number of parameters u ^k (control parameters). $$u\left(t\right)\approx u_{\mathit{app}}\left(t\right)=U^k\left(t,,u^k\right).t^k\le t\le t^{k+1}$$

des jeweiligen Intervalls als Variable des nichtlinearen Optimierungsproblems betrachtet wird. Durch geeignete Gleichungsbeschränkungen ist die Stetigkeit der approximierten Zustandstrajektorien zu sichern. Damit steigt die Dimension des nichtlinearen Optimierungsproblems an. Es ergibt sich jedoch eine beträchtliche Vereinfachung in der Verkopplung der Problemvariablen und zudem eine starke Strukturierung des nichtlinearen Optimierungsproblems. Daher sinkt der Lösungsaufwand in vielen Fällen beträchtlich, vorausgesetzt, die Problemstruktur wird im Lösungsalgorithmus geeignet ausgenutzt.Now, the state differential equation of the dynamic model can be numerically integrated and the target functionally evaluated, whereby the approximated time profiles are used instead of the control variables. As a result, the target function is obtained as a function of the control parameters u ^k k = 0, ..., K-1 . The Boundary conditions and the trajectory restrictions can also be understood as functions of the control parameters.
The optimal control problem is approximated in this way by a nonlinear optimization problem in the control parameters, where objective function calculation and constraint evaluation of the nonlinear optimization problem respectively require the numerical integration of the dynamic model taking into account the approximation approach of equation (34).
This limited nonlinear optimization problem can now be solved numerically, using a standard sequential quadratic programming (SQP) method, in which the solution of the nonlinear problem is determined by a series of linear-quadratic approximations.
The efficiency of the numerical solution can be increased considerably if, in addition to the control parameters of the interval k, also the initial state $$x^k\approx x\left(t^k\right).k=0.....K$$

of the respective interval is regarded as a variable of the nonlinear optimization problem. By appropriate equations restrictions the continuity of the approximated state trajectories has to be ensured. This increases the dimension of the nonlinear optimization problem. However, there is a considerable simplification in the coupling of the problem variables and also a strong structuring of the nonlinear optimization problem. Therefore, the solution effort in many cases drops considerably, provided that the problem structure is suitably exploited in the solution algorithm.

An additional significant reduction of the computational effort to solve the optimal control problem is achieved by an approximation by means of linearization of the system equations. In this case, the original nonlinear state differential equations and algebraic output equations (20) are linearized along a system trajectory (X _ref (t), U _ref (t)) which initially satisfies the arbitrary system differential equation and satisfies the state differential equations. $$\begin{array}{l}\mathrm{\Delta }\dot{x}=A\left(t\right)\mathrm{\Delta }x+B\left(t\right)\mathrm{\Delta }u\\ \mathrm{\Delta }y=C\left(t\right)\mathrm{\Delta }x\end{array}$$

The sizes Δ x, Δ denote u, Δ y are the deviations from the reference course of the respective size $$\begin{array}{l}\mathrm{\Delta }x=x-x_{\mathit{ref}}.\Delta u=u-u_{\mathit{ref}}.\Delta y=y-y_{\mathit{ref}}\\ {\dot{x}}_{\mathit{ref}}=a\left(x_{\mathit{ref}}\right)+b\left(x_{\mathit{ref}}\right)\cdot u_{\mathit{ref}}\\ y_{\mathit{ref}}=c\left(x_{\mathit{ref}}\right)\end{array}$$

The time-variant matrices A ( t ), B ( t ), C ( t ) result from the Jacobi matrices $$A\left(t\right)=\frac{\partial \left(a\left(x_{\mathit{ref}}\left(t\right)\right)+b\left(x_{\mathit{ref}}\right)\right)\cdot u_{\mathit{ref}}\left(t\right))}{\partial x_{\mathit{ref}}\left(t\right)}.B\left(t\right)=b\left(x_{\mathit{ref}}\left(t\right)\right).C\left(t\right)=\frac{\partial c\left(x_{\mathit{ref}}\left(t\right)\right)}{\partial x_{\mathit{ref}}\left(t\right)}$$

If now the optimal control task is formulated in the variables Δx, Δu , a limited linear-quadratic optimal control problem results. With a suitable choice of the starting functions U ^k , the state differential equation can be analytically solved on the basis of the associated equation of motion on each subinterval [t ^k , t ^{k + 1} ] , and the complicated numerical integration is dispensed with.
The optimal control task is thus approximated by a finite-dimensional quadratic optimization problem with linear equations and inequations restrictions, which can be solved numerically with a fitted standard method. The numerical effort for this, in turn, is significantly lower than in the case of the nonlinear optimization problem described above.
The described linearization approach is particularly suitable for the approximate solution of the optimal control problems in manual lever operation, since in this case, on the one hand, due to the shorter optimization horizon (time window [ t _o , t _f ]) the inaccuracies caused by the linearization have a lesser effect and, on the other hand, the ones calculated in the preceding periodical optimal control and state progressions suitable reference trajectories are available.

As a solution to the optimal control problem, the optimum time courses of both the control variables and the state variables of the dynamic model are obtained. These are switched on during operation with subordinate control as actuating and reference variables. Since the dynamic behavior of the crane is taken into account in these nominal functions, only disturbances and model deviations have to be compensated by the control.
In the case of operation without subordinate control, however, the optimum characteristics of the control variables are applied directly as manipulated variables.
Furthermore, the solution of the optimal control problem provides a prognosis of the path of the oscillating load, which can be used for extended measures for collision avoidance.

8 shows the flow chart for the calculation of the optimized control variable in fully automatic operation. This modifies module 37 of FIG. 3. Starting from the start and end points of the load movement determined by the set point matrix, the optimal control problem is defined by including the allowable range and technical parameters. The numerical solution of the optimal control problem provides optimal timing of the control and state variables. These are activated with subordinate control for load oscillation damping as actuating and reference variables. Alternatively, a realization without subordinate control - then with direct connection of the optimal control functions on the hydraulics - can be realized.

Fig. 9 shows the interaction of state reconstruction and calculation of the optimal control in the case of the hand lever operation. The state of the dynamic crane model is tracked using the available measured variables. By solving the optimal control problem, such time curves of the drive functions are determined which, starting from this current state, are reduced Load swinging brings the load speed to the setpoints specified by the hand lever.
A once calculated optimal control is not over the full time horizon [ t ₀ , t _f ] , but is constantly adapted to the current system status and the current setpoint values. The frequency of this adaptation is limited by the required computing time for recalculation of the optimal control.

10 shows exemplary results for optimal time courses of the control variables in fully automatic operation. A time horizon of 30 s was specified. The control functions are continuous functions of time with continuous 1st derivatives.

11 shows exemplary time profiles of control variables and controlled variables in simulated hand lever operation. The setpoint values for the load speed (the hand lever presettings) are varied in the form of staggered rectangular pulses. The update of the optimal control is done with a sampling time of 0.2 s.

Claims (15)

1. A crane or excavator for the transfer of a load (3), which is suspended at a load cable, having a slewing gear for the rotation of the crane or excavator, a luffing mechanism (7) for the setting upright or inclining of a boom (5) and a hoisting gear for the lifting or lowering of the load (3) suspended at the cable, having a track control (31),
   characterized in that,
   model based optimum control trajectories are calculated online in the track control (31) based on a non-linear model approach and are updated by feedback of state variables; and in that the starting values (U_outD, U_outA, U_outL, U_outR) of the track control (31) are used directlyor indirectly as input values in a feedback control for the position and/or speed of the crane (41) or excavator, with the objective functional of an optimum control problem taking into account, in addition to the desired reduction in load swing, the difference of the actual load speed from the desired speed predetermined by the hand lever positions as well as a penalized likelihood to moderate the load movement at the end of the optimization horizon so that the reference variables for the control in the track control (31) are generated so that a load movement with minimized swing deflections results.
2. A crane or excavator in accordance with claim 1, characterized by model based optimum control trajectories based on a model linearized by reference trajectories.
3. A crane or excavator in accordance with claim 1, characterized by model based optimum control trajectories based on a non-linear model approach.
4. A crane or excavator in accordance with any one of claims 1 to 3, characterized by model based optimum control trajectories while having feedback of all state variables.
5. A crane or excavator in accordance with any one of claims 1 to 3, characterized by model based optimum control trajectories while having feedback of at least one measured variable and estimating of the remaining state variables.
6. A crane or excavator in accordance with any one of claims 1 to 3, characterized by model based optimum control trajectories while having feedback of at least one measured variable and tracking of the remaining state variables by model based feed forward control.
7. A crane or excavator in accordance with any one of claims 1 to 6, characterized in that the track control (31) can be implemented as a fully automatic or as a semi-automatic control.
8. A crane or excavator in accordance with any one of the preceding claims, characterized in that a set point matrix (35) for the position and orientation of the load can be entered as an input value into the track control (31).
9. A crane or excavator in accordance with claim 8, characterized in that the set point matrix (35) comprises a start and finish point.
10. A crane or excavator in accordance with claim 7, 8 and/or 9, characterized in that the desired finish speed of the load can additionally be entered into the track control (31) by the position of a hand lever (34) in the case of semi-automatic operation.
11. A crane or excavator in accordance with claim 10, characterized in that the measured variables of the positions of crane and load can be detected via sensors and can be fed back into the track control (31) in the case of semi-automatic operation.
12. A crane or excavator in accordance with claim 10, characterized in that the positions of crane and load can be estimated in a module for model based estimation processes (43) and can be fed back into the track control system (31) in the case of semi-automatic operation.
13. A crane or excavator in accordance with any one of the preceding claims, characterized in that the output values (U_outD, U_outA, U_outL, U_outR) are first entered into an underlying feedback control with load swing damping.
14. A crane or excavator in accordance with claim 13, characterized in that the load swing damping comprises at least one track planning module, one centripetal force compensation device and at least one axis controller for the slewing gear, one axis controller for the luffing mechanism, one axis controller for the hoisting gear and one axis controller for the pivot mechanism.
15. A crane or excavator in accordance with any one of the preceding claims, characterized in that the movement track of the load can be fixed by the track control (31) such that predetermined free areas cannot be departed from by the swinging load.

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