Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 2, pp. 443-455, Warsaw 2008

EXPERIMENTAL VERIFICATION OF A METHOD OF FINAL

POSITIONING OF A LOAD FOR ROTARY CRANES

Andrzej Maczyński

Jerzy Płosa

Department of Mechanics and Computer Methods, University of Bielsko-Biała, Poland

e-mail: amaczynski@ath.bielsko.pl; jplosa@ath.bielsko.pl

In the paper, a method of final positioning of a load in slewing motion
for jib cranes is proposed.Themethod is based on the so called ”map of
basic functions”. The map stores drive functions of jib slewing motion
for particular slew angles chosen so that they form a kind of base. The
basic functions are determined for certain operational parameters (angle
of jib slew, time of slewing, crane radius, mass of the load, length of
the rope, etc.) using optimisationmethods. Drive functions for different
parameters, particularly for different slew angles, are calculated by a
crane control systemdirectly before slewing. Linear interpolation is used
for this task. The method proposed has been experimentally verified
using a physical model of a jib crane.

Key words: crane, positioning, experimental investigation

1. Introduction

Final positioning of a crane load in slewing motion means that at the end
of jib motion the load should stay motionless at the desired point. In real
conditions, this is difficult to achieve, butany significant limitation of final load
oscillations is welcome. Immobility of the load after slewing motion enables
next operations to be performed immediately by the operator. Effectiveness
of reload and assembly works and the level of both active and passive safety
also increases.

The problem of load positioning of jib cranes was considered bymany au-
thors (Hara et al., 1989; Sakawa andNakazumi, 1985; Sakawa et al., 1981; Ta-
nizumi et al., 1994). Often, an application of hoisting drummotion or motion



444 A. Maczyński, J. Płosa

changing the crane radius simultaneously with slewingmotion is proposed by
Abdel-Rahman and Nayfeh (2002), Bednarski et al. (1997), Kłosiński (2000).
Sometimes purposefulness of using additionalmechatronical systems is discus-
sed byBalachandran andFang (1999),Maczyński (2005). A close-loop control
system is assumed in the majority of proposed methods (Balachandran and
Fang, 1999; Kłosiński, 2000) that are designed for cranes with high capacity.
In this paper however, a simple method of final load positioning is presented.
It has been developed for cheap, small cranes, e.g. pillar ones. In themethod,
an open-loop control system based on a so-called ”map of basic slewing func-
tions” is assumed. The slewing functions of the jib (called the drive functions
from now on) for chosen slew angles and certain operation parameters are
determined bymeans of optimisation. Those functions would be permanently
stored in the crane control system (e.g. a PLC controller), forming the men-
tionedmap. The optimisation is performed on a simplified cranemodel which
ensures its significant numerical effectiveness. The drive functions for other,
intermediate slew angles would be calculated by the control system directly
before slewing bymeans of interpolation.

2. Dynamical model of a crane

The simplified crane model used for the optimisation task is presented in
Fig.1. It is assumed that the supporting structure, rope system and drives are
non-deformable. The equations ofmotion of the load can bewritten as follows

mLẍL = S
xG−xL
LL

mLÿL = S
yG−yL
LL

(2.1)

mLz̈L = S
zG−zL
LL

−mLg

where
mL – mass of the load
rL – vector of coordinates of the load, rL = [xL,yL,zL]

⊤

rG – vector of coordinates of the point G (head of the jib),
rG = [xG,yG,zG]

⊤

LL – length of the rope within the segment GL
S – force in the rope
ϕ – slew angle of the jib
ψ – jib slope angle.



Experimental verification of a method... 445

Fig. 1. Simplified model of a crane used for the optimisation task

There are four unknowns xL, yL, zL, S in these equations, and thus an
additional constraint equation is necessary

L2L = |GL|
2 =(xG−xL)

2+(yG−yL)
2+(zG−zL)

2 = const (2.2)

The force S can be calculated from the formula

S =
mL

LL

[

g(zG−zL)+(ẋG− ẋL)
2+(ẏG− ẏL)

2+(żG− żL)
2+

(2.3)

+ẍG(xG−xL)+ ÿG(yG−yL)+ z̈G(zG−zL)
]

With reference to Fig.1, the following relations occur

xG = dcosψcosϕ = d
′cosϕ yG = dcosψ sinϕ = d

′ sinϕ
(2.4)

zG = dsinψ = const

Including equations (2.3) and (2.4) in equations (2.2) and (2.1), differential
equations of motion for the simplified model of a crane are obtained.

3. Optimisation of drive functions

Let ϕw(t) denote the drive function of jib slewing. It is assumed that for
t ∈ 〈t0,T〉, the function ϕw(t) can be approximated by means of third-order
spline functions (Fig.2) and it can be expressed as follows



446 A. Maczyński, J. Płosa

ϕw(t)
∣

∣

∣

t∈〈ti−1,ti〉
= ai(t− ti−1)

3+ bi(t− ti−1)
2+ ci(t− ti−1)+di (3.1)

where

ti =
T

nd
i for i =0,1, . . . ,nd

ai,bi,ci,di – coefficients
nd – number of sections intowhich the time interval 〈t0,T〉

has been divided.

Fig. 2. Function ϕ
w
(t) approximated by spline functions

As thedecisive variables in the optimisation task, components of the vector
below have been assumed.
Components of the following vector

X = [ϕw,1,ϕw,2, . . . ,ϕw,nd−1]
⊤ (3.2)

where ϕw,i = ϕw(ti) for i =1,2, . . . ,nd−1, are chosen as thedecisive variables
for the optimisation task considered.
An objective function is defined in the form

F = C1
mL

2
v
2
LT +C2‖rLT −rLF‖

2 (3.3)

where
rLT = rL|t=T ,vLT =vL|t=T – vector of coordinates and velocity of

the load for t = T
rLF – vector of desired coordinates of the

load for t = T
C1,C2 – coefficients (weights).

Theobjective functiondefinedabovemeans that one can expect that at the
endof slewingmotion the load is at aparticular point in spaceand furthermore



Experimental verification of a method... 447

its kinetic energy isminimal.Therefore, precise formulationof theoptimisation
task can be expressed in the following terms: find theminimum of the functio-
nal F presented by equation (3.3) by selection of values ϕw,1, . . . ,ϕw,nd−1 that
are components of the vector X (in definition (3.2)). The problem of opti-
misation considered here is a problemwithout constraints. TheNelder-Meads
method (Wit, 1986) has been used for its solution. Like most optimisation
methods, this one is also sensitive to a selection of the initial approximation.
In the paper, it is assumed that the initial approximation of the vector X

X0 = [ϕw,1,0, . . . ,ϕw,nd−1,0]
⊤ (3.4)

is obtained based upon the formulae

ϕw,i,0 = ϕw(ti)=















8ϕw,max
T4

t
3(−t+T) when t ¬

T

2
8ϕw,max
T4

(t−T)3t+ϕw,max when t >
T

2

(3.5)

where ϕw,max is the final angle of slewing motion.
This function fulfills the following conditions

ϕw(0)= 0 ϕ̇w(0)= 0

ϕw

(T

2

)

=
1

2
ϕw,max

ϕw(T)= ϕw,max ϕ̇w(T)= 0

(3.6)

The thus chosen function makes it possible that courses of ϕw = ϕw(t) and
ϕ̇w = ϕ̇w(t) curves are smooth.
Values of the coefficients C1 and C2 (Eq. (3.3)) are determined during

numerical simulations. The main criterion of this determination is the best
quality of the load positioning at the end of slewing motion.

4. Determination of the drive functions for intermediate angles of

slewing

In this part of the paper, intermediate angles of slewingmean angles belonging
to an interval determined by base angles. Determination of drive functions for
an intermediate angle will be presented in an example. Let us assume that
drive functions for two basic slew angles are known, e.g. for 60◦ and 90◦.



448 A. Maczyński, J. Płosa

These functions are shown in Fig.3 respectively as the black and the grey line.
They have been calculated according to the method discussed in Section 3.
For these functions, the forms of vectors of decisive variables (3.2) obtained
as a result of the optimisation task are known. In Fig.3, the marks (squares)
placed on suitable curves correspond with elements of these vectors. In the
first step linear interpolation is used to determine the time of motion Tk for
a required (intermediate) slew angle (in our example for 75◦).

Fig. 3. Example of definition of the drive function for an intermediate slew angle

Next, the time interval 〈0,Tk〉 is divided into nd sections (in the example:
nd =5) and values of the time ti are determined according to

ti =
Tk

nd
i i =0,1, . . . ,nd

Usingoncemore the interpolation andknowledge of decisive variables for basic
angles 60◦ and 90◦, values of the”partial” angles for t1, . . . , tnϕ−1 and the slew
angle of 75◦ can be calculated. Figure 3 presents geometrical interpretation
of determination of the ”partial” angle for ti = t4. The set of partial angles
obtained in this way is an equivalent of the vector of decisive variables in the
optimisation task. Similarly as for the basic angles, the continuous form of
the drive function, thus the relationship φw = φw(t), can be expressed using
spline functions.



Experimental verification of a method... 449

5. Experimental stand

The proposed method of final load positioning in slewing motion has been
experimentally verified on a physical model. A schematic diagram of the test
stand is presented in Fig.4. A photograph of the stand is shown in Fig.5. The
model of a jib crane is themain part of the stand. Elements of the Sandiama-
nipulator owned by the Department ofMechanics and Computer Engineering
Methods, University of Bielsko-Biała, have been used to build the stand. The
jib (element 4 in Figs.4 and 5), made of aluminium, close profile is fixed to
themanipulator pedestal (element 3 in Figs.4 and 5). The jib can be fixed at
three different angles. The rope (element 6 in Figs.4 and 5) with a steel ball
(element 5 in Figs.4 and 5) modelling the load is suspended at the end of the
jib. The point of suspension can be freely moved along the jib.

Fig. 4. Schematic diagram of the test stand; 1 – PCwith ”Test Point” software and
DA card, 2 – controller, 3 – manipulator pedestal with servo-motor, gear and rate

generator, 4 – jib, 5 – load, 6 – rope

The slew of the manipulator pedestal and, simultaneously, the slew of the
jib are realized by means of a brushless, direct-current RTM ct type servo-
motor with three-phase winding and six poles. This motor is provided with a
brushless, direct-current rate generator with the Hall rotor position sensors.
This generator is used in the control system of electronic commutation. Speed
of themotor (max 4000r.p.m.) is reduced by harmonic gear type HDUCwith
the transmission ratio 1:158. Control of speed of the brushlessmotors relies on
changing the time duration (length) of a current impulse (Tunia, 1983). Such
motors are commonly used in CNCmachines andmanipulators.

Figure 6 shows a diagram of the control system of motor speed. Themain
elements of the control system are: ”Test Point” software, digital-analogue



450 A. Maczyński, J. Płosa

Fig. 5. Photograph of the test stand

card DA, controller, motor and direct-current rate generator. The software
”Test Point” ver. 3.1 and the digital-analogue Keithley DAS 1802 HR-DA
card have been installed on a PC (element 1 in Figs.4 and 5). As a controller,
the transistor controller SYD 106 TH (element 2 in Figs.4 and 5) has been
applied. This control system of motor speed enables realization of different
forms of kinematic inputs.

Fig. 6. Diagram of the control system of motor speed

The ”Test Point” software is a professional object-oriented system which
leverages multi-threading capabilities of operating systems. It enables:

• I/O services for analogue and digital cards,

• data analysis, filtration and conversion,

• graphical presentation of data.

It can be useful for control or diagnostics (monitoring) of various processess in
industry and research laboratories. In the present investigation, special appli-
cation hasbeen compiled in the ”TestPoint” software. Theapplication enables



Experimental verification of a method... 451

afile to be readwith adesired time course of rotational speed of themotor and
a suitable digital signal to be generated. Because drive functions determined
according to Sections 3 and 4 prescribe the time courses of slew angles of the
jib, they have to be first recalculated onto corresponding rotational speeds of
the motor.

The signal generated by the ”Test Point” application is converted by the
digital-analogue card DAS 1802 HR-DA to an analogue signal (voltage). The
voltage from theDA card is used by the transistor controller SYD 106 TH. It
is a special PID controller designed for speed control of brushlessmotors. The
signal from the rate generator is also the input to the controller. Replaceable
resistors and capacitors (installed on the controller) have beenmatched to the
motor at the Electrotechnical Institute inWarsaw.

6. Experimental investigations

The experimental investigations presented in the paper were carried out for
the following parameters: length of the jib (dimension d in Fig.1) 2.243m,
mass of the load 3.25kg, length of the rope 1.61m, angle of jib slope 57.5◦.
The experiments were performed for inputs listed in Table 1.

Table 1

Slew Duration of
Drive functions

angle motion [s]

1 70◦ 4.1
a) function according to formula (3.5)
b) optimised function

2 90◦ 4.4
a) function according to formula (3.5)
b) optimised function

a) function according to formula (3.5)
3 80◦ 4.3 b) optimised function

c) interpolated function

Graphs in Figs.7a,b present time courses of the optimised drive functions
of jib slewing and the functions determined according to initial approximation
formula (3.5), respectively for 70◦ and 90◦. Figure 7c shows thedrive functions
for the slew angle of 80◦ and, additionally, the function determined according
to Section 4.



452 A. Maczyński, J. Płosa

Fig. 7. Drive function for angle: (a) – 70◦, (b) – 90◦, (c) – 80◦

Fig. 8. Experimentally recorded final part of load trajectory for slew of 70◦ and
drive function calculated according to (3.5): (a) – case 1a, (b) – case 1b



Experimental verification of a method... 453

Fig. 9. Experimentally recorded final part of load trajectory for slew of 90◦ and
drive function calculated according to (3.5): (a) – case 2a, (b) – case 2b

Figures 8-10 present courses of load trajectories (final parts) for the analy-
sed cases of the jib slewing registered bymeans of a digital camera. The time
of exposure was 8s. The position of a red LED mounted in the centre of the
load was traced in these photographs. Three frames were executed for each
case in order to confirm reproducibility of the obtained results.

7. Summary

The obtained experimental results prove significant effectiveness of the pro-
posed method of the final load positioning in slewing motion of a jib crane.
This statement involves both basic and intermediate angles. It is important
to note that in the presented experiment, the basic functions have been cal-
culated fairly infrequently – the step between them is 20◦. The courses from
photographs show that final load oscillations have been reduced from nearly
50cm down to about 4cm in each analysed case.

Because the method requires only an open-loop control system, it can be
recommended for final load positioning in simple rotary cranes, e.g. pillar or
wall cranes. It can be also used in the case of small mobile cranes.



454 A. Maczyński, J. Płosa

Fig. 10. Experimentally recorded final part of load trajectory for slew of 80◦ and
drive function calculated according to (3.5): (a) – case 3a, (b) – case 3b, (c) – case 3c

References

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Experimental verification of a method... 455

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6. Maczyński A., 2005,Pozycjonowanie i stabilizacja położenia ładunku żurawi
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9. Tanizumi K., Hino J., Yoshimura T., Sakai T., 1994,Modelling of dyna-
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Eksperymentalna weryfikacja metody końcowego pozycjonowania

ładunku żurawi obrotowych

Streszczenie

W pracy przedstawiono metodę końcowego pozycjonowania ładunku żurawi wy-
sięgnikowychwruchuobrotowym.Bazuje onana tzw.mapie funkcji bazowych,wktó-
rej zapamiętywane są funkcje napędowe obrotu nadwozia dla wybranych ”bazowych”
kątówobrotu.Funkcje tewyznaczanesąnadrodzeoptymalizacji dla określonychpara-
metrów eksploatacyjnych (kąt obrotu nadwozia, czas obrotu, wysięg, masa ładunku,
długość liny). Funkcje napędowe dla innych parametrów, w szczególności dla inny
kątów obrotu, obliczane są przez układ sterujący bezpośrednio przed rozpoczęciem
ruchu.W tym celu stosowana jest interpolacja liniowa. Proponowanametoda została
zweryfikowana eksperymentalnie na stanowisku badawczym.

Manuscript received May 7, 2007; accepted for print January 18, 2008