Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 2, pp. 645-657, Warsaw 2016
DOI: 10.15632/jtam-pl.54.2.645

CABLE-SUSPENDED CPR-D TYPE PARALLEL ROBOT

Mirjana Filipovic
Mihajlo Pupin Institute, University of Belgrade, Belgrade, Serbia; e-mail: mira@robot.imp.bg.ac.rs

Ana Djuric
Wayne State University, Detroit, U.S.A; e-mail: ana.djuric2@wayne.edu

Thispaperdealswith theanalysis andsynthesis of anewly selectedCable-suspendedParallel
Robot configuration,namedCPR-Dsystem.Thecameracarrierworkspacehas the shapeofa
parallelepiped.TheCPR-D systemhas a unique Jacobianmatrix thatmaps the relationship
between internal and external coordinates. This geometric relationship is a key solution
for the definition of the system kinematic and dynamic models. Because of the CPR-D
system complexity, the Lagrangeprinciple of virtualwork has been adapted.Two significant
Examples have been used for the CPR-D system analysis and validation.

Keywords: cable-suspended parallel robot, camera observation, kinematics, dynamics

1. Introduction

Asystem for observation of aworkspacewithmoving objects has been developed to some extent
andwidely analyzed invarious researchareas aswell as for different applications. Similar systems
have been analyzed andmodeled as presented by numerous publications.
Thekinematicdesignof aplanar three-degree-of-freedomparallelmanipulatorwas considered

by Gosselin and Grenier (2011). Four optimal different design criteria were established and
analyzed.Atrajectoryplanningapproach for cable-suspendedparallelmechanismswaspresented
by Gosselin et al. (2012). A planar two-degree-of-freedom parallel mechanism was used in the
analysis. Carricato (2011) studied the kinematics and statics of under-constrained cable-driven
parallel robotswith less than six cables in crane configuration.Amotion controller for a sixDOF
tendon-based parallel manipulator (driven by seven cables) which moves a platform with high
speedwas introducedbyFang et al. (2004).Acontrol designof theCPRsystemswas investigated
by Kraus et al. (2013), and Avci et al. (2014). The workspace conditions and the dynamics of
the manipulator were described in details. Borgstrom et al. (2007) presented algorithms that
enabled precise trajectory control of theNetworked InfoMechanical Systems (NIMS), andunder
constrained three-dimensional (3D) cabled robot intended for use in actuated sensing. Several
prototypes of the wire-driven parallel robots with different actuation schemes were presented
by Merlet (2010). Two of them were evaluated through extensive tests and showed unexpected
kinematic problems.Thedetermination of thisworkspacewas an important issuebyGouttefarde
et al. (2006) since the cables can only pull and not push on the mobile platform.
Parallel cable-driven Stewart-Gough platforms consist of an end-effector which is connected

to themachine frame bymotor driven cables. Since the cables can transmit only tension forces,
at least m= n+1 cables are needed to tense a system having n-degrees-of-freedom. This will
cause a kinematical redundancy and leads to an (m−n)-dimensional solution space for the
cable force distribution presented by Bruckmann et al. (2007). The recent result from a newly
designed parallel wire robotwhich is currently under constructionwas presented byPott (2008).
It is used for developing a new technique for computation and transfer of its workspace to the
available CAD software. An auto-calibration method for over constrained cable-driven parallel



646 M. Filipovic, A. Djuric

robots using internal position sensors located in the motors was presented by Miermeister et
al. (2012). The wire-driven parallel robot presented by Higuchi et al. (1988) has attracted the
interest of researchers since the very beginning of the study of parallel robots. Oh and Agrawal
(2005) addressed the issue of control design for a redundant 6-DOF cable robot with positive
input constraints. Nonlinear dynamic analysis of the suspended cable system was carried out
with some sensible results presented by Duan (1998) that could be useful to the real engine-
ering of LSRT (Large Spherical Radio Telescopes). Integrated mechanical, electronic, optic and
automatic control technologies are employed tomake considerable improvement upon the same
system.Amultiple cable robotic crane designed byShiang et al. (2000) is used to provide impro-
ved cargo handling. This is one of a few papers dealing with flexible ropes. For the requirement
of trajectory tracking of the LSRT, a large fine tuning platform based on the Stewart platform
was presented by Su and Duan (2000a,b). The mathematical model for kinematic control was
developed with coordinate transformation, and dynamic analysis was made using a Jacobian
matrix with singularity analysis, which built a solid base for the tracking control. Kozak et al.
(2006) in their paper addressed the static analysis of cable-driven robotic manipulators with a
non-negligible cable mass. A cable suspended parallel robot was analyzed by Zi et al. (2008), in
which cables were utilized to replace links tomanipulate objects. It was developed from a paral-
lel and serial cable-driven robot. Therefore, a cable system with j end-effectors DOFs requires
at least (j+1) cables as shown by Hiller and Fang (2005). For three-translational motions of
the feed in the system, a four-cable-driven parallel manipulator was developed. The goal of Yao
et al. (2010) was to optimize dimensions of the four-cable-driven parallel manipulator to meet
the workspace requirement of the constraint condition in terms of cable tension and stiffness.

However, the same CPR can be used inmany different applications, but only some configu-
rations can be used for the workspace observation.

In this paper, we present a novel construction of the CPR system which is named CPR-D,
seeFigs. 1 and 2.The camera carrier hangs over ropes properly connected to four highest points,
i.e. four upper angles of the parallelepiped workspace. The system has been designed with only
three motors and two ropes for the maximum workspace, which means that the system is not
redundant.Thisworkspace is double bigger than theworkspace of similar systemswith the same
number of motors. The CPR-D system has more advantages in comparison with other similar
systems. This system is secured from falling during motion because it is constructed with two
parallel ropes, see Fig. 1.

Fig. 1. CPR-D in 3D space

Acamera workspace is an area where a camera canmove silently and continuously following
the observed object. A camera carrier moves freely in the space enabling the shooting of the
objects from the above. This gives a unique feeling to the event observer to watch objects
from the unusual proximity without disturbances. The observer will be very close to the action



Cable-suspended CPR-D type Parallel Robot 647

Fig. 2. CPR-D, top view

regardless of the size of the observed space. Motion of the ropes which carry the camera is
controlled. The ropes can uncoil or coil, which allow the camera to reach any position in the
space. The control system provides three-dimensional motion of the camera. The commands for
synchronizedmotion of each winch are provided by controlling themotion of each motor which
ultimately provides the three-dimensional continuous camera carrier motion. The gyroscopic
sensor that is installed in the camera carrier is stabilized to the horizon. The nature of this
system requires development of a newmethodology for calculation of its kinematic and dynamic
models, which will be used for building the system.

This work will be extended by implementing elastic properties of the ropes in the kinematic
and dynamicmodels. The research of elasticity dynamics for nonlinear systemswas done by the
following authors: Raskovic (1965), Rega (2004a,b), Hedrih (2010, 2012). In this paper, theCPR
model has been generated using the following assumptions: transverse vibrations of the ropes
are neglected and the ropes are unstretchable.

InSection2, adetaileddescriptionof a selectedCPRsystemtypeand itsmathematicalmodel
is given.Most of that Section is devoted to its kinematicmodel, which is directly involved in the
development of its dynamicmodel. Two cases of the system responses are analyzed for different
conditions in Section 3, while in Section 4 concluding remarks are presented.

2. Mathematical model of CPR-D system

In this research, one subsystem of the CPR family has been selected and analyzed in depth. A
graphical representation of that system, named CPR-D, is shown in Figs. 1 and 2. The camera
carrier of theCPR-D structure is guided through thework area of the parallelepiped shapewith
two ropes connected with three winches, each powered by a motor. The ropes coil or uncoil on
the winches of radius R1, R2 and R3. The motors rotate the winches directly, and the motor
shafts angular positions after gear boxes are θ1, θ2, θ3. This motion moves the camera in the x,
y, z Cartesian coordinates.



648 M. Filipovic, A. Djuric

Thefirst step towards the dynamicalmodel of theCPR-D is the development of its kinematic
model. The geometrical relationship between the lengths k, h, m, n and Cartesian coordinates
x, y, z is defined by the following equations

k=
√

x2+y2+z2 h=
√

(d−x)2+y2+z2

m=
√

(d−x)2+(s−y)2+z2 n=
√

x2+(s−y)2+z2
(2.1)

In Fig. 3, motions of motors 1 and 2 are depicted. Motor 1 (motor 2 as well) works so that it
wounds its corresponding rope from one side and unwounds from other side.

Fig. 3. Rope forces before motor 1, 2 and 3 and after motor 1 and 2

Motions of motors 1 and 2 toward the wall anchors (we call this line “before” motor) are
expressed with the following equations respectively

∆θ1

∆t
R1 =

∆h

∆t
+
∆m

∆t

∆θ2

∆t
R2 =

∆m

∆t
+
∆n

∆t
(2.2)

The third motor is used to wind up the two ropes about coil 3 in the k, h, m, n directions.
This motion produces winding or unwinding of both ropes at the same time. This can be seen
in Figs. 2 and 3. The winch used for winding the ropes has radius Ri, i= 1,2,3. The relation
between the thirdmotor motion changes∆θ3, and the lengths change∆k,∆h,∆m,∆n can be
expressed either with equation (2.3)1 or (2.3)2

∆θ3

∆t
R3 =

∆k

∆t
+
∆h

∆t
+
∆θ2

∆t
R2

∆θ3

∆t
R3 =

∆k

∆t
+
∆n

∆t
+
∆θ1

∆t
R1 (2.3)

Equation (2.4) is obtained by substituting equation (2.2)2 into (2.3)1, or equation (2.2)1 into
(2.3)2

∆θ3

∆t
R3 =

∆k

∆t
+
∆h

∆t
+
∆m

∆t
+
∆n

∆t
(2.4)

If the sampling time∆t is small enough then equations (2.2) and (2.4) can be expressed, respec-
tively, as

θ̇1R1 = ḣ+ ṁ θ̇2R2 = ṁ+ ṅ θ̇3R3 = k̇+ ḣ+ṁ+ ṅ (2.5)

By differentiating equations (2.1) and substituting them into equations (2.5), the relationship
between thevelocities of the cameracarrier in theCartesian space ṗ= [ẋ, ẏ, ż]T andthevelocities
of the internal coordinates φ̇= [θ̇1, θ̇2, θ̇3]

T can been obtained as following

φ̇=Jdṗ (2.6)



Cable-suspended CPR-D type Parallel Robot 649

This procedure is named KinCPRD-Solver (Kinematic Cable Parallel Robot D-type Solver). It
is clear that the JacobianmatrixJd [1/m] is a full matrix, and its elements beyond the diagonal
show strong coupling between the external and internal coordinates.
Thekinetic energyEk and thepotential energyEp of the camera carriermotionwithmassmc

are given in the following equations

Ek =
1

2
mcẋ

2+
1

2
mcẏ

2+
1

2
mcż

2
Ep =mcgz (2.7)

The gravitational acceleration is g=9.81m/s2. TheCPR-D system has threemotors, and their
mathematical model is expressed with vector equation (2.8) (Vukobratovic, 1989)

u=Gvφ̈+Lvφ̇+SvMd (2.8)

where u = [u1,u2,u3]
T is voltage, Gv = diag(Gvi) – motor inertia characteristic, Lv =

diag(Lvi) – motor damping characteristic, Sv = diag(Svi) – motor geometric characteristic,
Md – motor load torque

Gvi =
JriRri

CMi
Lvi =

RriBCi

CMi
+CEi Svi =

Rri

CMi
(2.9)

where: Rri [Ω] is rotor circuit resistance, CEi [V/(rad/s)] – back electromotive force constant,
CMi [Nm/A] – constant of torque proportionality, BCi [Nm/(rad/s)] – coefficient of viscous
friction, Jri [kgm

2] – moment of inertia for the rotor and the gear box.

Fig. 4. Rope forces carrying the camera

Vector equation (2.8) is based on Lagrange’s equation of motion. The angular positions of
the motors shafts θ1, θ2, θ3 are selected as generalized coordinates.
Themotor load torqueMd is defined with vector equation

Md = [F1R1,F2R2,F3R3]
T (2.10)

The load force Fd includes three components expressed in a vector form as Fd = [F1,F2,F3]
T.

This force is acting on the shaft of each motor, and its value depends on the external force F.
The external forceF= [Fx,Fy,Fz]

T represents the sum of the camera inertial forceFp which is
acting on the camera carrier described in equation (2.11)2 and the perturbation forcePp which
is disturbing the camera motion

F=Fp+Pp Fp =m(p̈+acc) (2.11)

Thevectoracc = [0,0,−g]
T represents the gravitational acceleration. Thenext step is todescribe

the dynamic balance between the forceFm in them rope direction and the external forceF, see



650 M. Filipovic, A. Djuric

Fig. 5. (a) Characteristic triangles in 3D space, (b) characteristic triangle in the (d-x)-m plane,
(c) characteristic triangle in the (s-y)-m plane, (d) characteristic triangle in the z-m plane

Figs. 5a-5d. Two similar right-angle triangles in the (d-x)-m plane, in the (s-y)-m plane, and
in the z-m plane are presented in 2D space, see Figs. 5b-5d, respectively. All of these triangles
can be seen in 3D space, which is shown in Fig. 5a. The hypotenuses of the GABx, GABy,
GABz triangles has length m, which is changeable during motion of the camera. The other
two sides of the GABx, GABy, GABz triangles have sizes d−x and k(s−y)z =

√

(s−y)2+z2,

s− y and k(d−x)z =
√

(d−x)2+z2, z and k(d−x)(s−y) =
√

(d−x)2+(s−y)2, respectively.
The component of the external force F in the x direction is Fx, in the y direction is Fy, in
the z direction is Fz. The projection forces Fx, Fy, Fz on the m direction are Fmx, Fmy, Fmz,
respectively, which can be seen in Figs. 5b-5d. The similarities of the two triangles in Fig. 5
produce the following relations

d−x

m
=
Fmx

Fx
Fmx =

d−x

m
Fx

s−y

m
=
Fmy

Fy

Fmy =
s−y

m
Fy

z

m
=
Fmz

Fz
Fmz =

z

m
Fz

(2.12)

The forceFm is a sum of the previously defined components and it is expressed in the following
equation

Fm =Fmx+Fmy +Fmz =
d−x

m
Fx+

s−y

m
Fy +

z

m
Fz (2.13)

The Lagrange principle of virtual work has been used to find the relation between the motor
load torqueMd and the external forceF

(Md)
T
φ̇=FTṗ (2.14)

By substituting equations (2.6) into (2.14), the following equations are generated

(Md)
T
Jdṗ=F

T
ṗ (Jd)

T
Md =F (2.15)

From (2.15)2, equation (2.16) can be expressed as

Md =
(

(Jd)
T
)−1
F (2.16)

Equation (2.16) cannot be directly applied to the system from Figs. 1 and 2 because of the
following reasons:



Cable-suspended CPR-D type Parallel Robot 651

• The system has two ropes in each direction. Equation (2.16) has been corrected using the
factor ♦=0.5.

• Motor 3 is used to synchronize winding or unwinding of the two ropes.

These will produce the adopted Jacobian matrix Jx♦d. The matrix Od [m] is generated, which
represents the torquemappingmatrix, as defined below

Od =
(

(Jx♦d)
T
)−1

(2.17)

The adapted Lagrange’s principle of virtual work has been used for solving the complex relation
between themotor load torqueMd (acting as a load on the first, second and thirdmotor shaft)
and external forces F (acting on the camera carrier)

Md =OdF u=Gvφ̈+Lvφ̇+SvOdF (2.18)

The torque mapping matrix Od [m] indicates that the system is highly coupled. The control
law is selected by the local feedback loop for the position and velocity of themotor shaft in the
following form

ui =Klpi(θ
o
i −θi)+Klvi(θ̇

o
i − θ̇i) (2.19)

whereKlpi is a position constant, andKlvi is a velocity constant for the motion control.
The comparison between previously published papers and this research is summarized as in

the following:

• The novel KinCPRD-Solver gives a relation between the internal and external coordinates
through the unique Jacobian matrix Jd.

• All threemotors in this systemare differently integrated in comparisonwith the previously
published systems.

• Most of thepreviously publishedpapers donot involve dynamics of themotor.TheCPR-D
system includes the motors which significantly influence the total system response.

• TheCPR-D system construction requires a novel dynamic relation between the load forces
which are acting on the camera and the forces in them direction.

• The previously published systems usedLagrange’s principle of virtual work for calculating
the relation between the external and internal forces in the original form. This system has
double ropes in all four directions, which requires an adaptation of the Lagrange principle
of virtual work.

3. Simulation results

The CPR-D system presented in Figs. 1 and 2 is modeled and analyzed by the software pac-
kage AIRCAMD. The software package AIRCAMD is used for validation of applied theoretical
contributions. This software includes three essential modules which are kinematic, dynamic and
motion control law solvers for the CPR-D system. The most important element of the CPR-D
system is the mathematical model of the motor which is an integral part of the software pac-
kage AIRCAMD. Through the simulation results, it is shown that the dynamic characteristics
of the motor significantly affect the response of the system and its stability. In order to make
the results comparable, simulation ismade for the same desired system parameters. The camera
carrier motion dynamics directly depends on the mechanism dynamic parameters. The camera
moves in the 3D space (x, y, z directions). The workspace is characterized by length d=3.2m,
width s= 2.2m and height v = 2.0m of the recorded field. The position of the camera carrier



652 M. Filipovic, A. Djuric

in the Cartesian space is p = [x,y,z] [m]. The starting point is postart = [0.3m,1.8m,−0.2m],
and the end point poend = [0.9m,1.2m,−0.9m], They are presented in Fig. 6a, while their re-
ference velocities are shown in Fig. 6c. In Fig. 6b, we show the reference composite velocity of
the camera carrier. The shape of the composite velocity is trapezoidal.

Fig. 6. The reference trajectorymotion of the camera carrier (a) position xo, yo, zo, (b) velocity
(maximum value: po

max
=0.494m/s, (c) velocity components ẋo, ẏo, żo (Examples 1, 2)

The motors are Heinzman SL100F type and gears are HFUC14-50-2A-GR+belt type. The
characteristics of themotors are:Rri =0.917Ω– rotor circuit resistance,CEi =3.3942V/(rad/s)
– back electromotive force constant, CMi =2.5194Nm/A – constant of torque proportionality,
BCi =0.0670Nm/(rad/s) – coefficient of viscous friction,Jri =1.5859kgm

2 –moment of inertia
for the rotor and the gear box.
The sample time is dt=0.0001s. The positional and velocity motion controller parameters

are Klpi = 4200 and Klvi = 130, respectively. Winches radii are Ri = 0.15m. The system
responses are comparable and therefore are shown in Table 1. The results for Example 1 are
presented in Figs. 7 and 10a, while for Example 2 are presented in Figs. 8 and 10b. Figure 7
(and Fig. 8 as well) has six pictures related to:

a) camera carrier position at the reference and the real frames,

b) motor shaft position at the reference and the real frames,

c) load force at the reference and the real frames,

d) deviation between the real and the reference trajectory of the camera carrier,

e) deviation between the real and the reference trajectory of the motor shaft positions,

f) control signals at the reference and the real frames.

Table 1.Comparison of two selected Examples

Example 1 2

Figure 7, 10a 8, 10b

Mathematical model of the system at the reference frame
(2.1)-(2.19) (3.1)-(3.5)

is defined by equations

Camera carrier is under the influence of the disturbance force yes yes

System at the reference frame is coupled yes no

The CPR-D system is designed for outdoor use. Because of that, we analyzed the system
behavior under the influence of wind impacts, determined as the force Fp = [100(sin(4πt) +
sin(32πt)),0,0]T, see Fig. 9. The force has a sine shape and operates only in the x direction,
while the components in the y and z directions are zero.

Example 1:Themotion responseof the camera carrier has oscillatory characteristics andangular
positions of all threemotors are caused by the sinusoidal disturbance force. There is a very good



Cable-suspended CPR-D type Parallel Robot 653

Fig. 7. Example 1

Fig. 8. Example 2

tracking of the desired trajectory at the camera carrier real frame and at the motor motion
real frame, until the moment when motor 2 enters the saturation in oscillatory manner, see
Fig. 7f. The first and the third motor do not enter the saturation at all. The force Fm has two
components of the same magnitude Fm1 = Fm2 acting in each rope in the m direction at the
reference and the real frames, which is presented in Fig. 10a.

Example 2: All system and control parameters are the same as in Example 1. This Example is
donewith one illogical assumption,which is that the systemat the reference frame is uncoupled.
In that case, the Jacobian matrix Jd⊕ has the diagonal form

φ̇=Jd⊕ṗ (3.1)



654 M. Filipovic, A. Djuric

Fig. 9. Perturbation force at the camera carrier (Examples 1, 2)

Fig. 10. Components of the force F
m
acting in them direction at the reference and the real frames,

(a) Example 1, (b) Example 2

Themathematical model of the system has the following form

u=Gvφ̈+Lvφ̇+SvMd⊕ (3.2)

Using the adapted Lagrange principle of virtual work, the relationship between the motor load
torqueMd⊕ and the external forceF has been given below

Md⊕ =Od⊕F (3.3)

The diagonal adopted Jacobianmatrix isJx♦d⊕. ThematrixOd⊕ is generated, which represents
the torquemappingmatrix, as defined below

Od⊕ =
(

(Jx♦d⊕)
T
)−1

(3.4)

The torque mapping matrix Od⊕ is diagonal like the Jacobian matrix Jd⊕. Substituting (3.3)
into equation (3.2) produces a dynamicmodel of the uncoupledCPR-D system at the reference
frame

u=Gvφ̈+Lvφ̇+SvOd⊕F (3.5)

In this case, the mathematical model of the CPR-D system at the reference frame is defined by
equations (3.1)-(3.5). At the real frame, the system is coupled and its kinematic and dynamic
models are defined by equations (2.1)-(2.19), see Fig. 8. The coupling characteristics are not
taken into the consideration at the reference frame. Due to that fact, the tracking of the referent
trajectory in the Cartesian space is not satisfactory, see Figs. 8a and 8d.
Thepositioncontrol lawproduces the ideal responseof themotor angularmotion, seeFigs. 8b

and 8e.
The forcesFm1 =Fm2 acting in each rope in them direction at the reference and real frames

are presented in Fig. 10b. Example 2 has an important theoretical meaning, because it confirms
the strong coupling between the external and internal coordinates.



Cable-suspended CPR-D type Parallel Robot 655

The presented results imply that the dynamics of the individualmotor significantly depends
of the selection of theCPR-Dstructure and its parameters.TheCPR-D ismodeled andanalyzed
by the software package AIRCAMD.

4. Conclusion

The highly authentic general mathematical model for the CPR-D system has been developed.
This model represents novel kinematic and dynamic solutions of the complex Cable suspended
Parallel Robot structure. TheCPR-D system is selected to carry the camera through four pivot
pointswhich produce a 3Dworkspace of a parallelepiped shape.The camera carrier is controlled
by two ropes in each of the three directions, and driven by threemotors. The kinematicmodel is
defined for themonitored system via the Jacobianmatrix. The generalized coordinates selected
for the CPR-Dmodel are angular positions of themotors θ1, θ2, θ3 named internal coordinates.
Cameramotion is defined in the Cartesian space, describedwith the x, y, z coordinates, named
the external coordinate system. The relation between the internal and external coordinate sys-
tems is described by the Jacobian matrix Jd. This relation represents the kinematic model of
the CPR-D system. The solution for the CPR-D kinematic structure has been found through
a novel procedure named KinCPRD-Solver (Kinematic Cable Parallel Robot D-type Solver)
which is developed and validated using two selected Examples. The relation between themotor
load torque and the force acting at the camera carrier is described by the Lagrange principle
of virtual work. This calculation shows that in this relation, the Jacobian matrix is involved.
Because of the construction complexity of this system, the Lagrange principle of virtual work
had to be adapted for two reasons. The software package AIRCAMD has been developed and
used for individual analysis of theCPR-Dmodel fromvarious aspects such as selecting different
workspace dimensions, camera carrier mass, external disturbances, choice of the control law,
reference trajectory, avoidance of singularity andmany other characteristics.
The future research will involve elastic ropes (type of nonlinear dynamic elasticity as de-

fined by Filipovic et al. (2007), Filipovic and Vukobratovic (2008a,b), Filipovic (2012) in the
mathematical model of the CPR system. Different CPR models previously developed will be
unified according to their similarities into a single reconfigurable model, using themethodology
presented byDjuric et al. (2010, 2012). Stability conditions, sensitivity analysis and singularity
analysis of the CPR-D systemwill be done in the future research.

Acknowledgment
This research has been supported by the Ministry of Education, Science and Technological Deve-

lopment, Government of the Republic of Serbia through the following two projects: Grant TR-35003
„Ambientally intelligent service robots of anthropomorphic characteristics”, byMihajlo Pupin Institute,
University of Belgrade, Serbia, and partially supported by the project SNSF Care-robotics project No.
IZ74Z0-137361/1by Ecole Polytechnique Federale de Lausanne, Switzerland.

Weare grateful toProf.Dr.KaticaR. (Stevanovic)Hedrih fromtheMathematical Institute,Belgrade

for helpful consultations during preparation of this paper.

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Manuscript received February 10, 2014; accepted for print October 21, 2015