Modelling of the gravity compensators in robotic manufacturing cells 
 
A. Klimchik*, Y. Wu*, S. Caro**, C. Dumas***, B. Furet*** and A. Pashkevich* 

*Ecole des Mines de Nantes, 4 rue Alfred-Kastler, Nantes 44307 France (Tel: +33-251-85-83-00; e-mail:  
alexandr.klimchik@mines-nantes.fr, yier.wu@mines-nantes.fr, anatol.pashkevich@mines-nantes.fr). 
**Centre National de la Recherche Scientifique, France (e-mail:stephane.caro@irccyn.ec-nantes.fr) 
*** University of Nantes, France,  (e-mail: claire.dumas@univ-nantes.fr, benoit.furet@univ-nantes.fr) 
Abstract: The paper deals with the modeling and identification of the gravity compensators used in 
heavy industrial robots. The main attention is paid to the geometrical parameters identification and 
calibration accuracy. To reduce impact of the measurement errors, the design of calibration experiments 
is used. The advantages of the developed technique are illustrated by experimental results . 
Keywords: industrial robot, calibration, gravity compensator, geometrical modeling. 

1. INTRODUCTION 
Currently, 
aeronautic 
industry 
requires 
high-precision 
machining of huge aircraft component made from the high 
performance materials. For these applications, robots are 
quite attractive due to their large workspace that can be also 
easily extended. Another considerable advantage is their 
capability to process the parts with complex shape and 
geometry. However, processing of these materials causes 
essential cutting forces that are generated by tool-workpiece 
interaction during the material removal. Since these forces 
decrease the machining accuracy, the robot manufactures try 
to improve the robot stiffness by increasing the link cross-
sections. This approach obviously leads to augmentation of 
robot link weights. So, the gravity forces applied to the 
manipulator components become non-negligible and also 
contribute to the position errors. To overcome this difficulty, 
the link weights are tended to be balanced by gravity 
compensators, which considerably complicate the stiffness 
modeling of these heavy manipulators. 
The problem of stiffness modeling for the heavy 
manipulators with gravity compensators has been in the focus 
of rather limited number of works. In contrast, for 
conventional 
serial 
manipulators 
without 
gravity 
compensators, the problem has been studied by a number of 
authors that considered both industrial and medical robots 
with essential compliance in the links and joints (Meggiolaro 
2005, Kövecses 2007). Relevant works are mainly based on 
the virtual joint method (VJM), which lumps the elastostatic 
properties of the robot components in virtual springs. To our 
knowledge, the stiffness modeling for the manipulators with 
gravity compensators has not been studied in detail yet. 
Currently, the main activity in this area focuses on the gravity 
compensator design (Takesue 2011, De Luca 2011). On the 
other hand, since the considered robots include closed loops 
induced by the compensators, some technique previously 
developed for the parallel manipulators can be adopted 
(Bouzgarrou 2004, Company 2005, Pashkevich 2011). 
This paper focuses on the geometrical and stiffness modeling 
of the spring-based gravity compensators that can be 
integrated in a VJM-based stiffness model of a serial 
manipulator (Klimchik 2012). The main attention is paid to 
the identification of the model parameters and calibration 
experiment planning. The developed approach is confirmed 
by the experimental results that deal with the industrial robot 
employed in manufacturing of large-dimensional aircraft 
components. 
To address these problems, the remainder of the paper is 
organized as follows. Section 2 presents the gravity 
compensator model. In Section 3, calibration methodology is 
presented. Section 4 proposes identification algorithms. 
Sections 5 is devoted to the experiment design. Experimental 
validation is presented in Section 6. Finally, Section 7 
summarizes the main contributions.  
2. GRAVITY COMPENSATOR MODEL 
The mechanical structure of the gravity compensator under 
study is presented in Fig. 1.  The compensator incorporates a 
passive spring attached to the first and second links, which 
creates a closed loop that generates the torque applied to the 
second joint of the manipulator. Corresponding model is 
presented in Fig. 2, where the most essential geometrical 
parameters are denoted as 
xa , 
y , 
 and the compensator is 
described by the spring compliance
ck  and the preloadi g 
0
a
L
 
n
s . This design allows us to limit the stiffness model 
modification by incorporating in it the compensator torque 
and adjusting the virtual joint stiffness matrix that here 
depends on the second joint variable 
 only. 
2q
The compensator geometrical model includes three node 
points P0, P1, P2, where two distances 
1
2
,
P P , 
0
2
,
P P  are 
constants and the third one 
0
1
,
P P  varies and depends on 
. 
Let us denote them 
2q
1
2
,
L
P P

, 
1
2
,
a
P P

, 
1
2P
,
s
P

. 
Besides, let us introduce the angles ,  and the distances 
xa  and 
, whose geometrical meaning is clear from Fig. 2. 
y
a
 
    
 
 
 
Using these notations, the variable s  describing the 
compensator spring deflection can be computed from the 
expression  
2
2
2
2
· · ·co
2
s(
a
L
)
s
a L
q





 
(1) 
which defines the function 

2
s q
. 
0P
2P
1P
01
P
02
P
 
Fig. 1. Gravity compensator of robot KUKA KR-270 TM  
xa
y
a
2q



L
0P
P
1P
2
a
ck
2
qk
sF
cF
lF
O
s
 
Fig. 2. Model of the gravity compensator  
This mechanical design allows to balance the manipulator 
weight for any given configuration by adjusting the 
compensator spring preloading. It can be taken into account 
by introducing the zero-value of the compensator length 
0s  
corresponding to the unloaded spring. Under this assumption, 
the compensator force applied to the node P1 can be 
expressed as follows 
0
·(
)
s
c
s
F
K
s


 
(2) 
where 
 is the compensator spring compliance. 
ck
Further, the angle  between the compensator links P0P1 and 
P1P2 (see Fig. 2) can be found from the expression 
2
sin
s·
(
ni
q
a s
)




 
(3) 
which allows us to compute the compensator torque 
c
M  
applied to the second joint 
0
·(1
)· · ·
(
/
sin
c
c
s
s
L
M
a
K




2)
q
2
q
 
(4) 
Upon differentiation of the latter expression with respect to 
2 , the equivalent stiffness of the second joint (comprising 
both the manipulator and compensator stiffnesses) can be 
expressed as: 
q
2
2
0
·
·
c
K
K
K
L
a





 
(5) 
where the coefficient  
2
2
0
2
2
2 sin
cos
co
·
(
s
)
(
)
(
q
s
a L
q
q
s
s















2)
q

 
(6) 
highly depends on the value of joint variable 
2  and the 
initial preloading in the compensator spring described by 
q
0s .  
Hence, using expression (5), it is possible to extend the 
classical stiffness model of the serial manipulator  


1
(F)
1
(F)T
C
θ
θ
θθ
θ
(
)




K
J
K
H
J
 
(7) 
by modifying the virtual spring parameters in accordance 
with the compensator properties. Here, 
C
θ  are stiffness 
matrices in the Cartesian and joint spaces respectively, 
θ  are corresponding Jacobian and Hessian matrices 
(for more details see Klimchik 2012). While in the paper this 
approach has been used for the particular compensator type 
(spring-based, acting on the second joint), the similar idea 
can be evidently applied to other compensator types. 
,
K
K
(F)
θ
θ
,
J
H
Summarizing this Section, it is worth mentioning that the 
geometrical and elastostatic models of a heavy manipulator 
with a gravity compensator should include some additional 
parameters (
xa , 
y , 
 and 
c
a
L
K , 
0s  for the presented case) 
that are usually not included in datasheets. For this reason, 
the following Sections focus on the identification of the 
extended set of manipulator parameters.  
3. MODEL CALIBRATION METHODOLOGY 
In contrast to the serial manipulator that can be treated as a 
principal mechanism of the considered robots, geometrical 
data concerning gravity compensators are usually not 
included in the technical documentation provided by the 
robot manufacturers. For this reason, this Section focuses on 
the calibration methodology of the geometrical parameters 
for the described above compensator mechanism (see Fig. 1). 
The geometrical structure of the considered gravity 
compensator is presented in Fig. 2. Its principal geometrical 
parameters are denoted as 
, 
L
xa , 
y , where 
a
·cos
xa
a


, 
·sin
ya
a


 (see notation in Section III). As follows from 
the figure, the identification problem can be reduced to the 
determination of relative locations of points P0 and P1 with 
respect to P2. 
It is assumed that the measurement data are provided by the 
laser tracker whose "world" coordinate system is located at 
the intersection of the first and second actuated manipulator 
joints. The axes Y, Z of this system are aligned with the axes 
of joints #1 and #2 respectively, while the axis X is directed 
to ensure right-handed orthogonal basis. To obtain required 
data, there are several markers attached to the compensator 
mechanism (see Fig. 3). The first one is located at point P1, 
which is easily accessible and perfectly visible (the center of 
the compensator axis P1 is exactly ticked on the fixing 
element). In contrast, for the point P0, it is not possible to 
 
    
 
 
 
locate the marker precisely. For this reason, several markers 
P0i are used that are shifted with respect to P0, but located on 
the rigid component of the compensator mechanism (these 
markers are rotating around P0 while the joint coordinate 
2  
is actuated). It should be noted that for the adopted 
compensator geometrical model (which is in fact a planar 
one), the marker location relative to the plane XY is not 
significant, since the identification algorithm presented in the 
following sub-section will ignore Z-coordinate.  
q
Using this setting, the identification problem is solved in two 
steps. The first step is devoted to the identification of the 
relative location of points P1 and P2. Here, for different 
values of the manipulator joint coordinates 
2
{
i , 
q
1, }
i
m

, 
the laser tracker provides the set of the vectors 
1
ip
 
describing the points that are located in an arc of the circle. 
After matching these points with a circle, one can obtain the 
desired value of 
 (circle radius) and the Cartesian 
coordinates 
2  of the point P2 (circle center) with respect to 
the laser tracker coordinate system. 
L
p
xa
y
a
2q

L
0P
2P
1P
a
O
02
P
01
P
1
R
2
R
2

1


,
,
x
y
L a
a
03
P
04
P
3
R
4
R
 
Fig. 3. Geometrical parameters of the gravity compensator 
and location of the measurement points labelled with markers  
The second step deals with the identification of the relative 
location of points P0 and P2. Relevant information is 
extracted from two data sets 

01
ip
 and 
 that are 
provided by the laser tracker while targeting at the markers 
P01 and P02. Here, the points are matched to two circle arcs 
with the same center (explicitly assuming that the 
compensator model is planar), which yields the Cartesian 
coordinates 
0  of the point P0, also with respect to the laser 
tracker coordinate system. Finally, the desired values 


02
ip
p
xa , 
y  
are computed as a projection of the difference vector 
2
 on the corresponding axis of the coordinate 
system. 
a
0


a
p
p
As follows from the presented methodology, a key numerical 
problem in the presented approach is the matching of the 
experimental points with a circle arc. It looks like a classical 
problem, however, there is a particularity here caused by 
availability of additional data 

2i
q
 describing relative 
locations of the points 
. This feature allows us to 
reformulate the identification problem and to achieve higher 
accuracy compared with the traditional approach.  

1
ip
4. IDENTIFICATION ALGORITHMS  
The above presented methodology requires solution of two 
identification problems. The first one aims at approximating 
of a given set of points (with additional arc angle argument) 
with an arc circle, which provides the circle center and the 
circle 
radius. 
The 
second 
problem 
deals 
with 
an 
approximation of several sets of points by corresponding 
number of circle arcs with the centers on the same rotation 
axis. Let us consider them sequentially. 
4.1  Algorithm #1 
To match the given set of points 
 with additional set of 
angles 

ip

iq
 with a circle arc, let us define the affine mapping 
i
i



p
Ru
t  
(8) 
where 
 denotes the set of reference 
points located on the unit circle whose distribution on the arc 
is similar to 
i , 
[cos
, sin
, 0]T
i
i
i
q
q

u
p
 is the scaling factor that defines the 
desired circle radius, 
 is the orthogonal rotation matrix,  
is the vector of the translation that defines the circle center. It 
worth mentioning that such a formulation has an advantage 
(in the sense of accuracy) comparing to a traditional circle 
approximation and it is a generalization of Procrustes 
problem known from the matrix analysis. 
R
t
Using equation (8), the identification can be reduced to the 
following optimization problem 



1
,
,
min
T
i
i
i
i
i
m
F











R t
p
Ru
t
p
Ru
t
 
(9) 
which should be solved subject to the orthogonality 
constraint 
T

R R
I . After differentiation with respect to t , 
the latter variable can be expressed as  
1
1
1
1
i
m
i
m
m







t
p
R
i
m
i

u  
(10) 
That leads to the simplification of (9)  to 



1
,min
m
i
r
T
i
i
i
i
F








R
p
Ru
p
Ru




 
(11) 
where 
1
1
1
1
;
i
i
i
i
m
i
i
i
i
m
m









p
p
p
u
u
u
m


 
(12) 
Further, differentiation with respect to  yields to 
1
1
m
T
i
i
i
i
m
i





p Ru
u u
T
i


 
(13) 
So, finally, after relevant substitutions the objective function 
can be presented as  
1
1
1
1
2
min
m
m
m
i
i
i
T
T
T
i
i
i
i
i
i
F






















R
p p
u u
p Ru




 
(14) 
where the unknown matrix 
 must satisfy the orthogonality 
constraint 
R
T

R R
I . Since the matrix 
 is included in the 
second term only, the problem can be further simplified to 
R


1
1
max
m
m
T
i
i
i
i
i
i
F
trace







R
p Ru
R
u p T




 
(15) 
 
    
 
 
 
and can be solved using SVD-decomposition of the matrix  
1
m
i
i
T
i



u p
U ΣV


 
(16) 
where the matrices 
 are orthogonal and 
 is the 
diagonal matrix of the singular values. Further, using the 
same approach as for the Procrustes problem, it can be 
proved that the desired rotation matrix can be computed as 
,
U V
Σ
T

R
V U  
(17) 
which sequentially allows to find the scaling factor  
defining the arc radius and the vector 
 defining the arc 
center. 
t
4.2  Algorithm #2 
The second problem aims at approximating of several point 
sets 
i
 by corresponding number of concentric 
circle arcs with the centers 
0  on the rotation axis n . It 
should be noted that the data set 


1 ,...,
k
ip
p
j
p

iq
 is not useful here, since 
the required angles 
i
 are not measured directly and 
cannot be computed without having exact compensator 
geometry. In this case, the objective function can be written 
in the straightforward way 





0
2
2
0
0
1
1
,
min
j
j
k
m
j
j
j
j
j
i
i
j
i
R
T
F
R







p
p
p
p
p

 
(18) 
where 
j
R  denotes the j-th ark radius and 
0
j
p  is the 
corresponding center point. However, for this formulation it 
can be easily proved that the optimization problem (18) does 
not lead to a unique solution. In fact, it gives the rotation axis 
passing through the center points 
0
j
p , which can be expressed 
as  
0
c
j
j



p
p
n  
(19) 
where 
 is the axis direction vector, 
c
p  is a point belonging 
to the axis, and 
n
j are corresponding scalar factors.  
To solve the problem (18), first the objective function F  can 
be differentiate with respect to 
2
j
R  that yields the following 
expressions for the arc radii 


2
1
0
1
m
j
j
j
T

0
j
j
i
i
i
R
m




p
p
p
p

min
s

 
(20) 
Further, after relevant substitution, the objective can be 
rewritten as  




1
1
2
2
k
m
j
j
i
i
T
c
j
j
i
F







p
n
p

 
(21) 
where 
1
1
1
;
m
m
1
j
j
j
j
j
j
i
i
l
i
i
i
i
i
l
T
T
m
s
m








p
p
p
p p
p p
j
j
l



 
(22) 
To solve the above mentioned ambiguity, additional 
objectives should be considered  
2
min
j
j
R
R 
 
(23) 
which leads to the following solution for the scalar parameter 


1
1
T
m
i
c
i
j
j
m






p
p
n  
(24) 
Further, after differentiation (21) with respect to 
  
c
p


1
1
1
1
1
2
k
m
k
m
j
j
i
i
i
i
j
i
c
j
i
T
j
j
s







p p
p
p



 
(25) 
one can compute the point on the desired rotational axis as  


1
1
1
1
1
2
k
m
k
m
1
j
j
c
i
i
j
i
j
i
T
j
j
i
i
s








p
p p
p



 
(26) 
The remaining unknown vector n  can be obtained from the 
orthogonality constraints 

0
T
i 

n
0
j
j
p
p
, 
1,
,
1,
i
m j


k  
that leads to the following optimization problem 




1
2
0
1
min
k
m
j
j
T
i
j
i
f





n
p
p
n

 
(27) 
that after substitution in it (19), (24) and (26) gives 


2
1
1
min
k
m
j
i
j
T
i
f



n
p n


 
(28) 
Further, differentiation (28) with respect to 
, the 
optimization problem reduces to the solving following 
homogeneous linear equation system   
n
1
1
k
m
j
j
i
i
j
i
T




p p
n
0

 
(29) 
Non-trivial solution of this system can be found using the 
singular value decomposition of the matrix 
1
1
k
m
j
j
i
i
j
i
T



p p

  
1
1
k
m
j
j
i
i
j
i
T




p p
U ΣVT

 
(30) 
where the vector 
 is the last column of the matrix 
. 
n
V
It should be mentioned that practical application of the latter 
expressions is essentially simplified by the adopted 
assumption 
concerning 
orientation 
of 
the 
reference 
coordinate system, where the direction of the rotation axis 
 
is close to Z-direction.  
n
Hence, the developed algorithms allows us to identify the 
compensator geometrical parameters 
, 
L
xa , 
y  that are 
directly related to the above mentioned rotation center points 
P0, P2 and corresponding radii. Below they will be applied to 
the processing of the experimental data.  
a
5  DESIGN OF CALIBRATION EXPERIMENTS 
The main idea of the calibration experiment design is a set of 
robot configurations (as well as marker locations) that ensure 
the best identification accuracy. The key issue here is the 
ranging of different plans in accordance with the prescribed 
performance measure.  
For the considered identification problem, the design 
variables are the set of angles 
 and the marker 
locations. The objective functions to be minimized are 
computed via the covariance matrix that describes the 
identification errors for the geometrical parameters 
 and 
 
to be estimated. Since two identification algorithms are 
independent, selection of the optimal configurations 

2i
q
L
a


2i
q
 
and marker locations can be considered sequentially. 
 
    
 
 
 
Assuming that each experiment includes the additive 
measurement errors in the Cartesian coordinates 
iε ,  
expression (13) allows us to present the variance of the 
parameter  in the following way 




2
1
1
1
var
T
T
T
T
i
i
m
m
m
i
i
i
i
i
i
i
E








u R ε
ε Ru
u u



 (31) 
where 
 denotes the expectation and the orthogonal 
matrix 
 defines the orientation of the base coordinate 
system. 
Following 
usual 
assumption 
concerning 
the 
measurement errors (independent identically distributed, with 
zero expectation and standard deviation 
(.)
E
R
2
 for each 
coordinate) that allows to present the covariance error matrix 
as 

T
i
i
2
E


I
ε ε
, the above expression (31) reduces to  



2
1
var
T
i
i
m
i





u u


 
(32) 
Further, it can be proved that 
1
i
m
T
i
i
m


u u


q
L
. So, the 
variance (32) does not depend on the angles 
2i . Thus, the 
identification accuracy for the parameter 
 depends on the 
number of experiments only. 
For 
the 
remaining 
geometrical 
parameter 
a , 
the 
identification error depends on the estimation precision of 
relative location of the points P2 and P0. Since relevant 
identification algorithms employ independent measurement 
data, the variance 
 can be computed as the sum of the 
traces of 
2
cov
 and 
0
cov
, where 
1t  and 
2  are the 
vectors of Cartesian coordinates for the points P2 and P0. 

var a
(
)
t
(
)
t
t
For the point P2, expression (10) leads to the following 
covariance matrix  



1
1
1
1
2
2
2
cov
m
T
i
m
m
m
i
i
i
i
i
i
i
m E












t
ε
ε
R
u
u R 
T
T

u
is simplification is based on the
(33) 
which can be further simplified down to  



1
2
2
1
2
1
cov
T
i
m
m
i
i
i
m
m









t
I
u
 
(34) 
Th
 above derived expressions 


1
2
1
T
i
i
m
m
i
i
E
m





ε
ε
I  and 
 and on the 
assumption that z-axis of the coordinate system is directed 
along the second joint axis. Hence, for the point P2, the 
optimization problem that is related to the design of 
calibration experiment, can be formulated as 


2
2 /
E 


m
q





2
2
2
2
1
2
1
cos
sin
min
i
i
m
i
i
m
i
q
q
F





 
(35) 
This problem should be solved taking into account joint 
limits of the industrial robot. In the case when the range of 
angles 
 is over 
2q
, it is possible to achieve zero value of 
this 
objective 
since 
equations 
1
2
cos
0
m
i
i
q



 
and 
1

 are solvable. It should be noted similar 
equations arise in calibration experiment design for some 
robots without gravity compensators and have been studied in 
details in our previous work (Klimchik 2011).  
2
sin q
0
m
i
i 

For the point P0, similar expression includes a set of the 
angles 
i that can be recomputed to the joint angles 
 
requires for the manipulator control (see Fig. 3). Here, it is 
reasonable to find optimal marker locations on the rigid part 
of the gravity compensator. It can be proven that using these 
assumptions, the design of experiment reduces to the 
following optimization problem  
2i
q




1
2
2
1
cos
sin
min
j
j
j
k
k
j
j
F










 
(36) 
where 
j
 are the angles around the point P0 between the 
compensator spring and j-th marker location. It is clear that in 
this case the best solution is produced by similar equations 
1
jcos 
0
j
k


 and 
1
j
, but contrary to (35), 
this problem can be easily solved by locating the markers on 
the opposite sides of the compensator rotation axis.  
sin
0
j
k


Thus, the calibration experiment design that produces the sets 
of the optimal manipulator configurations and the marker 
locations described by the variables 
 and 

2i
q

j

 
respectively, is reduced to the solution of the above presented 
trigonometric equations that allows us essentially increase the 
calibration accuracy. 
6  EXPERIMENTAL RESULTS 
To demonstrate efficiency of the developed technique, the 
experimental study has been carried out. The experimental 
setup employed the robot KR-270 and the Leica laser tracker, 
which allowed us to measure the Cartesian coordinates of the 
markers attached to the compensator elements with the 
accuracy of 10 μm (see Figs 1, 3). Six different manipulator 
configurations where considered that differed in the value of 
the joint angle 
2  and three markers has been used. The 
experimental data are presented in Table 1. 
q
These data has been processed using the developed 
identification algorithm presenting in the Section 4. The 
obtained values for the parameters of interest 
, 
L
xa , 
y  are 
given in Table 2, which also includes the identification errors 
computed using the Gibbs sampling technique.  
a
In addition, there were evaluated the elastostatic properties of 
the gravity compensator. Corresponding curves describing 
influence of the compensator on the equivalent stiffness of 
the manipulator joint (see eq. (6)), are presented in Fig. 4. 
They demonstrate essential non-linearity of the compensator 
impact throughout of the robot workspace, which, in 
addition, highly depend on the spring preloading 
0s . 
Table 1.  EXPERIMENTAL DATA  
P1 
P01 
P02 
q2 
[deg] 
x, [mm] 
y, [mm] 
x, [mm] 
y, [mm] 
x, [mm] 
y, [mm] 
-0.01 
-31.84 
183.86 
-872.10 
-125.38 
-813.50 
-255.59 
-30 
-118.44 
143.42 
-872.30 
-126.07 
-813.33 
-256.18 
-60 
-173.30 
65.12 
-872.50 
-109.90 
-825.09 
-244.64 
-90 
-181.76 
-30.14 
-868.43 
-78.20 
-844.66 
-219.04 
-120 
-141.45 
-116.82 
-858.90 
-47.60 
-859.43 
-190.44 
-145 
-78.10 
-165.47 
-852.53 
-33.68 
-864.66 
-176.01 
Table 2.  GEOMETRICAL PARAMETERS  
 
L, [mm] 
ax, [mm] 
ay, [mm] 
value 
184.72 
685.93 
120.30 
accuracy 
±0.06 
±0.70 
±0.69 
 
    
 
 
 
    
 
7. CONCLUSIONS 
-150
-120
-90
-60
-30
0
-0.05
0
0.05
0.1
0.15
0
0 [m]
s 
0
0.2 [m]
s 
0
0.2 [m]
s 
0
0.4 [m]
s 
0
0.6 [m]
s 
0
0.8 [m]
s 
2,[deg]
q

 
The paper presents a new approach for modeling and 
calibration 
of 
heavy 
industrial 
robots 
with 
gravity 
compensators. It proposes a methodology and data processing 
algorithms for the identification of the gravity compensator 
geometrical parameters. To increase the identification 
accuracy, the design of experiments has been used aimed at 
proper selection of the measurement configurations and 
marker point locations. The advantages of the developed 
techniques are illustrated by experimental study of the 
industrial robot Kuka KR-270, for which the model 
parameters of the gravity compensator have been identified. 
Fig. 4. Variation of the gravity compensator impact on the 
equivalent stiffness of the second joint  
ACKNOWLEDGMENT 
The work presented in this paper was partially funded by the 
ANR, 
France 
(Project 
ANR-2010-SEGI-003-02-
COROUSSO). The authors also thank Fabien Truchet, 
Guillaume Gallot, Joachim Marais and Sébastien Garnier for 
their great help with the experiments. 
Another set of experiments have been carried out to identify 
the elastostatic properties of the considered manipulator.  
There were considered 15 different configurations, which 
have been found taking into account physical constraints that 
are related to the joint limits, work-cell obstacles and safety 
reasons. To ensure identification accuracy for each 
configuration, the experiments were repeated three times. In 
total, 405 equations were considered for the identification, 
from which 7 physical parameters have been obtained. 
Corresponding values of the elastostatic parameters for the 
gravity compensator and for the manipulator are presented in 
Table 3, where ki denotes the i-th joint compliance. There 
were also computed the identification errors (using the Gibbs 
sampling technique). As follows from Fig. 5, the gravity 
compensator essentially reduces the equivalent compliance of 
the second joint (compared to the serial manipulator without 
the gravity compensator).  
REFERENCES 
Bouzgarrou B.C., Fauroux J.C., Gogu G., and Heerah Y. 
(2004). Rigidity analysis of T3R1 parallel robot with 
uncoupled kinematics. Proc. of the 35th International 
Symposium on Robotics, Paris, France 
Company O., Pierrot F., Fauroux J.-C. (2005). A Method for 
Modeling Analytical Stiffness of a Lower Mobility 
Parallel Manipulator. Proceedings of IEEE International 
Conference on Robotics and Automation, pp. 3232-3237. 
De Luca A., Flacco F., (2011). A PD-type Regulator with 
Exact Gravity Cancellation for Robots with Flexible 
Joints. In Proc. 2011 IEEE International Conference on 
Robotics and Automation, pp. 317-323 
Klimchik A., Wu Y., Caro S., Pashkevich A. (2001). Design 
of 
Experiments 
for 
Calibration 
of 
Planar 
Anthropomorphic 
Manipulators. 
In 
the 
2011 
IEEE/ASME International Conference on Advanced 
Intelligent Mechatronics (AIM), pp. 576-581 
Table 3.  ELASTOSTATIC PARAMETERS  
Parameter  
value 
accuracy 
kc, [rad×μm/N] 
0.144 
±0.031 (21.5%) 
s0, [mm] 
458 
±27 (5.9%) 
k2, [rad×μm/N] 
0.302 
±0.004 (1.3%) 
k3, [rad×μm/N] 
0.406 
±0.008 (2.0%) 
k4, [rad×μm/N] 
3.002 
±0.115 (3.8%) 
k5, [rad×μm/N] 
3.303 
±0.162 (4.9%) 
k6, [rad×μm/N] 
2.365 
±0.095 (4.0%) 
Klimchik A., Bondarenko D., Pashkevich A., Briot S., Furet, 
F. (2012). Compensation of tool deflection in robotic-
based Milling. The 9th International Conference on 
Informatics in Control, Automation and Robotics 
(ICINCO 2012),  pp. 113-122. 
-150
-120
-90
-60
-30
0
2.75
2.8
2.85
2.9
2.95
3
3.05 x 10
-7
2k
0
2k
- compliance without gravity compensator
- compliance with gravity compensator
2,[deg]
q
rad·m/N
Kövecses J., Angeles J., (2007). The stiffness matrix in 
elastically articulated rigid-body systems. Multibody 
System Dynamics, pp. 169–184. 
Meggiolaro M.A., Dubowsky S., Mavroidis C. (2005). 
Geometric and elastic error calibration of a high 
accuracy patient positioning system. Mechanism and 
Machine Theory, vol. 40, pp. 415–427. 
Pashkevich A., Klimchik A., Chablat D. (2011). Enhanced 
stiffness modeling of manipulators with passive joints. 
Mechanism and Machine Theory, vol. 46, pp. 662-679. 
Takesue N., Ikematsu T., Murayama H. and Fujimoto H. 
(2011). Design and Prototype of Variable Gravity 
Compensation Mechanism (VGCM). Journal of Robotics 
and Mechatronics, vol. 23, no. 2, pp. 249-257. 
 
Fig. 5. Compliance of equivalent non-linear spring in the 
second joint