Available online at www.sciencedirect.com 
 
Procedia Engineering 41 ( 2012 ) 1285 – 1291 
 
International Symposium on Robotics and Intelligent Sensors 2012 (IRIS 2012) 
 
Compliant motion control for handling a single object by two similar 
industrial robots 
Hanene Mkaouara, Olfa Boubakera* 
aNational Institute of Applied Sciences and Technology, Centre Urbain Nord, BP 676 - 1080 Tunis, Tunisia  
Abstract 
In this paper, we propose a compliant motion control strategy for handling a single object by two similar industrial robots. The dynamics 
of the object carried by the two robots is assimilated to the dynamics of a mass-spring-damper system described by a piecewise linear 
model (PWA). The coordination of the two robots is accomplished using a master slave synchronization approach dedicated for PWA 
systems, based on the Lyapunov theory, and solved via Linear Matrix Inequalities (LMIs). The performances of the proposed approach 
are proved by simulation results and compared to a related approach. 
 
© 2012 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Centre of 
Humanoid Robots and Bio-Sensor (HuRoBs), Faculty of Mechanical Engineering, Universiti Teknologi MARA. 
 
Keywords: Robot coordination, Compliant  motion, Master-slave synchronization, Lyapunov Theory, PWA systems. 
1. Introduction 
Several industrial tasks, demanding precision, require robot coordination. A specified operation consists of handling a 
single object [1]. Compliant motion tasks are manipulation tasks that  involve contacts between  the  manipulated object and  
the  robots in which  the  trajectory  of  the   manipulators  are modified  depending on the  contact forces [2]. Position/force 
control is usually used for solving such cooperation problems [3, 4, 5, 6, 7, 8]. In fact, the control in position can ensure a 
tracking of the desired trajectory of the constraint robot and the control in force ensures a specific desired behavior of the 
robot when it enters in contact with the handled object. Master slave synchronization can be considered as a promising 
solution for such problems [9, 10]. 
 
This paper proposes a master slave synchronization approach for solving a coordination problem of two similar robots 
handling a single object. The dynamics of the object carried by the two robots is assimilated to the dynamics of a mass-
spring-damper system described by a piecewise linear model (PWA). The master slave synchronization controller is based 
on the Lyapunov theory and solved via Linear Matrix Inequalities (LMIs). 
 
The paper will be organized as follows: The cooperation problem is presented in the second section. The third section 
presents the analogical spring-dumper-mass system. A master slave synchronization approach is then applied. The 
effectiveness of the proposed approach is shown by simulation results and compared to a related approach. 
 
 
* Corresponding author. Tel.: +216-98-950869. 
E-mail address: hanene.mkaouar@gmail.com 
Hanene Mkaouar and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1285 – 1291 
2. Problem formulation 
We consider in this paper the cooperation problem of two industrial robots handling a single object as shown by Fig. 1. 
The cooperation problem consists on handling a single object and displacing it through a compliance strategy. The control 
problem will be solved as a master slave synchronization problem of positions and velocities of the two robots as shown by 
Fig.2. 
 
 
 
 
Fig. 1. Two similar robots handling a single object  
Fig. 2. Controlled master-slave system 
3. System modeling 
To develop the master slave synchronization approach, an analogical system is chosen [9]. The analogue model shown 
by Fig.3 is a mechanical system composed of two masses m1 and m2 joined by a spring-dumper combination with a 
stiffness parameter k and a dumping coefficient c. Masses, springs and dumpers ensure programmable compliances between 
the robots and the environment (case of the damper c1), the robot and the object to be displaced (case of the dumper c and 
the spring k), or the object to be displaced and the second robot (case of the spring k2). The result is an elastic virtual 
behaviour that we want to give to the whole moving system thanks to the control law in order to prevent robots and object 
damaging. We assume, here, that the master robot is actuated by a variable force 
)
sin(
)
(
wt
A
t
u
d
=
and the slave’s one by 
the control force
)
(t
v
to be computed. 
 
Fig. 3. Analogical spring-dumper-mass master and slave systems. 
Using the Newton Euler formalism, the master dynamics are described by: 






−
−
+
−
−
=
+
−
−
−
−
−
=
)]
(
)
(
)
(
[
1
]
)
(
)
(
[
1
2
2
2
1
2
1
2
2
1
1
2
1
2
1
1
1
x
g
k
x
x
c
x
x
k
m
x
u
x
c
x
x
c
x
x
k
m
x
&
&
&&
&
&
&
&&
 
(1) 
 
 
 
 
Hanene Mkaouar and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1285 – 1291 
whereas the slave dynamics are described by: 






−
−
+
−
−
=
+
−
−
−
−
−
=
)]
(
)
(
)
(
[
1
]
)
(
)
(
[
1
2
2
2
1
2
1
2
2
1
1
2
1
2
1
1
1
y
g
k
y
y
c
y
y
k
m
y
v
y
c
y
y
c
y
y
k
m
y
&
&
&&
&
&
&
&&
 
(2) 
where g (x2)  and g (y2) are piecewise linear  functions described, respectively, by: 



<
=
>
−
=
2
2
2
2
2
2
2
2
0
)
(
)
(
d
x
if
x
g
d
x
if
d
x
x
g
 
(3) 



<
=
>
−
=
2
2
2
2
2
2
2
2
0
)
(
)
(
d
y
if
y
g
d
y
if
d
y
y
g
 
(4) 
 
The master and the slave systems can be, respectively, written in a piecewise linear form [11, 12, 13] as: 
2,1
=
+
+
=
j
b
Bu
x
A
x
j
j
&
 
(5) 
2,1
=
+
+
=
i
b
Bv
y
A
y
i
i
&
 
(6) 
where: 


















−
−
+
−
−
=
2
2
2
2
1
1
1
1
1
1
1
0
0
0
0
0
1
0
m
c
m
k
m
c
m
k
m
c
m
k
m
c
c
m
k
A
, 


















−
+
−
+
−
−
=
2
2
2
2
2
1
1
1
1
1
2
1
0
0
0
0
0
1
0
m
c
m
k
k
m
c
m
k
m
c
m
k
m
c
c
m
k
A
, 












=
0
0
0
0
1b
, 
















=
2
2
2
2
0
0
0
m
d
k
b
, 














=
0
0
1
0
1
m
B
 
[
]
4
3
2
1
x
x
x
x
x =
 and 
[
]
4
3
2
1
y
y
y
y
y =
, (
1
x ,
2
x ) and (
1
y ,
2
y ) are respectively master and slave positions of mass 
1
m  and 
2
m  and 
1
3
x
x
&
=
, 
2
4
x
x
&
=
, 
1
3
y
y
&
=
 and 
2
4
y
y
&
=
 are the velocities. 
4. Motion synchronization 
Consider the master slave PWA system described by: 



−
=
+
+
=
+
+
=
)
(
x
y
K
w
Bv
b
x
A
y
Bu
b
x
A
x
i
i
j
j
&
&
 
(7) 
where 
n
n
iA
×
ℜ
∈
, 
n
n
jA
×
ℜ
∈
, 
n
ib
ℜ
∈
 
n
jb
ℜ
∈
 are two constant matrices and two constant vectors, respectively. 
m
n
B
×
ℜ
∈
is the control matrix. K is a gain state matrix of appropriate dimension.  
Denote by 
j
Λ  and 
i
Λ the partition of the state-space into polyhedral cells defined respectively by the following 
polytopic description: 
{
}
0
Λ
<
+
=
j
T
j
j
h
x
H
x
 
(8) 
{
}
0
Λ
<
+
=
i
T
i
i
h
x
H
z
 
(9) 
where 
jr
n
j
H
×
ℜ
∈
 , 
1
×
ℜ
∈
jr
j
h
, 
ir
n
i
H
×
ℜ
∈
 and 
1
×
ℜ
∈
ir
jh
. 
 
The error dynamics between the master and the slave systems can be written as: 
ij
ij
i
b
x
A
e
BK
A
e
+
+
+
=
)
(
&
 
(10) 
where  
x
y
e
−
=
, 
,j
i
ij
A
A
A
−
=
 
j
i
ij
b
b
b
−
=
. 
 
Hanene Mkaouar and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1285 – 1291 
4.1. Theorem [14]: 
For a given decay 
0
1 >
α
 and for all pairs of indices
Q
j
i
∈
,
, if there exist constant symmetric positive definite matrix 
n
n
S
×
ℜ
∈
, constant matrix 
n
m
R
×
ℜ
∈
, diagonal negative definite matrices 
i
i r
r
ij
E
×
ℜ
∈
 and 
j
j r
r
ij
F
×
ℜ
∈
 and strictly negative 
constants 
ij
β  and 
ij
ξ
, such that the following LMIs: 
0
2
1
*
*
0
2
1
*
<
















ij
ij
T
j
ij
T
i
ij
ij
F
E
h
h
ξ
ξ
ξ
 
(11) 
0
2
1
*
*
*
*
*
2
1
*
*
*
*
0
0
2
*
*
*
0
0
0
2
*
*
2
1
2
1
*
2
1
0
,1
<


























−
−
−
−
−
−
∆
T
j
j
ij
ij
T
j
i
ij
T
i
i
ij
ij
ij
ij
j
j
i
i
j
i
ij
T
j
ij
ij
i
i
T
i
ij
ij
i
ij
i
h
h
F
h
h
h
h
E
F
E
M
H
M
H
H
H
I
h
b
M
SH
h
b
SH
A
ξ
ξ
ξ
β
ξ
ξ
 
T
ij
ij
ij
T
T
T
i
i
i
b
b
S
B
R
BR
SA
S
A
ξ
α
∆
1
,1
−
+
+
+
+
=
 
(12) 
are satisfied, then the master-slave synchronization error system (10) is globally asymptotically stable using the state gain 
matrix: 
1
−
= RS
K
 
(13) 
Proof: see [14] and for more details [15]. 
4.2. Simulation results 
Simulation results are conducted for the master and the slave systems (5) and (6) using the following parameters : 
1
,
100
2
1
=
=
m
m
, 
2
2 =
=c
c
, 
10
2 =
= k
k
, 
01
.0
2 =
d
, 
5.1
=
d
A
 and 
5.1
=
w
 and  for the initial conditions 
T
x
]
01
.0
01
.0
01
.0
1[
0 =
 and 
T
y
]1
01
.0
0
05
.0
[
0 =
. The polyhedral cells: 
{
}
2
3
1
d
x
x
≥
=
Λ
 and 
{
}
2
3
2
d
x
x
<
=
Λ
 are 
satisfying the polytopic description (8) and (9) where 
[
]T
H
0
1
0
0
1 =
, 
[
]T
H
0
1
0
0
2
−
=
,
2
1
d
h =
 and 
2
2
d
h
−
=
. 
 
Solving the LMIs (11) and (12) using the LMI toolbox of MatLab software for the parameter 
4
1
10−
=
α
 gives the state 
gain matrix (13) such as: 
]
0.2389
-
   
0.7830
    
0.4002
-
   
-1.7003
[
103
=
K
 
 
 
Simulation results shown by Fig.4, where the synchronization of the positions (
1
x ,
1
y ) and (
2
x ,
2
y ) and the velocities 
(
1
x& ,
1
y& ) and (
2
x& ,
2
y& ) of the masses m1 and m2 are shown, prove the efficiency of the proposed approach. On the other hand, 
the smoothness of the compliant control law is shown by Fig.5. 
Hanene Mkaouar and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1285 – 1291 
0
1
2
3
4
5
6
7
8
9
10
-0.5
0
0.5
1
1.5
Time (s)
Position
 
 
 Position of m1 : x1
 Position of m1 : y1
0
1
2
3
4
5
6
7
8
9
10
-1
0
1
2
3
Time (s)
Position
 
 
 Position of m2 : x2
 Position of m2 : y2
     
(a) 
0
1
2
3
4
5
6
7
8
9
10
-0.5
0
0.5
1
Time (s)
Speed
 
 
 Speed of m1 : x3
 Speed of m1 : y3
0
1
2
3
4
5
6
7
8
9
10
-1
0
1
2
Time (s)
Speed
 
 
 Speed of m2 : x4
 Speed of m2 : y4
 
(b) 
Fig. 4. Simulation results: (a) position synchronization of masses m1 and m2, (b) velocity synchronization of masses m1 and m2. 
0
1
2
3
4
5
6
7
8
9
10
-1000
-500
0
500
1000
1500
Time (s)
Control law
 
 
master control: u
Slave control: v
 
Fig.5. Compliant control law 
Hanene Mkaouar and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1285 – 1291 
4.3. Comparative study 
To derive a comparative study, the synchronization problem of the master and the slave systems (5) and (6) is achieved 
using the strategy proposed in [16] using the same parameters. The computed state gain vector is given by: 
]
9.8635
-
   
12.1954
-
    
6.5654
-
   
-35.2260
[
=
K
 
 
 
The synchronization errors relating the two approaches are shown by Fig.6. It is clear that the two approaches give 
comparable results. In fact both approaches allow asymptotic synchronization but even if error variance is larger using 
approach [15] for the case study specified in this paper, tighter stability criteria are guaranteed by the approach [16], as it 
was already proved in [14], since it based on Lyapunov theory. 
0
1
2
3
4
5
6
7
8
9
10
-1
-0.5
0
0.5
Time (s)
e1
 
 
0
1
2
3
4
5
6
7
8
9
10
-0.1
-0.05
0
0.05
0.1
0.15
Time (s)
e3
 
 
0
1
2
3
4
5
6
7
8
9
10
-1
0
1
2
3
Time (s)
e2
 
 
0
1
2
3
4
5
6
7
8
9
10
-1
-0.5
0
0.5
1
Time (s)
e4
 
 
 1st approach
 2nd appraoch
 1st approach
 2nd appraoch
 1st approach
 2nd appraoch
 1st approach
 2nd appraoch
 
Fig. 6. Synchronization error using first approach [15] and second approach [16] 
5. Conclusion 
In this paper, a compliant motion controller is proposed for handling a single object by two similar industrial robots. The 
coordination was accomplished using an analogue piecewise linear dynamical system and a master slave synchronization 
approach. The efficiency of the proposed approach was proved via simulation results and finally compared to a related 
approach. 
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