Available online at www.sciencedirect.com 
 
Procedia Engineering 41 ( 2012 ) 1292 – 1297 
 
International Symposium on Robotics and Intelligent Sensors 2012 (IRIS 2012) 
 
Robust Tracking Control for Constrained Robots  
Haifa Mehdi, Olfa Boubaker* 
National Institute of Applied Sciences and Technology, Centre Urbain Nord, BP 676 - 1080 Tunis, Tunisia 
 
Abstract 
In this paper, a novel robust tracking control law is proposed for constrained robots under unknown stiffness environment. The stability 
and the robustness of the controller are proved using a Lyapunov-based approach where the relationship between the error dynamics of 
the robotic system and its energy is investigated. Finally, a 3DOF constrained robotic arm is used to prove the stability, the robustness and 
the safety of the proposed 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: Position/force control; Robust control; Unknown stiffness environment; Constrained robot. 
1. Introduction 
Robust tracking control under unknown constrained environment is one of the main important issues in robotic field. A 
number of significant papers are proposed in this area. They include fuzzy logic based controllers [1, 2], neural networks 
based controllers [3, 4] and sliding mode controllers [5, 6]. In the most cases, these research papers are not concerned by the 
compromise between robustness and safety for the tracking control problem. Only few papers are devoted to such subject [7, 
8]. This paper is placed in this context. Furthermore, the controller proposed in this paper is simpler than those proposed in 
[7] and [8]. The stability and the robustness of the proposed approach are shown, by simulation results, on a robotic 
manipulator constrained to a circular trajectory. 
2. Problem formulation 
 
Consider a constrained robotic system with n degrees of freedom described by the dynamical model [9]: 
 
F
J
U
G
H
M
T)
θ
(
)
θ
(
)
θ
,θ
(
θ
)
θ
(
−
=
+
+
&
&&
                                                             (1) 
 
where 
n
R
,
,
∈
θ
θ
θ
&&
&
 are joint position, joint velocity and joint acceleration vectors, respectively. 
nxn
R
M
∈
)
θ
(
is the inertia 
matrix, 
n
R
H
∈
)
θ
,θ
(
&
 is the vector of centrifugal and Coriolis forces and 
n
R
G
∈
)
θ
(
 is the vector of gravity terms. 
n
R
U ∈
is 
the generalized joint force vector, 
p
R
F ∈
 is the vector of contact generalized forces exerted by the manipulator on the 
environment and p is the task space dimension. Let
p
R
X
X
∈
&
,
 the Cartesian position vector and the Cartesian velocity 
vectors of the robotic system defined, respectively, in the task space by: 
 
 
* Corresponding author. Tel.:+0-216-1-703-929; fax: .:+0-216-1-704-329. 
E-mail address: olfa.boubaker@insat.rnu.tn. 
Haifa Mehdi and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1292 – 1297 
 
 
)
θ
(
h
X =
                                                                                   (2) 
 
θ
)
θ
(
θ
)
θ
(
&
&
J
h
X
=
∂
∂
=
                                                                         (3) 
 
where 
p
n
h
R
→
R
:
)
θ
(
 is a vector of nonlinear  functions describing the forward kinematic model and 
pxn
R
J
∈
)
θ
(
 is the 
Jacobian matrix assumed to be full ranked. The temporal derivative of the Kinematic model (3) is given by: 
 
θ
)
θ
(
θ
)
θ
(
&&
&
&
&&
J
J
X
+
=
                                                                            (4) 
 
where 
p
R
X ∈
&&
 is Cartesian end-effector acceleration. 
Given a desired Cartesian position of the end-effector 
p
d
R
X
∈
 the control problem aims to ensure: 
 
0
lim
=
−
→
X
Xd
t
t
f
                                                                           (5)  
 
Notation: In the following, we will adopt the following notations: 
 
    
J
J
G
G
H
H
M
M
=
=
=
=
)
θ
(
    
,
)
θ
(
    
,
)
θ
,θ
(
    
,
)
θ
(
&
 
3. Robust position/force controller 
Theorem: 
 
The constrained robotic system described by the dynamical model: 
F
J
U
G
H
M
T
−
=
+
+
θ&&
                                                          (6) 
 is asymptotically stable under  the uncertain force model: 
)
X
X
)(
K
K
(
F
d
e
e
−
∆
+
=
                                                        (7) 
where
e
K and 
pxp
e
R
K ∈
∆
are the environment stiffness and the stiffness unknown uncertainty, respectively, and the robust 
control law is described by: 
G
sign
K
X
X
J
X
X
K
X
X
K
J
U
d
T
d
v
d
p
T
+
−
+
−
+
−
=
)
σ
(
σ
)
(
)]
(
)
(
[
&
&
&
&
                      (8) 
where σis a nonlinear function defined by:  
[
])
(
Λ
)
(
σ
X
X
X
X
C
d
d
−
+
−
=
&
&
                                                                 (9) 
for the constant vector
xp
R
C
1
∈
 and the positive constant matrices K  and 
pxp
R
∈
Λ
,  if there exist diagonal gain matrices 
pxp
v
p
R
K
K
∈
,
satisfying the following conditions:  



>
<
+
−
0
0
∆
v
e
p
e
K
K
K
K
                                                                (10) 
 
 
 
Haifa Mehdi and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1292 – 1297 
 
Remark: 
 
To satisfy safety of the robotic system and the environment, controller (8) is based on the uncertain force model (7). The 
nonlinear function 
)
σ
(
sign
 is used to ensure robustness by controlling at the same time constrained robot motion and 
constraint force. On the other hand, the nonlinear function is also able to limit the degradation of tracking performance 
occurring during saturation. 
 
Stability Proof: 
 
Consider for the constrained robot system (6) the error vector defined in the joint space by:  
d
θ
−
θ
=
Φ
                                                                               (11) 
and the error vector defined in the task space by: 
d
X
X
Y
−
θ
=
Φ
)
(
)
(
                                                                         (12) 
Under the unknown force model (7) and the robust control law (8) we can write that: 
0
)
(
sign
K
)
(
Y)
(
J
)
(
Y
K
)
(
J
)
(
Y
K
)
(
J
)
,
(
H
)
(
M
T
2
T
1
T
=
σ
σ
Φ
Φ
+
Φ
Φ
+
Φ
Φ
+
Φ
Φ
+
Φ
Φ
&
&
&
&&
 
                (13) 
where : 
               
v
2
e
e
p
1
K
K
K
K
K
K
=
∆
−
−
=
 
Following the some method presented in [10, 11] we can prove that: 
2
)
(
M
dt
d
)
,
(
H
n
1
i
i
i
Φ






Φ
∂
Φ
∂
Φ
=
Φ
Φ
∑
=
&
&
                                                      (14)                        
)
(
Y
K
)
(
J
)
(
P
1
T
Φ
Φ
=
Φ
∂
Φ
∂
                                                              (15)                        
)
(
sign
K
)
(
Y
)
(
J
)
(
Y
K
)
(
J
)
,
(
D
T
2
T
σ
σ
Φ
Φ
+
Φ
Φ
=
Φ
∂
Φ
Φ
∂
&
&
&
&
                               
 (16)               
Impose, now, to the system (13) to have a Lyapunov Hamiltonian function defined by: 
)
0
(
)
Φ
(
)
Φ
,
Φ
(
)
Φ
,
Φ
(
P
P
T
V
−
+
=
&
&
                                                          (17) 
The error system (13) is asymptotically stable if 
)
Φ
,
Φ
(
&
V
 satisfies the three conditions imposed by Lyapunov theorem 
[12]. For proving the first and the second conditions, we derive the same developments as those presented in [10]. The 
conditions 
0
K
K
K
e
p
e
<
+
−
∆
 are then well obtained. To prove the third Lyapunov condition, the derivative of the 
expression (17) gives: 
dt
dP
dt
dT
dt
dV
)
Φ
(
)
Φ
,
Φ
(
)
Φ
(
+
=
&
                                                           (18) 
 
From equations (15) and (16) we can write: 
Haifa Mehdi and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1292 – 1297 
 
(
)
)
,
(
)
(
 
2
)
(
)
(
,
Φ
Φ
Φ
+
Φ
Φ
Φ
=
Φ
Φ
Φ
+
Φ
Φ
Φ
=
Φ
Φ
&
&
&&
&
&
&
&&
&
&
H
M
dt
dM
M
dt
dT
T
T
T
T
                          (19)     
Furthermore, 
( )
)
(
)
(
)
(
1
Φ
Φ
Φ
=
Φ
∂
Φ
∂
Φ
=
Φ
Y
K
J
P
dt
dP
T
T
T
&
&
                                         (20) 
So, we can prove that:                     
(
)
( )
(
)
( )
( )
Φ
Φ
Φ
+
Φ
Φ
Φ
+
Φ
Φ
Φ
=
Φ
Φ
Y
K
J
H
M
dt
dV
T
T
T
T
1
,
,
&
&
&
&&
&
&
                                           (21) 
From (13) we can write:  
)
(
sign
k)
(
Y)
(
J
)
(
Y
K
)
(
J
)
(
Y
K
)
(
J
)
,
(
H
)
(
M
T
2
T
1
T
σ
σ
Φ
Φ
−
Φ
Φ
−
=
Φ
Φ
+
Φ
Φ
+
Φ
Φ
&
&
&
&&
                     (22)                        
Substituting the second member of (22) in (21) gives: 
(
)
)
(
sign
K
)
(
Y
)
(
J
)
(
Y
K
)
(
J
dt
,
dV
T
T
2
T
T
σ
σ
Φ
Φ
Φ
−
Φ
Φ
Φ
−
=
Φ
Φ
&
&
&
&
&
                            (23)                        
Using relations (3), (11) and (12) gives: 
  
(
)
( )
( )
)
(
sign
K
)
(
Y
Y
)
(
Y
K
Y
dt
,
dV
T
2
T
σ
σ
Φ
Φ
−
Φ
Φ
−
=
Φ
Φ
&
&
&
&
&
                               (24) 
The third Lyapunov condition is then verified if 
v
K  is positive definite. 
4. Simulation results 
Simulation results are carried out using a 3DOF robotic system using the physical parameter data given in [13, 14] for 
the constrained circular motion described by: 
  



π
−
π
=
π
−
π
=
)
t
2
t
3
sin(
76
.0
)
t(
y
)
t
2
t
3
cos(
76
.0
)
t(
x
3
2
d
3
2
d
 
 
for  
[
]
T
i
0
0
0
θ 0 =
[
]
T
id
π
π
π
θ
=
, 
0
0 =
t
 and 
s
t f
1
=
. For sufficient conditions (10), the numerical parameters are 
chosen as: 
[
]
10
10
Λ
diag
=
30
K =
,
[
]1
1
=
C
[
]
100
100
diag
Ke =
, 
[
]
50
20
∆
diag
Ke =
, 
[
]
500
300
diag
K p =
, 
[
]
350
200
diag
Kv =
. 
Fig.1 and  Fig. 2 show the evolution of the robot in the Cartesian space with respect to the constrained circular trajectory 
and the smooth profile of the robust control laws (8), respectively.  
Haifa Mehdi and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1292 – 1297 
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Position x (m)
Position y (m)
 
 
Desired position
Actual position
 
Fig. 1. End effector trajectory 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-30
-20
-10
0
10
20
30
time (s)
Control input (N.m)
 
 
U1
U2
U3
 
Fig. 2. Control laws 
 
Discussion: 
 
It ‘is clear that the current control algorithm can be implemented in many real applications. For example, the proposed 
approach can be relevant for rehabilitation device applications, see for example [10-11], and for bipedal and humanoid 
robots during the impact and double support phases, see for example [15-16].  In future investigations, a comparative 
analysis with the related works [7-8] are also planned. 
 
 
Haifa Mehdi and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1292 – 1297 
5. Conclusion 
This paper proposes a simple robust controller for motion tracking of constrained robots under unknown stiffness 
environment. The proposed approach takes care on the compromise between robustness and safety for the tracking control 
problem. The stability and the robustness of the controller are proved using a Lyapunov-based approach. Future 
investigations will concern the application of the proposed approach for humanoid and rehabilitation robotic devices. 
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