A robust control intended for a nonholonomic mobile robot is considered to guarantee good tracking a desired trajectory. The main drawbacks of the mobile robot model are the existence of nonholonomic constraints, uncertain system parameters and unmodeled dynamics. in order to overcome these drawbacks, we propose a robust control based on Lyapunov theory associated with slidingmode control, this solution shows good robustness with respect to parameter variations, measurement errors, noise and guarantees position and velocity tracking. The global asymptotic stability of the overall system is proven theoretically. The simulation results largely confirm the effectiveness of the proposed control.
1. Introduction
One of the basic issues in the field of mobile robotics is the running path. The trajectory tracking is to guide the robot through intermediate points to arrive at the final destination. This guide is done under a time constraint, ie, the robot must reach the goal within a predefined time. In the literature, the problem is treated as the continuation of a robot reference (virtual processor) which moves to the desired trajectory with a certain pace. The real robot must follow this virtual robot accurately and try to minimize the error in distance, varying its linear and angular velocities
[1

9]
.
There are lots of works on its tracking control. Their aims are mainly kinematic models; one method for dynamic models has been suggested
[1]
. In this case generally use linear and angular velocities of the robot (Fierro & Lewis, 1997; Fukao et al., 2000) or torques (Rajagopalan & Barakat , 1997; Topalov et al., 1998) as an input control vector
[2]
. The most authors determine the problem of mobile robot stability using nonlinear backstepping algorithm (Tanner & Kyriakopoulos, 2003) with steady parameters (Fierro & Lewis, 1997), or with the known functions (Oriollo et al., 2002)
[1

6]
. Other goals at the control architectures, the hybrid of the kinematic control, and the dynamic controller, the neural network controller, is proposed as some trajectory tracking methods
[3]
.
In this paper, first, a kinematic controller is introduced to the WMR. Second, the dynamic controller, PI then Lyapunov theory associated to a sliding mode control, is proposed to make the actual velocity of the mobile robot to reach the wheel velocity control desired.
2. Kinematic Model
Fig. 1
shows the typical model of a nonholonomic wheeled mobile robot. This last is operated by two independent wheels and with a passive wheel ensuring its stability. The posture of the WMR can be represented as
Error posture of a nonholonomic WMR
where the (
x
,
y
) is the center of mass (COM) position of the WMR in the world
X
−
Y
coordinate, and
θ
is the included angle between the
X
axis and
X
' axis representing the WMR
[5
,
7]
.
Know the derivative of the posture control
V
= (
v
w)
^{T}
is easy. A simple geometric consideration gives
What is written in matrix form
whith
The mobile robot is nonholonomic, this signify the wheels roll without slipping, ie
[3]
3. Dynamic Model
The dynamic equation of the WMR with ngeneralized coordinates
q
∈
R
^{n×1}
, and inputs
r
=
n
−
m
, can be expressed as
[1
,
3
,
7]
where
M
(
q
) ∈ 𝓡
^{n×m}
is a positive symmetric definite inertia matrix,
is the centripetal and coriolis matrix,
denotes the gravitational vector,
τ_{n}
∈ 𝓡
^{n×1}
is bounded unknown distrurbance,
B
(
q
) ∈ 𝓡
^{n×(n−m)}
denotes the input transformation matrix,
τ
∈ 𝓡
^{(n−m)×1}
is the control input vector,
A
∈ 𝓡
^{m×n}
is a matrix associated with the constraints.
The parameters in (6) are given as
I
and
m
are the moment of inertia and the mass of the WMR, respectively, the motors torques
τ_{r}
and
τ_{l}
act on the right and left wheels respectively
[1
,
7]
.
r
and
R
are the radius of the wheel and the distances between the two driving wheels, respectively.
Substituting (2) and its derivative in Eq. (5) premultiplied by
J^{T}
(
q
) , and without considering uncertainties and disturbances the Eq. (5) can be written as the following
[1
,
3
,
7
,
8]
The dynamic model of a WMR unicycle type is simplified and given by
where
4. Tracking Controller Design
The problem can be interpreted as consisting of the slave robot to robot reference, whose trajectory is given by t → [
x_{r}
(
t
),
y_{r}
(
t
)]. It is then desired to control the zero error vector [
x_{r}
(
t
) −
x
(
t
),
y_{r}
−
y
(
t
)] , where [
x_{r}
(
t
)
y_{r}
(
t
)
θ_{r}
(
t
)]
^{T}
denotes the coordinate vector generalized robot reference and [
x
(
t
)
y
(
t
)
θ
(
t
)]
^{T}
the vector of generalized coordinates of the real robot
[1

6]
.
For the tracking control problem, a timevarying reference mobile robot model is given as
[7]
where
and
denote the reference velocity and posture of the WMR,
J
(
θ_{r}
) is the Jacobean defined in the Eq. (4).
We define the error between the desired positions and orientations, and actual by
Characterization of the trajectory tracking.
We define
q_{e}
as following
T_{e}
is called the matrix
The vector of error variations can be expressed as :
5. Design of Hierarchical Controller
Fig. 3
illustrates the architecture of the controller designed to control a WMR.
Architecture robot controller
6. Kinematic Controller
Proposition :
Let the Lyapunov function candidate
where ; the derivative
L
_{0}
is given by
Eq. (11) gives
Then
By substitution in the Eq. (14)
The control law for the system to be stable is
, so we choose
Therefore the Eq. (14) can be rewritten as
For
must
k
_{1}
,
k
_{2}
and
We have opted for the following values
k
_{1}
= 10,
k
_{2}
= 5 and
k
_{3}
= 4 .
7. Dynamic Controller
We used in the present work two control technicals, first one is simple, it is a PI controller, the second is based on the nonlinear method.
The robot trajectory in XY plane
 7.1 PI control in dynamic level
To ensure that the movement of the WMR can follow the desired velocity controller generated by the kinematic, two dynamic controllers are introduced in this section. Both controllers are PI controllers responsible for providing left and right torques capable of powering the left and right wheels.
The parameters of both controllers are
k_{p}
=
k_{i}
= 2 .
 7.2 Simulation results
To show the effectiveness of the proposed controller, simulations were performed in MatlabSimulink. The examples chosen is that the tracking of a circular path.
The parameters of the robot used in the simulation are:
m
= 4
kg
,
I
= 2.5
kg
m
^{2}
,
R
= 0.15 m and
r
= 0.03
m
.
The actual initial posture of the mobile robot is
q
(0) = [3 1 180°]
^{T}
. The initial posture of the robot is defined by reference :
q_{r}
(0) = [0 0 0]
^{T}
.
PI controller gave us satisfactory results.
Figs. 5
,
6
and
7
show significant oscillations errors, dynamic errors,
Figs. 10
and
11
, are large. to reduce its peaks and errors, we replaced the PI power controller by Lyapunov controller.
Tracking errors trajectory e_{1}, e_{2} and e_{3}
The tracking errors in Xcoordinate
The tracking errors in Ycoordinate
The tracking θ_{r} and θ
The tracking error torque (τ_{r} − τ_{l}) (N*m)
Linear velocities
Angular velocities
 7.3 Lyapunov controller associated to a sliding mode control in dynamic level
In this section, we replaced the PI controller with a Lyapunov controller associated to a sliding mode control to improve the results to previous results.
The sliding surface is defined by selected
where
For a good pursuit of velocity, it is important to make the invariant surface
Ṡ
(
t
) = 0 and attractive
S^{T} Ṡ
< 0
where Δ
f
(
t
) present a parameter uncertainties and external disturbances.
Proposition:
Let the Lyapunov function candidate
where ; the derivative
is given by:
Eq. (8) gives
Eq. (19) becomes
If we put
We find
The Lyapunov condition is satisfied for
k_{a}
< 0 and
k_{b}
< 0 , where
k_{a}
= −100 and
k_{b}
= −1000
From the Eq. (17)
The derivative of Lyapunov becomes
If we put
We find
Substitution Eq. (23) in Eq. (27) gives
So for
, it is necessary that
And
, where
k_{c}
= −10 and
k_{d}
= −0,5 .
 7.4 Simulation results
The same initial conditions and the same parameters of the WMR are used in this section. To test the robustness of the proposed approach, we have proceeded to two tests. Initially, a large variation in the mass m was introduced, a 50% increase between t=30s → 50s. The second test is the robustness of the controller against the change of the radius r, a 10% reduction of r is applied between t=60s → 70s. The following figures illustrate these results.
Variation parameters: (a) Mass of the WMR; (b) Radius of the wheel
Figs. 13

20
shows clearly that the proposed control is widely robust against the parameters variations. By comparison, these results with previous results, we note that the oscillations of
e
_{1}
,
e
_{2}
and
e
_{3}
are reduced, which gives us a good improvement of dynamic and static errors, the same for the linear velocities and angular velocities. Tracking in ycoordinate is better and the dynamic error decreases,
Fig. 17
shows the
θ
follows
θ_{r}
with less oscillations. Employing nonlinear method for controlling the WMR gives a good trajectory tracking and velocity, and confirms the robustness of the control.
The robot trajectory in XY plane
Tracking errors trajectory e_{1}, e_{2} and e_{3}
The tracking errors in Xcoordinate
The tracking errors in Ycoordinate
The tracking θ_{r} and θ
The tracking error torque (N*m)
Linear velocities
Angular velocities
8. Conclusion
This paper focuses on the design of a nonlinear tracking controller for a nonholonomic mobile robot with unknown parameters, we proposed a controller based on Lyapunov theory associated with the sliding mode control. The stability of the system was proved by Lyapunov theory which satisfies a good performance tracking position control. Simulation results demonstrate that the proposed controller is effective.
BIO
Ghania Zidani was born in Batna, Algeria, on August 1971. He received B.Sc., M.Sc. degrees in Electronics from the University of Batna, Algeria, respectively in 1995 and 2009. Currently, she is a Teaching Assistant at the Electrical Engineering Institute at University of Ouargla. She is the member of the Research Laboratory LSPIE of Batna, Algeria. She works on her PhD thesis on the control of robot mobile at the University of Batna2, Algeria.
Saïd Drid was born in Batna, Algeria, in 1969. He received B.Sc., M.Sc. and PhD degrees in Electrical Engineering, from the University of Batna, Algeria, respectively in 1994, 2000 and 2005. Currently, he is full Professor at the Electrical Engineering Institute at University of Batna 2, Algeria. He is the head of the Energy Saving and Renewable Energy team in the Research Laboratory of Electromagnetic Induction and Propulsion Systems of Batna University. He is IEEE senior member of IEEE Power & Energy Society, IEEE Industry Applications Society, IEEE Industry Electronics Society, IEEE Vehicular Technology Society and IEEE Power Electronics Society. Currently, he is the vice chair of the PES chapter, IEEE Algeria section. His research interests include electric machines and drives, renewable energy. He is also a reviewer of some international journals.
Larbi ChrifiAlaoui received the Ph.D. in Automatic Control from the Ecole Centrale de Lyon. Since 1999 he has a teaching position in automatic control in Aisne University Institute of Technology, UPJV, CuffiesSoissons, he is the Head of the Electrical Engineering and industrial informatics Department. His research interests are mainly related to linear and nonlinear control theory including sliding mode control, adaptive control and robust control, with applications to electric drive and mechatronics systems.
Djemai Arar was born in Batna, Algeria, on July 1971. He received his DipIng. Degree in electronics (with honors) in 1995 from the Department of Electronics in the field of Control systems, university of Batna, Algeria. The Ph.D Degree in electronics and telecommunication in 2000 (with Magma Cum Laude) from the Polytechnic University of Bucharest, Romania. He is currently a full professor in the department of Electronics, Faculty of Technology, University of Batna 2, Algeria. His research interests include : Signal Processing, Control Systems, Sources Separation, Higher order statistics, Neural Networks and semi conductor devices. He is a member of Advanced Electronics Laboratory (LEA).
Pascal Bussy Professor of Mechanics at the University of Amiens. He was Director of Mechanics and CAO Laboratory from 1990 to 2000, then a researcher at the Laboratory for Innovative Technology at the University of Amiens. He is the team leader Mechatronics in 2006. He was director of the University Institute of Technology of Aisne between 1998 and 2003.
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