Hybrid and Multi-controller Architecture for Autonomous System
Application to the Navigation of a Mobile Robot
Amani Azzabi
1
, Marwa Regaieg
1
, Lounis Adouane
2
and Othman Nasri
1
1
Research Unit SAGE, ENISo, University of Sousse, BP 264 Sousse Erriadh 4023, Tunisia
2
Institut Pascal, Blaise Pascal University - UMR CNRS 6602, Clermont-Ferrand, France
Keywords:
Control Architecture, Mobile Robotics, Stability and Reachability Analysis, Hybrid Dynamic System, Hybrid
Automata, Interval Analysis.
Abstract:
This paper deals with the problem of unicycle mobile robot navigation in cluttered environments. It presents in
particular an approach which permits to verify the stability of the control architecture of mobile robot using the
reachability analysis. To perform this analysis, we consider the robot as a hybrid dynamic system. The latter
is modeled by an hybrid automata in order to verify the reachability property by using the interval analysis.
The simulation results validate the proposed control architecture.
1 INTRODUCTION
The navigation control of a mobile robot in cluttered
environment is a determining problem and is iden-
tified as among the priority field of research in the
robotics community. The main issues in this field is
how to obtain accurate, flexible and reliable naviga-
tion? In the proposed set-up, the mobile robot has for
mission to reach its target while avoiding any annoy-
ing obstacles. Thus, its main behavior is the attraction
toward the target and the Obstacle avoidance.
In the literature, a part of the community supposes
that the mobile robots use methods of path planning.
This means that the environment where it navigates is
totally or partially known. Thus, the robot leans on
a model of the world which is generally a map sup-
porting a planning. Among these methods, Vorono¨ı
diagrams, visibility graphs or artificial potential func-
tions include all the information about the task to be
achieved and the environment features (Santiago Gar-
rido and Jurewicz, 2011). The other part of the com-
munity admit that robot’s navigation is based only on
the capacity of the robot to answer to the perceived
stimuli using appropriate control law (called reactive)
which takes into account the robot’s constraints as
well as the local state of the environment (Luciano
C. A. Pimenta1 and Campos, 2006), (Egerstedt and
Hu, 2002).
Several control architectures based reactivemodes
are proposed in the literature. The conception of the
latter, called also behavioral architectures, is based on
several elementary controllers/behaviorsto be coordi-
nated: Selection of the actions (competitive architec-
tures) and the fusion of the actions (cooperativearchi-
tectures), (Adouane and Le-Fort-Piat, 2006), (Brooks,
1986). The work proposed in this paper studies the
stability of reactive control architecture, proposed in
(Adouane, 2009), for a unicycle robot.
Using the considered architecture, it is necessary
to guarantee the robot capacity to accomplish its mis-
sion while avoiding the obstacles. One of the solu-
tions consists in considering it as as hybrid dynam-
ical system whose behavior is modeled by a hybrid
automaton. As shown in (Luciano C. A. Pimenta1
and Campos, 2006), this approach of hybrid control
allows the coordination of the action of several mo-
bile robots and checks the reachability property. This
analysis consists in determining if a position or a con-
figuration can be reached by the system. Thus it
can check that no unwanted behavior of the system
will occur. To verify this property for dynamic sys-
tems, several approaches are proposed in the litera-
ture (J. Toibero and Kuchen, 2007). Among these
approaches, we adopt the reachability analysis using
the interval analysis (Ramdani et al., 2009). This ap-
proach, based on a hybrid automatons, allows find-
ing all the minimal and maximal trajectories of the
system, governed by Ordinary Differential Equations
(ODE), and to check according to these later if the
unwanted system configuration could occurs.
The rest of the paper is organized as follows: we
present briefly, in the section 2, the control architec-
491
Azzabi A., Regaieg M., Adouane L. and Nasri O..
Hybrid and Multi-controller Architecture for Autonomous System - Application to the Navigation of a Mobile Robot.
DOI: 10.5220/0005065404910497
In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 491-497
ISBN: 978-989-758-040-6
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

ture for the navigation of unicycle mobile robot in
a cluttered environment. After having modeled the
robot behavior by a hybrid automaton, we shall study
in section 3 the control architecture while considering
the reachability analysis using the interval analysis.
We present in section 4 the simulation results. Sec-
tion 5 concludes this paper with some prospects.
2 CONTROL ARCHITECTURE
FOR REACTIVE NAVIGATION
IN A CLUTTERED
ENVIRONMENT
We present in this section, the control/command ar-
chitecture of mobile robot developed in (Adouane,
2009).
2.1 Navigation in a Cluttered
Environment
The mobile robot has for mission to reach its target
while avoiding the obstacles met during its naviga-
tion. Therefore, its behavior is mainly the attrac-
tion toward the target and obstacle avoidance. Fur-
thermore, the navigation of the robot has to be safe,
smooth and fast.
The obstacles met during the robot navigation can
have different forms. In order that the robot can avoid
them, these obstacles are generally encompassed by
simple forms of circular or elliptic type (Adouane,
2009), (Adouane et al., 2011). After that, the ob-
stacle avoidance behavior based on limit-cycles can
be achieved (Kim and Kim, 2003), (Adouane, 2009),
(Adouane et al., 2011).
Figure 1: The used perceptions for mobile robot navigation.
In this work, the limit-cycles will be considered as
circular shape. In order to avoid collision, the robot
and the target will also be characterized or encom-
passed by circles. Several perceptions are necessary
for the navigation of the robot (cf. Fig. 1):
• D
ROi
distance between the robot and the obstacle
”i”,
• D
PROi
perpendicular distance between the line (l)
and the obstacle ”i”,
• D
TOi
is the distance between the obstacle i and the
target.
• R
R
, R
T
and R
Oi
are respectively the radius of the
robot, the target and the Obstacle
i
,
• for each detected obstacle
i
we define a circle of
influence with a radius of
R
Ii
= R
R
+ R
Oi
+ margin
The margin corresponds to a safety tolerance
which includes: the uncertainty of the perception,
control reliability and accuracy, etc.
2.2 Control Architecture
The proposed control architecture is represented by
the figure 2. This control architecture uses a proce-
dure for selecting a hierarchical action to manage the
switching between the controllers according to the en-
vironment perception and ensure the stability of the
overall control. Its objective is also to ensure safe,
smooth and fast robot navigation (Adouane, 2009).
The mechanism of selection activates the “obsta-
cle avoidance” controller if at least one obstacle is de-
tected. In order to understand the implemented hybrid
architecture of control to check the unicycle mobile
robot navigation, it is important to know its kinematic
model.
Figure 2: Control architecture for mobile robot navigation.
2.3 Model of the used Unicycle Robot
The well known kinetic model of a unicycle robot,
in a Cartesian reference frame (cf. Fig. 3), is given
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below:


˙x
˙y
˙
θ


=


cosθ −l
2
cosθ− l
1
sinθ
sinθ −l
2
cosθ+ l
1
sinθ
0 1


×

v
ω

(1)
with :
• x, y, θ: configuration state of the unicycle
robot at the point P
t
of abscissa and ordinate
(l
1
, l
2
) according to the mobile reference frame
(O
m
, X
m
, Y
m
) associated to the robot center.
• v: the robot’s linear velocity at the point ”P
t
”.
• ω : the robot’s angular velocity at the point ”P
t
”.
• (O
A
, X
A
,Y
A
) is the absolute reference.
m
O
1
l
2
l
x
y
θ
A
X
A
Y
A
O
m
X
m
Y
P
t
Figure 3: Robot configuration in a Cartesian reference
frame.
Knowing the model of the robot and the task to
achieve, we present the both controller ”Attraction to
the target” and ”Obstacle avoidance ”. The set of
these controllers will be synthesized using the Lya-
punov theorem (Benzerrouk et al., 2010).
2.3.1 Attraction to the Target Controller
This controller guides the robot toward the target po-
sition represented by a circle of center (x
T
, y
T
) and of
R
T
radius. As detailed in (Adouane, 2009), the used
control law is a control of position at the point P
t
of
coordinates (l
1
, 0). It is based on the configuration of
the robot position relative to the target, represented by
the errors e
x
and e
y
.
To guarantee that the center of the robot axis
reaches the target with asymptotic convergence, the
distance d =
q
e
2
x
+ e
2
y
must be smaller than R
T
.
The errors of position are given by the following
equations:
(
e
x
= x − x
T
e
y
= y − y
T
(2)
Thus:
(
˙e
x
= ˙x
˙e
y
= ˙y
(3)
For the stabilization of the error towards zero, a pro-
portional controller was used (Adouane, 2009):

v
w

= −K

cosθ −l
1
sinθ
sinθ l
1
sinθ

−1
×

e
x
e
y

(4)
with K > 0.
To study the asymptotic stability of the proposed
controller, let us consider the following Lyapunov
function: V
1
=
1
2
d
2
. Therefore, to guarantee this sta-
bility in the sense of Lyapunov, it is necessary that:
˙
V
1
< 0, so d
˙
d < 0, what is easily proven as long as
d 6= 0.
2.3.2 Obstacle Avoidance Controller
During the activation of this controller, the robot fol-
lows limit-cycle vector fields given by two differential
equations:
• For the clockwise trajectory motion:(cf. Fig 4(a))
˙x
s
= y
s
+ x
s
(R
2
c
− x
2
s
− y
2
s
)
˙y
s
= −x
s
+ y
s
(R
2
c
− x
2
s
− y
2
s
)
• For the counter-clockwise trajectory motion:(cf.
Fig 4(b))
˙x
s
= −y
s
+ x
s
(R
2
c
− x
2
s
− y
2
s
)
˙y
s
= x
s
+ y
s
(R
2
c
− x
2
s
− y
2
s
)
where (x
s
, y
s
) corresponds to the robot position ac-
cording to the center of the convergence circle (char-
acterized by an R
c
radius). These equations show the
direction of trajectories according to axes (x
s
, y
s
).
Figure 4: Shape possibilities for the used limit-cycles.
The control law associated with this controller al-
lows the robot to follow the trajectories of the limit-
cycles. The robot will be controlled according to its
center of coordinates (l
1
, l
2
) = (0, 0). The desired
robot orientation θ
d
is:
θ
d
= arctan(
˙y
s
˙x
s
) (5)
The error of orientation θ
e
is given by:
θ
e
= θ
d
− θ (6)
HybridandMulti-controllerArchitectureforAutonomousSystem-ApplicationtotheNavigationofaMobileRobot
493

The desired orientation is reached using the fol-
lowing control law:
w =
˙
θ
d
+ K
p
θ
e
(7)
with K
p
> 0.
The orientation error is given by the following dif-
ferential equation:
˙
θ
e
= −K
p
θ
e
. The asymptotic sta-
bility of the controller is verified through the follow-
ing Lyapunov function: V
2
=
1
2
θ
2
e
.
˙
V
2
is equal then to θ
e
˙
θ
e
= −K
p
θ
2
e
which is always
strictly negative (thus, asymptotically stable).
2.3.3 Hierarchical Action Selection Block
The block corresponding the selection of action man-
ages the switching between the controllers. It takes
a decision thanks to environment’s information, col-
lected by the sensors. The selection of action to be
made (“Attraction to the target” or “Obstacle avoid-
ance ”) depends on the distance between the perpen-
dicular distance to the line (l) and the distance be-
tween the obstacle and the circle of influence (cf. Fig
1).
• If (D
PROi
≤ R
Ii
) then the controller ”Obstacle
avoidance ” is activated.
• Else the controller ”Attraction to the target ” re-
main always active.
3 STABILITY OF THE CONTROL
ARCHITECTURE USING
REACHABILITY BY INTERVAL
ANALYSIS
In this section, we study the stability of the considered
reactive control architecture by checking the reacha-
bility property. That’s why, it is necessary to model
the robot behavior by a hybrid automaton.
3.1 Hybrid Automaton for the Robot’s
Control
An hybrid automaton handles a set of continuous dif-
ferential equations, models a dynamic system and de-
scribes its behavior. As mentioned in (Ramdani et al.,
2009), a hybrid automaton used to find the envelope
which frames, in a guaranteed way, the real robot’s
path, is generally considered as being one 6-uplet
H = (Q, X, P, F,T, RI) defined by:
⋄ Q is a discrete set of modes or situations;
⋄ X represents the continuous state space (of dimen-
sion n), the state vector is noted x and the initial
state is noted x
0
;
⋄ P = [p] = [p
, ¯p] represents the parameters’ admis-
sible bounded domain;
⋄ F = {( f
q
,
¯
f
q
), q ∈ Q}, f is the collection of vector
fields that surround all the possible dynamics;
⋄ T = {t
e
, e ∈ E} is the collection of the transitions’
moments where E ⊆ Q × Q is the set of transi-
tions;
⋄ RI = R
e
, e ∈ E is the collection of the functions
of update.
According to the case of our studied model, we
have:
⋄ Q = {q
1
, q
2
}={ Attraction to the target , Obstacle
avoidance} : the set of discrete states;
⋄ X = R
3
, z = (x, y, θ) ∈ X is the state vector;
⋄ P = {p
q
1
, p
q
2
} is the admissible bounded domain
of parameters;
⋄ T = {t
q
1
q
2
,t
q
2
q
1
} where
t
q
1
q
2
= (q
1
, D
PROi
≤ R
Ii
, Id, q
2
)
and
t
q
2
q
1
= (q
2
, D
PROi
> R
Ii
, Id, q
1
)
⋄ F = ( f
q
,
¯
f
q
), q ∈ Q where f is the collection of
vector fields that surround all the possible dynam-
ics.
The unicycle robot behaviorin a cluttered environ-
ment can be then modeled by the hybrid automaton of
the figure 5.
Figure 5: Hybrid automaton of the navigation task to be
performed by the robot.
The continuous dynamics of the robot in the dis-
crete state q
1
at the point P
t
of coordinates (l
1
, 0) is
defined by:
˙z = f
q
1
(z, u, p
q
1
) =





˙x = vcosθ − wl
1
sinθ
˙y = vsinθ − wl
1
cosθ
˙
θ = w
(8)
with z = (x, y, θ) ; u = (v, w) and p
q
1
= (l
1
, K).
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The continuous dynamics of the robot in the dis-
crete state q
2
which is controlled by its center of co-
ordinates (l
1
, l
2
) = (0, 0) is defined by:
˙z = f
q
2
(z, u, p
q
2
) =





˙x = vcosθ
˙y = vsinθ
˙
θ = w
(9)
with z = (x, y, θ) , u = (v, w) and p
q
2
= (K
p
).
The invariantregions associated with both discrete
states q
1
and q
2
are respectively (D
PROi
> R
Ii
) and
(D
PROi
≤ R
Ii
). The transition from a state to another
requires that the position and the orientation of the
robot verify the conditions of guard ((D
PROi
> R
Ii
)
and (D
PROi
≤ R
Ii
)). The update functions RI are con-
sider as the identity functions because the robot dy-
namics doesn’t present discontinuity during the cross-
ing of the transitions between the discrete states.
After modeling the robot behavior by an hybrid
automaton, we will verify the stability of the control
architecture considered using the reachability analy-
sis.
3.2 Principle of the Hybrid Bounding
Method
As indicated in (Ramdani et al., 2009), The method of
hybrid bounding is an approach of reachability anal-
ysis. It allows thus to find the envelope enclosing all
possible trajectories of the robot in order to study its
stability. For that purpose, we will detail the algo-
rithm used to calculate an over-approximation of the
reachable space.
The algorithm 1 of the hybrid bounding method
looks first for the initial discrete mode q through the
function Initialization ”Initialization”. This function
identifies the initial discrete state according to the
signs of all the partial derivatives of the vector fields
relative to the uncertain parameters. In our case, these
parameters are θ, x and y. Then, as long as the time
t doesn’t reach the final value t
nT
, a guarantee inte-
gration of the current mode is given by the algorithm
2 ”Integer-one-step” which makes a step of integra-
tion noted h. Next, according to the signs of partial
derivatives, the function ”switching” verifies if there
is a transition towards another mode on the interval
[x
j
, x
j+1
] and calculates the new mode q
′
. If it is the
case, then the boolean variable jump, that indicates
the existence of a transition, sets the boolean value to
1 and if not it takes the value 0. The switching from
a mode to another depends on the current mode. If it
is equal to 0 then it is enough to turn to the new mode
q
′
6= 0 and continues the integration. Else (q 6= 0), it
Algorithm 1: Main algorithm.
1 ⋄ Inputs : t
0
, t
nT
, F, [x
0
], [p]
2 ⋄ Outputs
: [ ˜x
0
], [ ˜x
1
],..., [ ˜x
nT
], [x
1
],..., [x
nT
]
3 Begin
4
j := 0 ;
5 q := Initialization ( f , [x
0
] , [p] ) ;
6 while ( j < nT) do
7 (h
j
, [x
j+1
], [ ˜x
j
]) :=
Integer-one-step
(q, f, F, t
j
, [x
j
], [p]) ;
8 ( jump, q
0
) := switching (q, f, [ ˜x
j
]) ;
9 if ( jump) then
10 if (q = 0) then
11 q := q
0
;
12 j := j+ 1
13 else
14 q := 0
15
end if
16 else
17 j:=j+1
18 end if
19 End
is necessary to repeat the integration on the mode q =
0 to save the guarantee of the frame for the transition
crossing.
Besides, the algorithm 2 ”Integer-one-step” de-
tailed below calculates on each step of integration,
a protected frame from solutions of uncertain differ-
ential equation. In fact, for the case of the mode
q = 0 the digital integration of the uncertain differ-
ential equation is given through the method of Taylor
Intervalle (MTI)in the line 5. This method contains
essentially two steps:
• a step of prediction which verify the existence and
the uniqueness of the solution.
• a step of correction which calculate the solution
[x
j+1
] at the moment t
j+1
= t
j
+ h
j
.
However, for the modes q 6= 0, firstly we select the
ODE bounding in the line 7 and the initial conditions
at the moment t
j
(ω(t
j
), Ω(t
j
)) are fixed to lines 8 and
9. Then, the digital integration of this EDO bounding
is executed in the line 10. Finally, to guarantee nu-
merically the obtained results, we use as in the line 5
the integration methods based on the models of Taylor
intervals to solve the ODE.
4 SIMULATION RESULTS
The stability of the proposed architecture is verified
using the hybrid bounding method. For that purpose,
HybridandMulti-controllerArchitectureforAutonomousSystem-ApplicationtotheNavigationofaMobileRobot
495

Algorithm 2: Integer-one-step.
1 ⋄ Inputs : q, F, t
j
, [x
j
], [p]
2 ⋄ Outputs
: h
j
, [x
j+1
], [ex
j
]
3 Begin
4
if q := 0 then
5 ( h
j
, [x
j+1
] , [ex
j
] ) := MTI( f , [x
j
] , [p] , t
j
);
6 else
7 ( f
q
, f
q
) := select frame( q , F ) ;
8 [ω
j
] := [x
j
]
;
9 [Ω
j
] := [x
j
] ;
10 ( h
j
, [ω
j+1
] , [Ω
j+1
] , [
e
Ω
j
] , [
e
ω
j
] ):= MTI( f
q
,
f
q
, [ω
j
] , [Ω
j
] , [p] , [p] , t
j
) ;
11 [ex
j
] := [
e
ω
j
,
e
Ω
j
] ;
12 [x
j+1
] := [ω
j+1
, Ω
j+1
] ;
13 end if
14 End
we used the MATLAB toolbox INTLAB and we sup-
pose that the data received from the robot sensors are
uncertain. Thus, the state vector components are rep-
resented by intervals translating this uncertainty on
the position (x et y) and on the orientation θ. The
execution of the hybrid bounding algorithm over the
period of simulation [0, 5]s gave the results of figures
6, 7 and 8.
The simulation results show the guarantee frame
of all the temporal trajectories of the robot. Indeed,
there is no intersection with hazardous areas and all
solutions do not leave the envelope during the period
of simulation. Therefore, these simulations prove the
stability of the robot and demonstrate the efficiency of
the reactive control architecture presented in section
2.
5 CONCLUSION
In this work, we were interested in the problem of mo-
bile robot navigation in a cluttered environment. In
particular, the stability of the dynamics of the robot
using reactive control architecture was studied. At
first, we modeled the set(continuous dynamics of the
robot and the discrete change of the used control laws)
by a hybrid automaton. Thereafter, we used an ap-
proach of reachability analysis based on the interval
method in order to verify the stability of the dynamic
system even in presence of uncertainty. This approach
is based on the calculation of an over-approximation
of the reachable space set of the robot (which corre-
Figure 6: Frame of all the possible solutions of x(t) due to
the hybrid bounding method.
Figure 7: Frame of all the possible solutions of y(t) due to
the hybrid bounding method.
Figure 8: Frame of all the possible solutions of θ(t) due to
the hybrid bounding method.
spond to all the points of the robot’s trajectory). The
idea was to find a guarantee frame of all the trajecto-
ries possible taking into account the uncertainties on
the data received by the sensors. Thus, the reacha-
bility analysis is handled by calling on to the inter-
val analysis. Finally, we validated the proposed so-
lution by simulations. The obtained result show that
the considered control architecture allows the robot to
have a smooth behavior while avoiding the obstacles
met throughout its navigation toward the target. Fu-
ture work will aims to test in real experimentation the
obtained results.
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496

ACKNOWLEDGEMENTS
This work was supported by LABEX IMobS3 Inno-
vative Mobility: Smart and Sustainable Solutions.
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