International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6583-6590
© Research India Publications. http://www.ripublication.com
Decentralized Non-Linear Control of Leader-Follower Formation of
Multiple Autonomous Mobile Robots
Azza El-Sayed Bayoumi Ibrahim
Department of Computers and Systems,
Electronics Research Institute, Giza, Egypt
E-mail: [email protected]
maintain the desired formation parameters (distance and
orientation) with the leader robot. The main critique of the
leader-follower approach is that the formation is depended on
the leader for achieving the goal and this over-reliance on a
single agent may be undesirable. In spite of these shortages,
the leader-follower architectures are particularly appreciated
for their simplicity and scalability [9].
The formation of a team of autonomous mobile robots with
nonholonomic constraints is considered as a nonlinear system.
Linearization techniques are used to deal with the
nonlinearities. Although the nonlinear formation system can
be linearized, there exist many problems in practice. Research
papers try to handle these problems as in [10,15, and 17], so
using a nonlinear control method is preferred. Sliding Mode
Control (SMC) is a non-linear systematic control method. Its
main advantage is the sliding motion exhibits complete
robustness to system uncertainties [12, 13]. Some previous
works utilized SMC to design the followers controllers
[11,14]. However, this control strategy has two main
drawbacks: the well-known chattering phenomenon and the
sensitivity of the system motion to disturbances during the
reaching phase. Thus, efforts have been made to minimize or
even remove the chattering and the reaching phase. A time
varying sliding mode control (TVSMC) method is suggested
to eliminate the reaching-phase [4]. In this method, the sliding
surface translates in the phase plane with constant slope
without rotating. The sliding surface should be chosen such
that the error along this surface vanishes.
In this paper, the formation controller is designed based on
TVSMC method. The fundamental idea is that the timevarying sliding surface passes through the initial system states
at the start of the motion and then moves towards a
predetermined time-invariant sliding surface with the form of
shifting and/or rotating [5]. Thus, the robustness of the
trajectory for the nominal system can be guaranteed from the
initial time instant [4]. Therefore, TVSMC technique can be a
very powerful solution to the problem of leader-follower
trajectory tracking control in the presence of disturbances.
Abstract
Control of the cooperative multi-robot system is one of the
most challenging problems. This paper investigates a novel
implement of integral time varying sliding mode control
(ITVSMC) on the leader-follower formation of nonholonomic
mobile robots with bounded control inputs. The objective of
the controller is to form up and maintain the follower robots in
tracking the leader motion. The controller parameters are
automatically computed using mathematical formulas. The
annoying chattering problem in the control law is solved using
the boundary layer technique. The effectiveness of the
suggested strategy is verified by simulating the system using
MATLAB/SIMULINK
software.
Simulation
results
demonstrate that the designed control strategy success in
getting suitable values of controller parameters resulting in
perfect performance for different system conditions. Also, it is
proved that the time-varying sliding surface technique is more
accurate even in the presence of external disturbances than the
conventional time-invariable sliding surface. Therefore,
ITVSMC technique is a good solution to the problem of
leader-followers trajectory tracking in the presence of
disturbances.
Keywords: Formation control; two-wheeled mobile robots;
Input constraints; time varying sliding mode control.
Introduction
In recent years, research on coordination control of multiple
mobile robots system has attracted many researchers attention
and shown its broad application prospect in the military, space
exploration, medical, service industry and other fields [2].
Cooperative/formation control appears because a group of
robots can accomplish a mission more efficiently over a single
robot. Differential drive wheeled mobile robots (WMR) are
usually employed in multi-robots applications, due to their
fast manoeuvring, low cost and simplicity [1].
Different strategies have been introduced to control the robots
formation [3], the most popular approaches are: (i) The
behaviour-based approach [6], where each vehicle is
described by several desired behaviours and the final control
is computed from a weighting of the each behaviour; (ii) The
virtual structure approach [7], where the entire agents are
treated as a single structure. The desired motion is assigned to
the virtual structure which trails trajectories for each member
in the formation to follow; (iii) The leader-follower approach
[8]. In this approach, one of the robots is selected as a leader,
while other robots are followers. The leader moves along a
predesigned collision free trajectory and the followers
Problem Formulation
Given a group of mobile robots (WMRs), one robot is selected
as a leader and the others are as followers. It is assumed that
the leader robot moves along a predefined collision-free
trajectory, the vector ( , ) can be measured and passed to
each follower. It is required to design a control law such that a
desired formation for the whole system can be achieved while
tracking the motion of the leader. The objective of the
controller of each follower is to find the values of its
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6583-6590
© Research India Publications. http://www.ripublication.com
translational, and rotational velocities (
) such that the
relative position errors to the leader tend to zero.
Leader-Follower Model
In an inertial reference frame, the configuration of the robot i
is given by the position of the centre between its two wheels
(xi,yi) and the angle θi of its heading direction. The heading
direction is the line perpendicular to the axes of wheel
rotations, and it is positive in the direction of forward motion.
The position of the robot is defined by qi = (xi,yi,θi ). The
kinematics of a robot "i" is described by Equation (1) and the
nonholonomic constraint, where the driving wheels are
assumed to roll purely and do not slip is expressed by
Equation (2), [16].
Controller Design
Based on the TVSMC method, the switching surface starts
moving uniformly (i.e. with a constant velocity) in the state
space and then it stops at a time instant > , where is the
initial time, the details of the method is well described by
Andrzej Bartoszewicz in [5]. As shown in Fig2. Once the
surface reaches the origin of the error coordinate frame, it
stops moving and remains fixed and time-invariant.
Where the translational velocity vi and the angular velocity ωi
are the input control signals.
Figure 2: Time Varying Sliding Surface phase portrait
A follower tracking errors and the errors rates are introduced
as:
, and
, and
(7)
To develop the ITVSMC approach, the integral-type time
varying sliding surface is defined twice times as declared in
Figure 2, for time t ϵ {
.
Figure 1: Sketch of leader-follower coordinated
(8)
Where, ci, Ai and Bi are constants. The selection of these
constants will be addressed later.
Since the considered surface stops moving at the time , for
any t ≥ it is fixed and can be described as follows:
The appropriate modelling of leader-follower formation is
derived in [11]. It is evident from Fig.1 that (xL,yL) represents
the leader’s position, and (xF,yF) for the follower’s position.
The angle between leader’s
and the x-axis is θL and the
angle between follower’s
and the x-axis is θF. The relative
distance between the leader and the follower is represented by
and it is inclination with the x-axis by the angle .
Therefore, the formation parameters can be expressed by Eq.
(3).
,
(3)
(9)
Therefore, two controllers are designed and switching
between them is done at time
or when the surfaces reach
the origin [4]. In the period of
, the sliding surface is
On the other hand, the relationships between velocities and
angles are obtained by Equations (4), (5), and (6).
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6583-6590
© Research India Publications. http://www.ripublication.com
fixed and described by equations (9). The controller is
designed as in the conventional SMC [12], the variable s is
differentiated and equated to zero to get the equivalent control
signal on the sliding surface.
(10)
(11)
Since the desired formation parameters described by
and
are chosen to be constants values,
, and
.
Then equations (4), (5), and (6) are substituting into the
resulting formulas, the control signals can be obtained with
the knowledge that
and
. Therefore,
equations (12) and (13) express the final control signal form.
Sliding Surface Parameters Design Subject to Input Signal
Constraint
The constants As, Bs and Cs are chosen in such a way that the
representative point of the system at the initial time belongs to
the switching surface and subject to a constraint on the input
signal, such that the maximum admissible values of the input
signals are u1max and u2max. It means that the following
inequalities should be held.
and
(17)
Where, u1max, u2max are constants, which satisfies the following
conditions:
and
(12)
(18)
Which imply that there exist such strictly positive constants
satisfy the following conditions.
(13)
To keep the system trajectory staying perfect on the sliding
surface despite the influences of the disturbances, the
reachability condition is essential. Such a condition is satisfied
by using the switching function sgn (si ),
is the switching
gain constant, should be chosen to compensate for the system
disturbances, in which the upper limit of the disturbances is
assumed to be known. The sgn (si ) function is always
replaced by sat (si / i ) to avoid occurring of fast oscillation,
where i is a small positive value which expresses the
boundary layer thickness, the mathematical description of it
is declared in Equation (14).
(19)
(20)
The optimal switching surface parameters are given by the
following formulas, considering
are strictly positive
constants, which represent the lower bound of the absolute
value of control signal multiplier, therefore, the complete
derivation is illustrated in [4].
,
,
(21)
(14)
Similarly, for the period
, the second control law is
designed, but using the sliding surfaces described by equation
(8). Using the same procedure, the control signals can be
obtained as:
(22)
Where
and
are the initial values of the linear and
the angular separation errors which depend on the initial
position of the follower and the leader vehicles. The switching
surface stops moving at the time instants,
,
computed
by Equations (21) and (22).
(15)
(16)
According to Equations (15) and (16), the control signals
values are dependent on the values of the parameters of the
sliding surfaces, As, Bs and Cs and, ks. Figure 3 illustrates the
block diagram of the overall controlled system of the leaderfollower approach.
Simulation Results
The effectiveness of the proposed algorithm can be verified by
a numerical simulation using Matlab/Simulink software. The
reference desired trajectory is given by y = g(x) =
sin(0.5x)+0.5x+1, and x( ) = . Four robots are used to form
an example of analysis, one robot act as a leader and three act
as followers. For a case study, the initial conditions of the
leader robot (xL(0), yL(0), θL(0)) = (3,2,0), and for a follower
(
,
) = (1.5,0.7,2 ) as in follower 3 in Table
[1]. The desired separation parameters are chosen as
and
. The parameters of the designed controller
are computed according to the proposed criteria such that
, and
and recorded in
table [1]. After several running of the simulator, it is found
that the suitable value of the bounded layer thickness is equal
Figure 3: The controlled system block diagram of one
follower
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6583-6590
© Research India Publications. http://www.ripublication.com
to 0.2 of the corresponding feedback gains (
. Figure
(4-a) shows that the follower robot can accurately track the
leader trajectory with very low steady state errors, the
corresponding error signals are depicted in Figure (4-b) and
(4-c). There is no reaching phases and the sliding mode is
starting from the initial time as shown in Fig. 5. The control
signals are drawn in Figure 6.
Figure 5: The behaviours of the sliding surfaces s1 & s2 based
on ITVSMC
Figure 4: (a) The Leader-Follower trajectories based on
ITVSMC. (b) The linear separation error signals; (c) The
angular separation error signals
Figure 6: The control signals u1 & u2 based on ITVSMC
The proposed algorithm is also tested in controlling a group of
robots. Two different cases are suggested.
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6583-6590
© Research India Publications. http://www.ripublication.com
In the first case, three followers are put in different initial
positions, and it is required from each follower to track the
leader with the same separation parameters. Simulation results
are recorded in table [1], and the system responses are
depicted in Figure (7-a). Referring to the figure, it is evident
that all followers track the leader trajectory precisely, and the
proposed methodology success in determining the appropriate
controllers’ parameters without trial and error method. Note
that the follower which starts far from the leader takes a
longer time to reach the desired situation.
performances are drawn in Fig.(7-b), As shown in the figure
the three followers track the leader correctly. The designed
controller for each follower is able to get precise values of its
parameters, which verify the powerful of the proposed
method.
Table 2: System parameters with no disturbances for different
desired positions, (Case2).
Case 2
Table 1: System parameters with no disturbances for different
followers initial positions, (Case1).
Case 1
parameters of
Controller (u1)
parameters of
Controller (u2)
Switching gains
Steady-state
errors
Desired Parameters
Follower 1 Follower 2 Follower 3
Initial conditions
A1
B1
C1
A2
B2
C2
K1
K2
e1s.s
e2s.s
5
0.9
2
0.6
0.78
0.4
-0.14
0.84
0.301
0.051
5.7e-10
4.2e-12
3.1
0.51
0.1
-0.25
2.7
0.01
-0.15
5.4
0.01
0.001
0
0
Follower 1 Follower 2 Follower 3
parameters of
Controller (u1)
1.5
0.7
-1.5
0.45
0.77
0.2
-0.12
1.09
0.5
0.4
-5e-11
2.27e-16
parameters of
Controller (u2)
Switching gains
Steady-state errors
A1
B1
C1
A2
B2
C2
K1
K2
e1s.s
e2s.s
3
0.5
-2
0.51
0.71
0.2
-0.14
1.19
0.301
0.051
-4.63e-9
1.1e-15
4
0.6
-3
0.57
0.63
0.1
-0.16
1.97
0.01
0.001
-1.8e-7
0
5
0.9
-4
0.45
0.47
-0.2
0.13
1.14
0.5
0.4
-4.2e-5
-5.6e-10
Figure (7-b): Multi-followers behaviours for different desired
relative position.
Comparing TVSMC with Conventional SMC methods
The SMC controller is also simulated, and the same previous
case study is used to test it. The follower trajectory designed
by the ITVSMC and the SMC is compared with the leader
trajectory and drawn in Fig. (8-a). In the case of the SMC
method the follower tracks the leader with steady state errors
equal to e1s.s = 0.0315, and e2s.s = 9.82e-8. But for TVSMC,
the error is avoided. The sliding surfaces are registered and
illustrated in Figure 9, which shows that the sliding surfaces
based on SMC take a longer time to reach the sliding mode
than the TVSMC.
Figure (7-a): Multi-followers behaviours for different initials
positions.
For the second case, the induced controllers are also examined
by another situation of the multiple followers. All followers
started from the same position and asked to follow the leader
with different separation parameters. All results and
parameters of this case are demonstrated in table2. The system
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6583-6590
© Research India Publications. http://www.ripublication.com
To demonstrate the robustness properties of the ITVSM from
the beginning, Both controllers are tested by the same case
study, but the system is suffered from external additive
disturbances and the results are drawn in Figure (8-b). It is
evident that a bad vibration appears on SMC trajectory in a
period which makes the system out of control over this period,
but the controlled system under TVSMC can withstand the
disturbances more than the SMC.
Moreover, the ITVSMC algorithm calculates the controller
parameters by itself. On the other hand, the SMC parameters
are adjusted manually by trial and error method which is a
tedious method resulting in wasting the efforts and consuming
the time.
Finally, the designed controller based on the ITVSMC is
faster, more accurate, more robust than the one designed
based on the SMC.
Figure 8: (a) Comparison bet. SMC & ITVSMC with nodisturbances. (b) The comparison with system disturbances
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6583-6590
© Research India Publications. http://www.ripublication.com
[2]
[3]
[4]
[5]
Figure 9: The behaviors of the sliding surfaces s1 & s2 based
on the SMC & ITVSMC
[6]
Conclusion
This paper introduces a framework for controlling a group of
autonomous two-wheeled mobile robots based on leaderfollower approach control in the presence of environmental
disturbances and control input constraint. A decentralized
cooperative control for each follower has been implemented
employing a time-varying sliding mode nonlinear controller.
Numerical simulations of the designed control strategy are
done. The suggested approach is tested using different status
of the system. The results show that the follower robots can
accurately track the leader trajectory with very low steady
state errors even there exist an additive bounded external
disturbances.
A comparison between the TVSMC and the conventional
SMC is carried out. It is proven that the time-varying sliding
surface can get faster error convergence and less steady state
error than the time-invariable sliding surface. Also, it is
noticed that the system behaviors based on the TVSMC are
more resistance to the external disturbances than the SMC.
Moreover, the time of converging to the equilibrium point is
lower than the conventional sliding mode control method. In a
case of SMC simulation, the parameters are changed to
choose the best ones after a process of trial and error.
However, this is not valid in the case of an actual system. But
the ITVSMC algorithm develops accurate calculations for the
controller parameters. A linear relation between the feedback
parameter and the thickness of the boundary layer is suggested
to avoid occurring of the chattering.
All these aspects make this approach highly attractive in
diverse application domains of mobile vehicles.
[7]
[8]
[9]
[10]
[11]
[12]
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