A Relative Measurement based Leader-follower Formation Control of
Mobile Robots
Yu. N. Zolotukhin, K. Yu. Kotov, A. S. Maltsev, A. A. Nesterov, M. A. Sobolev and M. N. Filippov
Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences,
pr. Akademika Koptyuga 1, Novosibirsk, 630090, Russia
Keywords:
Leader-follower Robot Formation, Formation Control, Desired Motion.
Abstract:
This paper deals with leader-follower formations of nonholonomic mobile robots and introduces a new non-
linear control method for the robot motion in formation. Proposed approach enables to track the target po-
sition and is based on using a forced movement along the desired trajectory in the state space. Moreover,
this approach requires only relative and local motion sensors data. Simulation results have demonstrated the
effectiviness and robustness of the proposed control shemes.
1 INTRODUCTION
During the last years there is an increasing interest
in the formation control and coordination of mul-
tiple mobile robots. Cooperative using of mobile
robots in the group is more efficient than the use of
a single robot in various tasks including search, ob-
servation, transport, rescue and military operations
(Schaub et al., 2000), (Smith et al., 2001), (Lawton
et al., 2003), (Burns et al., 2000).
At the present time there are three main ap-
proaches to tackle the robot formation control prob-
lem: leader-follower approach (Das et al., 2002), be-
havior based (Lawton et al., 2003) and virtual struc-
tures (Lewis and Tan, 1997). In this study, we use
the leader-follower approach proposing the division
of robot group members on leaders and followers.
The followers task is to maintain a desired distance
and orientation to the leader. The main drawback of
the leader-follower approach is that it depends heav-
ily on the leader for achieving the goal. However this
approach is appreciated for its simplicity and scalabil-
ity.
In contrast to approaches (Lawton et al., 2003),
(Consolini et al., 2008), (Zolotukhin et al., 2007),
where it is necessary to know the absolute position
and/or orientation of the leader for the determination
of desired follower position in the group, in this study
we use the approach in which follower position is de-
fined in terms of the relationship of robots in local
follower frame. This approach is more usable — in
the most cases the mobile robot in the group can deal
only with sensors data on the relative location of the
robots.
In leader-follower formation control, the most
widely used control technique is feedback lineariza-
tion based on the kinematics model of the system (De-
sai et al., 2001), (Min et al., 2009), (LIU Shi-Cai,
2007). The application of this approach is limited by
the complexity of the kinematic model and low ro-
bustness to external disturbances.
In this paper, the method of organization of forced
motion over a desired trajectory in the plane state
space is used to control homogeneous group of dif-
ferential drive mobile robots. It was developed by us
and successfully applied in a number of applications
(Zolotukhin et al., 2007), (Belokon et al., 2013).
Simulation results verify the effectiveness of the
proposed control system in the presence of measure-
ment noise and external perturbations.
The rest of this paper is organized as follows. In
Section 2 we recall the dynamic model of the mobile
robot and formulate the formation tracking control
problem. The proposed control schemes are presented
in Section 3. In Section 4 we show illustrative simu-
lation results and we conclude with some remarks in
Section 5.
2 PROBLEM STATEMENT
Let us consider the homogeneous group of mobile
robots, which of them have the following equations
310
Zolotukhin Y., Kotov K., Maltsev A., Nesterov A., Sobolev M. and Filippov M..
A Relative Measurement based Leader-follower Formation Control of Mobile Robots.
DOI: 10.5220/0005543103100313
In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 310-313
ISBN: 978-989-758-123-6
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

of motion (Lawton et al., 2003)
˙x
i
= v
i
sinφ
i
;
˙y
i
= v
i
cosφ
i
;
˙
φ
i
= w
i
;
˙v
i
=
F
i
m
i
;
˙w
i
=
M
i
J
i
.





















(1)
Here x
i
, y
i
are the position coordinates of the robot; v
i
,
w
i
are the linear and angular velocities; φ
i
is the angle
showing the moving direction of the vehicle relative
to the axis of ordinates (see Fig. 1); m
i
is the mass and
J
i
is the moment of inertia. The control inputs are the
linear propulsive force F
i
and the torque moment M
i
.
In what follows the point above the variable implies
its time derivative.
Vehicle control is realized by changing linear and
angular accelerations of the vehicle by the parameters
F
i
m
i
M
i
J
i
. We define a leader in the group that directs
the other members of the group and helps them to de-
termine their positions.
Let us assume that the leader motion along the
prescribed trajectory is defined by control algorithm
presented in (Zolotukhin et al., 2007). It is necessary
to indicate the place of group members against the
leader in order to describe their movement. We as-
sume that the navigation system of the follower can
determine parameters of the follower position against
the leader: d
i
— distance to the leader and α
i
— di-
rection to the leader relative to the direction of the
follower movement, Fig. 1. Such a data is a range
measuring data and can be provided from laser and in-
frared sensors, cameras. In this case, we can uniquely
specify the required coordinates for the vehicle and its
course in a group.
Let us set a task to the follower to move to the
target point T
0
, determined by the parameters d
i re f
,
α
i re f
of relative leader-follower arrangement, Fig. 1.
It will be shown that after finishing transients in con-
trol system, in stationary mode, follower and leader
courses are the same and thus this approach to deter-
mine follower position is equivalent to determine tar-
get position with using leader direction data. Hence, it
is unnecessary to apply additional control as in (Zolo-
tukhin et al., 2007) when follower approaches to the
target point. We define the error in the follower posi-
tion against the target point by values, Fig. 1a
E
τi
= d
i
cos(φ
i
+ α
i
) − d
i re f
cos(φ
i
+ α
i re f
);
E
ni
= d
i
sin(φ
i
+ α
i
) − d
i re f
sin(φ
i
+ α
i re f
).

(2)
Let us consider the stationary case of the movement,
when
˙
E
τi
=
˙
E
ni
= 0 and transients of target positioning
φ
i
(x
i
(t), y
i
(t))
α
i
V
i
Y
X
φ
0
V
0
(x
0
(t), y
0
(t))
α
i_ref
d
i_ref
φ
i
E
τ
E
n
d
i
T
0
Figure 1: Movement of the follower relative to target posi-
tion in the group.
are completed. Using the fact that
d
i
cos(φ
i
+ α
i
) = y
0
− y
i
;
d
i
sin(φ
i
+ α
i
) = x
0
− x
i

(3)
and with regard to motion equations (1) compute (2)
v
i
cos(φ
i
) = v
0
cos(φ
0
) + d
i re f
sin(φ
i
+ α
i re f
)
˙
φ
i
;
v
i
sin(φ
i
) = v
0
sin(φ
0
) − d
i re f
cos(φ
i
+ α
i re f
)
˙
φ
i
.
)
(4)
Given a constant course angle φ
i
or
˙
φ
i
= 0, from the
ratio of equations in (4) it is not difficult to show that
φ
i
= φ
0
. It follows that resulting follower position in
formation is independent of the definition of the target
position parameter α
i re f
— relative to the follower or
leader course.
3 CONTROL LAW SYNTHESIS
In accordance with the approach described in (Zolo-
tukhin et al., 2007), (Belokon et al., 2013) define the
functions
S
1i
=
˙
E
τi
+ k
e
E
τi
;
S
2i
=
˙
E
ni
+ k
e
E
ni
)
(5)
and require the following conditions to be satisfied
S
1i
= 0; S
2i
= 0. (6)
In this case, the errors (2) decrease exponentially,
with the time constants 1/k
e
.
Note that the absolute heading angle φ
i
is used in
the equations (2). If considering that x,y coordinate
system is a body-fixed system then we can suppose
φ
i
= 0 in equations (2)
E
τi
= d
i
cos(α
i
) − d
i re f
cos(α
i re f
);
E
ni
= d
i
sin(α
i
) − d
i re f
sin(α
i re f
).

(7)
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311

Thus, control of mobile robot can be relized only
on the basis of the relative leader-follower arrange-
ment data, defined by the parameters d
i
, α
i
.
The fulfillment of the conditions (6) can be pro-
vided by
d
dt
S
2
1i
≤ 0;
d
dt
S
2
2i
≤ 0. (8)
Let us strengthen conditions (8) by assuming that
˙
S
1i
= −α
1
S
1i
;
˙
S
2i
= −α
2
S
2i
. (9)
Here α
1
> 0, α
2
> 0 determine the time constants
1/α
1
, 1/α
2
with which S
1i
, S
2i
exponentially tend to
zero. Differentiating Eqs. (2) and substituting the
results to Eqs. (9) with taking into account (1) and
φ
i
= 0
F
i
m
i
= ¨y
0
− d
i re f
(
˙
φ
2
i
cos(α
i re f
) +
sin(α
i re f
)M
i
J
i
)−
− α
1
S
1i
− k
e
˙
E
τi
;
M
i
J
i
=
1
d
i re f
cos(α
i re f
)
(v
i
˙
φ
i
− ¨x
0
+
+ d
i re f
sin(α
i re f
)
˙
φ
2
i
+ α
2
S
2i
+ k
e
˙
E
ni
).

















(10)
Subject to uniform motion, the accelerations values
¨x
0
, ¨y
0
of the leader can be assumed to be zero, there-
fore when using the equations (10) it is necessary to
know the follower speed parameters v
i
and
˙
φ
i
. This
information can be provided by the inertial measur-
ing unit (accelerometers, gyroscopes) on a robot plat-
form.
We assume that the follower vehicle moves with
bounded linear velocity along smooth path and effects
due to rotation are expected to be small, hence we can
neglect measurement of angular velocity
˙
φ
i
= 0. The
equations (10) take the form
F
i
m
i
= −d
i re f
sin(α
i re f
)
M
i
J
i
− α
1
S
1i
− k
e
˙
E
τi
;
M
i
J
i
=
1
d
i re f
cos(α
i re f
)
(α
2
S
2i
+ k
e
˙
E
ni
).







(11)
In this case the control parameters are computed as
rather simple equations, however the control accuracy
is reduced.
4 SIMULATION RESULTS
The controlled object is an e-puck mobile robot with
a differential drive, which was designed for education
in engineering at the EPFL (Switzerland) (Mondada
et al., 2009). The discrete equations of the kinematic
and dynamic description of the robot have the form
x
k
i
= x
k−1
i
+ ∆tv
k
i
sinφ
k
i
;
y
k
i
= y
k−1
i
+ ∆tv
k
i
cosφ
k
i
;
φ
k
i
= φ
k−1
i
+ ∆tw
k
i
;
v
k
i
=
1
2a
0
(U
k
1i
+U
k
2i
);
w
k
i
=
1
2a
0
l
(U
k
1i
−U
k
2i
).























(12)
a)
b)
c)
Figure 2: Movement of the leader-follower group by using
control algorithm (11) : a) trajectory of the group move-
ment in the plane x, y (solid dashed line, follower target
track; solid black line, leader track; solid green line, fol-
lower track); b) control variables; c) deviation of the fol-
lower position from the target point.
ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics
312

where 1/(2a
0
) = 0.0652e-03, 2l = 0.05290 m; ∆t —
sampling time. Control inputs U
k
1i
+U
k
2i
, U
k
1i
−U
k
2i
cor-
respond to
F
i
m
i
and
M
i
J
i
. Position coordinates and orien-
tation x
k
i
, y
k
i
and φ
k
i
are corrupted by additive gaussian
noises σ
x
= 0.001 m, σ
y
= 0.001 m, σ
φ
= 0.05 rad.
The leader motion along the prescribed trajectory is
defined by control algorithm described in (Zolotukhin
et al., 2007).
The series of simulation results we carried out re-
flect the smooth trajectory of proposed control for the
follower Fig. 2. Follower reference or target trajec-
tory shown on Fig. 2 is calculated as
x
i re f
= x
0
− d
i re f
sin(φ
i
+ α
i re f
);
y
i re f
= y
0
− d
i re f
cos(φ
i
+ α
i re f
).

(13)
The control parameters are selected as k
e
= 10, α
1
=
α
2
= 20. The parameters of the follower position in
the group are d
1 re f
= 0.5 m, α
1 re f
= 1.046 rad.
The root-mean-square of the follower position de-
viations during the uniform motion is about 0.004 m,
Fig. 2c. The peak-to-peak amplitude of the control
signal oscillations is about 0.1 from the maximum al-
lowable value, Fig. 2b. Defining the follower target
position in the follower frame it is possible to provide
smaller required follower acceleration, Fig. 2a in con-
trast with results of (Zolotukhin et al., 2007).
5 CONCLUSIONS
In this paper, we have derived a robust control algo-
rithm for leader-follower formations of mobile robots.
The proposed controller does not need global sensor
for formation control and use only the relative mea-
surement of the motion states between robots. Simu-
lation results have demonstrated the efficiency of the
proposed methods even in the case of significant cur-
vature of the leader trajectory, and presence of mea-
surement noises. In the future, we intend to imple-
ment out approach experimentally on mobile robot
platform.
ACKNOWLEDGEMENTS
This research was funded by the Russian Foundation
for Basic Research (grant 15-08-03233).
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