IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 5, SEPTEMBER 2009 1173
Experiments of Formation Control With Multirobot Systems Using the
Null-Space-Based Behavioral Control
Gianluca Antonelli, Senior Member, IEEE, Filippo Arrichiello, Member, IEEE, and
Stefano Chiaverini, Senior Member, IEEE
Abstract—In this paper, the experimental validation of a be-
havior-based technique for multirobot systems (MRSs), namely,
the Null-Space-based Behavioral (NSB) control, is presented. The
NSB strategy, inherited from the singularity-robust task-priority
inverse kinematics for industrial manipulators, has been recently
proposed for the execution of different formation-control missions
with MRSs. In this paper, focusing on the experimental details,
the validation of the approach is achieved by performing dif-
ferent experimental missions, in presence of static and dynamic
obstacles, with a team of grounded mobile robots available at the
Laboratorio di Automazione Industriale of the Università degli
Studi di Cassino.
Index Terms—Formation control, mobile robots, multirobot sys-
tems (MRSs).
I. INTRODUCTION
I
N THE LATEST decades, the field of multirobot systems
(MRSs) has been object of widespread research interest
owing to the several advantages that such systems show with
respect to single autonomous vehicles and owing to the techno-
logical improvements that have allowed the interaction and the
integration among multiple systems. With respect to a single au-
tonomous robot or to a team of noncooperating robots, e.g., an
MRS can better perform a mission according to several perfor-
mance indexes, it can achieve tasks not executable by a single
robot such as moving a large object, or it can take advantages
of distributed sensing and actuation. Moreover, instead of de-
signing a single powerful robot, a multirobot solution can be
easier and cheaper, it can provide flexibility to task execution,
and it can make the system tolerant to possible robot faults. The
applications of MRSs may involve different fields like indus-
trial, military, and service robotics or research and study of bi-
ological systems. Moreover, they may concern largely different
kind of missions, e.g., exploration, box pushing, military opera-
tion, navigation in unstructured environment, traffic control, or
entertainment.
Most of the control approaches devised for autonomous
robots make use of biological inspiration. This kind of ap-
proaches began after the introduction of the robotics paradigm
of behavior-based control [9], [10], [13]. The behavior-based
paradigm has been useful for robotic researchers to examine the
social characteristics of insects and animals and to apply these
Manuscript received April 04, 2008. Manuscript received in final form July
24, 2008. First published April 14, 2009; current version published August 26,
2009. Recommended by Associate Editor F. Caccavale.
The authors are with the Dipartimento di Automazione, Elettromagnetismo,
Ingegneria dell’Informazione e Matematica Industriale, Università degli Studi
di Cassino, 03043 Cassino, Italy (e-mail: antonelli@unicas.it; f.arrichiello@
unicas.it; chiaverini@unicas.it).
Digital Object Identifier 10.1109/TCST.2008.2004447
findings to the design of MRSs. The most common application
is the use of elementary control rules of various biological ani-
mals (e.g., ants, bees, birds, and fishes) to reproduce a similar
behavior (e.g., foraging, flocking, homing, and dispersing) in
cooperative robotic systems. The first works were motivated by
computer-graphic applications; in 1986, Reynolds [25] made a
computer model for coordinating the motion of animals as bird
flocks or fish schools. This pioneering work inspired significant
efforts in the study of group behaviors [16], [19], [22], then in
the study of multirobot formations [11], [23], [29].
Apart from the behavior-based approaches, different analyt-
ical strategies to control MRSs have been proposed in literature.
These strategies may differ both for mathematical characteris-
tics and implementation aspects. A flatness-based theory aimed
at artificially coupling the motion of the vehicles is presented in
[24], while the use of control graphs to address the problem of
changing the platoon formation is discussed in [15]. In [12], a
formal abstraction-based approach to control a large number of
robots required to move as a group has been presented. This ap-
proach properly decouples the operational space into two con-
trol levels through a proper hierarchical subdivision, and it is
verified in simulation with a large number of robots. Reference
[27] focuses on formation motion feasibility of multiagent sys-
tems, i.e., it focuses on the algebraic conditions that guarantee
formation feasibility given the individual agent kinematics. Ref-
erence [18] presents experimental results of multirobot coordi-
nation controlled by a distributed control strategy based on a
circular-pursuit algorithm, while [28] presents experimental re-
sults of coalition formation of an MRS while simultaneously
performing heterogeneous missions.
Among the multiple approaches proposed in literature, a be-
havior-based approach, namely, the Null-Space-based Behav-
ioral (NSB) control, has been presented in [5] to control generic
robotic systems and in [7] and [8] to control MRSs. The NSB
approach takes the advantages of behavior-based approaches
in the reactivity to unknown or dynamically changing condi-
tions while, similarly to the analytical approaches, it presents a
rigorous mathematical formulation that permits to extrapolate
some analytical convergence properties. In [3], [4], [6], first ex-
perimental results with MRSs were reported. In this paper, em-
phasizing the experimental aspects, several missions with a pla-
toon of up to seven Khepera II mobile robots are collected and
discussed. Most of the presented experiments are accompanied
by a relative video that can be found at the Web page:
http://webuser.unicas.it/lai/robotica/video/.
II. NSB C
ONTROL
Generally, a mission involving several robots may require the
accomplishment of several tasks at the same time. A common
1063-6536/$26.00 © 2009 IEEE
Authorized licensed use limited to: Universita degli Studi di Cassino. Downloaded on August 27, 2009 at 00:30 from IEEE Xplore. Restrictions apply.
1174 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 5, SEPTEMBER 2009
approach is to decompose the overall mission of the system
in elementary tasks (or behaviors), to solve them as they were
working alone, and, finally, to combine the outputs of the single
tasks to obtain the motion commands to each robot. As dis-
cussed in [5], the NSB control differs from the other existing
methods in the behavioral coordination method, i.e., in the way
the outputs of the single elementary behaviors are assembled to
compose the final behavior. In particular, the NSB uses a geo-
metric hierarchy-based composition of the tasks’ outputs to ob-
tain the motion-reference commands for the robots that allow
the system to exhibit robustness with respect to eventually con-
flicting tasks. The basic concepts are recalled in the following,
while a complete description can be found in [8].
By defining as
IR the task variable to be controlled and
as
IR the system configuration, it is
(1)
with the corresponding differential relationship
(2)
where
IR is the configuration-dependent task Jacobian
matrix and
IR is the system velocity. Notice that repre-
sents the generic dimension of the specific task, while
depends
on the specific robotic system considered, e.g., in case of
mo-
bile robots
, and the term system configuration simply
refers to the robot positions.
An effective way to generate motion references
for the
vehicles starting from desired values
of the task function
is to act at the differential level by inverting the (locally linear)
mapping (2); in fact, this problem has been widely studied in
robotics (see, e.g., [26] for a tutorial). Notice that the desired
positions/velocities represent the input for the low-level con-
troller. A typical requirement is to pursue minimum-norm ve-
locity, leading to the least squares solution
(3)
where the pseudoinverse Jacobian
is elaborated as a matrix
that verifies the following properties:
where and are symmetric. However, when is a full-
rank lower-rectangular matrix, the pseudoinverse Jacobian is
simply elaborated as
(4)
At this point, the vehicle-motion controller needs a reference
position trajectory besides the velocity reference; this can be ob-
tained by time integration of
. However, discrete-time integra-
tion of the vehicle’s reference velocity would result in a numer-
ical drift of the reconstructed vehicle’s position; the drift can
be counteracted by a so-called closed-loop inverse-kinematics
(CLIK) version of the algorithm, namely
(5)
where
is a suitable constant positive-definite matrix of gains
and
is the task error defined as .
The NSB control intrinsically requires a differentiable ana-
lytic expression of the tasks defined, so that it is possible to com-
pute the required Jacobians. In detail, on the analogy of (5), the
single task velocity is computed as
(6)
where the subscript
denotes the th task quantities. If the sub-
script
also denotes the degree of priority of the task with, e.g.,
Task 1 being the highest priority one, in the case of three tasks
and according to [14] and [17], the CLIK solution (5) is modi-
fied into
(7)
where
is the identity matrix of proper dimensions. Remark-
ably, (7) has a nice geometrical interpretation. Each task velocity
is computed as if it were acting alone; then, before adding its
contribution to the overall vehicle velocity, a lower priority task
is projected onto the null space of the immediately higher pri-
ority task so as to remove those velocity components that would
conflict with it. Thus, the NSB control always fulfills the highest
priority task at nonsingular configurations. The fulfillment of the
lower priority tasks should be discussed in a case-by-case basis.
III. T
ASK-FUNCTION DEFINITIONS
Once recalled the basic concepts concerning the NSB ap-
proach, it is necessary to understand which task functions (or
elementary behaviors) can be defined and used to control an
MRS. Thus, in the following, different task functions expressing
global and local behaviors of the team are defined (more details
on the single task functions can be found in [5] and [8]).
A. Centroid
The centroid of a platoon expresses the mean value of the
vehicles’ positions. In a 2-D case, the task function is expressed
by
where is the position of the vehicle .
B. Variance
The task function for platoon 2-D variance
IR is de-
fined as
(8)
where
and are the current centroid coordinates.
Authorized licensed use limited to: Universita degli Studi di Cassino. Downloaded on August 27, 2009 at 00:30 from IEEE Xplore. Restrictions apply.
ANTONELLI et al.: EXPERIMENTS OF FORMATION CONTROL WITH MULTIROBOT SYSTEMS 1175
Fig. 1. Experimental setup available at the LAI, Università degli Studi di Cassino.
C. Rigid Formation
The rigid-formation task moves the vehicles to a predefined
formation relative to the centroid. The task function
IR
is defined as
.
.
.
where are the coordinates of the vehicle and are the
coordinates of the centroid.
D. Obstacle Avoidance
Obstacle avoidance is a crucial task to be followed by each ve-
hicle of the team. The corresponding task function is activated
only when the vehicle is approaching an obstacle, and it is de-
signed so that the vehicle slide around the obstacle. It is
where is the obstacle position and the symbol represents
the Euclidean norm.
IV. E
XPERIMENTS WITH A TEAM OF KHEPERA II
The NSB approach has been tested in different experimental
missions concerning the coordinated control of a team of
grounded mobile robots. The objective of the proposed experi-
ments is to coordinate the MRS by simultaneously controlling
some of its global parameters like the centroid position, the
robots spread in the environment, or the robot displacement. In
particular, the performed missions are as follows.
1) To handle the spread of the robots in the environment by
controlling their centroid and the variance around it; the
robots have to avoid eventual inter-vehicle hitting (two ex-
periments).
2) To impose to the robots a linear formation while avoiding
eventual obstacles and inter-vehicle hitting (three experi-
ments).
3) To impose to the robots a circular formation; to stress the
algorithm, the vehicles are commanded to switch their po-
sition with the opposite robot in the circle; in addition, a
moving obstacle is thrown into the robots cloud (one ex-
periment).
It is assumed that the robots have limited ranging capabilities
with respect to the presence of other robots or obstacles; the
corresponding obstacle-avoidance task, thus, is activated only if
the robot is close enough to another robot or obstacle.
A. Experimental Setup
The experimental setup available at the Laboratorio di Au-
tomazione Industriale (LAI) of the Università degli Studi di
Cassino is briefly discussed in the following. The setup is based
on a team of seven Khepera II (see Fig. 1), manufactured by
K-team [1], that are differential-drive mobile robots with a uni-
cyclelike kinematics of 8 cm of diameter. Each robot commu-
nicates via Bluetooth with a remote Linux-based PC, where a
Bluetooth dongle, building virtual serial ports, allows the com-
munication with up to seven robots. The remote PC implements
the NSB algorithm.
To allow the needed absolute position measurements, a vi-
sion-based system using two charge-coupled device cameras,
a Matrox Meteor-II frame grabber [2], and the self-developed
C++ image-processing functions have been developed. The ac-
quired images are 1024
768 RGB bitmaps. In particular, the
upper turrets of each robot have a set of colored LEDs that are
used to detect positions, orientations, and identification num-
Authorized licensed use limited to: Universita degli Studi di Cassino. Downloaded on August 27, 2009 at 00:30 from IEEE Xplore. Restrictions apply.
1176 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 5, SEPTEMBER 2009
Fig. 2. Mission
#
1. Snapshots of the (left) initial and (right) final robots configuration. The circles represent the desired and measured centroid positions and
variance. In the right plot the circles are superimposed.
bers of each robot. The position measurements are performed
at a sampling time of 100 ms while the estimation error has
an upper bound of
cm and . Moreover, the vision
system permits to identify static obstacles (e.g., the linear obsta-
cles in Fig. 1) or dynamic obstacles (i.e., a tennis ball) eventually
present in the environment. The measurements are sent over the
LAN to the Linux-based PC using the UDP/IP protocol.
Following the approach described in the previous sections,
the NSB elaborates the desired linear velocity for each robot
of the team. Being the Khepera unicycle-like robots, a heading
controller has been derived from the controller reported in [20]
to obtain the wheels’ desired velocities. Thus, the remote Linux-
based PC sends to each vehicle (through the Bluetooth module)
the wheels’ desired velocities with a sampling time of 120 ms.
The wheels’ controller (onboard of each robot) is a PID devel-
oped by the manufacturer. A saturation of 40 cm/s and 180
/s
has been introduced for the linear and angular velocities, respec-
tively. Moreover, the encoders’ resolution is such that a quanti-
zation of
cm/s and s are experienced.
The experimental results will be presented in the next sec-
tions resorting to a self-developed graphical simulator. The soft-
ware, developed in C under Linux (and that uses the OpenGL li-
braries), is used for debugging purposes and as graphical output
of the experimental data. To improve the experiments’ readings,
in the following, the snapshots always represent a graphical rep-
resentation of the experimental data.
B. Mission 1: Obstacle–Centroid–Variance
The first kind of mission concerns the possibility to control
the spread of the team of robots by the position of its centroid
and the variance around it. As an example, in Fig. 2, a team
of robots, starting from a random configuration, reaches a con-
figuration whose centroid and variance are given. In particular,
the circles are centered in the desired and current centroids with
radii, respectively, equal to the square roots of the desired and
current variances.
The global mission has been decomposed into three tasks,
where the order of the tasks represents their order of priority:
1) obstacle avoidance;
2) centroid;
3) bidimensional variance.
The corresponding gains are
where the subscript of the identity matrix denotes its dimension.
In the first experiment, a platoon of six robots, starting from
a random configuration, is commanded to move its centroid to
a constant desired configuration
cm, keeping
a desired variance of
cm . Fig. 3(a) shows
the path followed by the robots (thin lines) and the path of their
centroid (larger line); in gray, the starting positions, in black,
the final ones. Fig. 3(b) shows the centroid task-function values
during all the mission; the robots stop when the distance be-
tween the centroid position and its desired position is lower than
a threshold value. Fig. 3(c) shows the variance values with re-
spect to the centroid along the axes
and .
In the second experiment, a platoon of seven robots
is commanded to move its centroid to a constant de-
sired configuration
cm; three different
set points for the variance are consecutively assigned as
and cm .
Fig. 4(a) shows several steps of the mission execution, while
Fig. 4(b) and 4(c) shows, respectively, the centroid task-func-
tion values and the variance values along the axes
and
during all missions. It is worth noticing that the variance value
relates to the density of the robots in the team; thus, a small
variance value makes the robot stay close to the centroid and
a high value makes the robot spread in the environment. Obvi-
ously, the value
cm cannot be reached because
the obstacle-avoidance task function, i.e., the highest priority
task does not allow the vehicles to have a relative distance
lower than 12 cm; this desired value was commanded to test the
Authorized licensed use limited to: Universita degli Studi di Cassino. Downloaded on August 27, 2009 at 00:30 from IEEE Xplore. Restrictions apply.
ANTONELLI et al.: EXPERIMENTS OF FORMATION CONTROL WITH MULTIROBOT SYSTEMS 1177
Fig. 3. Mission
#
1, first experiment. (a) Paths followed by the robots during the mission. (b) Average task function: (dashed lines) Desired values of
and
(solid lines) real values of
. (c) Variance task function: (dashed lines) Desired values of
and (solid lines) real values of
.
Fig. 4. Mission
#
1, second experiment. (a) Snapshots: The circles around the robots represent their safety region, the last seconds path is reported. (b) Centroid
task function: (dashed lines) Desired values of
and (solid lines) real values of
. (c) Variance task function: (dashed lines) Desired values of
and (solid
lines) real values of
.
algorithm in a demanding situation. The corresponding video
is named f_variance.mpg.
C. Mission 2: Obstacle–Centroid Rigid Formation
In the second mission, the robots are commanded to reach a
linear rigid formation around the centroid (see Fig. 5). Three
experiments are reported: in the first one, the robots are com-
manded to assume a linear formation and then switch their rel-
ative positions; in the second experiment, the same mission is
executed in the presence of two linear static obstacles; and in
the last experiment, a moving obstacle (a tennis ball pushed by
hand) is thrown over the moving team of robots.
1) Rigid Formation
Change of Formation: The first exper-
iment of the second mission concerns a platoon of five robots
starting from a random configuration. The mission consists of
two steps: In the first step, the robots are commanded to assume
the configuration with a centroid of
cm and
to reach a linear configuration (rotated at
s with respect
to the axis
), where each robot has a distance from the others
of 30 cm; then, the second step of the mission consists of a po-
sition permutation, i.e., the robots are commanded to symmet-
rically invert their positions in the formation. Obviously, colli-
sions among the robots need to be avoided during all missions.
The priority of the three tasks implemented is as follows:
1) obstacle avoidance;
2) centroid;
3) rigid formation.
The matrix gains are
with cm.
Fig. 6 shows the experimental results; Fig. 6(a) shows some
snapshots where the last seconds path and the safety region
of the robots are highlighted. The first desired configuration is
reached at
s. At s, a new step input is com-
manded to the platoon by requiring a change in the robots’ con-
figuration while keeping the same desired centroid. It can be ob-
served from the plot of the centroid task function [Fig. 6(b)] and
Authorized licensed use limited to: Universita degli Studi di Cassino. Downloaded on August 27, 2009 at 00:30 from IEEE Xplore. Restrictions apply.
