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ronment. The formation is composed of a maximum
of four robots, where one of them, the conductor, is
equipped with a laser range-finder that allows it to
navigate in buildings and follow a path determined
from the map of the building, the goal point, and the
obstacles. The rest of robots have limited visibility,
that is, they only know their actual position.
Each robot determines its position in order to
maintain the formation with respect to the leader or
a displacement of the position of the leader, with a
“virtual” point. Also, they use a relative frame of ref-
erence, that is, they do not use an absolute positioning
system, only the robot with the laser uses an absolute
system in relation with the map that it uses to nav-
igate. And in relation with the communication capa-
bilities, the robots can share local and global informa-
tion to compensate for their limited visibility, in fact,
the global information shared among the robots is the
actual position of the robot with the laser, and the lo-
cal information is the actual position of each robot.
In the formation, the position of the robot follow-
ers can be controlled by the position of the conductor
if all the robots are arranged in a line formation. In
other cases, when the followers are positioned to the
right or left of the conductor, virtual points are added
to the system, as explained in (Chiem and Cervera,
2004). These virtual points are calculated by apply-
ing a displacement in the conductor position to the
left or to the right, depending on the desired forma-
tion. The followers, in this case, instead of following
the conductor must follow these virtual points. This
arrangement is shown in Figure 1.
In the approach here proposed to achieve the
maintenance of the formation, it is desired that robots
could perform the task using only the odometry in
conjunction with communication to estimate the po-
sition of the conductor with respect to each follower.
The conductor, by means of the localization task is
Figure 1: Formation using virtual points as reference for
followers.
Figure 2: Calculation of the local position of the conductor.
always correctly localized in the map, and it is as-
sumed that it will follow a predefined trajectory, with
a constant, known linear velocity. In order for another
robot to follow the conductor, the linear and angular
velocities need to be computed at each time step. It
must be noted that the linear velocity of the follower
robots is not constant, due to the different radius of
their respective trajectories or because their position
may be relatively further back or further forward in
the formation.
In order to calculate the linear and angular veloc-
ities that allow the followers to move following the
movement of the conductor and maintaining the for-
mation, it is necessary to construct the corresponding
Bezier curve that defines the trajectory to be followed
by the robot. To compute the Bezier curve two posi-
tions are needed, the current position of the follower
and the current position of the conductor in relation to
the follower.
The conductor, at each step sends its own posi-
tion to the followers. The conductor is always local-
ized in the map by means of the localization task, so,
when it requires information about its position, this is
done with the coordinates of the map. The followers
are not localized on the map, so the only information
they have available is their position on their own lo-
cal system which is determined by the origin that is
fixed by their initial position. So, the conductor must
transform their global position into a position in its
local system. In Figure 2, the necessary relationships
to transform the global position into a local position
can be seen.
From the figure, it can be deducted that to calcu-
late the position in the local system it is necessary to
know the origin of the trajectory in the global system
and the actual position of the robot in the global sys-
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