752 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 64, NO. 1, JANUARY 2017
next position p
j
, e.g., a function defined as
j
=1−
Δθ
j
π
where
Δθ
j
∈ [0,π] is the turning angle between the current moving
direction and next moving direction, e.g., if the robot moves
straight, Δθ
j
=0; if moving backward, Δθ
j
= π. Thus, Δθ
j
can be given as: Δθ
j
= |θ
j
− θ
c
| = | atan2(y
p
j
− y
p
c
,ψ
p
j
−
ψ
p
c
) − atan2(y
p
c
− y
p
c
,ψ
p
p
− ψ
p
p
)|. The current robot posi-
tion adaptively changes according to the varying environment
and previous positions of the robot along a zigzag path. After
the current position reaches its next position, the next position
is updated to a new current position. In a multirobot system
[33], each robot needs one NN, and all robots share the same
workspace information, each of which has its own NN updated
dynamically on the positions of the robot. The environmental
knowledge is also dynamically updated, which is sensed by
robots via sensors.
The following notations are utilized. Note that N
x
and N
y
are the discretized size of the Cartesian workspace.
P(x, y) The set of the entire discretized cell-based entire
workspace (WS) {(x, y), 1 ≤ x ≤ N
x
, 1 ≤ y ≤
N
y
} i.e., P(x, y)
EntireWS
.
P (x, y) The unknown position (x, y)1≤ x ≤ N
x
, 1 ≤
y ≤ N
y
.
¯
P (x, y) The covered position (x, y)1≤ x ≤ N
x
, 1 ≤
y ≤ N
y
.
ξ(x, y) Neural activity at unknown position (x, y).
P (x
c
,y
c
) The current neuron position (x
c
,y
c
).
P (x
n
,y
n
) The next neuron position (x
n
,y
n
) where activity
is maximal.
ξ
m
(x
n
,y
n
) The maximal neural activity at position (x
n
,y
n
).
I(x, y) External input to neuron P (x, y).
F(x, y) Indication of a cell’s status, 0 for uncovered 1 for
covered and 2 for obstacle.
n
r
The number of robots in the team.
n
ob
The number of detectable obstacles.
OB
l
(x, y) The lth obstacle.
S(x, y)
cov
Subspace of P(x, y)
EntireWS
that can be reached
by robots.
S(x, y)
ncov
Uncovered subspace of P(x, y)
EntireWS
.
R
i
The ith robot in the team, 1 ≤ i ≤ n
r
.
There are the following definitions in the sake of presentation
of the CAC algorithms.
Definition 1: Completeness of CAC.
Given the 2-D Cartesian workspace W of the cleaning robots,
and position P (x, y) of robot represented by a neuron P (x, y)
at time t, Completeness of CAC at a total time T is defined that
every nonobstacle neuron is covered and the current location
neuron P (x, y) has the following properties:
S(x, y)
cov
+
n
ob
l=1
OB = P(x, y)
EntireWS
. (3)
1) The external input to neuron P (x, y) is set to be zero,
I(x, y) ← 0.
2) The F(x, y), utilized to indicate the status of this cell, to
neuron P (x, y) is set to be one, F(x, y) ← 1.
Definition 2: Cooperativity of multiple robots.
The term “cooperate” is in the sense that multiple robots can
work together to achieve a common coverage mission more ef-
ficiently and more quickly. Given the 2-D Cartesian workspace
W of the cleaning robots, and position (m, n) of a robot repre-
sented by a neuron (m, n) at time t. Once one of the multiple
robots moves to position (m, n), the position should be marked
as the external input I
i
(m, n)=−E, indicated as a moving
obstacle, such that the following conditions are satisfied:
S(x, y)
cov
= S
1
(x, y)
cov
+ S
2
(x, y)
cov
+ ···+ S
n
r
(x, y)
cov
=
n
r
i=1
S
i
(x, y)
cov
(4)
S(x, y)
cov
= P(x, y)
EntireWS
−
n
ob
l=1
OB. (5)
Definition 3: Robustness of multiple robots.
The term “Robust” is in the sense that the multirobot system
does not fatally failed due to failure of a single robot. Given
the 2-D Cartesian workspace W of the cleaning robots. As
one of the multiple robots, for instance,R
i
, which moves to
position (m, n), fails, the position is marked as the external
input I
i
(m, n)=−E. The rest robots take the responsibility to
cover its portion. Hence, the rest area is covered as follows (see
Coverage Algorithm 2):
S(x, y)
cov
|
rest
=
n
r
i=1
S
i
(x, y)
cov
−S
2
(x, y)
cov
=
n
r
−1
i=1
S
i
(x, y)
cov
(6)
S(x, y)
cov
|
rest
= P(x, y)
EntireWS
−
n
ob
l=1
OB
−S
2
(x, y)
cov
. (7)
The feature of robustness of the swarm of robots is acquired
through the following theorems.
Theorem 1: The proposed method is robust as long as at least
one robot works. In other words, the multirobot system works
even if n
r
− 1 of the robots fatally fail.
Proof: It will be proven by Induction. The following state-
ment should be proven, with n
r
robots, R
i
, ∀ 1 ≤ i ≤ n
r
.
1) Induction Base:Leti be 1. Every robot should have at
least one supportive robot, which is applicable in real-
world applications. Hence, if a robot fatally fails, then its
supportive robot driven by the proposed ND can fulfill
the coverage mission.
2) Induction Step: Suppose it is known if i = ζ robots fa-
tally fail, the rest robots can complete the coverage mis-
sion. The statement for ζ +1 robots should be shown.
Since n
r
− ζ robots are capable of covering the entire
workspace, discussed previously, in the NNs, with the
neural activity propagation, the uncovered area in entire
workspace attracts the n
r
− ζ robots. The failed robots
are considered as obstacles, i.e., I(x, y), = −E.There-
maining n
r
− ζ −1 robots driven by the ND model can