Decentralized control of mobile robotic sensors for optimal coverage

and connectivity over an arbitrary unknown region*

Waqqas Ahmad

Member, IEEE

Abstract— The self deployment of a decentralized mobilesensor network has been addressed in this paper. The mobilerobotic sensors cover a bounded and connected two-dimensionalarbitrary unknown region. An algorithm with decentralizedcontrol law has been developed to self deploy the mobile roboticsensors into a triangular lattice pattern ensuring optimization.The optimization of deployment in terms of minimum numberof sensors ensuring 1-coverage and 6-connectivity(whereverpossible) has been considered during final deployment of mobilerobotic sensors over the arbitrary unknown region. A specialconsideration in continuation to the recent work has beentaken in order to remove the redundant placement of mobilerobotic sensors on the boundary of the region and to fullytrack the bounded smooth region. The concept of stronglyconnected quotient system of the decentralized control systemhas been used during line forming mode(s) inside the region.The algorithm is based on simple nearest neighbor rules derivedfrom swarm intelligence. A number of numerical simulationshave been done in order to validate the control algorithm.

I. INTRODUCTION

Mobile robotic sensors are of interest for use in military

and civil operations. They can be used for surveillance,

reconnaissance, maintenance, inspection and training [2, 10,

11]. The challenge in developing an autonomous multi-

robotic network is to plan efficient deployment. The idea

to deploy such sort of mobile robotic sensors is taken after

inspiring the motion of a flock of birds. Flocks organize

themselves in a well defined way, looking not only beautiful

but also being a unique and distinct phenomenon to study.

Furthermore, while each bird appears randomly placed, they

are significantly coordinating during motion [8]. The anal-

ysis of Vicsek model [9] has revealed that the decision of

movement of each of the agent is based on its own state and

state of the neighbors communicating with it.

The research work[1, 16-21, 24] demonstrates the decen-

tralized control of the mobile sensor networks based on

Gage’s classification[12] of the motion of mobile robotic

sensors as Blanket, Barrier and Sweep Coverage. The current

work is focusing on the blanket coverage. The idea is to

fully cover an arbitrary region with minimum number of

mobile sensors. The recent work[1] has already demonstrated

an algorithm for blanket coverage problem, but it places

redundant sensors on the boundary and it does not fully

track the boundary. The complete tracking of the boundary

provides network full information about the unknown region

*This work was supported by the Australian Research Council.W. Ahmad is with the School of Electrical Engineering and Telecom-

munications, University of New South Wales, Kensington Campus, Sydney,Australia. waqqas.ahmad3237666@gmail.com

to be covered. Now, the full tracking of the boundary of

the arbitrary region provides the saving of a number of

mobile robotic sensors. Also, this algorithm uses the concept

of the quotient system of the decentralized control system.

This quotient system deploys the strongly connected mobile

robotic senors during line forming mode inside the arbitrary

region to be covered.

The desired region R is said to be fully covered by the

mobile robotic sensor network if and only if every point

inside or at the boundary of R is fully sensed(covered)

by at least one mobile robotic sensor[22]. If any of the

two mobile robotic sensors in a network formation are

exhibiting q number of disjoint paths between each other,

the formation of the whole network is said to be q-connected

[2]. In the equilateral triangular lattice pattern formation,

a mobile robotic sensor node is connected with 6 other

nodes of mobile robotic sensors. Hence, the equilateral

triangular formation is believed to offer the maximum known

connectivity (i.e. 6 connected) so far and it is considered to

be the optimal solution to achieve coverage and connectivity

[2].We deploy the mobile robotic sensors in the so-called

triangular lattice pattern within the boundary of the region.

The triangular lattice pattern has been used to get the

asymptotic optimization in terms of minimum number of

mobile robotic sensor to cover the bounded set[2] defined

for the region.

II. PROBLEM STATEMENT

Let si(kT ) be the Cartesian coordinates and θi(kT ) be

the heading of the mobile robotic sensor measured counter

clockwise from the X-axis as shown in the Fig. 1.

Let vi(kT ) be the linear velocity of the sensor i and also

let u(p) be a vector function such that for any p ∈ R, we

have

u(p) = [cos(p) sin(p)]T (1)

for i = 1,2, . . . ,nThe kinematic equation of motion with the control inputs,

velocity vi and the heading angle θi, can be written as under;

si((k+1)T ) = si(kT )+T vi(kT )u(θi(kT )) (2)

for i = 1,2, . . . ,n, and for k = 0,1,2, . . .The first control constraint is |vi(t)| ≤ vmax for i =

1,2, . . . ,n, and for all t ≥ 0. Secondly, the initial heading

of each sensor satisfies θi(0) ∈ [0,π) for i = 1,2, . . . ,n.

2012 IEEE International Symposium on Intelligent Control (ISIC)Part of 2012 IEEE Multi-Conference on Systems and ControlOctober 3-5, 2012. Dubrovnik, Croatia

978-1-4673-4600-9/12/$31.00 ©2012 IEEE 31

The objective is to develop a decentralized control algo-

rithm to fully track and to cover a bounded and connected

unknown region (meeting assumptions) with minimum num-

ber of mobile robotic sensors ensuring 1-coverage and 6-

connectivity(wherever possible) as determined by the so-

called triangular lattice pattern [2] inside the region. Also,

the algorithm should not place redundant sensors to guide

or to direct the mobile robotic sensors during the line

forming mode(s). The control algorithm varies the blanket

coverage[1] in a sense that it fully tracks the unknown region

and it avoids the placement of unnecessary mobile robotic

sensors on the boundary of region to be covered.

A. Assumptions

1- A mobile robotic sensor has the ability to detect any

object within a disk of radius Rs > 0 for t ∈ [kT,(k+1)T ) and

k = 0,1,2, . . .. The disk of sensing range Rs is mathematically

defined as under;

Si,Rs(Kt) := p ∈ R2 : ||p− si(Kt)|| ≤ Rs (3)

where ‖ · ‖ denotes the Euclidean norm.

2- A mobile robotic sensor has the ability to communicate

with the neighbor(s) within a disk of radius Rc > 0 for t ∈[kT,(k+1)T ) and k = 0,1,2, . . .. The disk of communication

range Rc is mathematically defined as under;

Ci,Rc(Kt) := p ∈ R2 : ||p− si(Kt)|| ≤ Rc (4)

where ‖ · ‖ denotes the Euclidean norm.

3- In order to attain the triangular lattice formation as the

optimal deployment pattern[5], the sensing and communica-

tion range satisfies the bound, Rc/Rs ≥√

3[1].

4- There exists an infinite sequence of contiguous,

non-empty, bounded, time intervals [ki,ki+1] where i =0,1,2, . . . ,n with k0 = 0; such that for all [ki,ki+1] the graph

from the union of the collection G(kT ) ∈ P for all kT ∈[ki,ki+1], where i = 0,1,2, . . . ,n is connected[6].

5- The unknown R to be covered is bounded and con-

nected two dimensional region with smooth boundary ∂R.

The radius of curvature at any p ∈ ∂R ≥√

3Rs[4]. The

unknown region to be covered is not self intersecting.

III. CONTROL ALGORITHM

A. Interpolation Point

Let r1 be the first point of contact between all the sensors

and the region R. Let X and Y be the vectors(as shown in

Fig. 1) tangent and orthogonal to the point r1, respectively.

When a sensor is started from the point r1 and it follows

the smooth boundary ∂R of the arbitrary region, an interpo-

lation point pint will be stored in the mobile robotic sensor’s

memory meeting the following condition;

pinty := Kr1y where K = 1,2, ... (5)

i.e. any point p ∈ ∂R can be regarded as the interpolation

point if its yth co-ordinate is an exact multiple of the yth

coordinate of r1. However, numerical tolerance is considered

during the comparative decision making process.

−10 −8 −6 −4 −2 0 2 4 6 8 10

−16

−14

−12

−10

−8

−6

−4

−2

0

b7

a7

a6

a5

a4

a3

a2

b2

b3

b4

b5

b6

b1

Y

a1

L1

L3

L2

L4

L5

L6

L7

r1

r2

X

Fig. 1. Sensor movement (K(Rs) = 7).

B. Line Forming

A line forming mode is made if the length between any

two interpolation points(pint , ¯pint) parallel to the X will meet

the following condition;

||pint − ¯pint ||> dH(Rs) (6)

and the line formed is included inside the region.

The condition to check whether the line is included inside

or outside the region is based on the so-called Even-Odd test.

The direction of line forming mode, i.e. from left to right

or from right to left is decided based on the exploration of

second end point of a required line. However, if two end

points for a required line are explored at the same time, the

direction of line forming mode would be from left to right.

Let dH(Rs) be the function of distance between two

consecutive sensor nodes to be placed parallel to X, and

dV (Rs) be the function of distance between any two sensors

to be placed parallel to Y, i.e. it is the distance between two

consecutive lines. If the sensors are placed at the vertices of

an equilateral triangle, we can write an expression for the

function dV (Rs) using Pythagorean theorem,i.e.

dV (Rs) = Rs+√

Rs2 − (dH(Rs)/2)2 (7)

The distance between the corresponding interpolation

points ( e.g. aK ,bK ∈ ∂R) can be calculated as under;

DK =√

(bKx −aKx)2 +(bKy −aKy)2 (8)

where aKx,aKy and bKx,aKy are the x and y coordinates of

the points aK and bK , respectively and K = 1,2, . . ..We can find the optimal number of sensors required for a

line forming as under;

HK(Rs) := ⌈DK/dH(Rs)⌉. (9)

As we are deploying the sensor nodes for the region R

covered by the arbitrary boundary ∂R, any two or more

than two consecutive parallel lines LK−1,LK ,LK+1 might

have the equal length(DK). So, an additional sensor to avoid

the coverage hole and to retain the so-called triangular

lattice pattern might become the part of the above mentioned

32

equation. Hence, we can generalize the above mentioned set

of optimal number of sensor nodes as under;

HK(Rs) :=

⌈DK/dH(Rs)⌉⌈DK/dH(Rs)⌉+1 if DK = DK−1

and HK−1 ≤ HK

and |aKx −a(K−1)x|+ |bKx −b(K−1)x|= 0

(10)

The above condition shows that the additional sensor

would only be calculated when two or more than two

consecutive lines with same end points are equal.

We can define the K(Rs) number of parallel line segments

L K part of the region R such that these segments are

orthogonal to the vector Y as shown in Fig.1. Also, the

distance between each pair of these parallel lines is dV (Rs).The K(Rs) number of line segments LK can be defined by

LK = {q ∈ R : (r1 −q)TY ) = K(dV (Rs)) (11)

for K = 1,2, ....During tracking of the complete boundary segment

I (r1,r2), the unknown points aK and bK are formed by

the intersections of LK and ∂R such that (aK −bK)T X < 0.

Once the points aK and bK as shown in Fig.1 are identified

during boundary moving mode of sensors, it becomes easy

for the mobile robotic network to decide the optimal node

locations inside and/ or at the boundary of R.

During first line formation, the x-positions of H1(Rs)number of sensor nodes is decided as under;

L(1,w)x(Rs) :=

i f H1(Rs) ∈ 2z;∀z ∈ Z

D1

2± (2w−1)

dH(Rs)

2

for w = 1,2,3, . . . ,H1(Rs)

2i f H1(Rs) ∈ 2z+1;∀z ∈ Z

D1

2±wdH(Rs)

for w = 0,1,2,3, . . . ,H1(Rs)−1

2

(12)

If we place the sensors starting from the end point of the

line, there would be required an additional sensor to cover

the same length of the line. Eq. 12 finds the best optimal

positions to cover the line, i.e. it starts placing the sensors

from the imaginary middle point(not from the starting point

or from the end point of a proposed line as calculated in [1])

of the line as shown below;

Let |LK−1,w| be the number of positions of sensor nodes

placed on the (K − 1)th row, where w = 1,2, . . . ,HK−1(Rs)is the associated integer with the (K − 1)th row, and K =2,3, . . ..

Let L̄K−1,w̄ ∈ LK−1,w be the set of sensor nodes having

projection on LK . During Kth line formation, the quotient

system containing HK(Rs) number of mobile robotic sensors

can find the optimal positions by the below mentioned

computation.

L(K,w)x(Rs) :=

i f |L̄K−1,w̄| ∈ 2z;∀z ∈ Z

median((L̄K−1,w̄)x)± [wdH(Rs)]

for w = 0,1,2, . . .

i f |L̄K−1,w̄| ∈ 2z+1;∀z ∈ Z

median((L̄K−1,w̄)x)± [(2w−1)dH(Rs)

2]

for w = 1,2,3, . . .

(13)

The number of negative and/ or positive summations of

dH(Rs) is dependent on the achievement of points aK and bk

to the left and to the right of the inner most sensor position

calculated by the median of (L̄K−1,w̄)x. The loop breaks if

the calculated node position is reached or try to exceed the

x-coordinates of the end points aK and bk. So, we can also

limit theses line forming positions by introducing a saturation

function.

Sat[aK ,bK ](LK,w) =

aK if |LK,w| ≥ |aK |LK,w if |aK |< |LK,w|< |bK |

bK if |LK,w| ≥ |bK |(14)

We can see Eq.10 is providing the optimal number of

sensors. Also, Eq.13 and Eq.14 are providing the calculation

of the best positions to be attained by the mobile robotic

sensor. We can analyze the algorithm [1] does calculate the

boundary points aK and bK for the desired sensor positions.

But the above mentioned equations eliminate the unnecessary

need of sensor node positions at these boundary points. The

boundary is only occupied if there would be coverage hole

or a sensor position would be going outside the boundary

segment. So, the boundary limits are ensured by the use of

Eq.14.

As per the assumption, the radius of curvature at any

p ∈ ∂R ≥√

3Rs[4]. Let p1, p2 be any two points such that

p1, p2 ∈ ∂R. In order to ensure the coverage of I (p1, p2),the following condition needs to be satisfied.

||p1 − p2||=√

3Rs∀p1, p2 ∈ ∂R (15)

We define the following additional set of sensor locations

on the ∂R.

Bv(Rs) :={

bv ||pm − pm−1||>√

3Rs (16)

where v and m= 1,2,...

So, the above mentioned conditional set of sensor loca-

tions ensures the placements of sensor(s) at the boundary

segments(not covered by line forming sensors)if ||r1−a1||>√3Rs or if ||aK − aK−1|| >

√3Rs or if ||r2 − aK || >

√3Rs.

Similarly, If ||r1−b1||>√

3Rs or if ||bK −bK−1||>√

3Rs or

if ||r2−bK ||>√

3Rs, then necessary sensor locations become

part of the Eq.16.

Hence, we can define the complete set of sensor locations

by combining Eq.14 and Eq.16 as under;

P(Rs) := Bv(Rs)∪LK,w(Rs) (17)

33

Pm(Rs) := |P(Rs)| (18)

Due to inherent computation capability, mobile robotic

sensors can be self deployed on the Pm(Rs) number of sensor

node locations.

Let R ⊂ R2 be a bounded region to be covered by

nP(Rs) of sensors with the corresponding D(Rs) set of

sensor locations. Let Nm(Rs) be the minimum number of

sensors to cover R. The set of sensor locations D(Rs) is

asymptotically optimal for covering R if for any point p∈R,

there exists LK,w(Rs) ∈ D(Rs) such that

‖p−LK,w(Rs))‖ ≤ Rs, (19)

and

limRs→0

R2s nP(rs) = lim

Rs→0R2

s Nm(Rs). (20)

The following theorem can be stated with the derivations

of [5].

C. Theorem

The set of sensor locations P(Rs) is asymptotically optimal

for covering R(Rs).Proof: The main result of [5] states the optimal number of

circles required to cover a bounded plane point set. The

following step by step proof of the algorithm is based on

the main result of [5] and on the mathematical arguments of

the theorem presented in [1].

Let r1 ∈ ∂R be the first point formed by the contact

between a mobile sensor and the boundary of region R to

be covered. Let X ∈R2 be a unit vector tangent to the point

r1 ∈ ∂R. Let initial conditions of the mobile robotic sensor

dynamics(Eq. 2) be (si(0),θi(0)) with ‖si(0)− r1‖ ≤ Rc for

i = 1,2, . . . ,n.

As per the BM mode, all the sensors can travel along

I (a1,b1) above L1. Any sensor identifying a1 or b1 can

transmit a flag and the coordinates of these points to its

neighbors coming under its communication range Rc. By

the mobile sensor connectivity assumption, the very first

sensor reaching at a1 and transmitting the calculated op-

timal number of sensors(HK(Rs) where K=1) to acquire

the positions(calculated by L1,w, where w = 1,2, . . . ,H1(Rs))switches to LF mode, whereas the sensor detecting b1 after

moving along I (r1,b1) keeps continuing its BM mode.

From a1, the required number of sensors(H1(Rs)) are in

LF mode and move orthogonal to Y forming the first array

of mobile sensors along L1. As the first array of required

number of sensors shift to LF mode, the rest of the sensors

continue moving along I (r1,r2) to identify a2, i.e. the rest

of sensors are acknowledged to keep continuing BM mode

after crossing a1. By repeating the above procedure, this

calculated set of sensors form multiple layers(K(Rs)) parallel

to X and rest of the sensors turn back after covering and

fully tracking I (r1(Rs),r2(Rs)) in BM mode. The control

law in LF mode not only keeps the required number of

sensors(HK(Rs), where K = 1,2, ...) aligned in each layer of

sensor array, but also drives the sensors into the triangular

structure determined by LK,w(Rs).In above mentioned paragraph, we have shown the exis-

tence of parallel arrays during LF mode of mobile sensors.

Now, we can prove this argument layer by layer. Let N :={1,2, . . . ,n} and N0 ⊂N be the set of sensors already attached

to I (a1,b1) and |N0| be the number of elements in N0. In

order to prove the creation of first sensor array, let us assume

that b1 is discovered before a1. By the connectness property

imposed in the assumption and φ1(·) ≡ θ0, the update law

for the coordinate variable φi guarantee that

limk→∞

φi(kT ) = θ0, i = 2,3, . . . ,n, (21)

For i = 1, the sensor 1 acts as the leader[14] and rest

of the follower sensors for i = 2,3, . . . ,n ensure gain of the

consensus variable φi(·). Similarly, Mi(kT )→ aT1 Y by using

the fact that the sensor reached at a1 is the leader. So, we

get the following result by using φi(kT )→ θ0 and Mi(kT )→aT

1 Y ,

limk→∞

ci,i(kT ) = aT1 Y, i ∈ N\N0. (22)

Therefore, the existence of a line is guaranteed by the

conditions (21)–(22).

L1 = {p ∈ R2 : pTY = aT

1 Y (23)

such that a1 ∈ L1 and

limk→∞

d(si(kT ),L1) = 0, i ∈ N\N0, (24)

where d(si(·),L1) denotes the Euclidian distance between

the mobile sensor location, si(·) and the line L1. So, we can

say that the required number of sensors i, i∈N\N0, converge

to the point a1 which is the starting point of L1.

The control law shows the existence of a finite integer

I > 0 and a set {r(1)1 ,r

(1)2 , . . . ,r

(1)n−|N0|} which is a permutation

of the set N\N0 such that

qr(1)1

(kT )< qr(1)2

(kT )< .. . < qr(1)n−|N0|

(kT ) (25)

for all k ≥ I, where qr(1)1

(kT ) = aT1 X . As H1 number of

sensors are supposed to go into LF mode on L1, the rules

for Qi(kT ) that govern qr(1)1

(kT ), qr(1)2

(kT ), . . ., qr(1)H1

(kT ),

can then be written as a linear dynamic system

q(1)((k+1)T ) = A(1)q(1)(kT )+b(1)(kT ), for k ≥ I, (26)

where

q(1)(kT ) :=[

qr(1)2

(kT ) qr(1)3

(kT ) . . . qr(1)H1

(kT )]T

b(1)(kT ) :=[

(qr(1)1

(kT )+Rr(1)2

(kT ))/2 0 . . . 0 S/2]T

where A(1) is a square matrix having elements A(1)i, j ,1 ≤

i, j ≤ H1 −1, such that

A(1)i,i+1 = A

(1)i+1,i = 1/2, for 2 ≤ i ≤ H1 − 2; A

(1)1,1 = A

(1)2,1 =

A(1)H1−1,H1−1 = 1/2;

and A(1)i, j = 0, for all other i, j.

34

We have Rr(1)2

(·)≡ R̄r(1)2

= S for the first layer of sensor array.

By using the corollary[15, p. 514], i.e. If no eigenvalue

of a square matrix is repeated then that square matrix is

guaranteed with diagonalizability and the fact that A(1)1,1 =

A(1)2,1 = A

(1)n−1,n−1 = 1/2, it can be shown that

limk→∞

qr(1)i

(kT ) = qr(1)1

(kT )+ R̄r(1)2

+(i−2)S,

= aT1 X +(i−1)S,

(27)

for i = 1,2, . . . ,H1.

As limk→∞ d(si(kT ),L1) = 0, therefore

limk→∞

sr(1)i

(kT ) = a1 +S(i−1)X , for i = 1,2, . . . ,H1. (28)

The equation (28) remains valid with assumption that b1

is discovered before a1. However, only the required number

of sensors can switch back to LF mode once a2 and b2 is

achieved. Also, the revised algorithm does not necessarily

place a sensor at b1 or at a1 ∈ ∂R for the first layer.

Therefore, the existence of an integer k̄1 and a set

N(1) := {n(1)1 ,n

(1)2 , . . . ,n

(1)H1} ⊂ N\N0 (29)

is ensured for the first layer with the H1 number of sensors

on L1(H1), such that limk→∞ ‖sn(1)w(kT )−L1,H1−w+1‖= 0 for

w = 1,2, . . . ,H1.

The rest of mobile robotic sensors, n− |N0| −H1 do not

go into LF mode and travel along I (a1,a2) and I (b1,b2)in BM mode. There might be placement of sensors on

I (a1,a2) for full coverage of I (a1,a2) if ‖a1−a2‖>√

3Rs

and the same respective conditional placement of sensors

might be held for the sensors moving along I (b1,b2). Let

N1 be the sensors staying at I (a1,a2) and/ or at I (b1,b2)and |N1| be number of such mobile sensors.

The rest of sensors, i.e. n−|N0|−H1−|N1| will contribute

to form the second layer of sensors with H2 number of

sensors on L2 starting from a2 or from b2 . Let dv(Rs) is

the distance between the first and the second layer of mobile

robotic sensor arrays, i.e. (a1−a2)TY = dv(Rs). We can show

the existence of second layer staring from a2 or from b2 with

H2 number of sensors to be deposited on L2. Hence, there

exists an integer k̄2 and a set

N(2) := {n(2)1 ,n

(2)2 , . . . ,n

(2)H2} ⊂ N\(N0 ∪N(1)∪N1)

such that limk→∞ ‖sn(2)w(kT ) − L2,H2−w+1‖ = 0 for w =

1,2, . . . ,H2 if there is movement of mobile robotic sensors

from a2 to b2 or limk→∞ ‖sn(2)w(kT )− L2,w‖ = 0 for w =

1,2, . . . ,H2 if there is movement of mobile robotic sensors

from b2 to a2. As previously, some of the mobile sensors

might attain the position along I (b1,b2) and I (a1,a2)depending on the length of ‖b1 − b2‖ and ‖a1 − a2‖. Let

N2 be such sensors deposited on I (b1,b2) and I (a1,a2).Similarly, we can say that there always exist an integer k̄i

and a set N(K) where

N(1) := {n(1)1 ,n

(1)2 , . . . ,n

(1)H1} ⊂ N\N0, for K = 1;

N(K) := {n(K)1 ,n

(K)2 , . . . ,n

(K)HK

} ⊂ N\(K−1⋃

w=1

N(w)∪K−1⋃

w=0

N(w)

)

,

for K = 2,3, ... such that if there is movement of mobile

robotic sensors from aK to bK then

limk→∞

‖sn(K)w(kT )−LK,HK−w+1‖= 0, w = 1,2, . . . ,HK

or if there is movement of mobile robotic sensors from bK

to aK then

limk→∞

‖sn(K)w(kT )−LK,w‖= 0, w = 1,2, . . . ,HK ;

The movement of quotient system with HK(Rs) number of

strongly connected mobile robotic senors from aK to bK

or from bK to aK is dependent on the discovery of the

second end point required for a line forming mode. If bK is

discovered before aK , the HK(Rs) number of mobile robotic

sensors will be moving from aK to bK , and vice versa.

However, a default movement of mobile robotic sensors from

left to right or from right to left can be made upon discovery

of the end points at the same time.

We have the below mentioned set of mobile sensors that

spread along I (aK(Rs),bK(Rs)) after the formation of K(Rs)number of mobile sensor arrays.

NK(Rs) = N\(K(Rs)−1

⋃

w=1

N(w)∪K(Rs)−1⋃

w=0

N(w)

)

, (30)

Hence, it is proved that K(Rs) number of mobile sensor

arrays are formed into the triangular lattice pattern. Similarly,

we can prove that as k → ∞, the projected distance on X

caused by the pair of neighboring sensors on I (a1,b1)converges to

√3Rs/2.

Let {n(0)1 ,n

(0)2 , . . . ,n

(0)|N0|} ∪ N be the set of sensors

that are on I (a1,b1) such that aT1 X < q

n(0)|N0 |

(kT ) <

qn(0)|N0 |−1

(kT )< .. . < qn(0)2

(kT )< qn(0)1

(kT )< bT1 X(kT ), where

qn(0)w

= sT

n(0)w

(kT )X . By Eqs. (26)–(27), it is clear that

we have limk→∞ qn(0)w(kT ) = bT

1 X − w√

3Rs/2 (where,

w = 1,2, . . . , |N0|) as s =√

3Rs/2 in BM mode. Also,

limk→∞ |aT1 X −qn0

|N0|(kT )|<

√3Rs/2. As each pair of neigh-

boring sensors on I (a1,b1) is ensuring a mutual distance

less than or equal to√

3Rs, we can say the boundary segment

I (a1,b1) is fully covered and this segment ensures the va-

lidity of the assumption for the bounded region. Hence, this

argument can be taken as the general argument valid for any

of the boundary segment I in Bs (except I (aK(Rs),bK(Rs)))

having length between its end points greater than√

3Rs.

By modifying the matrix A(1) and the vector b(1) in (26)

and if (|NK(Rs)|+ 1)√

3Rs/2 ≥ ‖aK(Rs)− bK(Rs)‖ then it can

be shown that each pair of the neighboring sensor on

I (aK(Rs),bK(Rs)) will be having a projected distance on X

less than or equal to√

3Rs/2. So, BK(Rs) sensor locations

ensure the full coverage of all the boundary segments Bs.

Hence, it is shown that the set⋃K(Rs)

w=1 N(w) can provide

sensor locations LK,w(Rs) for mobile sensor arrays covering

inner area of the region and the set⋃K(Rs)

w=0 N(w) can provide

35

BK(Rs) number of sensor locations for the boundary cov-

erage of the region. So, the condition of the coverage with

triangular lattice pattern, i.e. i ∈ {1,2, . . . ,n} such that

limk→∞

‖si(kT )−LK,w(Rs)‖= 0. (31)

is fully satisfied.

D. Nearest Neighbor Rules

Some researchers have already defined nearest neighbor

rules and leader-follower scenario[7,23]. Theses rules have

been developed after studying the motion pattern of school

of fish, flocks of birds, etc. The mathematical form[7] of

these rules is as under;

Θi(kT ) :=1

|1+Fi(kT )|

(

φi(kT )+ ∑j∈Fi(kT )

φ j(kT )

)

(32)

for i = 2,3, . . . ,n, where |Fi(kT )| denotes the number of

elements in the set Fi(kT ), i.e., the number of neighbors.

The coordination variable φi(kT ) will be updated using the

rule

φi(k+1)T ) = Θi(kT ) (33)

There exists a scalar for the individual sensor i.

ci,i(kT ) = sTi (kT )u(Θi(kT )), (34)

Similarly, there exists a scalar for each neighbor of sensor

i.

ci, j(kT ) = sTj (kT )u(Θi(kT )), (35)

The update rule of the above mentioned scalar can be

written as under;

Mi(kT ) :=1

|1+Fi(kT )|

(

ci(kT )+ ∑j∈Fi(kT )

c j(kT )

)

(36)

The line to be followed by a sensor, ”i” can be mathemat-

ically written as under;

Li(kT ) = {p ∈ R2 : pT u(Θi(kT )) = si,i(kT )} (37)

E. Control Action

We use the same control action for sensor, ”i” of the past

work[1] described as under;

Let qii(kT ) be the projection of sensor i and qi

j(kT ) be the

projection of the neighboring sensor j on the line Li(kT ) at

time kT . Let r, l ∈ Fi(kT ) be the conditional sensors to the

right and to the left of sensor, ”i” such that,

qil(kT )< qi

i(kT )< qir(kT ). (38)

The coordinate update rule[1] is as under;

Qi(kT ) :=

(qil(kT )+qi

r(kT ))/2 i f l and r exists

(qil(kT )+qi

i(kT )+ s)/2 i f only l exists

(qir(kT )+qi

i(kT )− s)/2 i f only r exists

qii(kT ) i f l and r do not exist

(39)

where s = dV (Rs) in BM model and s = dH(Rs) for the

Line Forming Mode.

The velocity components[1] of sensor, ”i” parallel to and

orthogonal to Li(kT ) are as under;

v̄i(kT ) := (Qi(kT )−qii(kT ))/T,

v̂i(kT ) := (Mi(kT )− ci,i(kT ))/T,(40)

The decentralized control law[1] is described as under:

vi(kT ) =√

v̄i(kT )2 + v̂i(kT )2;

θi(kT ) =

{

Θi(kT )+βi(kT )−π/2, if v̂i(kT )≥ 0

Θi(kT )−βi(kT )−π/2, if v̂i(kT )< 0,

(41)

where βi(kT ) := cos−1(v̄i(kT )/vi(kT )), for i= 1,2, . . . ,n and

k = 0,1,2, . . . .

IV. SIMULATION

Fig. 2 shows the formation of triangular pattern with com-

munication range, whereas Fig. 3 shows the coverage with

sensing range. The simulation results have been obtained

with the following parameters;

Rs = 1.5,Rc =√

(3)Rs,dV (Rs) = 3Rs/2,dH(Rs) =√

(3)Rs/2

V. CONCLUSIONS AND FUTURE

RECOMMENDATIONS

A. Conclusions

The algorithm ensures the complete tracking and full

coverage of an arbitrary region while avoiding the redundant

placement of sensors on the boundary of the region to be

covered. There exists one saving of mobile robotic sensor

for each of the line to be formed as shown in Fig. 2(iv) and

Fig. 3(iv).

We can compare the recent blanket coverage algorithm[1]

with this work in the following way;

In the recent work[1], the mobile robotic sensors do

necessarily stay at the interpolation points; a1,a2, ...,aK and

b1,b2, ...,bK to guide or to direct the upcoming sensors

for a required line forming mode, but this work removes

the redundant placement of mobile robotic sensors at the

boundary of the region. Thus, it saves one mobile robotic

sensor on each of the line and K number of sensor nodes

are saved after full coverage of the region.

In the recent work(Fig. 3, Pg. 1260 of [1]), there is tracking

of the boundary of the region R from a1 to b1 , from b1 to

b2, from b2 to a2, from a2 to a3, from a3 to b3... So, it

does leave the untracked boundary from a1 to a2 , from b2

to b3... These portions of the untracked boundary segments

36

−6 −4 −2 0 2 4 6

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Optimal Blanket Coverage120Rs=1.5

X−axis

Y−

axis

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

x−axis

y−

axis

(i) (ii)

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

x−axis

y−

axis

−8 −6 −4 −2 0 2 4 6 8 10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0Optimal Area Coverage with117 sensors and Rs=1.5

x−axis

y−

axis

(iii) (iv)

Fig. 2. Simulation Results of Triangular Blanket Coverage with 6-Connectivity

−8 −6 −4 −2 0 2 4 6 8 10

−14

−12

−10

−8

−6

−4

−2

0

X−axis

Y−

axis

−10 −5 0 5 10 15

−14

−12

−10

−8

−6

−4

−2

0

X−axis

Y−

axis

(i) (ii)

−10 −5 0 5 10 15

−14

−12

−10

−8

−6

−4

−2

0

X−axis

Y−

axis

−10 −5 0 5 10 15

−14

−12

−10

−8

−6

−4

−2

0

X−axis

Y−

axis

(iii) (iv)

Fig. 3. Simulation Results of Blanket Coverage over Non-Convex Region

37

i.e., I (a1,a2),I (b1,b2), ... might leave uncertainly in es-

timating the boundary of the region. However, this work

completely tracks the boundary of the region in order to

avoid possible coverage hole.

B. Future Recommendations

The future work can be done for sharp edges of the

boundary with variable radius of curvature. The control law

can be revised to reject the disturbances and to avoid the

collisions among mobile robotic sensors.

REFERENCES

[1] T. M. Cheng and A. V. Savkin, ”Decentralized control of mobilerobotic sensor networks for asymptotically optimal self-deploymentin blanket coverage”49th IEEE Conference on Decision and Control,Atlanta, GA, USA, pp.1257-1262, 2010.

[2] X. Bai,S. Kumar, D. Xuan, Z. Yun and T. Lai, ”Deploying WirelessSensors to Achieve Both Coverage and Connectivity”, MobiHoc 06,Florence, Italy, pp.131-142, 2006.

[3] F. Santoso, ”Sub-Optimal decentralized control algorithms for blanketand k-barrier coverage in autonomous robotic wireless sensor net-works”IET Communications, vol.4, issue 17, pp. 2041 - 2057, 2010.

[4] W. Hengartner and G. Schober, ”A remark on level curves for domainsconvex in one dimension” Applicable Analysis, vol.3, issue 1, pp.101-106, 1973.

[5] R. Kershner, ”The number of circles covering a set”American Journal

of Mathematics, vol. 61, pp. 665-671,1939.

[6] T. M. Cheng and A. V. Savkin, ”Decentralized control of mobilerobotic sensor network for deployment in corridor coverage”48th IEEE

Conference on Decision and Control, Shangai, China, pp. 7897-7902,2009.

[7] A. Jadbabaie, J. Lin, and A.S. Morse, ”Coordination of groupsof mobile autonomous agents using nearest neighbour rules”,IEEE

Transactions on Automatic Control, vol. 48, issue 6, pp. 988-1001,2003.

[8] C. W. Reynolds, ”Flocks, Herds, and Schools: A Distributed Be-havioural Model”14th annual conference on computer graphics and

interactive techniques, vol. 21, issue 4, New York, USA, pp. 25-34,1987.

[9] A. V. Savkin, ”Coordinated collective motion of groups of autonomousmobile robots: Analysis of Vicsek’s model”IEEE Transactions on

Automatic Control,vol. 49, issue 6, pp. 981- 982, 2004.

[10] D. W. Gage, ”Many-Robot MCM Search Systems,” Proceedings of Au-

tonomous Vehicles in Mine Counter measures Symposium, Monterey,CA, USA, pp. 9-55, 1995.

[11] E. Garcia and P. G. De Santos, ”Mobile-robot navigation withcomplete coverage of unstructured environments,” Robotics and Au-

tonomous Systems, vol. 46, pp. 195-204, 2004.

[12] D. W. Gage, ”Command control for many-robot systems,”Proceedings

of the 19th Annual AUVS Technical Symposium, vol. 4, Hunstville,Allabama, USA, 1992, pp.22-24.

[13] R. Lyengar, K. Kar, and S. Banjerjee, ”Low Coordination Topologiesfor Redundancy in Sensor Networks” MobiHoc 2005, New York, USA,pp. 332-342, 2005.

[14] A. Jadbabaie, J. Lin and A.S. Morse, ”Coordination of groups ofmobile autonomous agents using nearest neighbour rules”IEEE Trans-

actions on Automatic Control, vol. 48, issue 6, pp. 988-1001, 2003..

[15] C. D. Meyer, ”Matrix Analysis and Applied Linear Algebra”SIAM,

Philadelphia, 2000.

[16] H. Teimoori and A. V. Savkin, ”A Biologically Inspired Method forRobot Navigation in a Cluttered Environment”Robotica, vol. 28, issue5, pp. 637-648, 2010.

[17] T. M. Cheng and A. V. Savkin, ”A distributed self deploymentalgorithm for the coverage of mobile wireless sensor networks”IEEE

Communications Letters, vol. 13, issue 11, November 2009.[18] T. M. Cheng and A. V. Savkin, ”Decentralized control for mobile

robotic sensor network self-deployment: Barrier and sweep coverageproblems”Robotica vol. 29, issue 2, pp. 283-294, 2011.

[19] A. S. Matveev and A. V. Savkin, ”Estimation and Control overCommunication Networks”, Birkhauser, Boston, 2009.

[20] A. V. Savkin and H. Teimoori, ”Decentralized Navigation of Groupsof Wheeled Mobile Robots With Limited Communication”, IEEE

Transactions on Robotics, vol. 26, issue 6, pp.1099-1104, 2010.[21] T. M. Cheng, A. V. Savkin and F. Javed, ”Decentralized control of a

group of mobile robots for deployment in sweep coverage”,Robotics

and Autonomous Systems, vol. 59, issue 7-8, pp. 497-507, 2011.[22] S. A. R. Zaidi, M. Hafeez, S. A. Khayam, D.C. Mclernon, M.

Ghogho and K. Kim, ”On Minimum Cost Coverage in WirelessSensor Networks”,43rd Annual Conference on Information Sciences

and Systems, Baltimore, MD, USA, pp. 213-218 , 2009.[23] F. Bullo, J. Cortes and S. Matinez, ”Distributed control of robotic net-

works: a mathematical approach to motion coordination algorithms”,Princeton University Press, Princeton and Oxford, 2008.

[24] A. V. Savkin, F. Javed and A. S. Matveev, ”Optimal Distributed Blan-ket Coverage Self-Deployment of Mobile Wireless Sensor Networks”,IEEE Communications Letters, vol. 16, issue 6, pp. 949 - 951, 2012.

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