Sensor Deployment and Relocation: A Unified Scheme - CiteSeerX

Month 200X, Vol.21, No.X, pp.XX–XX

J. Comput. Sci. & Technol.

Sensor Deployment and Relocation: A Unified Scheme Michele Garetto1, Marco Gribaudo1 , Carla-Fabiana Chiasserini2 and Emilio Leonardi2 1 Dipartimento

di Informatica, Universit` a di Torino, Torino, Italy

2 Dipartimento

di Elettronica, Politecnico di Torino, Torino, Italy

E-mail: {garetto,marcog}@di.unito.it; {chiasserini,leonardi}@polito.it Received MONTH DATE, YEAR. Abstract Sensor networks are envisioned to revolutionize our daily life by ubiquitously monitoring our environment and/or adjusting it to suit our needs. Recent progress in robotics and low-power embedded systems has made possible to add mobility to small, light, low-cost sensors to be used in teams or swarms. Augmenting static sensor networks with mobile nodes addresses many design challenges that exist in traditional static sensor networks. This paper addresses the problem of topology control in mobile wireless networks. Limitations in communication, computation and energy capabilities push towards the adoption of distributed, energy-efficient solutions to perform self-deployment and relocation of the nodes. We develop a unified, distributed algorithm that has the following features. During deployment, our algorithm yields a regular tessellation of the geographical area with a given node density, called monitoring configuration. Upon the occurrence of a physical phenomenon, network nodes relocate themselves so as to properly sample and control the event, while maintaining the network connectivity. Then, as soon as the event ends, all nodes return to the monitoring configuration. To achieve these goals, we use a virtual force-based strategy, which proves to be very effective even when compared to an optimal centralized solution. We assess the performance of our approach in presence of events with different shape, and we investigate the transient behavior of our algorithm. This allows us to evaluate the effectiveness and the response time of the proposed solution, under various environmental conditions. Keywords

1

Distributed algorithms, Mobile networks, Sensor relocation, Wireless sensor networks

Introduction

ject tracking, and disaster relief.

A flurry

of robot systems with sensing capabilities are Wireless sensor networks are well suited

being designed with scopes such as underwa-

for applications like habitat and environment

ter monitoring [1], human movement detection,

monitoring, exploration of hostile areas, ob∗

This work is supported by the Italian Ministry for Scientific Research (MIUR), through the MEADOW project

2

J. Comput. Sci. & Technol., Month 200X, Vol.21, No.X

and detection/relief of chemical plumes or fires.

lution that is able to meet all of the above re-

Sub-Kilogram Intelligent Tele-Robots (SKIT)

quirements. Our network system is composed of

are also under study to build multi-agent ex-

numerous mobile sensor-actuator nodes that au-

ploratory systems for asteroids exploration [2],

tonomously organize and react to triggers from

while mini-robots (1/4 cubic inch and weighing

the environment. The specific problem we ad-

less than an ounce) with communication capa-

dress here is how to enable these nodes to both

bilities are being developed at the Sandia Na-

self-deploy and relocate in a distributed fashion.

tional Laboratories [3] to locate and disable land

Traditionally, network deployment [6] is per-

mines.

formed at the initial stage of the network

In this paper, we study mobile wireless net-

functioning, to obtain the desired geographi-

works composed of wireless devices that, besides

cal coverage or spatial sensor density. For the

featuring sensing, computing and communica-

particular applications that we consider, self-

tion capabilities, also integrate actuators [4].

deployment of mobile sensor-actuators is neces-

For the sake of concreteness, we consider the

sary since node placement cannot be performed

case of a sensor-actuator network whose task

manually and accurately, due to possible hostile

is to detect fires in a forest area and counter-

environmental conditions and/or to the large

act them by sprinkling water or other fire re-

number of nodes to deploy. The goal in our case

tardants [4]. In this case, exact deployment of

is to form a connected network with roughly

the sensor-actuators may not be possible due

the same node density and continuous cover-

to the characteristics of the environment. Also,

age, starting from an initial random topology

localization techniques based on GPS may not

or from a configuration in which nodes are pro-

be available due to the high cost (see for exam-

gressively released in the environment from a

ple [5] for the requirements that GPS receivers

single point in the area. Relocation, instead,

must meet to properly work in a forest envi-

is needed when the network has to react to a

ronment). Upon occurrence of a fire outbreak,

physical phenomenon [7], which in our applica-

sensor-actuators should be able to surround the

tion may be an environmental disaster (event-

event area and track the movement of the fire

based relocation). Here, we do not deal with

front, while keeping themselves at a safe dis-

fine-grained relocation aimed at stopping a cov-

tance, from which they can control the fire with-

erage hole created for example by a node fail-

out being damaged. Finally, the node density

ure [8,9], but focus on event-based relocation,

should be increased in proximity of the fire front

where the node positions and density have to

so as to contain its expansion.

be adapted to properly sample and control a

In this work, we present a distributed so-

large-scale event.

M. Garetto et al.: Sensor Deployment and Relocation: A Unified Scheme

3

When developing solutions for deployment

ticular, [11] presents a centralized algorithm to

and relocation of sensor-actuator networks, it is

be executed by a cluster head, which has per-

clear that one must obey to the system con-

fect knowledge of sensor positions and is able

straints imposed by the application and the

to orchestrate the movement of all sensors to-

employed technology.

In the case of mobile

ward the desired locations. A distributed ap-

nodes, the current trends in robotics push to-

proach is proposed in [6], where three iterative

wards small, light, low-cost devices to be used in

protocols for sensor deployment are presented.

teams or swarms; it follows that mobile sensor-

Sensors move from high- to low-density zones so

actuator networks are subject to limitations

as to increase coverage; at each protocol itera-

in computation, memory and energy capabili-

tion, coverage holes are detected by sensors us-

ties and that distributed solutions are required.

ing Voronoi diagrams. The scheme in [6], how-

Several papers have addressed the problem of

ever, requires sensors to have knowledge of their

mobility-assisted deployment and relocation in

absolute position.

sensor networks. However, we are the first to

More related to our work are [12,13,14] in

propose a unified, distributed algorithm that

which distributed algorithms based on virtual

enables the network nodes to self-deploy for en-

repulsive forces exchanged by sensors are con-

vironmental monitoring, relocate upon an event

sidered for the sake of network deployment

occurrence, and return to the monitoring con-

only (i.e, no event-based movement). However,

figuration as the event ends. Also, the solution

the solutions in [12,13] require sensors to have

we define has the following features: (i) it does

knowledge of their absolute position, while the

not require the use of absolute node localiza-

scheme in [14] is applicable only to deploy sen-

tion; (ii) it meets the constraints on the devices’

sors over a limited area with well defined bound-

computation, memory and energy capabilities;

aries. Finally, in [15] the focus is on a sen-

(iii) it provides real-time response to events as

sor deployment scheme that provides workload

well as coordination among nodes as emergent

balancing, besides coverage; global information

behavior (swarm intelligence [10]).

about the sensor positions, however, is still required.

2

Related Work

With regards to self-deployment, we emphasize that our solution, in contrast to previ-

Movement-assisted deployment, driving mo-

ous work, does require neither a network-wide

bile sensor-actuators from an initial random

knowledge of the node location nor the use of

configuration to some desirable network state,

GPS. Moreover, we aim at obtaining a network

has been addressed in [6,11,12,13,14,15]. In par-

featuring a desired node density (not maximum

4


coverage). By achieving a regular tessellation of

the relative positions of the neighboring nodes

the network area with assigned inter-nodal dis-

are needed. Furthermore, we provide a unified,

tance, we attain our goal, while providing also

distributed algorithm which enables sensors to

load balance.

self-deploy, as well as move upon an event oc-

In the context of event-based relocation, a pioneer work is [7], where the aim is to relocate

currence and return to the monitoring configuration once the event ends.

sensors so as to approximate the spatial distribution of events taking place within the network area. Our objective is different: we do not want to approximate the event distribution, rather we want to relocate nodes in response to an oc-

3

Goals and Assumptions In this section we state more precisely the

goals and assumptions of our work, and introduce some notation and definitions.

curred event so as to provide the necessary sampling, tracking and control functionalities, and

3.1

Goals

then restore the network configuration used for normal monitoring after the event ends. In [8] sensor relocation mainly aims at filling coverage holes created by a single node that fails or moves away. The proposed scheme assumes an initial grid structure for the network

Our objective is to design a distributed solution that allows nodes to automatically relocate themselves so that the following functionalities are provided: (i) Self-deployment.

Starting from any

and the presence of grid-head nodes. Our goal

(connected) initial deployment of the

and assumptions are different from the one pro-

nodes, the network achieves a target

posed in that work. We do not address the prob-

structure at the equilibrium, called mon-

lem of filling a hole created by a single sensor

itoring configuration. As monitoring net-

that becomes unavailable; rather we deal with

work topology, ideally we would like a con-

event-based relocation, that typically requires

nected tessellation of the network area,

the simultaneous movement of several nodes.

with continuous coverage and a desired

Also, our algorithm starts from any initial de-

node density ρm . In particular, we have

ployment of the nodes and no sensor has to be

selected a regular triangular tessellation

selected for playing special logical roles.

for the following two reasons: (i) among

To conclude, we emphasize that our solution

the regular tessellations is the one provid-

differs from previous work since nodes are not

ing the highest connectivity degree; (ii)

required to learn their absolute location within

results in [16] show that, when the node

the area, nor the location of the event; only

communication range is not smaller than


√

5

3 times the sensing range (which is typ-

communication range, and to detect any event

ically the case), a triangular tessellation

having effect on at least one spatial point within

maximizes the network area covered by a

its sensing range.

given number of nodes (conversely, it re-

We consider the triangular tessellation to

quires the minimum number of nodes to

be our network monitoring configuration, and √ we assume Rc ≥ 3Rs [16]. One of the goals

cover a given network area).

achieved by our self-deployment procedure is to (ii) Reaction to physical phenomena and environmental disasters.

When an

event occurs, the nodes detecting the event move toward its location. We require nodes to relocate around the area where the event takes place, yielding a de-

yield a triangular tessellation featuring the desired monitoring node density ρm . With this goal in mind, each node (not on the border) should be surrounded at the equilibrium by six nodes placed at the vertices of a regular hexagon of radius Dm , with:

sired sensor density in its proximity. Network connectivity is maintained throughout the nodes relocation. As the event is no longer detected, nodes move back to the network monitoring configuration (although they are not required to return to their exact initial positions).

3.2

Assumptions and Definitions

2 Dm =

2 1 √ = ρm sin 60◦ 3ρm

(1)

Indeed, by geometric considerations, one can see that a triangular tessellation with internodal distance equal to Dm corresponds to a node density: ρm = 1/Am , where Am is twice the area of an equilateral triangle of side Dm . Note that ρm should be such that Dm < Rc . Given (1), hereinafter we use interchangeably inter-nodal distance and node density.

We consider a network composed of N mo-

After deployment, we want the network to

bile sensor-actuator nodes, which are deployed

be able to react to physical phenomena or en-

over a bidimensional geographical region. Let

vironmental disasters (event-based relocation).

us denote with xi (t) the position of node i at

Note that nodes may become aware of such

time t, and with dik (t) the Euclidean distance

events either because they detect a variation

between nodes i and k at time t.

in the standard values measured by their sens-

Network nodes can move at a maximum

ing device (e.g., temperature variation in case

speed of vmax , and have common sensing range

of fire), or because they are alerted by other

Rs and communication range Rc . A node is

nodes through the transmission of notification

able to communicate with the nodes within its

messages. For simplicity, in the following we

6


only consider the first case.

this reason, it has also been largely used in the

In the presence of an event, sensors have

literature (see [17] and the references therein).

to relocate themselves so as to provide a new

Note that a high value of si (t) corresponds ei-

desired node density. Similarly to the case of

ther to an event of moderate intensity happen-

self-deployment, the new density is achieved by

ing close to node i, or to a very catastrophic one

specifying a target inter-node distance dTik (t)

occurring far away from the node position. In

(i, k=1, . . . , N).

both cases, some action is required by the nodes

We assume that the target

inter-node distance is a function of the event intensity sensed by each pair of nodes.

detecting the event.

By

Let us denote with Smin the minimum inten-

denoting with si (t) the intensity of the event

sity for detecting an event, and with Smax the

sensed by node i at time t, we have: dTik (t) =

maximum intensity a node can measure with-

f (si (t), sk (t)). In general, f (·) should be chosen

out damage for the device. We model the de-

such that the higher the event intensity detected

tection of an event by the generic node i at

by a node, the higher the target node density

time t by defining a function ψi (t), which can

(i.e., the smaller the target inter-node distance

take three values: {−1, 0, 1}. If no event is

to be used).

detected at time t, then ψi (t) = 0.

When

We model the event intensity as follows. In

the node i senses an event intensity si (t) s.t.:

order to simplify the presentation, let each event

Smin ≤ si (t) < Smax , then ψi (t) = 1. Finally,

have an epicenter (the case of events occurring over an extended area will be considered later).

if si (t) ≥ Smax , ψi (t) = −1. The latter condition allows us to model a node moving away

The signal sensed by the generic node i at time

from the event location to avoid a device dam-

t is given by:

age. Note that, dependently on the value of si (t) =

α(t) die (t)β

(2)

Smax , an empty area can be created around the event epicenter. We denote with De (De ≥ 0)

where die (t) is the Euclidean distance between

the distance from the epicenter at which an in-

the node position xi (t) and the epicenter loca-

tensity of Smax is sensed by a node, i.e., De is

tion at time t, α(t) is the energy emitted by the

the minimum distance that must be kept from

event at the epicenter at time t, and β is the de-

the event.

caying exponent of the event energy. We chose to use this sensing model because, in spite of its

4

The Distributed Algorithm

simplicity, it is able to represent various types of signals (e.g., radio, acoustic, etc.) and physical phenomena (e.g., heat, temperature); for

The proposed distributed algorithm exe-

7


cutes both node deployment and event-based re-

events, only exchange and friction forces act on

location, using a unified approach. Nodes are

the nodes, leading the network toward the de-

moved by virtual forces [6,11,,12,13,11,14], de-

sired monitoring configuration. Potential forces

termined only by the relative position of neigh-

instead come into play when an event occurs.

boring nodes and by the presence of an event.

We characterize the three components below.

We remark that, although the idea of virtual forces has been already exploited in many previous works, our specific implementation and goals differ in many ways from existing propos-

• Exchange forces are established between pairs of nodes which are sufficiently close to each other. Given two nodes, say i and ˆ (t) = xk (t)−xi (t) the k, we denote with x ik

dik (t)

als found in the literature. In the following we

normalized vector parallel to the segment

assume that only one event occurs at a time and

linking i to k.

that events do not overlap in time; later we will discuss the case where the event area enlarges over time and the case where multiple events take place simultaneously. 4.1

Sensor

Let Ii (t) be the set of nodes acting with forces on i. A node k belongs to Ii (t) if the distance between i and k does not exceed the communication range:

Movement

and

Virtual

Forces

dik (t) < Rc and there are no other nodes k ′ shielding

At time t each sensor-actuator is moved by

the action of k on i. Node k ′ shields k if k ′

a virtual force that depends on the position of

is closer to i and the angle formed by the

the other nodes and on the event location. The

ˆ ik (t) is smaller than ˆ ik′ (t) and x vectors x

trajectory of node i at time t is driven by the

600 , i.e.,

Newton motion law:

dik′ (t) < dik (t) and

d2 xi (t) = F i (t) dt2

ˆ ik (t) > cos 600 ˆ ik′ (t) · x x

where F i (t) is the virtual force acting on i at

ˆ ik (t) is the scalar product ˆ ik′ (t) · x where x

time t. F i (t) can be defined as the sum of three

ˆ ik (t). We ˆ ik′ (t) and x between vectors x

components:

emphasize that, based on the above conditions, at most six nodes concurrently

F i (t) =

F ei , (t)

+

F pi (t)

+

F fi (t).

(3)

act on node i at each time instant (i.e.,

F ei (t) is the resultant of the exchange forces

|I(i, t)| ≤ 6, ∀t).

among mobile nodes, F pi (t) is the potential force

The force that node k acts on node i de-

and F fi (t) is the friction force. In absence of

pends on the distance between i and k:

8


it may be either attractive or repulsive.

where P and Q are positive constants

Given the monitoring distance Dm , the

that do not depend on the event intensity

force is repulsive if dik (t) < Dm , and be-

and such that Q ≫ P . Note that, ac-

comes attractive when dik (t) > Dm :

cording to (5), a far away sensor-actuator

X (dik (t) − Dm ) ˆ ik (t) (4) F ei (t) = Gik (t) x dik (t)(Rc −dik (t)) k∈Ii (t)

In (4), in absence of potential forces the parameter Gik (t) = G is a constant whose value determines the speed of the nodes, while in presence of potential forces Gik (t)

detecting the event will be attracted toward the event location, while nodes sensing an event intensity greater than Smax (ψi (t) = −1) will be pushed away so as to make nodes locate at a distance not smaller than De from the epicenter.

is dynamically updated to achieve the de-

Finally we remark that, if the event ex-

sired node density in the event proximity.

ˆ ie (t) is tends over a closed finite region, x

Note that: F ei (t) → +∞ (i.e., attractive

force) as dik (t) → Rc , while F ei (t) → −∞ (i.e., repulsive force) as dik (t) → 0, and F ei (t) = 0 if dik (t) = Dm . In absence

of potential forces, exchange forces push

the normalized vector parallel to the segment linking node i with the nearest point inside the event region. • Friction forces are essential to make nodes stop around the equilibrium configuration

nodes to form at equilibrium a regular tri-

defined by potential and exchange forces,

angular tessellation of the area, each node

reduce the overall motion, and stabilize

(not on the border) being surrounded by

the system. Motion of node i is affected

six neighbors at distance Dm .

by two friction forces, a static Coulomb

• Potential forces F pi (t) are positional forces that attract/repel nodes toward/from the ˆ ie (t) event epicenter. By denoting with x the normalized vector parallel to the segment linking node i with the event epicenter e, the potential force acting on i at time t can be written as:    0 if ψi (t) = 0   F pi (t) = ˆ ie (t) Px if ψi (t) = 1     −Qˆ xie (t) if ψi (t) = −1

(5)

friction plus a viscous friction. The static Coulomb friction acts while i is still, opposing to the motion. The viscous friction acts while node i is moving, producing a force proportional to its instantaneous speed:  p e  ˆ − min k x (t), F (t) + F (t)  0 0 i i    d x (t) if dti = 0 F fi (t) =      −kv dxdti (t) otherwise

ˆ 0 (t) is the normalized vector parwhere x


9

allel to the resultant of exchange and po-

Recall that we want sensor-actuators to re-

tential forces, and k0 and kv are the static

locate around the event with a target density

and the viscous friction coefficient, respec-

(i.e., inter-nodal distance dTik (t)), while preserv-

tively [18].

ing network connectivity. This is achieved as follows. Each node i determines, based on the

4.2

The Algorithm

Next we describe how virtual forces act on nodes during deployment and relocation.

event intensity sensed by itself and by neighborˆ ie (t) directed ing nodes, the normalized vector x toward the event, and modulates the intensity of the exchange forces by dynamically adapt-

Behavior during self-deployment. Recall that, starting from a random topology, the monitoring configuration must be achieved. To this end, nodes use a common target inter-node distance, equal to Dm . Sensor-actuators are therefore moved by exchange forces, whose direction and intensity are determined by the current value and the monitoring value of the inter-

ing Gik (t) so as to reach, at the equilibrium, the target inter-nodal distance. More specifically, Gik (t) is computed exploiting a first order Proportional Integrative (PI) controller driven by the error between the current distance dik (t) and the target value dTik (t) [19]. Notice that our solution does not require global knowledge of the absolute position of the event.

nodal distance. Clearly, since no event is taking place, potential forces are null. Self-deployment terminates when all nodes have reached an equilibrium configuration in which the force F i (t) acting on each node i is null.

Behavior when an event ends. As soon as an event ends, the potential forces become zero and the nodes reset Gik (t) to the constant value G. Sensor-actuators are therefore moved by exchange forces only, as during self-deployment,

Behavior in the absence of an event. In absence of events the network maintains the equilibrium configuration reached during selfdeployment.

and the monitoring network configuration is restored. 4.3

Implementation Issues To implement our algorithm, we consider

Behavior in the presence of an event.

time to be discretized into fixed slot intervals.

When an event takes place, a potential force is

Within each interval, every node computes the

generated which triggers the nodes’ movement.

virtual forces acting on itself, thus determining

The monitoring configuration is therefore bro-

the direction and the speed of its movement dur-

ken and exchange forces arise again.

ing that time slot, independently of the other

10


nodes. Virtual forces with a too small inten-

5

Performance Evaluation

sity are neglected to avoid useless movements, while the node speed is clamped by the maxi-

In this section, first we analyze the performance of our algorithm during self-deployment;

mum value vmax .

we assume that no events occur during this phase, although our algorithm could handle To compute the exchange forces, each node

events even during the initial deployment.

must be aware of the relative positions, in terms

Then, we consider the event-based relocation,

of distance and direction, of the nodes located

and focus on the most critical case in which one

inside its communication range.

or more events take place outside the area oc-

This infor-

mation can be obtained by a sensor through

cupied by the sensors.

standard techniques, such as those based on the measurement of the RSSI and the direction of arrival (DoA) [20] of the signals received from the neighbors. Clearly, depending on the employed techniques, the estimates may be affected by significant errors; we will account for such inaccuracies while evaluating the performance of our scheme.

5.1

Self-deployment

We first show a typical example of how the system behaves when we start from a random topology. We fix the sensing range Rs = 1. The monitoring distance Dm is assumed to be √ equal to 3, corresponding to the triangular tessellation maximizing the area fully covered by the network (i.e., all points within the net-

In addition, to cope with events, nodes must acquire the direction of the event epicenter with respect to their current position. Since we are dealing with physical phenomena, this information can be acquired either independently by each sensor locally estimating the direction along which the sensed signal si (t) exhibits the steepest variation, or through explicit signaling by the sink. The latter approach has the advantage to trigger the movement also of those sensors which, being too far away from the event epicenter, are not able to directly sense it.

work are sensed by at least one node). We consider N = 400 mobile sensor-actuators that are initially placed uniformly at random within a √ 2 3/2 (the expected size of disk of area NDm the area covered by the triangular tessellation), as shown in the left plot of Figure 1.


11

Figure 1: Initial and final locations of N = 400

must be larger than zero so as to stop the system

nodes: initial random placement in a disk ac-

movement. Indeed, some dissipative forces are

cording to the uniform distribution (left); mon-

required to gradually reduce the internal energy

itoring configuration after deployment (right).

of the system, allowing it to stabilize around an

The communication range Rc is set to 3

equilibrium point (if kv = 0 the system oscil-

(the system behavior is not really affected by

lates indefinitely around one or more equilib-

this parameter, provided that it is sufficiently

rium point, see [21]). We found that 0.1 is a

larger than Dm ). We run our algorithm with

suitable value for kv in all considered scenar-

G = 0.01, kv = 0.1, vmax = 0.01, k0 = 1e-5,

ios, thus we will no longer discuss the impact

and we assume no measurement errors. After

of this parameter.

490 time stepsAn animated picture of the tran-

small amount of static friction (i.e., k0 > 0)

sient behavior is available at [21], all nodes stop

is also necessary to bring the system to a com-

moving, and the system reaches the monitor-

plete stop within finite time. Note that, using a

ing configuration depicted in the right plot of

minimum force threshold that prevents a node

Figure 1.

from starting to move in response to arbitrarily

We observe that the final configuration is close to a triangular tessellation, although the

Finally, we found that a

small forces is a reasonable design choice to save energy.

structure is not perfect and presents some ir-

To assess the quality of our algorithm, we

regularities. However, the ‘holes’ that naturally

consider the following metrics: i) the network

appear as a consequence of a random initial de-

coverage achieved after the system reaches the

ployment (see left plot of Figure 1) are stopped,

monitoring configuration; ii) the time and en-

and the whole area is covered almost uniformly.

ergy required to reach the monitoring configu-

A few observations about the choice of pa-

ration (transient behavior). While evaluating

rameters. For a given value of vmax (that de-

these metrics, we discuss the effect of a few pa-

pends on the device technology), the duration

rameters and the impact of measurement errors.

of the transient essentially depends on the intensity of the forces exchanged by the nodes, which

5.2

Coverage

is proportional to G. In our example, we chose

Since the algorithm does not achieve a

a quite large value of G, so that nodes react

perfect triangular tessellation, it is important

rapidly to changes in their neighborhood and

to quantify how the equilibrium configurations

tend to move at the maximum possible speed of

reached by the algorithm differ from the ideal

vmax . We investigate the effect of G in more de-

triangular tessellation (which provides the opti-

tail in the following. The viscosity coefficient kv

mal coverage).

12

J. Comput. Sci. & Technol., Month 200X, Vol.21, No.X G = 0.001 - no errors G = 0.01 - no errors G = 0.001 - errors G = 0.01 - errors

ues of G (namely, 0.01 and 0.001), as well as the

triangular lattice random deployment network area

presence or not of measurement errors. Mea2

100

surement errors are modelled as follows: dis-

1.6 90

1.4

85

1.2

80

1

75

0.8

70

0.6

65

0.4 1.2

1.4

1.6

1.8 Dm

2

2.2

Network Area

Coverage Percentage

1.8 95

2.4

tance errors follow a uniform distribution with zero mean and maximum value equal to 5% of Dm [22], while angle measurements are independently introduced according to a uniform distribution in the range ±10◦ [20]. In presence of errors, the static friction coefficient k0 has to be

Figure 2: The fraction of area covered (left y axes) and the network size (right y axes) as functions of the monitoring distance Dm , for

chosen so that the worst possible error in the location of neighbors does not make a sensor resume moving.

different node deployments. In Figure 2 we show the percentage of the area covered by at least one node as a function of the monitoring distance Dm (the sensing range is fixed to 1), for several deployment strategies and N = 400. The thick solid line

Coverage Percentage Deficit

6

no errors errors

5 4 3 2 1

corresponds to the regular triangular tessella-

0 1

1.2

1.4

tion, for which the coverage percentage is 100% √ up to Dm = 3 = 1.732; after this point the coverage decreases for increasing values of Dm , and can be computed using elementary geometry. The lower dotted line with error bars corresponds to the coverage achieved if nodes are just randomly deployed over the area and do

1.6

1.8

2

2.2

2.4

Dm

Figure 3: Difference between the fraction of area covered through a perfect triangular tessellation and our algorithm. Results are plotted versus the monitoring distance Dm for G = 0.001, with and without errors.

not move from their initial location. The cov-

In Figure 3 we present the difference (in per-

erage percentage is computed using a Monte-

centage) between the fraction of area covered

Carlo technique and averaging the results over

by the triangular tessellation and our algorithm.

30 experiments (error bars refer to 95%-level

We set G = 0.001, and we consider the two cases

confidence intervals). In between the above two

with and without measurement errors. Again,

extreme cases there are results related to four

error bars refer to 95%-level confidence inter-

variants of our algorithm: we consider two val-

vals.

13


Looking at Figures 2 and 3, we observe that

5.2.1 Transient Behavior

the difference in performance of our solution with respect to a perfect triangular tessellation

To evaluate the performance of the algorithm

is very small, i.e., about 5% in the worst case as

during the transient phase, we consider both the

shown in Figure 3. Also, the algorithm performs

amount of time taken to reach the equilibrium,

slightly better (i.e., it is closer to the perfect

and the energy spent by nodes to move from

triangular tessellation) when errors are present

their initial location to their final position. The

and when G is small (see Figure 2). Note that

first metric is meaningful if one wants to mini-

the difference in performance with and without

mize the duration of the deployment phase; the

errors is not caused by the inaccuracy affect-

second metric is probably more relevant, since

ing the simulation results, as confirmed by the

sensors usually have severe energy limitations.

confidence interval reported in Figure 3. Intu-

By assuming that the energy required to move a

itively, the impact of errors can be explained

node is proportional to the traveled distance [8],

considering the equivalent effect played by tem-

we can focus on the total movement performed

perature in simulated annealing: local vibra-

by the nodes to reach the final configuration.

tions increase the probability of avoiding spuri-

We compare our distributed algorithm with

ous equilibrium points thus resulting in a more

an optimal centralized one, i.e., the algorithm

regular monitoring configuration. A small G

that instructs each sensor-actuator to move in

also reduces the responsiveness of the system

such a way that the aggregate movement per-

away from the equilibrium point, helping to ex-

formed by all nodes to reach the final topol-

plore more patiently the solution space. Note

ogy is minimized.

that, to obtain a coverage percentage closer to

duces to finding the minimum weighted match-

100%, one could just reduce the monitoring dis-

ing (mWM) of a bipartite N x N graph, where

tance Dm (e.g., use Dm = 1.4). However, a

N vertices correspond to the initial locations of

perfect coverage cannot be always guaranteed,

the nodes, and N vertices correspond to their

moreover this strategy has the important draw-

location in the final configuration. We draw an

back of reducing the size of the network area: in

edge between any vertex in the first set to any

Figure 2 the curve labeled ‘network area’ (right

vertex in the second set, with associated weight

y axes) actually serves as a reminder that the √ network area (normalized to 1 at Dm = 3) is

equal to the Euclidean distance between the cor-

proportional to the square of Dm , thus a trade-

putes which node goes where, i.e., the identity

off exists between coverage percentage and net-

of the location where each node has to go so as

work size.

to minimize the aggregate movement. Finally,

The optimal solution re-

responding locations. Then, the mWM com-

14


down the system dynamics using a small G:

till it reaches its final position.

by doing so, the total movement (and thus

Total Movement

each node moves at the maximum speed vmax

350

350

300

300

250

250

200

200

150

150

100

100

50 0

100

200 Time

also the achieved percentage of covered area is higher (see Figure 2).

50

algorithm - G = 0.01 MWM

0

the node energy expenditure) is reduced, and

0 300

400

5.3

algorithm - G = 0.001 MWM 0

400

800

1200

1600

2000

Time

Event-based Relocation

We now evaluate the performance of our algorithm in relocating the nodes when one or

Figure 4: Comparison of the proposed algorithm with the centralized, optimal one that minimizes the total movement performed by the nodes. We consider the example scenario with N =

more physical events are detected. We consider the event intensity to have a circular symmetry around the epicenter, although this is not a requirement for our algorithm, and we set the parameter P of the potential force to 0.01.

400 and initial topology as shown in the left plot

Recall that our goal is to make sensor-

of Figure 1. To obtain a fair comparison, the

actuators move toward an event, and possibly

final configuration for the mWM solution is as-

surround it, while achieving a target node den-

sumed to be the one achieved by our algorithm

sity (i.e., inter-nodal distance dTik (t)) and pre-

at the equilibrium (e.g., for G = 0.01, the one

serving network connectivity. In the following,

shown in the right plot of Figure 1). In Figure

we take the target density to be proportional to

4, we compare our algorithm with the central-

the local intensity of the event (2). More specif-

ized, optimal one for two different values of G,

ically, we assume that the desired density on the

namely 0.001 and 0.01. When G = 0.01, our

frontier of the event (i.e., where si (t) = Smax )

algorithm produces about twice the total move-

is W times larger than the monitoring density

ment and takes about twice the time to reach

ρm , and it decays as the event intensity. Thus,

the final configuration, with respect to the opti-

the target density computed by node i at time

mal solution. When G = 0.001, the total move-

t is:

ment produced by our algorithm is reduced (still 75% more of the total movement obtained with

ρTi (t) = ρm

si (t) 1 + (W − 1) Smax

(6)

mWM), but the duration of the transient is sig-

To compute the target inter-nodal distance

nificantly increased (note the different x scale).

dTik (t), each sensor-actuator i first collects the

We conclude that, when the deployment du-

event intensity sk (t) sensed by its neighbor k.

ration is not crucial, it is convenient to slow

Then, assuming a perfect triangular tessellation

15


(1) and using the target density defined in (6),

that all nodes are able to detect the event since

it derives dTik (t) as:

it starts, i.e., si (t) ≥ Smin ∀i, t.

dTik (t) =

1 √ 3

1 1 + ρTi (t) ρTk (t)

20

initial 40

We start by considering the case of a single

0

30

event, occurring after the network deployment,

-10

and evaluate the performance of our algorithm both at the equilibrium and during the transient phase. We assume that the event epicenter is located outside the initial area in which nodes are

Y

10

-20 -30 -40

final -50 -50

-40

-30

-20

-10

0

10

20

X

deployed (for a single event whose epicenter is located inside the initial area, even better re-

Figure 5: Initial and final configuration ob-

sults than those presented here are obtained).

tained by the algorithm in case of event epi-

The case of an event area that enlarges over

center at [−30, 30], W = 2, β = 2 and De = 6.

time and the case of multiple events overlap-

Looking at Figure 5, we observe that the re-

ping in time are considered at the end of the

location takes place successfully, and nodes sur-

section.

round the event getting denser as the distance from the epicenter decreases. To verify that the

5.3.1 At the Equilibrium

target density is indeed achieved, we compute

Figure 5 shows the topology obtained

the theoretical number of nodes, NT (d), that

through self-deployment and the final topology

are expected to fall within distance d from the

achieved by our relocation algorithm, for the

epicenter. NT (d) can be obtained by integrat-

same parameters of the scenario in Figure 1

ing the target density defined above, over a ring

(N = 400 nodes). The event occurs at position

with inner radius De and outer radius d. Figure

[−30, −30], and is characterized by α(t) = 1

6 shows the good match between the number

and β = 2 (see the event model in (2)). The

of nodes counted on a single experiment within

parameter W in the target density is set to 2,

disks of increasing radius d, and the correspond-

while Smax is such that the minimum distance

ing theoretical value NT (d), for different combi-

De from the epicenter at which the nodes can

nations of W and β. We observe that poten-

be located is equal to 6. Note that, assuming an

tial forces tend to greatly compress the nodes

event with circular symmetry, the event frontier

in proximity of the event; it follows that the lo-

turns out to be a circle of radius De , centered

cal adaptation of the parameter Gik (t) in the

at the epicenter. Furthermore, here we assume

expression of the exchange forces (4) is indeed

16


necessary to obtain the desired density.

350 300 Number of Sensors

node, and use this point as the event epicenter

β = 1 - W = 3 - algorithm β = 1 - W = 3 - target β = 2 - W = 2 - algorithm β = 2 - W = 2 - target W = 1 - algorithm W = 1 - target

400

imum distance from the current position of the

for the sensor. Again, the event is characterized by α(t) = 1 and β = 2, while the tar-

250

get density W is set to 2. As can be observed,

200

relocation occurs successfully also in this case:

150 100

sensors completely surround the event, meeting

50

the desired node density. Note that the sen-

0 0

5

10

15

20

Distance from Epicenter

Figure 6: Number of nodes within distance d from the epicenter (in the final configuration),

sors maintain the safe distance De = 6 from the nearest point along the segment on which the event takes place.

for different values of W and β. Comparison between algorithm performance and theoretical target values. 20 15

5.3.2 Transient Behavior 10

Y

5 0 -5

The performance of our relocation algorithm

-10

during the transient phase is compared against

-15

the results provided by the optimal, central-

-20 -20

-15

-10

-5

0

5

10

15

20

ized mWM solution (as in Section 5.3.2). We

X

consider the same scenario as for Figure 5. To Figure 7: Final locations of N = 400 nodes

get a fair comparison, the initial and final net-

when the event taking place in the area is not

work configurations are assumed to be the same

characterized by rotational symmetry.

for both our algorithm and the mWM solution.

Figure 7 shows the final topology of the sen-

Also, by setting De = 6, the mWM trajecto-

sor network when the event is not characterized

ries are constrained to pass at a distance not

by rotational symmetry. In particular, we con-

smaller than 6 from the event epicenter; this

sider an event which occurs along a segment,

implies that the optimal trajectory to reach a

from point (-6,0) to point (6,0). In this case,

position behind the event consists of two seg-

at each time step we consider, for each sensor,

ments connected by a piece of arc belonging to

the point along the segment which has the min-

the circular event frontier of radius 6.

17


the shortest path toward a final position located

Number of Sensors

400 algorithm mWM 350

d < 18

behind the event.

300 250

So far we have assumed that all nodes can

d < 12

detect the event since it begins. However, our

200 150

d

Sensor Deployment and Relocation: A Unified Scheme - CiteSeerX

Sensor Deployment and Relocation: A Unified Scheme - CiteSeerX

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