Chapter 4 Our Proposed Schemes
4.1 Problem Statement
In this chapter, we formally define the problem to solve and describe our proposed schemes. The problem is as follows. Given N randomly deployed mobile directional sensors with sensing range Rs, communication range Rc, and sensing angle α in a given target sensing region, we must maximize sensor coverage with a minimal amount of required moving distance.
To address the above problem, we make the following assumptions, some of which are similar to those made in [22]:
1) All directional sensors have the same sensing range (Rs), communication range (Rc), and sensing angle α, where α . Directional sensors within Rc of a sensor are called the sensor’s neighboring nodes. The communication range is equal to twice the sensing radius.
2) Directional sensors can move to arbitrary positions via GPS, but their sensing directions are not rotatable.
3) Each directional sensor knows its location information, which includes sensing region, sensing radius, sensing angle, and field of view. Each sensor can also obtain the location information of its neighboring sensors.
4) The target region is on a two-dimensional plane with no obstacles, but the boundary of the target region is regarded as a wall-like obstacle.
5) We do not consider the energy consumption of the directional sensors, but we view the distance moved as proportional to energy consumption—i.e., the more distance moved, the more energy consumed.
4.2 Improving the Coverage of the Virtual Force
Algorithm
To maximize area coverage, we present multiple forces for directional mobile sensors.
More specifically, the resultant forces are composed of four computing schemes, namely the Centroid Push Auxiliary Point (CPA) scheme, Centroid Push Centroid (CPC) scheme, Voronoi Point Pull Centroid (VPPC) scheme, and Neighbor’s Repulsion (NR) scheme. Our proposed computing schemes are all based on Virtual Force.
4.2.1 Centroid Push Auxiliary Point (CPA) Scheme
Each directional sensor calculates the auxiliary points of its sensing sector; these auxiliary points are then regarded as the virtual points by which the sensor can check whether auxiliary points are covered by a neighboring sensor. We use the TIS test to define how auxiliary points are covered. Then, at the beginning of each round, each sensor computes the auxiliary points of a directional sensor. If a sensor has the same overlapping area with a neighbor, the sensor’s auxiliary points are covered by the neighbor. The neighbor’s centroid pushes the sensor’s auxiliary points covered by the neighbor of the sensing sector. For example, as shown in Figure 4.1, if Si has overlap with Sj and APi3 is
covered by Sj then Si computes a vector from Gj to APi3.
Si
Sj
Auxiliary point
APi2
APi1
APi3
Directional sensor
The direction of Virtual force Sj exert on Si .
Gj
Gi
Figure 4.1 Example of the repulsion model in which Sj exerts a repulsive force on Si
The calculation of the sensor’s repulsive force can be expressed as
(3)
Each sensor keeps the force temporarily when each sensor does all operations over.
4.2.2 Centroid Push Centroid (CPC) Scheme
In the CPC scheme, each directional sensor calculates the centroid of its sensing sector; the centroids are regarded as corresponding virtual points. We also use the TIS test to define the size of the overlapping regions. The area of the sensing sector SA is equally divided into M small areas, which we assume to be points (given large enough M). At the beginning of each round, each sensor computes its overlapping area size (or OAS) with its neighbors. Next, each sensor finds which neighbor has the same overlapping area. If a sensor has the same overlapping area with a neighbor, but its auxiliary points are not covered by any neighbors within its sensing range, then the directional sensor uses its centroid to calculate the force of the neighbor’s centroid on its own centroid. For example, as shown in Figure 4.2, if Si has overlap with Sj but APi1, APi2, and APi3 aren’t covered by
Sj then Si computes a vector from centroid Gj to centroid Gi.
Si Sj
Auxiliary point
Directional sensor
The direction of Virtual force Sj exert on Si.
Gj
Gi
Figure 4.2 Example of the repulsion model in which Sj exerts a repulsive force on Si
The calculation of the sensor’s repulsive force can be expressed as
(4)
Each sensor keeps the force temporarily when each sensor does all operations over.
4.2.3 Voronoi Point Pull Centroid (VPPC) Scheme
After implementing the above two schemes (i.e., CPA and CPC), we feel results are still insufficient. To obtain more accurate forces for each sensor, each sensor uses its centroid to simulate a Voronoi polygon with its neighbor’s centroid, then each sensor finds its own Voronoi polygon. At the same time, we use VOR and centroid methods to define the attracting point that will attract the sensor’s centroid (this point is abbreviated ATP). If a sensor finds ATP by using VOR, the sensor uses its Voronoi polygon to find the vertex that is farthest from the sensor’s centroid, as shown in Figure 4.3(a). If a sensor finds ATP by using the centroid approach, the sensor uses its Voronoi polygon’s centroid as ATP, as shown in Figure 4.3(b). We pick one of these methods to compute the attracting force for
each sensor. For example, Si finds its Voronoi polygon with its neighbor’s centroid and computes its ATPi using this polygon’s vertices. Gi is attracted by ATPi, as shown in Figure 4.4.
The Farthest point of Voronoi polygon Centroid of Voronoi polygon
G G
Sensor of Centroid
Figure 4.3 Two approaches to determining attracting points (ATPs): (a) using VOR;
(b) using the centroid approach
Gi
Si
ATPi Attracting
point
Directional sensor
The direction of Virtual force APT attract Si.
Sensor of centroid
Figure 4.4 Example gravitational model in which ATPi attracts Gi
The calculation of the sensor’s attracting force is expressed as
(5)
Each sensor keeps the force temporarily when each sensor does all operations over. If a sensor does not have any neighbors to simulate its Voronoi polygon in next round, then the sensor will not perform this step.
4.2.4 Neighbor’s Repulsion (NR) Scheme
In the NR scheme, we address the case in which no neighbors have any overlap with the sensor. When we have a higher density of random deployment in the target region, the uncovered area is usually very small in the neighborhood of the sensor; in other words, the sensor increases its chance that it has an increased number of neighbors. In the above schemes, if a sensor gets the same force, then the sensor has substantial movement as a result of the combined forces. In this case, the sensor moves away from the original overlapping area, but the sensor may be able to move to cover the same area as other neighbors, as shown in Figure 4.5.
Sensor has overlapping area
Sensor doesn't have overlapping area
Sensor gets the forces
60 60
(a) Initial deployment in the case
(b) Resultant of forces (c) After moving
Si Si Si
Figure 4.5 The case in which a sensor still has an overlapping area after moving
Si
Gi
Sj
Gj
Auxiliary point
Directional sensor
The direction of Virtual force Sj exert on Si .
Figure 4.6 The case involving a neighbor’s repulsive force
The calculation of the sensor’s repulsive force is expressed as
(6)
Each sensor keeps the force temporarily when each sensor does all operations over.
4.2.5 Moving Algorithm
In this subsection, we propose a means to improve the coverage of the virtual force algorithm to maximize sensor coverage. Instead, of using the CPA, CPC, VPPC, and NR schemes to guide the directional sensor movements toward a new position, the virtual force scheme employs the repulsive force between auxiliary points that surround the sensing sector of the directional sensor, and thereby use the neighboring directional sensors as a basis of movement.
The fundamental idea of our virtual force scheme is to repel sensor nodes from one another such that sensor nodes will spread from dense to sparse areas. We assume APCij is the set of auxiliary points of Si that are covered by its neighboring sensor Sj. is the
vector from Gj to . is the distance between S j and APik. SA is the size of the sensing sector. is the size of the overlapping area of Si (i.e., the sensor collected information of each neighbor’s location, sensing angle, and direction, and the sensor can use TIS for simulation in a two-dimensional array).
If sensor Si has overlapping coverage with Sj (i.e., ), then we can divide the problem into the following two cases for overlapping coverage: (1) an auxiliary point of Si is covered by Sj; and (2) Si without any auxiliary points is covered by Sj. If sensor Si does not have overlapping coverage with Sj (i.e., ), then Si does not do anything. The virtual force exerted by Sj on Si is denoted as FRij, and the repulsion model is defined as
if sensor hasoverlappingcoveragewith and if sensor has overlappingcoveragewith
and and
Next, we survey the case of overlapping coverage as in [33]. If a sensor has more overlapping area, the sensor cannot move away from the overlapping area. We define threshold to raise the accuracy of direction. If a sensor’s threshold is greater than in the first case above, the force of the second case is substituted for the first case. If a sensor’s threshold is less than in the first case above, the force of the first case does not change. We assume the threshold of Si is i; the calculation of a sensor’s threshold is then expressed as
i = (7)
We also propose a move-back scheme to prevent the sensor’s sensing region from being outside the target region. If the auxiliary point of a sensor is out of the target region,
the sensor should move to the new position until its auxiliary point is located on the boundary of the target region, as shown in Figure 4.7. In Figure 4.7(a), we observe that auxiliary points A and B are outside the target region, so we let these auxiliary points inside the target region. By doing so, the directional sensor will find the farthest auxiliary point to boundaries b1 and b2. APi2 is the farthest point to boundary b1 and APi3 is the farthest point to boundary b2. The directional sensor will therefore move down the distance between auxiliary point APi2 and boundary b1, and auxiliary point APi2 will lie just on the boundary of the target region, as shown in Figure 4.7(b).
Auxiliary point APi3 is also outside the target region, so the directional sensor will move right the distance from APi3 to boundary b2. APi3 is also located on the boundary of the target region, as shown in Figure 4.7(c). In Figure 4.7(d), the sensing sector of the directional sensor will fall inside the target region after it has moved to the new position.
Target region New position to move Directional sensor Moving direction
APi3
APi2
(a) Before move-back scheme.
(b) Move virtual point A back to the target
region.
(d) After move-back scheme.
b1 b1 b1
b2
b2
(c) Move virtual point B back to the target
region.
b2
b1
APi2
APi3 APi3
APi3
APi2
APi2
Figure 4.7 Example of the move-back scheme in which a sensor (shown in (a)) moves down (shown in (b)) and right (shown in (c)) to result in location (d)
Improving the coverage of the virtual force algorithm will stop when it achieves the maximum number of rounds of the algorithm. The complete procedure of our Virtual Force
Moving algorithm is shown in Figure 4.8.
Improving the coverage of the virtual force algorithm Notation:
APCij, APik, , FR ij, CR, , : defined above Ni: the neighbor of sensor Si
moving vector of Si VPPC of Si
NR of Si
Max_Round: pre-defined maximum number of rounds Procedure:
(1) Enter discovery phase:
(1.1) set timer to be discovery interval and enter Moving phase upon timeout (1.2) broadcast hello after a random time slot
(2) Enter moving phase:
(2.1) set timer to be discovery interval and enter discovery phase upon timeout (2.2) Compute the overlap and useful neighbors
(2.2.1) for each Si, compute and i
(2.3) Compute the force with ≠ 0 of Si
(2.3.1) (2.3.2) for each Sj in Ni
If sensor Si has overlapping coverage with Sj and APCij ≠ Ø and APik APCij
If i
FRij = = + FRij
Else
APCij Ø Endif
Endif
If sensor Si has overlapping coverage with Sj and APCij Ø and APik APCij
FRij =
= + FRij
Endif
(2.3.3) for each Si, find and = + +
(2.3.4) perform move-back scheme End for
(2.4) Max_Round --
(2.5) Done when Max_Round =0
Figure 4.8 Procedures of our ICVFA algorithm