Mobile sensor nodes (MSNs) are equipped with locomotive can move around after having been deployed. They are equipped with limited energy. A large portion of energy is drained during the traversal. In order to extend the life time of a MSN, the traveling distance must be minimized. Region of interest (ROI) is covered with multiple MSNs using coverage based pattern movement. When a group of MSNs are deployed to cover a given ROI, all the deployed MSNs should travel an approximately equal distance. Otherwise, the MSN which travels longer distance depletes more energy compared to the MSN which travels a shorter distance. In this work we show that, ROI partition plays great role in hole free coverage and makes the MSNs have optimized movement cost with fault tolerant support.
Mobile sensor nodes are a particular class of wireless sensor nodes. They are equipped with locomotive. Some classes of MSNs are equipped with location identification devices along with locomotive. Like static wireless sensor nodes MSNs are also encapsulated with sensor unit, power supply unit, data processing unit, data storage, and data transmission units [
In mobile sensor networks sensor node mobility plays a key role in the execution of the application. Mobile sensor networks are extremely valuable in situations where traditional deployment mechanisms fail or not suitable.
Static sensor nodes do not move once they are deployed. Predetermined positioning of static wireless sensor nodes is flexible only when the sensor field is small and human friendly. Placement of static wireless sensor nodes might not be possible in large and in situations like hazardous and disastrous fields. MSNs that can move around after having been deployed are the most suitable in such situations.
An algorithm or a mechanism which helps a single or a group of MSNs to participate in predefined coverage related operations in sensor field is defined as mobile traversal algorithm (MTA). In this work sensor field to be monitored is rectangular in shape; it is refereed as region of interest, which is geographically partitioned into polygons in tessellation fashion. MSNs are moved along the vertices or the center points of the polygons.
Physical phenomenon within the boundaries of the polygon is monitored by the MSNs which are placed at the vertices of it or at the center points of the polygons. On completion of predetermined operations at a polygon, MSNs are moved to occupy the vertices or center points of the horizontally or vertically adjacent polygon for predetermined operations. This procedure is repeated till the entire ROI is covered.
Research in the area of coverage pattern based movement using MSNs is very useful in detection of objects in hazardous sensor fields, where human intervention might be not possible.
This paper is organized as follows. In Section
Coverage hole free placement and MSNs traversal patterns are two different branches of research. ROI partition makes the MSNs have coverage hole free sensing and an efficient traversal pattern makes the MSNs deplete less amount of energy during their traversal. An efficient MTA should focus on both the aspects.
Sensing area of sensor is circular area, limited by the radius of sensing range
This problem is very similar to wellknown traveling salesman problem (TSP), where the traveling salesman visits each city exactly once starting from source and reaches destination with shortest path. On completion of predetermined operation at a polygon, MSNs are moved to occupy their positions at neighboring polygon. Each polygon here can be represented as city and the total distance traveled by the MSNs to occupy their positions in neighboring polygon can be treated as edge weight.
Khan et al. [
To two cover a sensor field, Purohit et al. [
To two cover a rectangular ROI, D’Souza and Santoshi [
To provide dynamic degree of coverage D’Souza and Santoshi [
MTA described in [
Mobile sensor nodes are electromechanical devices, combination of many subcomponents like locomotive, location devices, sensing unit, and communication unit, battery/power management. A MSN may get failed after initial deployment due to failure of any one of the components or unexpected environmental conditions. A MSN with any failed component is not suitable for further traversal, even though its remaining components are functioning well. A MSN which fails to coordinate with remaining deployed MSNs is treated as failed node [
The proposed MTA focuses on both the aspects namely coverage hole free sensing and optimized movement cost. In this work a rectangular ROI of length
Mobile traversal algorithm is similar to finding the Hamiltonian path in ROI. Single vertex or a set of vertices in ROI represented as a vertex in Hamiltonian path.
The problem is to define a triangular grid graph. Let
In a triangular grid graph
If
If
In order to form
An equilateral triangle can be formed by connecting first two vertices of bottom row and one from its adjacent row. To make a disjoint equilateral triangle, two vertices are chosen which are horizontally adjacent to the previous triangle and connected with third vertex which is adjacent to both the vertices. Repeating this procedure
Triangular grid graph.
According to Wang [
Triangle based MSNs traversal pattern.
The number of equilateral triangles which are crosssectionally adjacent to one other, in same orientation, is called trihexagonal column denoted as THC. As shown in Figure
In the row order, the equilateral triangles are inverted alternately. The trihexagonal rows are numbered as THR 1 to THR
Traversal pattern with three active MSNs will be referred to as triangle based MSN traversal pattern. In triangle based MSN traversal pattern three active MSNs are traversed along the vertices of equilateral triangles. In this traversal pattern three MSNs form an equilateral triangle at every sensing point. Sensing point is a location where single or group of MSNs participate in predefined operations like coordination, sensing, and other necessary activities. The MSN movements are controlled using the method of triangulation or any suitable method supported by the GPS and location identification devices or signal from static beacons.
Source or initial sensing points in triangular pattern is
In Figure
In a triangular grid graph
In column order equilateral triangles are in the same orientation. Length of side of the equilateral triangle is
In the row order successive equilateral triangles are in inverted orientation. We define diagonal distance as the distance between a vertex of equilateral triangle and its corresponding vertex in horizontally adjacent equilateral triangle. This distance is
Figure
Triangular grid graph redrawn as rectangular grid graph.
The traveling salesman problem (TSP) is defined as follows. Let
This forms a rectangular grid graph and it is connected. The
A MSN may get failed after initial deployment. A failed MSN is not suitable for further traversal. A MSN which fails to coordinate with remaining deployed MSNs is treated as failed node. A MSN may fail at any location and at any row, including the first sensing point and the last sensing point of a row in the ROI.
The coordinates of the positions of MSNs can be calculated from the satellite using GPS if the sensor field is open. If the sensor field to be monitored is closed, signals from static beacons can be used to identify the location [
Every traversal pattern has starting and ending sensing points. Starting point is represented as
In a triangular grid graph
If
If the total number of equilateral triangles in a row is odd, then total number of vertices is also odd. The odd number of vertices is partitioned into disjoint sets of each set containing 2 vertices. This makes a single vertex remain unconnected along with some disjoint sets.
As shown in Figure
Triangular grid graph partitioned as subgraphs.
Traversal pattern with two active MSNs will be referred to as line based traversal pattern. As shown in Figure
Line based MSNs traversal pattern.
We define the resulting graph as
In graph
We apply the concept of graph theory for identifying a Hamiltonian path in graph
Hamiltonian path in graph
In the line based traversal pattern starting point is
In a triangular grid graph the horizontal and diagonal distance between two adjacent vertices is the same.
Triangular grid graph is made up of equilateral triangles. Distance between any two vertices is the same. Hence the theorem is proved.
As shown in Figure
Failure of MSN in line based pattern.
Let the location where a single MSN fails in line based pattern be
Total distance
For experimental purpose, a rectangular ROI of length
The ROI is covered using
To provide complete coverage along the boundaries of ROI, the values of
In the first phase of simulation no failure case is considered. In triangle based traversal pattern all three MSNs are used to cover the entire ROI. In line based pattern two active MSNs are used to cover the ROI and in case of vertex based traversal single MSN is deployed to cover the entire ROI.
In triangle based traversal pattern, the ROI contains
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16) Scan ROI in Line based traversal pattern
(17) Call : Traversal pattern
(18)
(19)
(20)
(21)
(22)
(23)
(24) Scan ROI in Vertex based traversal pattern
(25) Call : Traversal pattern
(26)
(27)
(28)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
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(12) Perform coordination and Predetermined operations at sensing point
(13)
(14)
(15) Move MSNs to next column
(16)
(17)
(18)
Simulation is carried on different values of sensing range
The values of total distance traveled by the MSNs in different traversal patterns are tabulated in Table
Total distance traveled by MSNs in triangle, line, and vertex based traversal.







17.38  609286  356568  41.48  352143  1.24 
19.76  531844  311543  41.42  307129  1.42 
21.74  485068  284535  41.34  280129  1.55 
24.17  438289  257525  41.24  253130  1.71 
27.20  391506  230513  41.12  226130  1.90 
31.11  344719  203497  40.97  199131  2.15 
36.34  297924  176475  40.77  172132  2.46 
39.67  266669  158427  40.59  154099  2.73 
48.56  219840  131383  40.24  127093  3.27 
62.60  172975  104313  39.69  100084  4.05 
From Table
As the sensing range increases the total distance traveled by the MSNs in line based traversal shows a decrease in distance traveled compared to triangle based traversal pattern.
When the value of sensing range is 17.38 units, vertex based traversal shows a reduction of 1.24
From Table
Total distance traveled by MSNs in triangle, line, and vertex based traversal.
The relation between
From (
All MSNs have equal amount of battery power at the time of deployment. In Table
Energy depletion, residual energy in different traversal patterns of MSNs.









17.38  352143  178284  50.63  49.38  230275  57.73  42.27 
19.76  307129  155771  50.72  49.28  177281  57.72  42.28 
21.74  280129  142267  50.79  49.21  161689  57.72  42.28 
24.17  253130  128762  50.87  49.13  146096  57.72  42.28 
27.20  226130  115256  51.97  49.03  130502  57.71  42.29 
31.11  199131  101748  51.50  48.90  114906  57.70  42.30 
36.34  172132  88237  51.26  48.74  99308  57.69  42.31 
39.67  154099  79213  51.40  48.60  88889  57.68  42.32 
48.56  127093  65691  51.69  48.31  73280  57.66  42.34 
62.60  100084  52156  52.11  47.89  57658  57.61  42.39 
When sensing range is 17.38 units, distance traveled by a single MSN in triangle based traversal is 230275 units, and in line based traversal is 178284 units, in vertex based traversal is 352143 units. At this value of sensing range, MSN in triangle based traversal shows a reduction in distance traveled by 50.63% and MSN in line based traversal shows a reduction of 57.73% in total distance traveled compared to the total distance traveled by the MSN in vertex based traversal.
Similarly at this value of sensing range, residual energy in MSN in triangle based and line based traversal is 49.38% and 42.27%, respectively. This shows that MSN which travels in triangle based traversal depletes less amount of energy during traversal and saves more energy compared to two other patterns.
When the sensing range is 62.60 units, distance traveled by the MSN in triangle, line, and vertex based traversal is 57658 units, 52156 units, and 100084 units. At this stage MSN in triangle based traversal shows a reduction of 52.11% in total distance traveled and in line based traversal it shows a reduction of 57.61% in total distance traveled compared to vertex based traversal. At this value of sensing range, residual energy in MSN in triangle and line based traversal is 47.89% and 57.61%, respectively.
On an average MSN in triangle based traversal shows 51.15% of reduction in total distance traveled and energy saving of 48.69% and in line based traversal it shows a reduction of 57.69% in total distance traveled and an energy saving of 42.19% compared to vertex based traversal.
In second phase of simulation MSNs with failure are considered. Total number of vertices in ROI is equally divided into three groups. First set of vertices are traversed using three MSNs by triangle based traversal pattern. Second group of vertices are covered using two MSNs in line based traversal pattern and remaining ROI is covered by single MSN in vertex based traversal pattern.
When the sensing range is 17.38 units, total distance traveled by the MSNs in tolerant traversal is 558391. At this value of sensing range the total distance traveled by the MSNs in triangle based traversal pattern is 609286 and in line based traversal is 356568.
When the sensing range is 62.60 units, total distance traveled by the MSNs in triangle based traversal is 172975 units. At this value of sensing range the total distance traveled by the MSNs in fault tolerant traversal is 160482.
Total number of vertices in ROI is
Total distance traveled by the MSNs in fault tolerant traversal.







17.38  11700  203327  237672  117391  558391 
19.76  8976  177281  207650  102422  487354 
21.74  7440  161624  189640  93464  444729 
24.17  6048  146168  171627  84390  402186 
27.20  4800  130502  153612  75439  359554 
31.11  3696  114813  135592  66502  316908 
36.34  2736  99417  117566  57398  274382 
39.67  2244  88889  105526  51458  245874 
48.56  1512  73426  87476  42393  203295 
62.60  924  57470  69397  33614  160482 
Distance traveled by the MSNs in fault and nonfault tolerant traversal.
Experimental results of fault tolerant and nontolerant traversal are tabulated in Table
Comparison of distance traveled by the MSNs in triangle based traversal, fault tolerant traversal, and line based traversal.







17.38  609286  558391  8.35  356568  36.14 
19.76  531844  487354  8.37  311543  36.07 
21.74  485068  444729  8.32  284535  36.02 
24.17  438289  402186  8.24  257525  35.97 
27.20  391506  359554  8.16  230513  35.89 
31.11  344719  316908  8.07  203497  35.79 
36.34  297924  274382  7.90  176475  35.68 
39.67  266669  245874  7.80  158427  35.57 
48.56  219840  203295  7.53  131383  35.37 
62.60  172975  160482  7.22  104313  35.00 
On the other hand, when the sensing range is 17.38 units, line based traversal shows a reduction of 36.14%. At this value of sensing range the value of fault tolerant traversal is 558391 and line based traversal is 356568. When the sensing range is 62.60 units, total distance traveled in fault tolerance traversal is 160482 units and in line based traversal is 104313 units. It shows a reduction of 35.00%. As the sensing range increases from 17.38 units to 62.60 units, the percentage of reduction decreases from 36.14% to 35.00%.
The total distance traveled by the MSNs in fault tolerant system is the sum of the distance traveled by the MSNs in triangle based travel
Let
The relation between the
But from the experimental results tabulated in Table
In fault tolerant coverage based pattern movement with three MSNs, the energy depletion per MSN is less, due to traversal pattern designed for uniform energy depletion. But the total distance traveled by all the MSNs is more. When a single MSN is considered, the energy depletion is more and it travels less distance compared to other traversal patterns. In vertex based traversal or single MSN based traversal the fault tolerance is not applicable. Fault tolerance is applicable only when the deployed MSNs are more than one. In realtime systems, application with fault tolerance is highly preferred.
In this work ROI is divided into regular hexagons for coverage based pattern mobile sensor node movement. MSNs are moved along the centre point of the regular hexagons. Total distance traveled by the MSNs in fault tolerant and nonfault tolerant traversal is calculated. The value of total distance traveled by the MSNs in fault tolerant traversal is less compared to the total distance traveled by the MSNs in nonfault tolerant system. In the future we propose to extend our research to random deployment based mobile sensor node movement.
In this section we have shown two algorithms in abstract form. Total number of MSNs which are in working condition are referred to as
Value of number of active MSNs is updated regularly and returned on the event of failure of MSNs as shown in Algorithm
Failure locations, starting points, and ending points of traversal patterns are controlled using internal algorithms with the help of signals from static beacons or GPS. Conversion of coordinates from triangle based traversal pattern to line based traversal pattern and from line based traversal pattern to vertex based traversal patterns is done implicitly.
The authors declare that there is no conflict of interests regarding the publication of this paper. Authors of the paper do not have a direct financial relation that might lead to a conflict of interests for any of the authors.