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Article

Mobile Anchor Route Scheduling with an Iterative Sensor Positioning Algorithm in Wireless Sensor Networks

by
Shih-Chang Huang
* and
Yi-Kai Wang
Department of Computer Science and Information Engineering, National Formosa University, Yunlin 632, Taiwan
*
Author to whom correspondence should be addressed.
Appl. Sci. 2023, 13(1), 22; https://doi.org/10.3390/app13010022
Submission received: 3 October 2022 / Revised: 15 December 2022 / Accepted: 16 December 2022 / Published: 20 December 2022
(This article belongs to the Special Issue Advances in Wireless Communication Technologies)
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Abstract

:
Locating deployed sensor nodes in wireless sensor networks is a crucial issue for general environmental monitoring applications. Most current positioning technologies use stationary anchors to compute the positions of sensor nodes. However, placing the static anchors in the monitoring area is not a trivial job, especially in hazardous environments. Applying a mobile anchor is a good solution. This paper proposes an iterative sensor positioning (ISP) algorithm that uses the neighboring-positioned nodes to assist the mobile anchor in locating the unknown nodes. In addition, a mobile anchor route scheduling method, named MARS, is also proposed to shorten the traveling distance of the mobile anchor. MARS uses a projection mechanism and the ISP algorithm to reduce the traveling distance. Simulation results indicate that MARS can reduce the traveling distance of the mobile anchor by about 21–49% compared with the SCAN method. In the unfavorable scenario of the projection mechanism, MARS still provides the mobile anchor with a traveling distance that is 34% less than the SCAN method. The mobile anchor, when applying the MARS method, is able to locate all unknown nodes in only 45% to 70% of the traveling distance required by the one using the SCAN method in the scenario favorable to the ISP algorithm, while still reducing the traveling distance by 15% to 21% in the scenario unfavorable to the ISP algorithm. In the non-ideal signal scenario, the positioning size of the positioning area of the ISP algorithm when using the MARS method is about 53% to 74% less than when using the Triangulation algorithm.

1. Introduction

Many wireless positioning technologies have been proposed for locating deployed sensor nodes. These positioning techniques can be classified into range-based methods and range-free methods. Well-known range-based methods include Received Signal Strength Indicator (RSSI) [1,2,3,4,5,6,7,8], Time of Arrival (ToA) [9,10,11], and Time Difference of Arrival (TDoA) [12,13,14]. Widely used range-free methods include Point in Triangle (PiT) [15,16,17], Approximate Point in Triangle, APiT [15,16,17], DV-Hop [18,19,20], and Angle of Arrival (AoA) [21,22,23,24]. All RSSI, ToA, TDoA, and DV-Hop methods use the strength of the received signal to estimate the distance between the sender and receiver. The AoA needs specific hardware to identify the angle of the incoming signal.
The above methods use stationary reference points to perform wireless positioning. Placing the unmoving reference points is limited by the terrain status. Placement is inflexible and difficult to maintain when the deployment area is in a hazardous location. Additionally, estimating the distance on the basis of the received signal strength may introduce errors when the environment includes radio interference. The accuracy when employing this type of wireless positioning is dependent on the number of reference points, because positioning the target using wireless signals depends heavily on internal factors such as path loss, the shadowing effect, the Doppler effect, and multipath delay. However, the cost of one reference point is much higher than that of one deployed sensor. Increasing the number of reference points therefore also implies increased costs when building the reference points.
Therefore, some studies have used a mobile anchor traveling randomly within the deployment area to locate the unknown sensor nodes [25]. Due to the fact that the sensors can appear anywhere, the mobile anchor needs to be able to guarantee that its radio signal on its traveling route is able to cover all locations in the deployment area. The most typical traveling route is the straight-line mode, which scans the deployment area in order to obtain three non-collinear positions in order to locate the sensor [26,27,28].
A localization algorithm with a mobile anchor node based on trilateration (LMAT) [29] is proposed to solve the problem of collinear points. LMAT uses a mobile anchor moving in the scan mode with a trilateration trajectory in the deployment area. The mobile anchor periodically broadcasts its current position in order to locate the unknown nodes. A similar method combines the LMAT and SCAN algorithms, referred to as SLMAT [30], to achieve a compromise between positioning accuracy and energy consumption. Another scanning method that can be used to decrease the length of the movement path of the mobile anchor is SCAN-ET [31]. SCAN-ET firstly divides the deployment area into several smaller zones. The mobile anchor then uses the SCAN method to determine its travel path in each of the zones. If any of the smaller zones contains unknown nodes, the mobile anchor will move along the sides of the triangle in this zone, broadcasting messages at each of the vertices of the triangle. Methods similar to the SCAN methods, but with variations in the movement path, include the H-curve [32], M-curves [33], NHexCurves [34], and Z-curve [35] methods.
The movement path of the mobile anchor is designed to follow a curved path [36,37] in order to solve the problem of the collinearity of the three points. The curved path has a minimal number of straight-line segments, resulting in much less collinearity occurring during localization, as, for example, when using the CIRCLES movement trajectory, the S-CURVES movement trajectory, or the spiral trajectory.
The above methods depend heavily on the mobile anchor to locate the unknown nodes. The traveling distance of the mobile anchor can be reduced if the unknown nodes whose positions have been determined are able to act as reference points to help the mobile anchor dynamically change its subsequent movement direction, decreasing the traveling distance required compared to the SCAN trajectory. Therefore, this paper proposes an iterative positioning algorithm (ISP) that uses the positioned nodes as reference points in cooperation with the mobile anchor to locate the unknown nodes. A mobile anchor route scheduling method using the ISP algorithm, named MARS, is also proposed to reduce the traveling distance of the mobile anchor.
The rest of this article is structured as follows. Section 2 presents related works that use the mobile anchor to locate the unknown nodes. Section 3 describes in detail the mechanism by which the ISP algorithm and the MARS method dynamically determine the mobile anchor’s next movement direction to reduce the traveling distance of the mobile anchor. Section 4 presents the results of the evaluation of the proposed method in comparison with existing methods. Finally, Section 5 presents the conclusions.

2. Related Works

The mobile anchor needs to approach an unknown node to receive the radio signal while performing the positioning. Ssu et al. had the mobile anchor use the random waypoint model to travel within the deployment area [25]. However, this method is inefficient when the size of the deployment area in which positioning is to be carried out is known in advance. Himanshu et al. used a mobile anchor to locate the target nodes on the basis of received signal strength (RSS) and angle of arrival (AoA) [38]. They also applied artificial intelligence technologies such as particle swarm optimization (PSO), hybrid PSO (HPSO), and the firefly algorithm (FA) to optimize the positioning of the target nodes. However, their study was focused on the positioning of the targets, and it did not address methods for scheduling the route of the mobile anchor.
The simplest way for the mobile anchor to efficiently scan the whole area is to schedule a route with less overlapping of the radio signal. To this end, Koutsonikolas et al. proposed the SCAN method [26]. In this method, a single mobile anchor travels along a single dimension, for example, in the direction of the x-axis or y-axis. Let the communication range of the sensor nodes be R. The resolution of the mobile anchor’s SCAN trajectory, which indicates the distance between two neighboring segments, should be 2R at most. Furthermore, all sensors will be able to receive the beacon signal emitted by the mobile anchor. When an unknown node receives the mobile anchor’s beacon signal from more than three different locations, range-based positioning technology is applied to compute the location of the unknown node.
Using the SCAN trajectory introduces the problem of beacon collinearity in cases when the resolution is greater than the transmission range. To solve this problem, Koutsonikolas et al. also proposed the DOUBLE SCAN trajectory and the HILBERT trajectory. The DOUBLE SCAN trajectory requires the mobile anchor to scan the deployment area along both the x-axis and y-axis directions. This comes at the cost of doubling the traveling distance required by the mobile anchor. The HILBERT trajectory divides the two-dimensional space into 4n squares. The centers of those squares are connected using 4n line segments. The length of each line segment is the edge length of one square. This comes at the cost of the additional time required to schedule the connection of these line segments and the additional travel distance compared to when using SCAN.
To solve the problem of three-point collinearity, Ou et al. varied the resolution of the mobile anchor’s SCAN trajectory to increase the chances that the beacon signal would be overhead [28]. Renuka et al. also proposed a similar method with the use of two mobile anchors moving simultaneously in opposite directions in order to scan the deployment area [27]. Han et al. proposed a localization method for the mobile anchor based on trilateration (LMAT) [29]. LMAT uses a mobile anchor that scans the deployment area in a trajectory shaped like an equilateral triangle and periodically broadcasts its current position to locate the unknown nodes. Han et al. proposed an enhanced method employing the localization method for the mobile anchor based on trilateration (LMAT) and the SCAN algorithm, referred to as SLMAT [30], in order to reduce the size of the triangular trajectory used by LMAT. SLMAT computes the visiting points using LMAT in order to guarantee that the regular triangle formed by the beacons is able to cover all of the unknown nodes. These points will be collinear, and SLMAT postpones the positioning operation of the mobile anchor until the mobile anchor is moving on the neighboring line segments.
Javad et al. proposed the Z-curve method [35], which is an extension of the HILBERT method. When using the Z-curve method, the deployment area is divided into multiple squares according to the radio range. Then, each square is cut into four sub-squares. The centroid of these four sub-squares creates a path in the form of the letter Z. The paths between the squares are linked using a letter Z Hamilton path as the trajectory for the mobile anchor. Many similar methods have been proposed in which the deployment area is divided into small areas at the beginning and then different linking strategies are used in the four sub-squares, for example, the M-curves [33], H-curve [32], and NHexCurves [34] methods.
Tian et al. proposed an algorithm that integrates the SCAN trajectory into a structure with multiple squares, referred to as SCAN-ET [31]. SCAN_ET divides the deployment area into several small zones. The mobile anchor uses the SCAN method to travel through every zone. If the mobile anchor detects any unknown nodes, it moves along the edges of the triangle in this zone and broadcasts messages at each of the vertices of the triangle.
Some studies have scheduled the trajectory of the mobile anchor as a curve in order to solve the problem of three-point collinearity [36,37]. The curve path has a minimal number of straight-line segments, resulting in much less collinearity occurring during the localization process. For example, the CIRCLES trajectory, proposed in [27], consists of a sequence of concentric circles with their center in the deployment area. One potential problem with the use of CIRCLES is coverage of corners. If the deployment area is a square, CIRCLES is not able to effectively cover the four corners. Thus, the S-CURVES trajectory schedules an ‘S’ curve to mitigate the problem of corner coverage. Hu et al. proposed the mobile anchor centroid localization (MACL) method [37]. MACL places the mobile anchor in the center of the deployment area and drives it in a spiral trajectory. MACL also has the problem of corner coverage.
Li et al. proposed a DeteRministic bEAcon Mobility Scheduling (DREAMS) algorithm [39]. The mobile anchor first locates a sensor by random movement. Subsequently, depth-first traversal (DFT) is performed. The mobile anchor uses the track of the DFT graph to locate unknown nodes. The sensors are used to create the DFT in the DREAMS algorithm. However, this method does not take into consideration the communication range of the sensors. Therefore, it is unsuitable in scenarios where the communication range is limited.
Considering that unknown nodes may be located anywhere in the deployment area, most of the above methods schedule the trajectory for the mobile anchor in accordance with the information on the size of the deployment area in which positioning is to be performed and the communication range of sensors. In this paper, a trajectory scheduling method is proposed in which the nodes whose locations have already been determined are used to help the mobile anchor dynamically change the default scheduled trajectory used in the SCAN method, thus reducing the traveling distance.

3. Proposed Method

This section gives details on the proposed methods. The preliminaries and assumptions are presented in the first subsection. Next is the methodology of the iterative sensor positioning algorithm. The main concept of the mobile anchor route scheduling (MARS) method is presented in the third subsection. In the last subsection, a non-ideal signal scenario is evaluated.

3.1. Preliminaries and Assumptions

This paper assumes that the unknown nodes and the mobile anchor have the same communication range, R. The deployment area is a rectangle with N unknown nodes placed in it. The unknown nodes are stationary and can exchange messages with their neighbors if they are within the communication range. A mobile anchor, which can accurately get its location, will travel in the deployment area to position all unknown nodes. To simplify the illustration of the main idea of the proposed methods, we assume the radio signal strength can accurately map to the distance. In the latter, we release this constraint to evaluate the positioning accuracy.
The mobile anchor continuously broadcasts the beacon signal while it moves. When the unknown node enters the radio range of the mobile anchor for the first time, it commences periodically replying to the beacon signal from the mobile anchor, and it will continue to do so until the beacon signal disappears. The mobile anchor continuously monitors the unknown node and records the locations at which reply messages were received for the first time and the last time from the unknown node. As shown in Figure 1, the mobile anchor M records the positions of A and B, which are similar to the reference points used in triangulation. The distances between these two positions and the actual location of the unknown node z correspond to the communication range, R. On the basis of these two points, it can be determined that node z must be at one of the locations z or z’. To eliminate the fake point z, the mobile anchor needs to change its movement direction to solve the problem arising from the collinearity of the three points.

3.2. Iterative Sensor Positioning Algorithm

This paper proposes an iterative sensor positioning (ISP) algorithm to be used in cooperation with the mobile anchor. The ISP uses the positioned nodes as reference points to reduce the positioning load of the mobile anchor. Reference points for each unknown node can quickly be obtained from position-aware neighboring nodes, instead of needing to obtain every reference point from the mobile anchor. Algorithm 1 presents the pseudocode of the ISP algorithm. Every unknown node exchanges its position information with its neighbors within communication range. When an unknown node receives the beacon signal from the mobile anchor, it reports its neighbors’ position information to the mobile anchor. The mobile anchor uses the trilateration method to compute the location of the unknown node. The corresponding pseudocode is in lines 23 to 33 in Algorithm 1. The mobile anchor also records the positions at which each unknown node receives the beacon signal from the mobile anchor for the first time and the last time. The corresponding pseudocode is in lines 35 to 40 in Algorithm 1.
There are three possible cases in which the ISP algorithm is able to directly locate the unknown nodes. In the first case, the unknown node X already has three position-aware neighbors, A, B, and C. In this case, the positions of these neighbor nodes are used to compute the location of the unknown node X via triangulation, as shown in Figure 2a. The mobile anchor is able to remove the node from the positioning list. The corresponding pseudocode is in lines 2 to 5 in Algorithm 1. In the second case, the unknown node X has two position-aware neighbor nodes, A and B. The mobile anchor only has to record one non-collinear reference point p for X in this case. Then, the positions A, B, and p are used to compute the location of X via triangulation, as shown in Figure 2b. The corresponding pseudocode is in lines 6 to 9 in Algorithm 1.
In the third case, the unknown node has just one position-aware neighbor, A. Generally, the mobile anchor has to record two reference points, {p, q}, before it applies triangulation to compute the location of X. The corresponding pseudocode is in lines 11 to 14 in Algorithm 1. However, the ISP algorithm adds the following strategy in order to speed up the positioning task. When the mobile anchor moves along the path of the current trajectory, if it detects position-aware neighbor A before the unknown node X, the mobile anchor is able to directly position the unknown node X without the use of the second reference point. The corresponding pseudocode is in lines 15 to 22 in Algorithm 1.
Algorithm 1. The ISP algorithm.
M: Mobile anchor.
R: Communication range.
U: Set of unknown nodes.
W: Set of nodes that have been discovered but not positioned.
D: Set of positioned nodes.
(g, g’) = IntSecPoints(r1, r2): Find intersection points (g, g’) from reference points {r1, r2}.
Lo(x) = Trilateration(r1, r2, r3): Trilateral positioning function uses the reference points {r1, r2, r3} to get the location of node x.
M.pos: The current location of the mobile anchor.
dS(gi, gi’): A function selects the {gi, gi’}, which is on the side of the movement direction of the mobile anchor.
ISP(U,W,D)
1. When M detects node ui ∈ U or received information from detected nodes{
2.  If ui has more than 3 neighbors {da, db, dc,...} ∈ D
3.   //three position-aware neighbors
4.   Lo(ui) = Trilateration(Lo(da), Lo(db), Lo(dc));
5.   D←D∪{ui}; U←U \{ui};
6.  If ui has 2 neighbors {da, db}∈ D
7.   //two position-aware neighbors
8.   Lo(ui) = Trilateration(Lo(da), Lo(db), M.pos);
9.   D←D∪{ui}; U←U \{ui};
10.  If ui has 1 neighbors {da} ∈ D
11.    If  ui ∈ W  // ui has been detected by M before
12.    ui.p2 = M.pos;
13.    Lo(ui) = Trilateration( Lo(da), ui.p1, ui.p2);
14.    D←D∪{ui}; W←W \{ui}; U←U \{ui};
15.    Else
16.    If M detects node da before reach M.pos and Lo(da) != M.pos
17.     {ui.g, ui.g’} = IntSecPoints(da, M.pos);
18.     Lo(ui) = dS(ui.ga, ui.gb);
19.     D←D∪{ui}; U←U \{ui};
20.    Else
21.     ui.p1 = M.pos;
22.     W ← W∪{ui};
23.  If ui has 0 neighbors
24.   If  ui ∈ W
25.     If ui has two reference points
26.     Lo(ui) = Trilateration(M.pos, ui.p1, ui.p2);
27.     D←D∪{ui}; W←W \{ui}; U←U \{ui};
28.     Elseif ui has one reference point
29.     ui.p2 = M.pos;
30.     {ui.g, ui.g’} = IntSecPoints(ui.p1, ui.p2);
31.   Else
32.    ui.p1 = M.pos;
33.    W←W∪{ui};
34. }
35. When a point g* ∈ u*|u*∈W and g*∈{u*.g, u*.g’} and |g*M.pos| = R {
36.  If g* = u*.g and M detects the signal of node u*
37.   Lo(u*) = u*.g;
38.  Else
39.   Lo(u*) = u*.g’;
40.  D←D∪{u*};W←W \{u*};U←U \{u*};
41. }
For example, the mobile anchor M finds node A before node X while it moves right, as shown in Figure 2c. The first time the mobile anchor detects X at p, the position of A and p can indicate two possible locations: t, the actual location of X, and a false one, f. Here the mobile anchor can directly identify that t is the actual location, rather than f, because M did not detect node X before it arrived at the location p. However, if M does not detect the signal of neighbor A before that of the unknown node X, M will still need to record two reference points.

3.3. Mobile Anchor Route Scheduling Method (MARS)

In this section, the proposed mobile anchor route scheduling (MARS) method based on the ISP mechanism is described in detail. MARS includes a projection mechanism and works in cooperation with the ISP algorithm to improve on the SCAN method. MARS consists of a movement phase and a projection phase. The mobile anchor receives messages from the detected unknown nodes and records the area covered by its radio signal during the movement phase. Let Lu and Ld be the upper and lower boundaries of the covered area. In the beginning, the mobile anchor moves from left to right, as in the SCAN method. When the mobile anchor reaches the right boundary of the deployment region, the Lu of the covered area corresponds to the upper boundary of the deployment region. The Ld of the covered area is a line parallel to Lu. The distance between Lu and Ld is 2R.
During the projection phase, the unknown nodes detected during the movement phase are projected vertically onto the line Ld. Because the mobile anchor always employs a straight movement trajectory when using the SCAN method, this results in the collinear problem of the three reference points, whereby two possible points are determined by the mobile anchor in the straight-line segment of the movement trajectory for each detected unknown node. MARS projects these two possible points vertically onto the Ld.
When the mobile anchor finishes a line segment of the SCAN movement trajectory and reaches the right boundary of the deployment area, the mobile anchor shifts to the next line segment, which is also line Ld, but in the reverse direction. If there are any projected points on Ld, the mobile anchor will move directly to the closest projection point, instead of shifting vertically to the next parallel line segment, as it does when using the SCAN method. If there are no projection points on Ld, this indicates that there are no unknown nodes in the current scanned area. The mobile anchor does not need to scan overlapping coverage to perform positioning. Rather, it is able to directly shift to the next line segment of the SCAN movement trajectory.
Figure 3 presents a small example of the application of MARS. M is the mobile anchor, and {x, y, e, s, w} are unknown nodes, as shown in Figure 3a. {E2, E9, Ea, E3, E4, Eb, Ec, E5, E6, Ed} is the traveling trajectory of M when using the SCAN method. The mobile anchor using MARS starts moving along the line segment {E2, E9}. When M reaches location x1, it detects the signal of unknown nodes x and y. Subsequently, their signal vanishes at x2 and y2, respectively. When M reaches location E9, it finds that both x and y have only two reference points. M performs the projection phase from here to obtain the locations x” and y” on the line segment {Ea, E3}. Then, M moves toward the closest projection location, y”. At location y3, M detects the signal of the unknown node y and determines the actual position of node y.
As M moves continuously toward the location y” on the line segment {Ea, E3}, shown in Figure 3b, it detects two unknown nodes, s and e, at the locations s1 and e1, respectively. Both nodes s and e report that they have a position-aware neighbor y. When applying the ISP algorithm, more reference points are needed for positioning. When M moves to location w1 on the path segment {y”, E3}, it detects the unknown node w and receives the signal of node x again. It is at this point that M is able to determine the actual position of node x.
After obtaining the position of node x, node e collects two position-aware neighbor nodes {x, y}. On the basis of the reference point e1, M is able to locate node e, shown in Figure 3c. As M continuously moves toward the location E3 on the line segment {y”, E3}, node s receives a third reference point at s2. When the position of node s has been determined, the unknown node w will have three position-aware neighbors, and its position can be reported to the mobile anchor by its connected neighbors. Therefore, the mobile anchor does not need to schedule a route to locate it.
When M reaches location s2, there are no other unknown nodes in the line segment from E3 to s2 in the region □E2E3E9Ea waiting to be positioned. Thus, M moves directly from s2 to E4 in order to start scanning the line segment {E4, Eb}. Furthermore, the mobile anchor M detects no unknown nodes when it moves from E4 to Eb. Therefore, it moves directly from Eb to Ed, skipping the line segment {Ec, E5}, and starts to scan the line segment {Ed, E6}, shown as a dashed line in Figure 3a, because it is certain that there is no unknown node located in the region □E4EbEcE5. As M reaches E6, the mobile anchor M has already scanned the total deployment area, and the positioning procedure is terminated.

3.4. The Case of Non-Ideal Signal

In practical cases, radio coverage is not able to achieve a perfect disk shape because environmental factors, such as temperature, moisture, and obstacle, may influence the broadcast radio signal of the mobile anchor. In this section, the impact of the unstable fluctuation of radio signals on the positioning of unknown nodes will be considered. Fluctuations in radio signal cause the communication range of the mobile anchor to be shorter than the theoretical value R. Let the error be ε. The actual distance will fall in the range {R-ε, R} when the mobile anchor first detects the signal of an unknown node. The positioning results obtained by using triangulation to compute the location of unknown nodes become a region, instead of a point. The ISP algorithm is used to apply the range-free positioning algorithm coverage area pruning positioning system (CAPPS) [40] to compute the positioning area instead of using triangulation on the basis of three reference points. All neighboring-position-aware nodes of an unknown node will serve as reference points in CAPPS. In addition, the mobile anchor will record all the locations of all unknown node radio signals as they appear and disappear.
Figure 4 illustrates the main idea of the CAPPS. The example and the explanation were extracted from the content provided in [40]. Let Ω be the sensor nodes that detect the target, Ωc be the set of vertices comprising the convex hull of the set Ω, and Tg be the centroid of the coordinates of the sensor nodes in the set Ω. We have sets Ω = {r, s, t, u, v, w, x, y, z} and Ωc = {u, v, w, x, y, z} shown in Figure 4a. The set Ω is employed to compute Tg. Its location is represented by the circle located in the positioning area shown in Figure 4b. For the sensor nodes in Ωc, their one-hop neighbors are Πu = {a, b, c, d}, Πv = {b, c, d, e, f}, Πw = {f, g, h}, Πx = {h, i, j}, Πy = {k, l, m, n}, and Πz = {a, b, p}. Therefore, Π = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, p}.
For the sensor node u, its sorted one-hop neighbors are stored in the set Π*u = {d, a, b, c}. Let Aud be the intersection area of the two circles centered at sensor nodes u and d, each with radius D * = | u d ¯ | . There is a sensor node c enclosed in Aud, as shown in Figure 4c. Therefore, the sensor node d is removed from the set Π*u. Next, the intersection area Aua of the circles centered at sensor nodes u and a, each with radius D * = | u a ¯ | , is evaluated as shown in Figure 4d. The sensor node a is removed because the sensor node b lies within the area Aua. The sensor node b is also removed for the same reason, as shown in Figure 4e. Finally, the sensor node c is checked, and no sensor node is found to be enclosed in the area Auc. Thus, the sensor node c is kept in the set Π*u in order to prune the positioning area, as shown in Figure 4f. The same procedure is applied to all βi in the set Ωc to retrieve all Π i * | ∀ β i ∈ Ω c . The final set of sensor nodes for pruning the positioning area is Γ = {c, e, i, k, m, p}, as shown in Figure 4g. The positioning area pruned by the sensor nodes in the set Γ is shown in Figure 4h.
CAPPS employs the position-aware nodes and the reference points collected by the mobile anchor that are able to detect an unknown node in order to obtain the approximate location of that unknown node. Next, CAPPS uses the sensor nodes that are not able to sense an unknown node in order to refine the possible area. The centroid point (CP) mechanism employs the position-aware nodes or reference points that can receive signal from an unknown node and their one-hop neighbor nodes that cannot detect the unknown node to find the approximate location of the unknown node. CAPPS uses the CP mechanism to filter the points in order to compute the possible area of the unknown node, thus refining the size of the positioning area.
Among the existing methods that use a mobile anchor to locate the unknown nodes, both the unknown nodes and the mobile anchor use the same communication range. Thus, the proposed i-SCAN and MARS methods also apply this parameter. However, if the communication range is not the same, the smaller of the two communication ranges between the unknown nodes and the mobile anchor must be applied in order to achieve a uniform communication range for positioning. It is also necessary to use the minimum communication range when computing the upper and down boundary of the covered area. Therefore, it can be ensured that the communication range of the mobile anchor is able to cover all unknown nodes as the mobile anchor is moving along its trajectory.
In addition, almost all methods using a mobile anchor to position the unknown nodes set the deployment area as a rectangle. If this is not the case, the practical area can be enclosed with an outer rectangle before applying mobile anchor positioning algorithms. Therefore, these positioning methods can also be used in non-rectangular scenarios.

4. Simulation

This section presents the simulation results. First, we consider an ideal radio signal to evaluate the traveling distance of the mobile anchor. Next, we estimate the impact of the non-ideal radio signal on positioning accuracy. The simulation compares the proposed MARS method with the existing SCAN, SLMAT, SCAN-ET, and NHexCurve methods. In the following evaluation, we also combine the SCAN method with the ISP algorithm to locate unknown nodes, denoted as i-SCAN, to show the benefits of the ISP. We implement the simulation program using the C# program.

4.1. Environment Setup

This paper considers a scenario with a flat surface. There is a mobile anchor and multiple unknown nodes. The mobile anchor is equipped with a precise GPS component to accurately obtain its location information when traveling within the deployment area. The unknown nodes are randomly deployed in the deployment area, waiting for positioning. They are stationary after being deployed. The mobile anchor knows the number of unknown nodes at the beginning. Each unknown node has a unique identity and has the same communication range as the mobile anchor.
The deployment area is considered to be a 600 m × 600 m square, with no obstacles or sources of radio interference. The numbers of unknown nodes evaluated are {20, 25, 30, 35, 40, 45, 50, 55, 60, 65}. The evaluated communication ranges are R = {50 m, 75 m, 100 m, 125 m, 150 m}. The movement step of the mobile anchor is set to 1 m. In the non-ideal signal case, the evaluated signal-to-distance errors in RSSI are ε = {0–3%, 0–4%, 0–5%}. This will influence the distance evaluation when the ISP algorithm is used to locate the unknown nodes. All simulation results are the average of 100 random scenarios.

4.2. Numerical Results

Figure 5 shows the traveling distances of the mobile anchor with different numbers of unknown nodes. In this experiment, the communication range R is 100 m, and we calculated the traveling distance of the mobile anchor when the mobile anchor locates all unknown nodes. The average traveling distances of the mobile anchor when using the SCAN, SCAN-ET, SLMAT, and NHexCurve methods do not explicitly increase or decrease with an increasing number of unknown nodes, because the mobile anchor in these methods follows a prescheduled route. The traveling distance when using the NHexCurve method is the worst, because its movement trajectory creates a larger radio signal overlap area to improve the positioning. SLMAT also creates a larger radio signal overlap area than the SCAN method to solve the collinear problem when positioning an unknown node.
The traveling distance of the mobile anchor when using SCAN-ET is better than when using the SCAN method when there are fewer unknown nodes. This is because the mobile anchor moves to the vertices of the triangle in a zone that can immediately position the unknown nodes. This makes it possible to reduce the total traveling distance in the next line segment used by the SCAN method to find the third point to solve the collinear problem. However, the SCAN-ET becomes slightly worse than the SCAN method when the number of unknown nodes is higher than 55, because the mobile anchor has a greater opportunity to move to the vertices of the triangle in each zone.
By applying the ISP algorithm, the mobile anchor’s average traveling distances when using the i-SCAN and MARS methods decreases with increasing numbers of unknown nodes. When using the i-SCAN method, the mobile anchor obtains more available reference points from neighboring-positioned nodes, such that the traveling distance decreases with increasing numbers of unknown nodes. The traveling distance of the mobile anchor is decreased by about 200 m to 1500 m when using i-SCAN compared with when using the original SCAN method. MARS uses the projection mechanism to prevent the mobile anchor from regularly following the traveling trajectories of the SCAN method, enabling the mobile anchor to further reduce its traveling distance by 400 m to 600 m compared to when using i-SCAN. In the following, we will show that the mobile anchor using MARS is able to exhibit shorter traveling distances when the unknown nodes are densely deployed.
Figure 6 shows the traveling distances of all of the evaluated methods with different communication ranges. The number of unknown nodes in this experiment is 40. The mobile anchor is able to cover a greater area when using a long communication range. This also implies that the mobile anchor needs to travel less distance to be able to cover the entire deployment area. Therefore, the traveling distances of all methods decrease with increasing communication range.
Because the moving trajectories of the mobile anchor in the SLMAT and NHexCurve methods generate more overlapping of coverage, their mobile anchor’s traveling distances are still longer than that when using the SCAN method. The mobile anchor when using NHexCurve has a shorter traveling distance than that when using SLMAT when the communication range is 150 m. The mobile anchor when using NHexCurve is able to achieve a reduction in the number of horizontal scans when using a long communication range R. The mobile anchor’s traveling distances when using SCAN-ET and SCAN are very close. SCAN-ET exhibits clearly better results than SCAN when using a short communication range. The reason for this is that the mobile anchor is able to immediately position the unknown nodes after visiting the triangle’s vertices in a given zone. Therefore, the traveling distance of the mobile anchor can be reduced for the next line segment used by the SCAN method to find the third non-collinear reference point.
Long communication ranges help the mobile anchor to find more neighbors while traveling along a line segment. With the assistance of neighboring-position-aware nodes contributed by the mobile anchor, a mobile anchor with a long communication range will be able to find more neighbors when traveling along a line segment of its trajectory. With the assistance of neighboring-position-aware nodes contributed by the ISP algorithm, the gap in the mobile anchor’s traveling distance between the i-SCAN and SCAN methods increases with increasing communication range. The reduction in traveling distance compared with when using the SCAN method is about 10% to 25%. The projection mechanism in the MARS method is able to further reduce the traveling distance of the mobile anchor. The reduction in traveling distance compared to that required when using the SCAN method is about 24% to 34%. When using the MARS method, the benefits of the projection mechanism can be best observed when using a short communication range. This advantage becomes minor with increasing communication range, as evidenced by the improved traveling distance compared with that when using i-SCAN. Compared to the traveling distance when using i-SCAN, there is about a 17% reduction in traveling distance with a communication range of 50 m, and 11.6% with a communication range of 150 m.
Next, the percentage of unknown nodes successfully positioned under the constraint of a limited traveling distance of the mobile anchor was evaluated, as shown in Figure 7. This simulation is similar to the evaluation in which the mobile anchor had finite energy. In this experiment, the maximum traveling distances of the mobile anchor evaluated are {500, 1000, 1500, 2000, 2500}. The number of unknown nodes is 40, and the communication range is 100 m.
Other than SCAN-ET and NHexCurve, none of the methods are able to position unknown nodes when the maximum traveling distance is 500 m. By applying the straight line as the initial path for scanning the deployment area from left to right, the mobile anchor is able to detect unknown nodes but is not able to position them because of the collinear problem. SCAN-ET and NHexCurve do not use a straight line for the mobile anchor’s initial path, as in the SCAN method. When the mobile anchor detects any unknown nodes, the mobile anchor in these two methods is able to quickly locate the positions of the unknown nodes.
When the maximum traveling distance is extended to 1500 m, the mobile anchor is able to quickly locate the discovered unknown nodes when using the i-SCAN and MARS methods, which use the ISP algorithm. Their positioning ratios for the unknown nodes are 28% and 45%, respectively. The ratios obtained when using the SCAN-ET and NHexCurve methods, which exhibit better positioning ratios than the other methods at 500 m and 1000 m, increase moderately, with ratios of 34% and 13%. When the maximum traveling distance is 2000 m, the advantage of using the ISP algorithm becomes obvious. The positioning ratios when using i-SCAN and MARS are 65% and 83%, respectively. The mobile anchor when using MARS is able to scan more areas by reducing its traveling distance compared to the i-SCAN method. However, the ratios obtained when using SCAN-ET and NHexCurve are still lower than 50%. At the maximum traveling distance of 2500 m, the ratios of successfully positioned unknown nodes when using the SCAN-ET, NHexCurve, i-SCAN, and MARS methods are 56%, 35%, 80%, and 94%. Here, the ratio of NHexCurve is 45%, which is worse than the results obtained when using the SCAN method.
The following experiments evaluate the dense and sparse deployment of unknown nodes. In the dense deployment case, the topology of the deployed unknown nodes is a strongly connected graph. In the sparse deployment case, the unknown nodes are not able to find any neighbor nodes within their communication range. The unknown nodes in these two cases are still randomly deployed. This simulation only uses the deployment results that satisfy the requirements of sparse and dense deployment.
Figure 8a presents the results obtained for the dense deployment scenario. Without obtaining the information from the position-aware neighbor nodes, the traveling distances of the mobile anchor when using the SCAN, SLMAT, NHexCurve, and SCAN-ET methods slightly increase with increasing numbers of unknown nodes. The mobile anchor still has a greater traveling distance when using both SLMAT and NHexCurve than when using the other methods, because they regularly follow the prescheduled trajectory, which is longer than that of the SCAN method. SCAN-ET exhibits better results than the others because it smartly determines whether the mobile anchor needs to visit the vertices of a triangle in a given zone. Conversely, the traveling distances of the mobile anchor when using i-SCAN and MARS decrease with an increasing number of unknown nodes. The ISP algorithm uses the positioned nodes to reduce the number of unknown nodes that need to be located by the mobile anchor. i-SCAN receives the advantages of the ISP algorithm, and as such, the traveling distance of the mobile anchor is better than that obtained when using the other SCAN-like methods. The traveling distance is only about 68% to 84% of that obtained when using the SCAN method. MARS uses the ISP algorithm and smartly determines whether to change the prescheduled scan trajectory employed by the SCAN method. Therefore, MARS exhibits outstanding results compared to the other methods. The traveling distance of the mobile anchor is only about 45% to 70% of that obtained when using the SCAN method.
Figure 8b presents the results for the sparse deployment scenario. The mobile anchor scans almost the whole deployment area following its prescheduled trajectory in the SCAN, SLMAT, and NHexCurve methods, because the unknown nodes are scattered everywhere. Similarly, the ISP algorithm also cannot leverage its advantages here. This is evidenced by the almost identical traveling distances of the mobile anchor when using the SCAN and i-SCAN methods. The traveling distance of the mobile anchor when using the SCAN-ET method depends on whether the mobile anchor detects unknown nodes in a zone. Therefore, its results are better than those obtained with SCAN and i-SCAN. The traveling distance of the mobile anchor is about 10% to 3.7% of that when using the SCAN method. MARS still achieves better results than other methods, even though the advantage of the ISP algorithm is not applicable. MARS uses the projection mechanism to determine whether the mobile anchor can skip the next line segment of the movement trajectory employed by the SCAN method in order to help the mobile anchor shorten its traveling distance. The traveling distance of the mobile anchor is about 15% to 21% of that when using the SCAN method.
Table 1 shows the positioning size of the positioning area for the non-ideal signal scenario. The maximum communication range in this simulation is 100 m, and the number of unknown nodes is 40. The error ratio increases with increasing size. The positioning area of the SCAN, SLMAT, NHexCurve, and SCAN-ET methods are about 17 m2–20 m2, 14.72 m2–19.26 m2, 15.35 m2–19.38 m2, and 14.45 m2–19.4 m2, respectively. These methods use the triangulation method to locate unknown nodes. The sizes of the positioning areas when using these methods—SLMAT, NHexCurve, and SCAN-ET—are very similar, because the three reference points in the triangulation method they selected are the vertices of an isosceles triangle.
Similarly, the positioning size of the positioning areas of i-SCAN and MARS are very similar, because they use the CAPPS method to compute the size of the positioning area. CAPPS uses multiple reference points to prune the smaller overlapping coverage area in order to refine the positioning accuracy. The positioning size of the positioning areas of i-SCAN and MARS methods are 5.14 m2–7.9 m2 and 4.31 m2–7.47 m2, respectively. Their areas are about 53% to 74% smaller than when using the SCAN method.
The positioning procedure in all of the compared methods is able to directly obtain the data for the reference points or positions. Thus, the time complexity is O(1). Although the ISP algorithm shows that each unknown node needs to scan the set of unpositioned nodes, the unknown node provides this information by exchanging it with its neighbors. Their differences are the procedure for scheduling routes in the deployment area. In an N × N deployment area with a communication range R, SCAN, i-SCAN, SCAN-ET, and MARS need to scan N R 2 line segments, that is, O(N). SLMAT and NHexCurve consider the equilateral triangle, so they need to scan N R 2 line segments, which is also O(N).

5. Conclusions

This paper proposed an iterative sensor positioning (ISP) algorithm that uses the neighboring-positioned node to assist the mobile anchor in locating the unknown nodes and a mobile anchor route scheduling method, named MARS, using the ISP algorithm. MARS uses the projection mechanism to reduce the traveling distance of the mobile anchor. The simulation results show that the mobile anchor in MARS is able to reduce the traveling distance by 21–49% compared to when using the SCAN method. When the communication range of the mobile anchor is 150 m, which corresponds to the unfavorable scenario for the projection mechanism, the traveling distance using the MARS method is still better than that using the SCAN method. The traveling distance is 34% less than that when using the SCAN method. In the dense deployment scenario that is favorable for the ISP algorithm, applying the ISP algorithm to the mobile anchor using the MARS method made it possible to locate all unknown nodes in only 68% to 84% of the traveling distance required when using the SCAN method. If the MARS method is applied directly to the mobile anchor, the traveling distance is 45% to 70%. In the sparse deployment scenario, which is unfavorable for the ISP algorithm, the mobile anchor using the MARS method was still able to achieve a 15% to 21% reduction in traveling distance compared to when using the SCAN method. In the non-ideal signal scenario, CAPPS was applied in MARS to improve positioning accuracy. The size of the positioning area was about 53% to 74% smaller than when using the Triangulation algorithm.
The proposed MARS method uses the trajectory of the SCAN method to improve the scheduled route. In future work, we will extend the MARS to random waypoints. Without the limitation of the straight-line movement strategy, MARS should be able to provide a shorter route.

Author Contributions

Conceptualization, S.-C.H. and Y.-K.W.; methodology, S.-C.H.; software, Y.-K.W.; validation, S.-C.H. and Y.-K.W.; formal analysis, S.-C.H.; investigation, S.-C.H. and Y.-K.W.; resources, S.-C.H.; data curation, S.-C.H. and Y.-K.W.; writing—original draft preparation, S.-C.H.; writing—review and editing, S.-C.H.; visualization, S.-C.H. and Y.-K.W.; supervision, S.-C.H.; project administration, S.-C.H.; funding acquisition, S.-C.H. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by National Science and Technology Council project of Taiwan, Grant Nos. MOST 111-2221-E-150-022.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. The mobile anchor records the reference points for the unknown node z.
Figure 1. The mobile anchor records the reference points for the unknown node z.
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Figure 2. Three cases in which the ISP algorithm is able to directly locate an unknown node: (a) the unknown node has three position-aware neighbors; (b) the unknown node has two position-aware neighbors; (c) the unknown node has one position-aware neighbor.
Figure 2. Three cases in which the ISP algorithm is able to directly locate an unknown node: (a) the unknown node has three position-aware neighbors; (b) the unknown node has two position-aware neighbors; (c) the unknown node has one position-aware neighbor.
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Figure 3. MARS with ISP algorithm: (a) the projection of MARS in the first line segment; (b) the unknown node has one position-aware neighbor; (c) the unknown node has two position-aware neighbors.
Figure 3. MARS with ISP algorithm: (a) the projection of MARS in the first line segment; (b) the unknown node has one position-aware neighbor; (c) the unknown node has two position-aware neighbors.
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Figure 4. Positioning procedure of CAPPS (Reprinted with permission from Ref. [40]. 2018, Huang, S.C.; Li, F.) (a) The coverage areas of the non-target-detecting sensor nodes {c, i, m, p}. (b) The sensor nodes in the set Ω are used to compute Tg, illustrated by the circle located at the positioning area. The one-hop neighbors of the sensor node u are identified, and the sorted sequence is stored in the set Π*u = {d, a, b, c}. (c) The first sensor nodes d of the set Π*u are verified. The sensor node c is enclosed within the intersection area Aud. Thus, the sensor node d is removed from the set Π*u. (d–f) The other sensor nodes a, b, and c in the set Π*u are verified. (g) The set of remaining sensor nodes is used for pruning the positioning area Γ = {c, e, i, k, m, p}. (h) The pruned positioning area obtained by applying the sensor nodes in the set Γ.
Figure 4. Positioning procedure of CAPPS (Reprinted with permission from Ref. [40]. 2018, Huang, S.C.; Li, F.) (a) The coverage areas of the non-target-detecting sensor nodes {c, i, m, p}. (b) The sensor nodes in the set Ω are used to compute Tg, illustrated by the circle located at the positioning area. The one-hop neighbors of the sensor node u are identified, and the sorted sequence is stored in the set Π*u = {d, a, b, c}. (c) The first sensor nodes d of the set Π*u are verified. The sensor node c is enclosed within the intersection area Aud. Thus, the sensor node d is removed from the set Π*u. (d–f) The other sensor nodes a, b, and c in the set Π*u are verified. (g) The set of remaining sensor nodes is used for pruning the positioning area Γ = {c, e, i, k, m, p}. (h) The pruned positioning area obtained by applying the sensor nodes in the set Γ.
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Figure 5. Traveling distances with different numbers of unknown nodes (R = 100 m).
Figure 5. Traveling distances with different numbers of unknown nodes (R = 100 m).
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Figure 6. Traveling distances with different communication ranges (40 nodes).
Figure 6. Traveling distances with different communication ranges (40 nodes).
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Figure 7. The percentage of positioned unknown nodes with limited traveling distance.
Figure 7. The percentage of positioned unknown nodes with limited traveling distance.
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Figure 8. The traveling distance in the dense and sparse deployment scenarios. (a) Dense deployment scenario, which is the scenario favorable for the ISP algorithm. The topology of the deployed unknown nodes is a strongly connected graph. Each unknown node can be connected to any other node. (b) Sparse deployment scenario, which is the scenario unfavorable for the ISP algorithm. The unknown nodes are not able to find any neighbor nodes within their communication range.
Figure 8. The traveling distance in the dense and sparse deployment scenarios. (a) Dense deployment scenario, which is the scenario favorable for the ISP algorithm. The topology of the deployed unknown nodes is a strongly connected graph. Each unknown node can be connected to any other node. (b) Sparse deployment scenario, which is the scenario unfavorable for the ISP algorithm. The unknown nodes are not able to find any neighbor nodes within their communication range.
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Table
Table 1. The size of the positioning area for different non-ideal signal error ratios (m2).
Table 1. The size of the positioning area for different non-ideal signal error ratios (m2).
Error RatioSCAN [26]SLMAT [30]NHexCurve [34]SCAN-ET [31]i-SCANMARS
0~3%1714.7215.3514.455.144.31
0~4%1916.6416.6816.596.265.32
0~5%2019.4219.3819.47.97.47
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Huang, S.-C.; Wang, Y.-K. Mobile Anchor Route Scheduling with an Iterative Sensor Positioning Algorithm in Wireless Sensor Networks. Appl. Sci. 2023, 13, 22. https://doi.org/10.3390/app13010022

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Huang S-C, Wang Y-K. Mobile Anchor Route Scheduling with an Iterative Sensor Positioning Algorithm in Wireless Sensor Networks. Applied Sciences. 2023; 13(1):22. https://doi.org/10.3390/app13010022

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Huang, Shih-Chang, and Yi-Kai Wang. 2023. "Mobile Anchor Route Scheduling with an Iterative Sensor Positioning Algorithm in Wireless Sensor Networks" Applied Sciences 13, no. 1: 22. https://doi.org/10.3390/app13010022

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