Sage Journals: Discover world-class research

Abstract

Wireless indoor positioning systems are susceptible to environmental distortion and attenuation of the signal, which can affect positioning accuracy. In this article, we present a two-stage indoor positioning scheme using a fuzzy-based algorithm aimed at minimizing uncertainty in received signal strength indicator measures from reference nodes in wireless sensor networks. In the first stage, the indoor space is divided into several zones and a fuzzy-based indoor zone-positioning scheme is used to identify the zone in which the target node is located via zone splitting. In the second stage, adaptive trilateration is used to position the target node within the zone identified in the first stage. Simulation results demonstrate that the proposed two-stage fuzzy rectangular splitting outperforms non-fuzzy-based algorithms, including K-Nearest Neighbors–based localization, and traditional triangular splitting schemes. We also developed an expanded positioning scheme to facilitate the selection of a positioning map for large indoor spaces, thereby overcoming the limitations imposed by the size of the positioning area while maintaining high positioning resolution.

Keywords

Wireless sensor networks fuzzy inference system received signal strength indicator trilateration technique positioning

Introduction

The Internet of things (IoT)¹ is opening up opportunities to build smart cities in which traffic congestion, parking, street lighting, waste, irrigation, and urban noise are monitored in real time and managed more effectively. Wireless sensor networks (WSNs) provide the eyes and ears of the IoT, bridging the real world and the digital world. This has led to considerable investment and research in WSNs, which are essentially distributed network systems comprising a large number of sensor nodes interconnected wirelessly. WSNs can be used in a variety of applications, such as environmental monitoring, military surveillance, medical monitoring, and object tracking.² However, all of these applications require positioning information.

Indoor location-based services (LBSs) are expanding rapidly.³ The global positioning system (GPS) is widely used outdoors;⁴ however, it does not perform well indoors, due to the blocking of radio waves by structures. Furthermore, indoor environments are prone to interference from moving bodies, multi-path effects, and shadowing. This has necessitated the development of positioning systems specific to indoor environments.^5,6

Most indoor positioning systems employ multiple independent, decentralized sensors in the form of a WSN, using the received signal strength indicator (RSSI) as the basis for positioning.^7–9 Most existing RSSI-based positioning systems use scene analysis for positioning, wherein the environment is divided into several zones and an RSSI database is established for each zone. The current RSSI value at an unknown location is compared with those stored in the database, and the pattern that most closely matches the user’s pattern is returned as an estimate of the user’s location. The number of zones determines the amount of time required to build the database as well as the degree of complexity involved in matching. The zone approach does not provide high positioning resolution.

In this study, we developed a novel two-stage indoor positioning scheme. The first stage involves scene analysis designed specifically for small-scale indoor environments based on a signal propagation channel model constructed in advance. RSSI data from reference nodes is distributed according to power level and the monitored space is split into zones. A fuzzy inference system (FIS) algorithm is used to improve the accuracy of zone localization and sharpen zone-positioning estimates. RSSI data from the reference nodes are used as inputs to determine the location of the target node based on fuzzy rules. The proposed positioning system is able to determine locations without the need for training. Furthermore, unlike existing systems based on fingerprinting algorithms, the proposed system does not require numerous measurement points for the construction of a radio map.^10,11

Positioning resolution is further enhanced in the second stage. Coordinate positioning is performed according to the position of zones determined in the first stage using an adaptive trilateral positioning technique. We also developed an expansion scheme to enable the application of the proposed two-stage positioning method to large-area environments without sacrificing positioning resolution. This is achieved by extending the original rectangular area to form a larger rectangular area, using a few simple estimates to maintain positioning resolution.

The remainder of this article is organized as follows. Section “Related works” describes some related works. Section “Zone-positioning” details the FIS applied to zone-based location as well as various zone-splitting methods for first-stage localization. Section “Coordinate positioning” describes coordinate positioning in the second stage. Section “Extended positioning” outlines the extended positioning method. Section “Simulation results and analysis” provides simulation results and discusses the performance of the positioning system in a variety of situations. Conclusions are drawn in section “Conclusion.”

Related works

The techniques commonly used to calculate positions within an enclosed space are classified according to the sensor technology they employ: time of arrival (ToA), time difference of arrival (TDoA), angle of arrival (AoA), and RSSI. Unlike the first three methods, RSSI is output by most off-the-shelf sensor nodes, leading to its wide-scale application in localization systems. Fingerprinting is the most effective solution for RSSI-based indoor location sensing. Fingerprinting is generally divided into two phases: training and online positioning.¹² A variety of location-detection algorithms have been developed, including probabilistic methods, deterministic methods, and neural networks. K-Nearest Neighbors (KNN) is the most popular RSS-based location-fingerprinting approach used to determine indoor locations.¹³ The methods presented in the above-mentioned studies provide adequate estimation accuracy; however, they require considerable quantities of training data. Furthermore, computational overhead is high and distance-based learning lacks specificity.

Fuzzy logic (FL) is widely used in inference systems to imitate human behavior and make decisions. These systems are referred to as FISs (fuzzy inference systems).¹⁴ FIS has been used to improve accuracy in WSN localization. Chenji and Stoleru¹⁵ proposed an FL-based approach to mobile node localization in sparse networks with few available anchors. In that study, FIS provided a 24%–40% improvement in localization accuracy. In Abdelhadi and Anan,¹⁶ a fuzzy system for collaborative feedback communication in WSNs reduced error in localization estimates to less than 5%. In this study, we employed an FIS with a triangular membership function for input (RSSI) and output (zone) to improve the accuracy of zone localization.

Most indoor localization methods are based on a signal propagation model in which distance is estimated using RSSI data. Most authors have adopted the log-distance path loss model for the calculation of distance.^17,18 Achieving a realistic simulation environment requires that we take into account the effects of shadowing, path loss, multipath effects, noise, and interference. Thus, we use the equation of the log-distance path loss model as follows

$\bar{PL} (d) [dB] = \bar{PL} (d_{0}) + 10 n \times \log (\frac{d}{d_{0}}) + X_{σ}$ (1)

where $\bar{PL} (dB)$ is received power and $d_{0}$ is close-in distance. “Close-in distance” is a technical term used in radio engineering indicating a reference point for measurements of radio field strength, usually close to the transmitter. $\bar{PL} (d_{0})$ is the received power when $d$ is within close-in distance, $d$ is the distance between the transmitter and the receiver, $n$ is the path loss exponent, and $X_{σ}$ is a zero-mean Gaussian distributed random variable with standard deviation $σ$ between 3 and 10 dB. In this article, this variable is used to simulate instability in the received RSSI value. Standard deviation can be adjusted to meet a variety of indoor environments. Table 1 lists the simulation parameters employed in the channel model.

Table 1.

Channel model simulation parameters.

Parameter	Value
Transmission power	2 mW
Transmission antenna gain	1
Received antenna gain	1
Carrier frequency	2.4 GHz
Close-in distance	0.1 m
Path loss exponent (n)	1.8
Standard deviation parameter	3, 5, 7, 9 dB

Zone-positioning

The first stage of the proposed method involves zone-positioning. In the following, we discuss three zone-splitting methods: traditional triangular splitting, fuzzy triangular splitting, and fuzzy rectangular splitting. We adopted the FIS to enhance the efficiency of zone-positioning.

Trianglar splitting

Figure 1 illustrates the proposed triangular splitting method in which the simulation area is divided into four triangular zones of equal size. The position of the reference nodes can greatly affect the accuracy of positioning, particularly indoors. Locating reference nodes in four corners (as a square) can improve positioning accuracy.¹⁹ However, it is not always possible to distinguish the four triangular zones when the reference nodes are located in four corners. We therefore included an additional reference node in the center of the square by which to assess the zones. After sorting the RSSI values from the five reference nodes from large to small, we select the three strongest for zone-positioning. Large-area zones permit only low-resolution positioning, whereas dividing the space into a larger number of zones increases RSSI instability at zone boundaries.

Figure 1.

Triangular zone-splitting method.

Fuzzy triangular splitting method

The triangular splitting method produces large zones with limited accuracy. In Figure 2, the number of zones was increased to eight. The shape of the zones is non-uniform; therefore, fuzzy inference^15,16,20 was used to improve positioning accuracy.

Figure 2.

Fuzzy triangular splitting method.

Fuzzy rectangular splitting method

Existing fuzzy triangular splitting methods have been shown to improve localization precision; however, improvements are still required with regard to zone identification, due to the fact that the central node covers four zones. Thus, we re-divide the indoor environment into nine zones, as shown in Figure 3. This splitting method reduces uncertainty in the central area while enhancing the resolution of zone-based positioning.

Figure 3.

Fuzzy rectangular splitting method.

Fuzzy inference

The FIS used in this article provides an alternative approach to zone identification. The fuzziness of RSSI values is represented in the membership function. Inference based on fuzzy rules brings the process of estimation in line with human thinking patterns. Figure 4 presents a flow chart of zone-positioning in conjunction with fuzzy inference.

Figure 4.

Flowchart of zone-positioning in conjunction with fuzzy inference.

In our FIS, the RSSI value of the target and five reference nodes are used as input parameters. The RSSI value is converted into a membership function between 0 and 1 using fuzzification and mapped to an input fuzzy set, which is defined according to the power level of the RSSI value. RSSI power levels from the corner reference node (A) and central reference node (E) are, respectively, presented in Figures 5 and 6.

Figure 5.

RSSI power level at reference node A.

Figure 6.

RSSI power level at reference node E.

In the fuzzy set of input parameters, we selected a triangle as the shape of the membership function due to its low complexity (i.e. enabling rapid computation). The number of membership functions is determined by the input RSSI power levels, which are defined as six fuzzy sets (L1, L2, L3, L4, L5, and L6), as shown in Table 2. Figure 7 presents the fuzzy set of input parameters for the RSSI value of reference node A. Since the RSSI value in S6 and S7 is similar, we placed these two power levels in the same fuzzy set (L6). Reference nodes A, B, C, D are in the four corners of the indoor space range; therefore, the fuzzy sets of these four reference nodes are identical. Figure 8 presents the fuzzy set of input parameters for the RSSI value of central reference node E.

Table 2.

Fuzzy membership function for RSSI power level.

Fuzzy set	RSSI power level	RSSI range (dB m)
L1	S1	0 to −11
L2	S2	−9 to −16
L3	S3	−13 to −19
L4	S4	−17 to −21
L5	S5	−20 to −23
L6	S6, S7	−24 to −26

RSSI: received signal strength indicator.

Figure 7.

RSSI fuzzy set of reference node A.

Figure 8.

RSSI fuzzy set of reference node E.

In this article, we based the number of fuzzy sets in the output parameters on the total number of zones. From Figures 2 and 3, fuzzy triangular splitting resulted in eight zones, whereas fuzzy rectangular splitting resulted in nine zones. Figures 9 and 10 present the fuzzy sets of output zone positions obtained using fuzzy triangular splitting and fuzzy rectangular splitting, respectively.

Figure 9.

Fuzzy set of output zone positions obtained using fuzzy triangular splitting.

Figure 10.

Fuzzy set of output zone positions obtained using fuzzy rectangular splitting.

In the following, we summarize the structure of the FIS used to identify the zone in which the target is located:

Fuzzifier: triangular membership function;

Fuzzy Inference Rule: Mamdani fuzzy rules;

Fuzzy Inference Engine: max–min operation;

Defuzzifiction: center of gravity.

Simulation results are presented in section “Simulation results and analysis.”

Coordinate positioning

In the following, we describe the coordinate positioning method used in the second phase, which involves selecting reference points for trilateration based on the results of zone-positioning in the first phase in order to improve positioning accuracy.

Distance measurement using linear interpolation

Trilateration is a method used to identify the absolute or relative location of a point through the measurement of distances based on the geometry of circles, spheres, or triangles. Thus, the accuracy of the initial distance information is important. Most previous methods use RSSI values to perform range finding.^21–23 The computational complexity of distance calculation can be reduced by translating curves in the nonlinear channel model into linear equations through linear interpolation, as shown in Figures 11 and 12.

Figure 11.

Schematic illustration showing channel model.

Figure 12.

Distance measuring based on linear interpolation.

In Figure 11, the blue curve represents the original channel model, and the red dots indicate RSSI values measured every 0.5 m. The curve in the channel model indicates that an increase in distance reduces the slope of the curve, particularly at distances exceeding 3 m. This leads to a drop in resolution, which reduces positioning accuracy. Thus, we sought to keep the range of the indoor channel model to within 3 m.

We define an RSSI value as a set of right-angled triangles based on its corresponding known distance, as shown in Figure 12. The bevel (red segment) connected to the right triangle in each segment is a simplified channel model, each of which represents a set of linear equations. The concept of linear interpolation is to utilize the slope (m) between two arbitrary points on the same line. We use equation (2) to represent the relationship, as follows

$m = \frac{RSS I_{2} - RSS I_{1}}{D_{2} - D_{1}} = \frac{RSS I_{Measure} - RSS I_{1}}{D_{Estimate} - D_{1}}$ (2)

Equation (3) can be obtained by rearranging equation (2) as follows

$D_{Estimate} = D_{1} + \frac{RSS I_{Measure} - RSS I_{1}}{RSS I_{2} - RSS I_{1}} \times (D_{2} - D_{1})$ (3)

where $D_{Estimate}$ is the estimated distance value.

Adaptive trilateral positioning

Trilateration is based on the distance between the three reference points and the target point, wherein equations are derived for the three circles. The position of the target point is obtained by solving these equations.^24,25 However, in practical applications, distance estimates are prone to error, such that the three circles fail to intersect at a point, that is, a solution cannot be derived for the three equations. In this article, we sought to overcome location error in trilateral positioning by developing an adaptive approach using the relative position of any two circles.

Three circles intersecting within a common area

First, we find the intersection points of any two circles and identify the intersection point closest to the center of the other circle, which results in three intersection points (red points in Figure 13). We then draw a triangle using these points. Finally, we estimate the coordinates of the target point (centroid of the triangle, as represented by black dot in Figure 13).

Figure 13.

Three circles intersecting within a common area.

Two circles intersecting within a circle

First, we find the intersection points of two circles (e.g. A&C, C&B), and then we identify the two intersection points closest to the separate circles (A&B: red dots). We then calculate the center of the mass (black dot), which gives us the coordinates of the target point, as shown in Figure 14.

Figure 14.

Two circles intersecting in a circle.

Two circles intersecting with one separate circle

We use the intersection of two intersecting circles to calculate the centroid points. We then assign a weighting value between the centroid point and the center of the separate circle to obtain the coordinates of the target node, as shown in Figure 15. This situation commonly occurs when the target node is closer to the location of reference node A. In this situation, we would, respectively, use equations (4) and (5) for weighting1 and weighting2

$Weighting 1 = \frac{R_{A}}{R_{A} + R_{B}}$ (4)

$Weighting 2 = \frac{R_{B}}{R_{A} + R_{B}}$ (5)

Figure 15.

Two intersecting circles with a separate circle.

Selection mechanism used for multi-set positioning reference points

The precision of distance values and reference point coordinates are crucial to the effectiveness of trilateration. In this article, we use the results obtained in the first stage to identify the zone of the target node, which is then used as the basis location for selection of a combination of reference nodes. Zone selection affects the choice of reference points as well as the positioning results. Selecting only a single-set reference point can lead to the selection of an erroneous combination of reference points, which can increase positioning error. Thus, we developed a mechanism for the selection of multi-set positioning reference points, using a combination of multi-trilateration methods. Figures 16 –18 present a schematic illustration of the process in which a combination of reference nodes is selected in the second positioning stage using rectangular zone splitting in conjunction with fuzzy inference.

Figure 16.

Reference node selection method: Zone2.

Figure 17.

Reference node selection method: Zone1.

Figure 18.

Reference node selection method: Zone5.

In Figure 16, under the assumption that the target point is in $Zone 2$ , the first set of default reference nodes is $Δ ABE$ . To take into account the possibility that zone identification was erroneous, we also selected $Δ ABC$ and $Δ ABD$ to assist in positioning. The four zone locations (Zones 4, 8, 6, 2) in Figure 16 produce three sets of reference node combination, which can be adjusted for adaptive trilateral positioning.

In Figure 17, under the assumption that the target point is in $Zone 1$ , the first set of reference node combinations is $Δ ABC$ . To take into account the possibility that zone identification was erroneous, we considered the distance estimate between the target node to reference nodes B and C. If the estimated distance to node B were greater than the estimated distance to node C, then we would identify the target node toward the top-left position. This would lead to the selection of a second reference node combination $Δ ACE$ for use in positioning. If the estimated distance to node B did not exceed the estimated distance to node C, then we would identify the target node toward the lower-right, such that $Δ ABE$ would be selected as the second combination set. The four zone locations in Figure 17 (Zones 1, 3, 7, 9) have two sets of reference nodes for adaptive trilateral positioning.

In Figure 18, we assume that the target node is in $Zone 5$ . The range of this zone is close to the reference node in the center of the area; therefore, reference node E is prioritized for selection. The selection of a second reference node is based on the estimated distance between the target points and the reference nodes in the four corners, respectively.

We divide the square average into four small triangular blocks under the assumption that the estimated distance to reference node A is less than the estimated distance to reference node D, and further that the estimated distance to reference node B is less than the estimated distance to reference node C. Under these conditions, the target node would be toward the lower side, such that triangle $Δ ABE$ would be selected as the combination for the first set of default reference nodes, as indicated by the blue triangle in Figure 18. To account for the possibility that the zone was identified erroneously, we, respectively, selected the large triangles $Δ ABC$ and $Δ ABD$ as second and third sets of reference nodes, as indicated by the green and orange triangles in Figure 18. This makes it possible to adjust the position of the four small triangular blocks in $Zone 5$ , each of which has three sets of reference nodes.

Figures 19 and 20 present schematic illustrations showing the selection of reference nodes in the second stage of triangular zone splitting in conjunction with fuzzy inference.

Figure 19.

Reference node selection method: Zone1–Zone4.

Figure 20.

Reference node selection method: Zone5–Zone8.

In Figure 19, we assume that the target node is in $Zone 1$ , such that the first default combination of reference nodes is $Δ ABE$ . To reduce the likelihood of erroneous zone identification, we, respectively, select $Δ ABC$ and $Δ ABD$ for positioning in $Zone 1$ to $Zone 4$ , and all three reference nodes combinations can be adjusted adaptively for trilateral positioning, as shown in Figure 19.

In Figure 20, we assume that the target node is in $Zone 5$ , such that its geographic location is toward the center of triangle $Δ ABC$ . Thus, triangle $Δ ABC$ is the default combination of reference nodes. We then identify the reference nodes to be selected by estimating the distance from the target node to reference nodes B and C, respectively.

When the estimated distance to reference node B exceeds that of reference node C, then we can be sure that the target node is toward the top-left. We therefore select triangle $Δ ACE$ as the second reference node combination, as indicated by the green triangles in Figure 20. If the estimated distance to reference node B is less than that to reference node C, then we can be sure that the target node is toward the lower-right. We therefore select the small triangle $Δ ABE$ as the second reference node combination indicated by the orange triangle in Figure 20. In this image, $Zone 5$ to $Zone 8$ all have two sets of positioning reference nodes.

Extended positioning

Limitations in the channel model restrict the proposed two-stage fuzzy inference positioning method to a range of 3 m × 3 m.

We therefore developed an extended positioning method to overcome this limitation. If we consider a 3 m² area as a unit map, the two-stage positioning method can apply in each unit map. We treat a 3 m × 3 m area as a small-scale map and then extend/combine it to create a large-scale 6 m × 6 m area containing 13 reference nodes for positioning. This large-scale area is divided into five small-scale positioning maps (3 m × 3 m), as shown in Figure 21.

Figure 21.

Expanded positioning environment.

In the resulting large indoor space, positioning can be divided into two steps. The first step involves identifying the small-scale map in which the target node resides. In the second step, we use the five reference nodes in that particular map as a benchmark for two-stage fuzzy inference positioning. This makes it possible to enlarge the positioning range to match the actual dimensions of the indoor space while maintaining the original resolution.

The first step involves using the RSSI determination of the reference node in Table 3 to identify the correct positioning map. We begin by comparing the RSSI values of reference nodes $Nod e_{1}$ , $Nod e_{6}$ , $Nod e_{9}$ , and $Nod e_{12}$ in the four corners of the expanded positioning range.

Table 3.

Unit map detection.

RSSI decision rules	Unit map
1. $RSS I_{1} > RSS I_{6} and RSS I_{9} > RSS I_{12}$	Map B
2. $RSS I_{5} > RSS I_{8} and RSS I_{5} > RSS I_{11}$
3. $RSS I_{5} > RSS I_{4}$
1. $RSS I_{1} > RSS I_{6} and RSS I_{9} > RSS I_{12}$	Map G
2. $RSS I_{11} > RSS I_{13} and RSS I_{11} > RSS I_{5}$
3. $RSS I_{11} > RSS I_{4}$
1. $RSS I_{1} < RSS I_{6} and RSS I_{9} < RSS I_{12}$	Map R
2. $RSS I_{8} > RSS I_{5} and RSS I_{8} > RSS I_{13}$
3. $RSS I_{8} > RSS I_{4}$
1. $RSS I_{1} < RSS I_{6} and RSS I_{9} < RSS I_{12}$	Map P
2. $RSS I_{13} > RSS I_{11} and RSS I_{13} > RSS I_{8}$
3. $RSS I_{13} > RSS I_{4}$
1. $RSS I_{4} > RSS I_{5}$	Map Y
$And RSS I_{4} > RSS I_{8}$
$And RSS I_{4} > RSS I_{11}$
$And RSS I_{4} > RSS I_{13}$

RSSI: received signal strength indicator.

For example, the situation where $RSS I_{1} > RSS I_{6}$ and $RSS I_{9} > RSS I_{12}$ corresponds to the two maps on the left (map B and map G), the reverse relationships would correspond to map R and map P on the right.

We can then use the RSSI values of $Nod e_{5}$ and $Nod e_{11}$ to perform the next topological assessments.

In the situation where $RSS I_{5} > RSS I_{8}$ and $RSS I_{5} > RSS I_{11}$ , the target node is located at the bottom-left of map B. In the situation where $RSS I_{11} > RSS I_{13}$ and $RSS I_{11} > RSS I_{5}$ , the target node is in the top-left of map G. Finally, we compare the RSSI value of reference $Nod e_{4}$ in the central map to the RSSI value of $Nod e_{5}$ . Thus, in the situation where $RSS I_{5} > RSS I_{4}$ , we can be sure that the final topological position is in the lower-left corner of map B. Conversely, if $RSS I_{5} < RSS I_{4}$ , then the target node is located in the central region of map Y.

Table 3 lists the decision rules of the RSSI values for the five unit maps. We could use the decision rules to determine which map should be used further. When using the extended positioning method, the unit map of the target node is selected in the first step. This map is then used to perform two-stage fuzzy inference positioning. This approach makes it possible to extend the range of detection while maintaining high positioning resolution.

Simulation results and analysis

In this section, we present the results of simulations using the proposed indoor positioning system. The simulations were conducted using several custom functions as well as standard functions of MATLAB^®. We implemented the proposed channel model using RSSI values simulated by a mathematical model. Gaussian random variables $X_{σ}$ were added to the channel model to simulate the effects of shadowing, path loss, multipath, noise and interference typical of actual indoor environments. This inevitably undermined the stability of the received RSSI values. We set the standard deviation parameter at 3, 5, 7, and 9 dB m and then analyzed positioning accuracy in indoor environments.

Efficiency analysis: first-stage zone-positioning

We began by investigating the effects of the standard deviation parameter in the first stage of zone-positioning. We randomly deployed 100 sampling points throughout the indoor environment. Figures 22 –24 present the results indicating the percentage of correct positioning using the three splitting methods with four standard deviation values. The percentage of correct estimates is calculated as follows (6)

$Correcrt zone positioning estimates = \frac{P_{c o r r e c t}}{P_{T o t a l}} \times 100 %$ (6)

where $P_{Total}$ is the total number of sampling points deployed in the indoor environment, and $P_{correct}$ represents the number of sampling points with the correct zone estimates. These figures indicate that when using any of the three zone-splitting methods, the percentage of correct zone estimates decreases with an increase in standard deviation and achieves convergence with an increase in the number of sampling points.

Figure 22.

Percentage of correct zone-positioning estimates obtained using triangular splitting with various standard deviation values.

Figure 23.

Percentage of correct zone-positioning estimates obtained using fuzzy triangular splitting with various standard deviation values.

Figure 24.

Percentage of correct zone-positioning estimates obtained using fuzzy rectangular splitting with various standard deviation values.

Figure 25 and Table 4 compare the percentage of correct zone estimations obtained using the three methods when positioning sampling points = 255. These results indicate that regardless of the splitting method, the percentage of correct zone estimates decreased with an increase in standard deviation. Rectangular zone splitting in conjunction with fuzzy inference achieved correct zone estimates of approximately 73%–82%, while triangular zone splitting in conjunction with fuzzy inference achieved correct zone estimates of approximately 46%–53%. Conventional triangular zone splitting achieved correct zone estimates of 83%–95%, due to the smaller number of zones (i.e. fewer chances to be wrong).

Figure 25.

Comparison of percentage of correct zone-positioning estimates using three splitting methods with four levels of standard deviation.

Table 4.

Comparison of percentage of correct zone-positioning estimates using three splitting methods.

Standard deviation (dB m)	Triangle zone splitting (%)	Fuzzy triangular splitting (%)	Fuzzy rectangular splitting (%)
3	95	53	82
5	91	52	80
7	86	49	77
9	83	46	73

In the above simulations, the three splitting methods did not generate the same number of splitting areas or areas of the same size. Thus, it is not fair to calculate the percentage of correct zone-positioning estimates from the results in Table 4. Nonetheless, we can see that the rectangular splitting method in conjunction with fuzzy inference generated a larger number of splitting zones as well as a higher percentage of correct zone-positioning estimates.

Equation (7) was used to calculate the improvement provided by the fuzzy triangular splitting method over the fuzzy rectangular splitting method

$R_{i} = \frac{E_{F u z z y S q r} - E_{F u z z y T r i}}{E_{F u z z y T r i}} \times 100 %$ (7)

where $E_{FuzzySqr}$ represents the average percentage of correct zone-positioning estimates obtained using fuzzy rectangular splitting, and $E_{FuzzyTri}$ represents the results obtained using fuzzy triangular splitting.

As shown in Tables 4 and 5, fuzzy rectangular splitting outperformed fuzzy triangular splitting by 53.8%–58.7%. Thus, it would be reasonable to expect that the selection of fuzzy rectangular splitting for first-stage processing would achieve superior results with regard to zone localization.

Table 5.

Improvements afforded by fuzzy rectangular zone splitting compared to fuzzy triangular splitting.

Standard deviation (dB m)	Improvements afforded by fuzzy rectangular splitting over triangular splitting $(R_{i})$ (%)
3	54.7
5	53.8
7	57.1
9	58.7

In the previous simulation, we examined two zone-splitting methods based on fuzzy inference. In an indoor environment, fuzzy rectangular splitting results in higher resolution and superior positioning performance. The above results were averaged from all of the regions; however, we also investigated the performance of these methods in specific regions. This was achieved by deploying 50 sampling points in each zone (to ensure a sufficient number of samples), and then applying the rectangular splitting method with fuzzy inference with the aim of identifying zones in which positioning performance was not up to par. Figure 26 presents the average positioning performance in each zone.

Figure 26.

Positioning performance of fuzzy rectangular splitting method in nine specific zones.

As shown in Figure 26, in the case of $σ = 3$ and $σ = 5$ , the percentages of correct assessments in the nine zones were 70% or greater. However, in the case of $σ = 7$ and $σ = 9$ , positioning performance was good in zones 1, 3, 5, 7, and 9, but poor in zones 2, 4, 6, and 8, due to the lack of reference nodes.

Analysis of coordinate positioning in second stage

In the following, we analyze coordinate positioning performance in the second stage. We begin by using the linear interpolation distance measuring method to reduce the complexity of the curve channel model. Table 6 lists the parameter setting used in linear interpolation. Table 7 presents a comparison of mean error and standard deviation imposed by the linear interpolation distance measuring method under four levels of standard deviation.

Table 6.

Parameter settings used in linear interpolation.

Standard deviation parameter of the channel model	3, 5, 7, 9 dB m
Measurement spacing	0.1 m
Measuring range	0.1–3 m
The number of RSSI values collected per measurement	10

RSSI: received signal strength indicator.

Table 7.

Linear interpolation ranging and actual distance mean error and standard error.

Standard deviation parameter $σ$ (dB m)	Mean error (cm)	Standard error (cm)
3	2.7	2.1
5	2.1	1.8
7	2.2	1.3
9	3.8	2.4

In Table 7, the observed mean error induced by linear interpolation within a distance of 3 m was 2–4 cm under four levels of standard deviation, whereas the standard error was controlled to within 2 cm. These results demonstrate the efficacy of the proposed linear interpolation method in reducing computational complexity and converting RSSI values into distance values.

In the following, we discuss the performance of the proposed adaptive trilateral positioning method (second stage) under four levels of standard deviation.

For the sake of consistency and completeness, we employed data compiled in this study to evaluate the proposed fuzzy splitting method and KNN-based localization. The KNN method is based on comparative searches of fingerprint profiles aimed at selecting the K closest samples with the aim of minimizing discrepancies between the query RSS sample and the profiled ones. For the KNN-based localization method, we collected fingerprint samples at 36 positions uniformly distributed throughout an area of 3 m² using the parameter value K = 4.

Simulations were conducted using various zone-splitting methods to determine the mean error and standard error in the coordinate positions of 225 randomly selected sampling points.

We began by defining the error of the ith sample point ( $P_{error} (i)$ ) and the mean error for sampling points ( ${\bar{P}}_{error}$ ) as follows

$P_{error} (i) = \sqrt{{(x_{real} - x_{est} (i))}^{2} + {(y_{real} - y_{est} (i))}^{2}}$ (8)

${\bar{P}}_{error} = \frac{1}{N} \sum_{i = 1}^{N} \sqrt{{(x_{real} - x_{est} (i))}^{2} + {(y_{real} - y_{est} (i))}^{2}}$ (9)

where N is the total number of estimates. In this simulation, N was set to 255. $(x_{real}, y_{real})$ are the precise coordinates of the ith sampling point, and $(x_{est} (i), y_{est} (i))$ are the estimated coordinates.

The standard error in the N sampling points is derived as follows

$σ_{error} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(P_{error} (i) - {\bar{P}}_{error})}^{2}}$ (10)

Trilateral positioning allows for the selection of single-set or multi-set positioning methods. Tables 8 and 9 present a comparison of these methods.

Table 8.

Mean error in coordinate positioning under four levels of standard deviation.

Method		Standard deviation
		Mean error (cm)
		$σ = 3$	$σ = 5$	$σ = 7$	$σ = 9$
KNN-based method		34.5	40.2	44.6	50.8
Single	Traditional triangular splitting	33.3	41.7	48.9	55.9
	Fuzzy triangular splitting	41	42.4	46.4	46.8
	Fuzzy rectangular splitting	30.9	33.3	34.8	38.8
Multiple	Fuzzy triangular splitting	38.5	39.2	41	43.1
Multiple	Fuzzy rectangular splitting	22.6	26.7	29.1	35.7

KNN: K-Nearest Neighbors.

Table 9.

Comparison of standard error in coordinate positioning under four levels of standard deviation.

Method		Standard deviation
			Standard error (cm)
		$σ = 3$	$σ = 5$	$σ = 7$	$σ = 9$
KNN-based method		35.2	40.1	42.3	50.1
Single	Traditional triangular splitting	37.3	42.5	44.5	50.9
	Fuzzy triangular splitting	40.6	39.2	41.1	36.8
	Fuzzy rectangular splitting	26.1	27.8	29.3	30.7
Multiple	Fuzzy triangular splitting	32.8	34	35.4	37.8
Multiple	Fuzzy rectangular splitting	18.3	20.1	21.9	24.6

KNN: K-Nearest Neighbors.

Simulation results demonstrate that the proposed scheme outperforms the non-fuzzy positioning-based algorithms followed by the KNN-based and then by the traditional triangular splitting method. The results in Tables 8 and 9 show that fuzzy rectangular splitting outperformed the other methods, regardless of whether it was implemented in a single set or multiple sets. Multiple-set implementations of fuzzy rectangular as well as fuzzy triangular splitting exceeded their single-set counterparts. In the case of σ = 3, the mean error of multiple set fuzzy rectangular splitting was 8.3 cm less than the standard error obtained using a single set. However, in the case of σ = 9, larger fluctuations in RSSI reduced the difference to only 3.1 cm. From Table 9, we can see that rectangular splitting used in conjunction with fuzzy inference and multiple sets of triangles can control the standard error to within 25 cm, which is far lower than that of other methods. Conventional triangular splitting reduces the mean error value to 10.7 cm with σ = 3. The areas created using triangular splitting with fuzzy inference are not equal. In an environment with higher standard deviation, triangular splitting with fuzzy inference outperforms the conventional splitting method.

Single-set implementations of all three zone-splitting methods resulted in average error exceeding 30 cm as well as a significant increase in the standard deviation, compared to trilateral positioning in multiple sets. We can also see that the mean error of the two methods combined with FIS has better performance than that of the traditional triangular splitting method in the second stage. The mean error and standard deviation imposed by two-stage fuzzy rectangular splitting were lower than those of two-stage triangular splitting. Our simulation results indicate that coordinate positioning would be optimized by selecting fuzzy rectangular splitting for the first stage and adaptive trilateral positioning for the second stage.

Extended positioning

In the following, we describe the extension of a 3 m² area into a 6 m² area with 13 reference nodes and five maps, as shown in Figure 21.

First, we determine which unit map should be used. Two-stage fuzzy rectangular splitting is applied to each map and 50 sampling points were deployed in each zone.

As shown in Figure 27, if a target node is within a non-overlapping area, the percentage of correct estimations approaches 100%. As shown in Figure 28, if the target node lies in an overlapping area, map selection becomes important. Differences in the relative position of zones in different maps of the same location produce differences in the positioning error, which can be used for two-stage positioning. Thus, we focused on areas of topological overlap in our analysis of positioning performance within different maps.

Figure 27.

Topological distribution of non-overlapping areas.

Figure 28.

Topological distribution of overlapping areas.

We used the channel model with $σ = 5$ in our analysis of maps in the central overlapping area using the fuzzy rectangular splitting method. As shown in Figure 29, four sampling points were placed in each zone, with each sample point covering two maps simultaneously. Two-stage positioning was applied to each sample 10 times. The mean error and standard deviation in positioning are presented in Tables 10 –17.

Figure 29.

Positions of sampling points in overlapping range.

Table 10.

Positioning results from first overlapping zone.

Coordinates of sampling points	Map	First-stage positioning results	Second-stage positioning results
Coordinates of sampling points	Map	First-stage positioning results	Mean error (cm)	Standard error (cm)
(1.75, 1.75)	B	Zone 5	11.3	5
(1.75, 1.75)	Y	Zone 1	13.8	7.8
(2.25, 1.75)	B	Zone 6	9.8	3.6
(2.25, 1.75)	Y	Zone 1	14.6	5
(1.75, 2.25)	B	Zone 8	10.3	4.6
(1.75, 2.25)	Y	Zone 1	15.9	5.3
(2.25, 2.25)	B	Zone 9	13.5	4.1
(2.25, 2.25)	Y	Zone 1	17.2	5.5

Table 11.

Positioning results from second overlapping zone.

The coordinate of sampling points	Map	First-stage positioning results	Second-stage positioning results
The coordinate of sampling points	Map	First-stage positioning results	Mean error (cm)	Standard error (cm)
(2.75, 1.75)	B	Zone 6	23.6	7
(2.75, 1.75)	Y	Zone 2	25.5	7.9
(3.25, 1.75)	R	Zone 4	19.3	7.7
(3.25, 1.75)	Y	Zone 2	20.6	8
(2.75, 2.25)	B	Zone 9	15.8	8.1
(2.75, 2.25)	Y	Zone 2	12.2	5.5
(3.25, 2.25)	R	Zone 7	17.5	5.7
(3.25, 2.25)	Y	Zone 2	12.5	5

Table 12.

Positioning results from third overlapping zone.

Coordinates of sampling points	Map	First-stage Positioning results	Second-stage positioning results
Coordinates of sampling points	Map	First-stage Positioning results	Mean error (cm)	Standard error (cm)
(3.75, 1.75)	R	Zone 4	11.4	6.3
(3.75, 1.75)	Y	Zone 3	17.7	9.2
(4.25, 1.75)	R	Zone 5	11.1	5
(4.25, 1.75)	Y	Zone 3	16.1	6.2
(3.75, 2.25)	R	Zone 7	15.4	10
(3.75, 2.25)	Y	Zone 3	23.1	13.1
(4.25, 2.25)	R	Zone 8	10	5.7
(4.25, 2.25)	Y	Zone 3	15.4	5.7

Table 13.

Positioning results from fourth overlapping zone.

Coordinates of sampling points	Map	First-stage positioning results	Second-stage positioning results
Coordinates of sampling points	Map	First-stage positioning results	Mean error (cm)	Standard error (cm)
(1.75, 2.75)	B	Zone 8	21.3	13
(1.75, 2.75)	Y	Zone 4	21.6	13.6
(2.25, 2.75)	B	Zone 9	16.5	7.2
(2.25, 2.75)	Y	Zone 4	12.6	5.2
(1.75, 3.25)	G	Zone 2	19.3	7.5
(1.75, 3.25)	Y	Zone 4	21	7.3
(2.25, 3.25)	G	Zone 3	19	4.9
(2.25, 3.25)	Y	Zone 4	10.8	4

Table 14.

Positioning results from fifth overlapping zone.

Coordinates of sampling points	Map	First-stage positioning results	Second-stage positioning results
Coordinates of sampling points	Map	First-stage positioning results	Mean error (cm)	Standard error (cm)
(3.75, 2.75)	R	Zone 7	15.1	6.2
(3.75, 2.75)	Y	Zone 6	9.8	3.1
(4.25, 2.75)	R	Zone 8	18.8	6.4
(4.25, 2.75)	Y	Zone 6	21.1	6.7
(3.75, 3.25)	P	Zone 1	17.6	7.8
(3.75, 3.25)	Y	Zone 6	11.2	4
(4.25, 3.25)	P	Zone 2	18.5	6.8
(4.25, 3.25)	Y	Zone 6	19.1	7.2

Table 15.

Positioning results from sixth overlapping zone.

Coordinates of sampling points	Map	First-stage positioning results	Second-stage positioning results
Coordinates of sampling points	Map	First-stage positioning results	Mean error (cm)	Standard error (cm)
(1.75, 3.75)	G	Zone 2	11.1	3.1
(1.75, 3.75)	Y	Zone 7	13.6	7.5
(2.25, 3.75)	G	Zone 3	17.4	7.3
(2.25, 3.75)	Y	Zone 7	23.4	18.2
(1.75, 4.25)	G	Zone 3	12.4	5.8
(1.75, 4.25)	Y	Zone 7	14.2	9.1
(2.25, 4.25)	G	Zone 6	11.6	6.4
(2.25, 4.25)	Y	Zone 7	17.3	9.6

Table 16.

Positioning results from seventh overlapping zone.

Coordinates of sampling points	Map	First-stage positioning results	Second-stage positioning results
Coordinates of sampling points	Map	First-stage positioning results	Mean error (cm)	Standard error (cm)
(2.75, 3.75)	G	Zone 3	18.7	7
(2.75, 3.75)	Y	Zone 8	12.1	2.4
(3.25, 3.75)	P	Zone 1	13.6	9.5
(3.25, 3.75)	Y	Zone 8	11.2	4.9
(2.75, 4.25)	G	Zone 6	17.7	4.5
(2.75, 4.25)	Y	Zone 8	19.7	6.6
(3.25, 4.25)	P	Zone 4	18.1	5.9
(3.25, 4.25)	Y	Zone 8	18.3	8.5

Table 17.

Positioning results from eighth overlapping zone.

Coordinates of sampling points	Map	First-stage positioning results	Second-stage positioning results
Coordinates of sampling points	Map	First-stage positioning results	Mean error (cm)	Standard error (cm)
(3.75, 3.75)	P	Zone 1	14.5	7.6
(3.75, 3.75)	Y	Zone 9	20.9	12.2
(4.25, 3.75)	P	Zone 2	11	7.7
(4.25, 3.75)	Y	Zone 9	19.5	7.7
(3.75, 4.25)	P	Zone 4	10.2	4.5
(3.75, 4.25)	Y	Zone 9	18.5	6.3
(4.25, 4.25)	P	Zone 5	13.3	7.4
(4.25, 4.25)	Y	Zone 9	17.4	7.5

In the simulation results presented in Tables 10 to 17, the map of sampling points marked in color was obtained using the proposed method. The fact that positioning performance is highest in overlapping areas demonstrates the efficacy of the proposed unit map. According to the above data results, we mark in accordance with the color of the suitable positioning map for each sampling point, as shown in Figure 30.

Figure 30.

Optimal positioning of maps in overlapping area.

As shown in the figure, if the sample point is within 1 m from the central reference node, then we can select central map Y for two-stage positioning to improve positioning accuracy. Otherwise, a neighboring map could be selected to improve positioning performance. Such cases are termed “single topological preferred positioning.”

According to Tables 11, 13, 14, and 16, eight sampling points are marked using black triangles in Figure 30. The difference in mean error between two neighboring maps is no more than 2 cm, and the difference in standard deviation error is no more than 3 cm. Therefore, the location of these eight sampling points can be selected from one of the two neighboring maps. We achieved 100% correct positioning in non-overlapping areas. This method can be applied to large indoor areas while enabling the selection of the best map for two-stage positioning and retaining the advantages of high positioning resolution.

Conclusion

This article analyzes the performance of two-stage (coarse/fine) indoor WSN positioning schemes using a variety of zone-splitting methods in conjunction with fuzzy inference. The RSSI between the reference node and target node is used to estimate the zone without the need for additional expensive hardware. The proposed method is fast and does not require training or a large number of measurement points for the formulation of a database. In the first stage, we use zone splitting to estimate the position of a target node within a zone. We employed fuzzy rectangular splitting as well as fuzzy triangular splitting to enhance the accuracy of zone-positioning within an indoor environment. Simulation results demonstrate that fuzzy rectangular splitting outperforms fuzzy triangular splitting in zone-positioning estimates in a variety of indoor environments.

As long as the zone containing the target is correctly identified, advanced trilateration can be used to achieve the goal of two-stage indoor location. In the second stage of localization, a combination of reference nodes is used to formulate a triangulation algorithm based on the position of the zone identified in the first stage. If the zone identified in the first stage is correct, then the selection of a reference node combination can enhance positioning accuracy. If the zone identified in the first stage is incorrect, then the selection mechanism can opt for an alternative combination of reference nodes to reduce positioning error.

We also developed an extended positioning scheme to expand the applicability of this approach from the original rectangular area (3 m × 3 m) to a large rectangular area (6 m × 6 m). The proposed scheme identified the best rectangular map for implementation using a two-stage positioning method. The selection of an appropriate map also helps to retain high positioning resolution, even within an expanded environment.

In the future, we hope to apply the indoor positioning method proposed in this article to indoor environments of various sizes, from small homes to large indoor shopping malls.

Footnotes

Handling Editor: Wenbing Zhao

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) received no financial support for the research,authorship,and/or publication of this article.

References

Boman

Taylor

Ngu

. Flexible IoT middleware for integration of things and applications. In: IEEE international conference on collaborative computing: networking, applications and worksharing, Miami, FL, 22–25 October 2014, pp.481–488. New York: IEEE.

Xiong

Wang

et al . Application research of WSN in precise agriculture irrigation. In: International conference on environmental science and information application technology, Wuhan, China, 4–5 July 2009, vol. 2, pp.297–300. New York: IEEE.

Dey

Hightower

de Lara

et al . Location-based services. IEEE Pervas Comput 2010; 9: 11–12.

Wang

Zhang

. Research on the incomplete constellation GPS positioning algorithm aided by altitude. In: Proceedings of the international conference on electrical and control engineering (ICECE), Wuhan, China, 25–27 June 2010, pp.2974–2977. New York: IEEE.

Kharidia

Sampalli

et al . HILL: a hybrid indoor localization scheme. In: Proceedings of the 10th international conference on mobile ad-hoc and sensor networks (MSN), Maui, HI, 19–21 December 2014, pp.201–206. New York: IEEE.

Shin

Lee

et al . Enhanced weighted k-nearest neighbor algorithm for indoor Wi-Fi positioning systems. In: Proceedings of the 8th international conference on computing technology and information management (ICCM), Seoul, South Korea, 24–26 April 2012, vol. 2, pp.574–577. New York: IEEE.

Pires

Gracioli

Wanner

et al . Evaluation of an RSSI-based positioning algorithm for wireless sensor networks. IEEE Lat Am T 2011; 9: 830–835.

Ahmad

. Localization of wireless sensor networks using a mobile beacon. In: Proceedings of the IEEE international conference on pervasive computing and communication workshops (PerCom workshops), Sydney, NSW, Australia, 14–18 March 2016, pp.1–3. New York: IEEE.

Wang

Yang

Zhao

et al . Bluetooth positioning using RSSI and triangulation methods. In: Proceedings of the IEEE consumer communications and networking conference (CCNC), Las Vegas, NV, 11–14 January 2013, pp.837–842. New York: IEEE.

10.

Yang

Chen

Cui

et al . Probabilistic-KNN: a novel algorithm for passive indoor-localization scenario. In: Proceedings of the IEEE 81st vehicular technology conference (VTC Spring), Glasgow, 11–14 May 2015, pp.1–5. New York: IEEE.

11.

Yin

Cai

et al . Neural-network-assisted UE localization using radio-channel fingerprints in LTE networks. IEEE Access 2017; 5: 12071–12087.

12.

Yang

Zhen

et al . An area zoning based WLAN location system. In: Proceedings of the international communication conference on wireless mobile and computing (CCWMC), Shanghai, China, 7–9 December 2009, pp.437–440. New York: IEEE.

13.

Feng

WSA

. Received-signal-strength-based indoor positioning using compressive sensing. IEEE T Mobile Comput 2012; 11: 1983–1993.

14.

Salazar

Aguilar

Licea

. Estimating indoor zone-level positioning using Wi-Fi RSSI fingerprinting based on fuzzy inference system. In: Proceedings of the international conference on mechatronics, electronics and automotive engineering (ICMEAE), Morelos, Mexico, 19–22 November 2013, pp.178–184. New York: IEEE.

15.

Chenji

Stoleru

. Toward accurate mobile sensor network localization in noisy environments. IEEE T Mobile Comput 2013; 12: 1094–1106.

16.

Abdelhadi

Anan

. A fuzzy-based collaborative localization algorithm for wireless sensor networks. In: Proceedings of the international conference on collaboration technologies and systems (CTS), Denver, CO, 21–25 May 2012, pp.152–156. New York: IEEE.

17.

Alwarafy

Sulyman

Alsanie

et al . Receiver spatial diversity propagation path-loss model for an indoor environment at 2.4GHz. In: Proceedings of the 6th international conference on network of future (NOF), Montreal, QC, Canada, 30 September–2 October 2015, pp.1–4. New York: IEEE.

18.

Parwekar

Reddy

. An efficient fuzzy localization approach in Wireless Sensor Networks. In: Proceedings of the international conference on fuzzy systems (FUZZ), Hyderabad, India, 7–10 July 2013, pp.1–6. New York: IEEE.

19.

Chen

Yang

Yao

et al . Research on ZigBee localization reference node layout scheme. China Science and Technology Information 2013; 12: 89.

20.

Yue

Maple

et al . Three-dimensional localisation for wireless sensor networks using probabilistic fuzzy logic. In: Proceedings of the international conference on fuzzy system and knowledge discovery (FSKD), Shenyang, China, 23–25 July 2013, pp.396–401. New York: IEEE.

21.

Jianwu

. Research on distance measurement based on RSSI of ZigBee. In: Proceedings of the international colloquium on computing, communication, control, and management (ISECS), Sanya, China, 8–9 August 2009, vol. 3, pp.210–212. New York: IEEE.

22.

Larranaga

Muguira

Lopez-Garde

et al . An environment adaptive ZigBee-based indoor positioning algorithm. In: Proceedings of the international conference on indoor positioning and indoor navigation (IPIN), Zurich, 15–17 September 2010, pp.1–8. New York: IEEE.

23.

Xiong

Qin

Zeng

. A distance measurement wireless localization correction algorithm based on RSSI. In: Proceedings of the 7th international symposium on computational intelligence and design (ISCID), Hangzhou, China, 13–14 December 2014, vol. 2, pp.276–278. New York: IEEE.

24.

Yang

Liu

. Beyond trilateration: on the localizability of wireless ad hoc networks. IEEE ACM T Network 2010; 18: 1806–1814.

25.

Roh

Choe

Jung

et al . Position estimation of Landmark using 3D point cloud and trilateration method. In: Proceedings of the 11th international conference on ubiquitous robots and ambient intelligence (URAI), Kuala Lumpur, Malaysia, 12–15 November 2014, pp.298–302. New York: IEEE.

Indoor positioning system for wireless sensor networks based on two-stage fuzzy inference

Abstract

Keywords

Introduction

Related works

Zone-positioning

Trianglar splitting

Fuzzy triangular splitting method

Fuzzy rectangular splitting method

Fuzzy inference

Coordinate positioning

Distance measurement using linear interpolation

Adaptive trilateral positioning

Three circles intersecting within a common area

Two circles intersecting within a circle

Two circles intersecting with one separate circle

Selection mechanism used for multi-set positioning reference points

Extended positioning

Simulation results and analysis

Efficiency analysis: first-stage zone-positioning

Analysis of coordinate positioning in second stage

Extended positioning

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References