Robust Localization Algorithm Based on the RSSI Ranging Scope

Abstract

Wireless signal can be easily influenced by the environment in the propagation process. The signal propagation model is the most appropriate model for current indoor environment to ensure the ranging accuracy based on received signal strength indicator (RSSI). In this paper, we propose a robust localization algorithm based on the RSSI ranging scope by which the RSSI ranging error caused by using a fixed parameter in signal propagation model is dramatically eliminated. Our contributions in this paper are twofold. First, the influence of RSSI ranging error on positioning accuracy is well discussed in detail in the scope of the wireless signal propagation model. Second, we develop a robust localization algorithm which creates a one-to-one mapping between the RSSI value and the distance scope based on the value scope of path loss exponent in the signal propagation model. Simulation results indicate that the proposed localization algorithm based on the RSSI ranging scope is robust under different environments, when the real path loss exponent is difficult to measure accurately.

1. Introduction

Wireless sensor networks (WSNs) are one fundamental component of the Internet of Things (IOT). The localization capability of WSNs has attracted more and more attention, because it can be applied for localization and tracking of targets in the IOT [1]. In the literature, localization approaches for WSNs can be roughly divided into two categories: range-based and range-free [2]. Range-based localization approaches utilize time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), and received signal strength indicator (RSSI) to measure the distance between two nodes [3, 4]. Range-free localization approaches only utilize connectivity and proximity to estimate the region where a node stays [5–8]. Therefore, compared with range-free localization approaches, range-based localization approaches are more accurate at the expense of more source consumption.

Because all of commercial wireless chips are capable of computing received signal strength, RSSI based localization algorithms have been widely used [9–13]. Firstly, RSSI based localization algorithms take advantage of the less signal propagation loss model to measure the distances from a node to the beacons. Then, the trilateral algorithm, the least square algorithm, or the maximum likelihood estimation algorithm could be applied to calculate the coordinates of the unknown nodes. It is obvious that the accuracy of a RSSI based localization algorithm is decided by the wireless signal propagation loss model. In most of the previous works, the parameters of the model are calculated by measuring the RSSI values between the beacons [14–16]. Since wireless signal can be easily influenced by the environment in propagation process, signal propagation characteristics will be different at the same time in different regions or at different instants of time in the same region. It is hard to measure the parameters of wireless signal propagation loss model accurately [17–19]. Therefore, it is inaccurate that all nodes in the localization area use fixed and same parameters of wireless signal propagation loss model to measure the distances. There are other related works: [3, 4] utilize range-based localization with ranging errors; [5–7] use a sensor array to detect locations of intrusion objects.

In this paper, we firstly analyze the influence of the RSSI ranging error on positioning accuracy in great depth. Then, we propose a robust localization algorithm based on the RSSI ranging scope. In order to eliminate the error caused by using the fixed and same parameter of the signal propagation model, our proposed algorithm creates a one-to-one mapping between the RSSI value and the distance scope based on the value scope of the parameter in the signal propagation model. Moreover, we conduct extensive simulations to evaluate the proposed algorithm. Based on the evaluation results, the proposed algorithm is greatly adaptable to a dynamic environment.

The remainder of this paper is organized as follows. Section 2 discusses the influence of the RSSI ranging error on positioning accuracy. Then, the robust localization algorithm is proposed in Section 3. Simulation results are provided in Section 4, and Section 5 concludes the paper.

2. Analysis of RSSI Based Ranging

In a RSSI based ranging algorithm, a node applies RSSI measurements to estimate its distances from the beacons, by using a known signal propagation model. Since signal attenuation is often related to the specific environment, the signal propagation model, the most appropriate model for the current environment, is essential to ensure the ranging accuracy. Currently, the shadowing model is widely used to model wireless signal propagation loss [20], which is expressed as

\begin{matrix} P_{r} (d) = P_{r} (d_{0}) - 10 n lo g_{10} (\frac{d}{d_{0}}) + X_{σ}, \end{matrix}

(1)

where d and

d_{0}

denote the real distance and the reference distance, respectively;

P_{r} (d)

and

P_{r} (d_{0})

denote the received signal power in dBm at the real distance and the reference distance, respectively; n is the path loss exponent and

X_{σ}

is a random variable representing the noise in the measured

P_{r} (d)

. For most of the indoor applications,

d_{0} = 1

meter and

P_{r} (d_{0})

is calculated by free space path loss formula. The noise

X_{σ}

is incurred by both time varying and time-invariant sources. In order to model the random effects of shadowing appropriately, we assume that

X_{σ}

is a Gaussian distributed random variable with zero mean and variance

σ^{2}

. The path loss exponent n is determined by the environmental variables and surrounding structure. It is easy to see that the ranging error is mainly caused by the influence of the physical environment. Thus, the simplified shadowing model is used as follows:

\begin{matrix} RSSI (d) = A - 10 n lg d, \end{matrix}

(2)

where

RSSI (d)

is the received signal power at the distance d and A is the received signal power of the receiver from a transmitter one meter away. In (2), the parameters A and n are closely related to the specific hardware and environment. In particular, the path loss exponent n is dynamically varied within a certain range depending on the characteristics of the environment [21], as shown in Table 1.

Table 1

Path loss index n under different environments.

The environment	Path loss exponent n
Office	1.4~2.5
Corridor	1.9~2.5
Stairs and balconies	1.4~2.4
Park	2.7~3.4
Fences	4.6~5.1
Alley	2.1~3.0

If A and n are accurate in the current indoor environment, then the RSSI based ranging is perfect. In most of the RSSI based ranging algorithms, the parameters of the signal propagation model are calculated by a set of online or offline RSSI measurements between the beacons. Of course, online RSSI measurements consume computation and communication. However, this is not the case in a practical environment where the signal propagation model is extremely difficult to estimate [22]. The parameters of the estimated model may not accurately reflect the real radio channel of an indoor environment at any particular time and space.

3. Robust Localization Based on the RSSI Ranging Scope

3.1. The Influence of RSSI Ranging Error on Positioning Accuracy

In the localization area, assume that the known coordinate of the ith beacon $B_{i}$ is $(x_{B i}, y_{B i})$ , the coordinate of the unknown node T is $(x, y)$ , ${RSSI}_{i}$ is the RSSI measurement between the unknown node T and the ith beacon $B_{i}$ , and $d_{i}$ is the estimated distance between the unknown node T and the ith beacon $B_{i}$ obtained by (2). In most of the RSSI based localization algorithms, the trilateral algorithm, the least squares algorithm, or the maximum likelihood estimation algorithm is applied to calculate the coordinate of the unknown node. In real cases, three beacons are enough to locate an unknown node.

Assume that there are three beacons $B_{i}$ ( $i = 1,2, 3$ ) and all the estimated distances $d_{i}$ ( $i = 1,2, 3$ ) are accurate. Then, the coordinate of the unknown node T can be obtained by equations as follows:

\begin{matrix} {(x - x_{B 1})}^{2} + {(y - y_{B 1})}^{2} = d_{1}^{2}, \\ {(x - x_{B 2})}^{2} + {(y - y_{B 2})}^{2} = d_{2}^{2}, \\ {(x - x_{B 3})}^{2} + {(y - y_{B 3})}^{2} = d_{3}^{2} . \end{matrix}

(3)

According to (3), there is a localization circle for each beacon

B_{i}

. Three localization circles intersect at one point

(x, y)

. By using the prior two equations minus the third equation, respectively, we can derive the following equations:

\begin{array}{l} 2 (x_{B 3} - x_{B 1}) x + 2 (y_{B 3} - y_{B 1}) y \\ = x_{B 3}^{2} - x_{B 1}^{2} + y_{B 3}^{2} - y_{B 1}^{2} + d_{1}^{2} - d_{3}^{2}, \\ 2 (x_{B 3} - x_{B 2}) x + 2 (y_{B 3} - y_{B 2}) y \\ = x_{B 3}^{2} - x_{B 2}^{2} + y_{B 3}^{2} - y_{B 2}^{2} + d_{2}^{2} - d_{3}^{2} . \end{array}

(4)

Expressing (4) in matrix form, we can further get

\begin{array}{l} [\begin{array}{l} x \\ y \end{array}] & = & {[\begin{array}{l} 2 (x_{B 3} - x_{B 1}) 2 (y_{B 3} - y_{B 1}) \\ 2 (x_{B 3} - x_{B 2}) 2 (y_{B 3} - y_{B 2}) \end{array}]}^{- 1} \\ \cdot [\begin{array}{l} x_{B 3}^{2} - x_{B 1}^{2} + y_{B 3}^{2} - y_{B 1}^{2} + d_{1}^{2} - d_{3}^{2} \\ x_{B 3}^{2} - x_{B 2}^{2} + y_{B 3}^{2} - y_{B 2}^{2} + d_{2}^{2} - d_{3}^{2} \end{array}] . \end{array}

(5)

Wireless signal propagation process can be easily influenced by building materials and types, including even people in real indoor environment. Since signal propagation characteristics will be different at the same time in different regions or at different instants of time in the same region, it is hard to accurately estimate the parameters of the wireless signal propagation loss model. The RSSI measurement is disturbed by channel noise, obstacles, and other shadowing effects. As a result, the distance estimation is inaccurate by using same and fixed parameters of the wireless signal propagation loss model. Therefore, instead of utilizing the real distances $d_{i}$ , it is reasonable to use noisy estimations ${\hat{d}}_{i}$ , and (3) can be written as

\begin{matrix} {(\hat{x} - x_{B 1})}^{2} + {(\hat{y} - y_{B 1})}^{2} = {\hat{d}}_{1}^{2} \\ {(\hat{x} - x_{B 2})}^{2} + {(\hat{y} - y_{B 2})}^{2} = {\hat{d}}_{2}^{2} \\ {(\hat{x} - x_{B 3})}^{2} + {(\hat{y} - y_{B 3})}^{2} = {\hat{d}}_{3}^{2}, \end{matrix}

(6)

where

(\hat{x}, \hat{y})

is the estimated coordinate of the unknown node T. Then, we can correspondingly get

\begin{array}{l} [\begin{array}{l} \hat{x} \\ \hat{y} \end{array}] & = & {[\begin{array}{l} 2 (x_{B 3} - x_{B 1}) 2 (y_{B 3} - y_{B 1}) \\ 2 (x_{B 3} - x_{B 2}) 2 (y_{B 3} - y_{B 2}) \end{array}]}^{- 1} \\ \cdot [\begin{array}{l} x_{B 3}^{2} - x_{B 1}^{2} + y_{B 3}^{2} - y_{B 1}^{2} + {\hat{d}}_{1}^{2} - {\hat{d}}_{3}^{2} \\ x_{B 3}^{2} - x_{B 2}^{2} + y_{B 3}^{2} - y_{B 2}^{2} + {\hat{d}}_{2}^{2} - {\hat{d}}_{3}^{2} \end{array}] . \end{array}

(7)

Using (7) minus (5), we can get the localization error of the unknown node T as shown below:

\begin{array}{l} [\begin{array}{l} \hat{x} - x \\ \hat{y} - y \end{array}] & = & {[\begin{array}{l} 2 (x_{B 3} - x_{B 1}) 2 (y_{B 3} - y_{B 1}) \\ 2 (x_{B 3} - x_{B 2}) 2 (y_{B 3} - y_{B 2}) \end{array}]}^{- 1} \\ \cdot [\begin{array}{l} ({\hat{d}}_{1}^{2} - d_{1}^{2}) - ({\hat{d}}_{3}^{2} - d_{3}^{2}) \\ ({\hat{d}}_{2}^{2} - d_{2}^{2}) - ({\hat{d}}_{3}^{2} - d_{3}^{2}) \end{array}] . \end{array}

(8)

Based on (8), it shows that the localization error of the trilateral algorithm is related to the coordinates of the beacons and the RSSI ranging errors. If we assume that the coordinates of the beacons and the RSSI ranging errors are independent, the localization error of the trilateral algorithm is determined by the RSSI ranging errors when the beacons are deployed properly. Thus, this paper only studies the influence of the RSSI ranging error on positioning accuracy, which is expressed as

\begin{array}{l} E & = & [\begin{array}{l} ({\hat{d}}_{1}^{2} - d_{1}^{2}) - ({\hat{d}}_{3}^{2} - d_{3}^{2}) \\ ({\hat{d}}_{2}^{2} - d_{2}^{2}) - ({\hat{d}}_{3}^{2} - d_{3}^{2}) \end{array}] \\ = & [\begin{array}{l} (10^{A - {RSSI}_{1} / 5 n_{1}} - 10^{A - {RSSI}_{1} / 5 n_{r 1}}) - (10^{A - {RSSI}_{3} / 5 n_{3}} - 10^{A - {RSSI}_{3} / 5 n_{r 3}}) \\ (10^{A - {RSSI}_{2} / 5 n_{2}} - 10^{A - {RSSI}_{2} / 5 n_{r 2}}) - (10^{A - {RSSI}_{3} / 5 n_{3}} - 10^{A - {RSSI}_{3} / 5 n_{r 3}}) \end{array}], \end{array}

(9)

where $n_{i}$ and $n_{r i}$ ( $i = 1,2, 3$ ) are the path loss exponent in ranging process and the real path loss exponent in the current environment, respectively.

For each beacon $B_{i}$ , $n_{i}$ may be smaller than or larger than $n_{r i}$ in the localization process. It is difficult to evaluate the value of E, according to (9). To simplify the analysis, assume that $n_{i}$ is either smaller than or larger than $n_{r i}$ for all beacons. Figure 1 shows the region division based on the RSSI value when $n_{i} < n_{r i}$ for all beacons. ${\hat{d}}_{i}$ calculated by (2) is larger than the real distance $d_{i}$ because of $n_{i} < n_{r i}$ for all beacons. Then, a crossing region C is generated by three localization circles. Therefore, the unknown node T is supposed to be located in the crossing region C. On the contrary, if $n_{i} > n_{r i}$ for all beacons, ${\hat{d}}_{i}$ calculated by (2) is less than the real distance $d_{i}$ so that there is no crossing region among three localization circles as shown in Figure 2, and obviously it is difficult to determine the region where the unknown node T lies. If the $n_{i}$ of each beacon is either smaller than or larger than $n_{r i}$ and each ${\hat{d}}_{i}$ is either less than or greater than $d_{i}$ , a crossing region or one or two crossing points will appear. If one or two crossing points appear, it is difficult to determine the region where the unknown node T lies. If a crossing region appears, the unknown node T lies in the crossing region just like Figure 1.

Figure 1

Region division based on the RSSI value when $n < n_{r}$ .

Figure 2

Region division based on the RSSI value when $n > n_{r}$ .

3.2. The Localization Algorithm Based on the RSSI Ranging Scope

As mentioned earlier, due to the inaccuracy of the path loss exponent, distances based on the RSSI ranging may have errors. Therefore, the estimated coordinate obtained by the trilateral algorithm will not be accurate. To get higher positioning accuracy, we need to adopt the signal propagation loss model which is more appropriate for the current indoor environment. The signal propagation is easily influenced by environmental factors; hence it is extremely difficult to develop a graph between the RSSI value and the distance. Since the value of n is varied within a certain range depending on the characteristics of the environment, a localization algorithm based on the RSSI ranging scope is proposed. When it is hard to measure the path loss exponent n accurately, a value scope of n is measured instead of using a fixed value. If a fixed value or a value scope of n is given, a one-to-one mapping between the RSSI value and the distance can be created.

Assume that the value scope of the real path loss exponent $n_{r}$ in the current environment is known, which is expressed as $n_{r} \in [n_{r 1}, n_{r 2}]$ . The distance scope can be calculated as follows:

\begin{matrix} {RSSI}_{i} = A - 10 n_{r 1} lg d_{i 1} \\ RSS I_{i} = A - 10 n_{r 2} lg d_{i 2}, \end{matrix}

(10)

where

d_{i 1}

and

d_{i 2}

are the estimated distances when the path loss exponents are

n_{r 1}

and

n_{r 2}

, respectively. The greater the path loss exponent is, the smaller the estimated distance is. In the localization process, three distance scopes between the unknown node T and the beacon

B_{i}

are needed. The reason of using three distance scopes instead of two distance scopes is explained as follows. For a RSSI based localization algorithm, such as the trilateral algorithm, least squares algorithm, or maximum likelihood estimation algorithm, three distance scopes are needed at least. The more the distance scopes that we use are, the more accurate the estimated coordinate will be. By creating a one-to-one mapping between the RSSI value and the distance scope, two distance scopes can build a crossing region. But when more distance scopes are used, the estimated coordinate will be more accurate.

Being different from the localization algorithm based on the RSSI ranging value, $n_{r} \in [n_{r 1}, n_{r 2}]$ in the localization algorithm is based on the RSSI ranging scope, and then the real distance is expressed as $d_{r i} \in [d_{i 2}, d_{i 1}]$ . Figure 3 shows the region division based on the RSSI ranging scope method. Each beacon $B_{i}$ determines a circle area $C_{i}$ according to its own ranging scope $[d_{i 2}, d_{i 1}]$ . Therefore, the unknown node T lies in the crossing region $C = C_{1} \cap C_{2} \cap C_{3}$ , which is generated by three circle areas. Obviously, the localization error depends on the area of the crossing region C. In the context of RSSI ranging scope, in fact, the crossing region C could also be determined by any two of the three circles ( $C_{1}$ , $C_{2}$ , and $C_{3}$ ). However, three ranging scopes would give better localization estimation than any two ranging scopes. Thus, we still use three beacons in the proposed localization algorithm which is based on the RSSI ranging scope. Comparing with Figures 1 and 3, we observe that the area of the crossing region C given by the localization algorithm based on the RSSI ranging scope is smaller and this indicates that the localization estimation is more accurate. Figure 1 shows the region division based on RSSI value when $n_{i} < n_{r i}$ . If $n_{i} > n_{r i}$ , it is impossible to get the region where the unknown node $T_{i}$ lies. Therefore, in the localization algorithm based on the RSSI ranging value, we take $n = n_{r 1}$ , while in the localization algorithm based on RSSI ranging scope, we take $n_{r} \in [n_{r 1}, n_{r 2}]$ . The $d_{i 1}$ in Figure 3 equals the $d_{i}$ in Figure 1. The crossing region given by the range scope scheme is smaller than that given by the range value scheme, because of the $d_{i 2}$ .

Figure 3

Region division based on the RSSI ranging scope.

In RSSI based ranging algorithms, the parameters of the signal propagation model should be calculated at first. Considering that it is hard to measure the path loss exponent n accurately, a value scope of n is measured instead of a fixed value in the proposed algorithm. It is more simple to obtain a value scope $[n_{r 1}, n_{r 2}]$ than a value n. The value scope $[n_{r 1}, n_{r 2}]$ can be calculated using a set of online or offline RSSI measurements between the beacons.

Both localization algorithms based on the RSSI ranging value and based on the RSSI ranging scope need to calculate the estimated coordinate $(\hat{x}, \hat{y})$ of the unknown node T after obtaining the crossing region C. For the consideration of the hardware and energy consumption of sensor nodes, the algorithm to compute the estimated coordinate should not be too complex. Assume that there is a group of boundary points of the crossing region C; $(x_{A i}, y_{A i})$ is the coordinate of the ith boundary point $A_{i}$ in the group. For example, there are three boundary points in Figure 1 while six boundary points in Figure 3. In order to decrease the complexity of the algorithm, we take the average of the coordinates of all $A_{i}$ as the estimated coordinate $(\hat{x}, \hat{y})$ :

\begin{matrix} \hat{x} = \frac{1}{a} \sum_{i = 1}^{a} x_{A i}, \hat{y} = \frac{1}{a} \sum_{i = 1}^{a} y_{A i} . \end{matrix}

(11)

4. Performance Evaluation

This section presents performance evaluation of localization algorithms based on the RSSI ranging value and based on the RSSI ranging scope. The experimental environment is depicted as in Figure 4. The localization area is defined as an equilateral triangle and each side is 50 m. Three beacons are deployed at three peaks of the equilateral triangle. There are $m = 10$ unknown nodes randomly deployed in the localization area. In the simulation, we have $A = - 40$ dB. The real path loss exponent $n_{r}$ is random and $n_{r} \in [n_{r 1}, n_{r 2}]$ in ranging between the unknown node $T_{i}$ and the beacon $B_{i}$ . If $n > n_{r}$ , it is impossible to get the region where the unknown node $T_{i}$ lies for the localization algorithm based on the RSSI ranging value, as shown in Figure 2. Therefore, we take $n = n_{r 1}$ .

Figure 4

Simulation scenario.

The following equation is used to evaluate the performance of two localization algorithms:

\begin{matrix} error = \frac{1}{m} \sum_{i = 1}^{m} \sqrt{{(x_{i} - {\hat{x}}_{i})}^{2} + {(y_{i} - {\hat{y}}_{i})}^{2}}, \end{matrix}

(12)

where

(x_{i}, y_{i})

and

({\hat{x}}_{i}, {\hat{y}}_{i})

are the real and estimated coordinates of the unknown node

T_{i}

, respectively.

4.1. Analysis of Localization Errors Caused by n

Table 2 shows the average localization errors of two localization algorithms obtained by conducting simulations 50 times while the value scope $[n_{r 1}, n_{r 2}]$ of the real path loss exponent varied and $n_{r 2} - n_{r 1} = 0.1$ . We can observe that the average localization errors of the two localization algorithms both decrease while the real path loss exponent increases. Meanwhile, the average localization errors of the proposed localization algorithm are always less than that of the localization algorithm based on the RSSI ranging value. When $n_{r} \in [2.0,2.3]$ , the differences of the average localization errors between two localization algorithms are more than ten centimeters. When $n_{r} \in [4.0,4.3]$ , the differences of average localization errors between two localization algorithms are only a few centimeters.

Table 2

Comparison of average localization errors under different environments.

Value scope $[n_{r 1}, n_{r 2}]$	Average localization error (m)
Value scope $[n_{r 1}, n_{r 2}]$	Based on RSSI ranging value	Based on RSSI ranging scope
$[2.0,2.1]$	2.0296	1.8809
$[2.1,2.2]$	1.8878	1.6882
$[2.2,2.3]$	1.8774	1.6539
$[3.0,3.1]$	1.3120	1.2533
$[3.1,3.2]$	1.2324	1.1366
$[3.2,3.3]$	1.2540	1.1344
$[4.0,4.1]$	0.9824	0.9241
$[4.1,4.2]$	0.9532	0.9029
$[4.2,4.3]$	0.8976	0.8228

Next, we evaluate the relationship between the average localization errors and the uncertainty of $n_{r}$ . The larger the value of $n_{r 2} - n_{r 1}$ is, the more uncertain the real path loss exponent is. Table 3 shows the average localization errors of two localization algorithms obtained by conducting simulations 50 times while the value of $n_{r 2} - n_{r 1}$ is varied. The localization errors of two localization algorithms both increase dramatically, when the uncertainty of $n_{r}$ increases. Meanwhile, the localization error of the proposed localization algorithm is always smaller than that of the localization algorithm based on the RSSI ranging value. When $n_{r 2} - n_{r 1} = 0.05$ , the differences of average localization errors between two localization algorithms are only a few centimeters. When $n_{r 2} - n_{r 1} = 0.15$ , the differences of average localization errors between two localization algorithms are more than ten centimeters, even dozens of centimeters.

Table 3

Influence of average localization errors by uncertainty of $n_{r}$ .

Value scope $[n_{r 1}, n_{r 2}]$	Average localization error (m)
Value scope $[n_{r 1}, n_{r 2}]$	Based on RSSI ranging value	Based on RSSI ranging scope
$[2.0,2.05]$	0.9824	0.9241
$[2.0,2.15]$	3.2402	2.8233
$[3.0,3.05]$	0.6650	0.6227
$[3.0,3.15]$	1.9950	1.7742
$[4.0,4.05]$	0.5055	0.4687
$[4.0,4.15]$	1.4439	1.3478

Above simulation results show that the localization algorithm based on the RSSI ranging scope is more accurate and robust, when real path loss exponent is uncertain in a dynamic environment.

4.2. Analysis of Localization Errors Caused by A

Table 4 shows the average localization errors of two localization algorithms obtained by conducting simulations 50 times while A changes. We can observe that the average localization errors of both localization algorithms have not been changed much while A changes from −30 dB to −50 dB. As a result, we can see that localization errors are mainly caused by the path loss exponent n in RSSI based localization algorithms.

Table 4

Influence of average localization errors by A.

	Based on RSSI ranging value			Based on RSSI ranging scope
	$[2.0,2.1]$	$[3.0,3.1]$	$[4.0,4.1]$	$[2.0,2.1]$	$[3.0,3.1]$	$[4.0,4.1]$
$A = - 30$ dB	2.0358	1.3120	1.0130	1.8288	1.2533	0.9041
$A = - 35$ dB	1.9945	1.3422	0.9353	1.8360	1.2084	0.8783
$A = - 45$ dB	1.9959	1.3430	0.9760	1.8119	1.2401	0.9030
$A = - 50$ dB	2.0140	1.2763	0.9570	1.8324	1.1734	0.8781

4.3. Analysis of Localization Errors Caused by $X_{σ}$

The simplified shadowing model described by (2) is used in the above analysis. Now we evaluate the average localization errors caused by $X_{σ}$ in (1). Thus, (2) is changed as follows:

\begin{matrix} RSSI (d) = A - 10 n lg d + X_{σ} . \end{matrix}

(13)

It has been mentioned that

X_{σ}

is a Gaussian distributed random variable with zero mean and variance

σ^{2}

. In order to reduce the influence of

X_{σ}

, we collect N RSSI values and take the average RSSI values to calculate the distance in the simulation:

\begin{matrix} aveRSS I_{i} = \frac{1}{N} \sum_{j = 1}^{N} RSS I_{i j}, \end{matrix}

(14)

where

RSS I_{i j}

represents the jth RSSI measurement between the unknown node T and the ith beacon

B_{i}

Tables 5, 6, and 7 show the average localization errors of two localization algorithms obtained by conducting simulations 50 times while σ is varied. We can observe that the average localization errors of two localization algorithms both increase while σ increases. Meanwhile, the average localization errors of two localization algorithms both decrease while N increases. The larger the N value is, the more the communication cost is.

Table 5

Average localization errors when $σ = 2$ .

	Based on RSSI ranging value			Based on RSSI ranging scope
	$N = 10$	$N = 20$	$N = 30$	$N = 10$	$N = 20$	$N = 30$
$[2.0,2.05]$	2.7066	1.9780	1.7273	2.4404	1.8072	1.5765
$[3.0,3.05]$	1.8179	1.3136	1.1581	1.6647	1.2318	1.0620
$[4.0,4.05]$	1.3967	1.0390	0.9111	1.3101	0.9700	0.8659

Table 6

Average localization errors when $σ = 3$ .

	Based on RSSI ranging value			Based on RSSI ranging scope
	$N = 10$	$N = 20$	$N = 30$	$N = 10$	$N = 20$	$N = 30$
$[2.0,2.05]$	3.8944	2.9132	2.3792	3.5750	2.6580	2.1749
$[3.0,3.05]$	2.7612	1.9230	1.7061	2.5733	1.7630	1.5703
$[4.0,4.05]$	1.9373	1.4356	1.2589	1.8622	1.3421	1.1788

Table 7

Average localization errors when $σ = 4$ .

	Based on RSSI ranging value			Based on RSSI ranging scope
	$N = 10$	$N = 20$	$N = 30$	$N = 10$	$N = 20$	$N = 30$
$[2.0,2.05]$	5.1253	3.5954	3.2229	4.7282	3.3114	2.9478
$[3.0,3.05]$	3.4091	2.4401	2.0789	3.2215	2.2582	1.9424
$[4.0,4.05]$	2.5783	1.8487	1.5225	2.4659	1.7497	1.4399

5. Conclusion

In this paper, we mainly discuss the relationship between the parameters of the signal propagation model and RSSI ranging accuracy. Since it is difficult to measure the parameter, a robust localization algorithm based on the RSSI ranging scope is proposed. Instead of using the fixed parameter in the signal propagation model, the proposed algorithm created a one-to-one mapping between RSSI value and the distance scope based on the value scope of the path loss index. Simulation results demonstrate that the localization algorithm based on the RSSI ranging scope is more accurate and robust under different environments, when the signal propagation model, which is more appropriate for the current indoor environment, is extremely difficult to estimate.

Footnotes

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China under Grant no. 61203218, the Natural Science Foundation of Jiangsu Province under Grant no. BK2011318, and the Natural Science Foundation of China under Grant no. 61373098.

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Robust Localization Algorithm Based on the RSSI Ranging Scope

Abstract

1. Introduction

2. Analysis of RSSI Based Ranging

3. Robust Localization Based on the RSSI Ranging Scope

3.1. The Influence of RSSI Ranging Error on Positioning Accuracy

3.2. The Localization Algorithm Based on the RSSI Ranging Scope

4. Performance Evaluation

4.1. Analysis of Localization Errors Caused by n

4.2. Analysis of Localization Errors Caused by A

4.3. Analysis of Localization Errors Caused by X σ

5. Conclusion

Footnotes

Conflict of Interests

Acknowledgments

References

4.3. Analysis of Localization Errors Caused by $X_{σ}$