An Improvement of DV-Hop Localization Algorithm Based on Cyclotomic Method in Wireless Sensor Networks

Abstract

Location information is one of the crucial and essential elements for monitoring data in wireless sensor networks. The distance vector-hop (DV-Hop) localization algorithm is of practical importance in improving its localization performance. To achieve global optimization, a DV-Hop algorithm based on the cyclotomic method and weighted normalization, also known as CMWN-DV-Hop, is nominated in this paper. Therefore, the segmentation and weighting factors are introduced and normalized. The weighted recursive least-squares (WRLS) algorithm is chosen to compute the coordinates of the unknown nodes. The effects of various factors on this algorithm are tested, including the number of nodes, the anchor node ratio, and the communication radius. The simulation results show that the proposed algorithm has a super performance in reducing the localization error.

1. Introduction

As a new type of information acquisition system, wireless sensor networks (WSNs) have superiority in high flexibility, fault tolerance, self-control, and rapid deployment [1]. WSNs have a wide range of applications, including military counter-terrorism [2], precision agriculture [3], and healthcare [4]. A fundamental and hot topic in WSNs is localization. In recent years, many scholars have carried out numerous investigations and studies, mainly because many relevant algorithms have been presented in the literature. These algorithms are categorized into two types [5]: range-free and range-based localization. In all these algorithms, a few nodes, called anchor nodes, are equipped with a global positioning system (GPS) module and are configured to discriminate their position [6]. Range-free localization algorithms have low hardware requirements for nodes. They are more suitable for large-scale sensor networks, for example, multi-sequence positioning (MSP) [7], approximate point-in-triangulation test (APIT) [8], DV-Hop [9], etc. A novel framework for the performance analysis of cooperative network synchronization is proposed [10]. A quantitative description of the asymptotic local optimization performance is achieved by introducing generalized random walks with matrix-valued “pseudo-probabilities” in [11].

The DV-Hop is a localization algorithm designed by Dragos Niculescu et al. [9]. It has received much attention because of its low hardware requirements for nodes and simple implementation. The disadvantage is that hop distances are used directly instead of straight-line distances, which makes the ranging error larger. The average hop distance of anchor nodes was corrected by Tang et al., but the error in the estimation of the unknown node position was not considered [12]. Xingjuan Cai et al. introduced a weight model to improve the accuracy but ignored the intrinsic connection between weights and errors [13]. Li et al. used geometric features to correct the shortest distance of zigzag paths but did not consider the distribution density of nodes [14]. Shahzad et al. removed some anchor nodes that were far from unknown nodes but did not make good use of the effective information of anchor nodes [15]. In order to improve the accuracy of location, we propose an elaborated DV-Hop based on the cyclotomic method and weighted normalization (CMWN-DV-Hop). The error sources of the original DV-Hop localization algorithm are analyzed and evaluated. For implementing the global optimization, the DV-Hop algorithm is optimized and revised by utilizing the recursive algorithm of the cyclotomic method, introducing the weighting factor and the weighted recursive least squares. The simulation compares the localization performance of CMWN-DV-Hop and the other four algorithms under three factors.

The article is structured as follows: In Section 2, we summarize the previous studies in detail in Section 2. Section 3 illustrates the sources of error in DV-Hop, the cyclotomic method, and the recursive algorithm for least-squares parameter estimation. Section 4 introduces the theory of CMWN-DV-Hop. Section 5 covers simulation, curve analysis, and capacity evaluation. Section 6 presents the conclusions of the paper.

2. Related Works

Many scholars have conducted detailed research on the DV-Hop algorithm [16]. Different improvement schemes have been proposed for the original algorithm, and good progress has been made. The DV-Hop algorithm had three phases. Phase 1: The minimum number of hops between the unknown node and the anchor node was calculated [17]. Phase 2: The estimated distance was calculated by multiplying the minimum number of hops and the average distance per hop [18]. Phase 3: The specific coordinate of the unknown node was calculated based on the trilateration localization mechanism or the excellent likelihood appraisal method [9,19].

Considering the localization error, many researchers have optimized phases 1 and 2 of DV-Hop. For example, Li et al. [20] optimized the minimum number of hops by increasing the communication radius (R) and setting two communication radii. A medium access control (MAC) method based on the Chinese remainder theorem (CRT) protocol sequence has been proposed for DV-Hop localization [21]. To overcome the low accuracy and high energy consumption, the enhanced weighted centroid DV-Hop (EWCL) algorithm was proposed by Kaur et al. [22]. The weighting factor in the EWCL algorithm was a function of hop count, average hop distance, and transmission radius.

Other scholars have also optimized phase 3 of DV-Hop. Recently, with intelligent computing showing excellent performance in various complex optimization problems, various nature-inspired schemes have been presented [13,23]. For instance, a DV-Hop improvement algorithm based on bacterial foraging optimization was raised in the literature by Zhou et al. [24], and the algorithm was a nature-inspired intelligent parallel search algorithm. The application of the DV-Hop algorithm based on the gray-wolf optimization technique in 2D and 3D WSNs has been noticed [25]. A hybrid chaotic strategy firefly swarm optimization algorithm was proposed based on chaotic variation and chaotic inertia weight update [26]. In the literature [27], the proposed algorithm eliminates communication from one of the steps by calculating hop size at unknown nodes. The hybrid DV-Hop algorithm combined with cuckoo search [28], also known as CS-DV-Hop, showed better localization performance than DV-Hop [29]. In the literature [30], the DV-Hop algorithm based on the bacterial foraging algorithm (BFA) and glow-worm swarm optimization (GSO) (BFO-GSO) was described in detail. The BFO algorithm was a new evolutionary algorithm inspired by the foraging behavior of Escherichia coli. A coupled scheme in the light of the bacterial foraging algorithm and the glow-worm swarm optimization algorithm was chosen for this improved algorithm. This scheme had tremendous convergence speed and BFO local search capability; however, due to the use of BFO-GSO, computation time was increased slightly.

It is well known that localization algorithms for WSNs need to consider practical application scenarios. New characteristic factors need to be considered for specific complex deployment environments (such as indoor environments, coal mine tunnels, canyon terrain, and lake terrains [31,32,33]) and their corresponding node topologies. The non-dominant sorting genetic algorithm (NSGA-II) has advantages [34]. Consequently, a multi-objective DV-Hop based on NSGA-II, also known as NSGA-II-DV-Hop, has been proposed in the literature [35]. Furthermore, detailed simulations and evaluations were performed in four application contexts, and the experimental results show that NSGA-II-DV-Hop significantly outperformed the original DV-Hop.

3. DV-Hop and Error Analysis

3.1. DV-Hop Localization Algorithm

The DV-Hop algorithm is a localization algorithm designed using the principle of distance vector routing. Overall, the DV-Hop algorithm is divided into three phases, which are described in detail below.

Phase 1: The minimum number of hops is determined between the unknown node and each anchor node. The anchor node floods its information grouping to the neighbor nodes, including the location information of the anchor node and the initial hop count of 0 [36,37]. Groups with larger hops of the same anchor node are ignored. Upon receiving a packet of the anchor node, the neighbor node attaches a hop value to the packet [38]. Then add 1 to the hop value and forward it to the neighbor node [39,40]. As shown in Figure 1, the packet broadcast by the anchor node is propagated through the network in approximately concentric circles.

Phase 2: The actual distance between unknown nodes and anchor nodes is accurately and quickly estimated. Consequently, given the total number of hops recorded and the coordinates of the locations of the other anchor nodes, we estimate the average valid distance per hop [41].

$${\mathrm{Hopsize}}_i=\frac{{\displaystyle {\sum }_{i\ne j}{‖{\mathbf{X}}_i-{\mathbf{X}}_j‖}_2}}{{\displaystyle {\sum }_{i\ne j}{\mathrm{Hop}}_{ij}}}$$

(1)

In the above formula, ${\mathbf{X}}_i={[x_i,y_i]}^T$, $x_i$ and $y_i$ are the horizontal and vertical coordinates of the anchor node, individually. ${\mathrm{Hop}}_{ij}$ is the hop count for the respectively shortest path of anchor nodes i and j [42]. Subsequently, while the anchor node broadcasts the calculated average distance, each unknown node merely stores the initial average distance it receives. In a network with numerous nodes or a wide node deployment, unnecessary communication between nodes can be reduced by setting a time to live for the packets. Consequently, the distance from anchor node i to the unknown node is more precisely obtained [39].

$$d_{i,k}=h_{i,k}·{\mathrm{Hopsize}}_i$$

(2)

where $h_{i,k}$ is the hop value of the anchor node i and the unknown node k.

Phase 3: The trilateral localization method, or multilateral localization method, is an essential mechanism to acquire the complete information in more detail. The diagram of the multi-edge positioning method is legibly illustrated in Figure 2.

There are n anchor nodes in the deployment environment. The equation is formed as shown below.

$$\left\{\begin{array}{c}{(x_1-x)}^2+{(y_1-y)}^2=r_1^2\\ ⋮\\ {(x_n-x)}^2+{(y_n-y)}^2=r_n^2\end{array}\right.$$

(3)

The anchor node position is $(x_i,y_i),i=1,2,3,\cdots$. $r_i$ is the measured distance value from the unknown node to each anchor node. Subtract from the 1st equation in sequence with the nth equation.

$$\left\{\begin{array}{c}{x}_1^2-x_n^2-2(x_1-x_n)x+y_1^2-y_n^2-2(y_1-y_n)y=r_1^2-r_n^2\\ ⋮\\ x_{n-1}^2-x_n^2-2(x_{n-1}-x_n)x+y_{n-1}^2-y_n^2-2(y_{n-1}-y_n)y=r_{n-1}^2-r_n^2\end{array}\right.$$

(4)

The square term is eliminated to obtain a system of simultaneous equations in n − 1 dimensions.

$$\mathbf{A}X=b$$

(5)

The matrices A and b are concerned.

$$\mathbf{A}=2\left[\begin{array}{cc}\begin{array}{c}{x}_n-x_1\\ ⋮\\ x_n-x_{n-1}\end{array}& \begin{array}{c}{y}_n-y_1\\ ⋮\\ y_n-y_{n-1}\end{array}\end{array}\right]$$

(6)

$$b=\left[\begin{array}{c}{r}_1^2-r_n^2-x_1^2+x_n^2-y_1^2+y_n^2\\ ⋮\\ r_{n-1}^2-r_n^2-x_{n-1}^2+x_n^2-y_{n-1}^2+y_n^2\end{array}\right]$$

(7)

Since there is a predetermined error in the actual measurement and $E$ is an n−1 dimensional error vector, the system of (5) should be expressed as:

$$\mathbf{A}X+E=b$$

(8)

To ensure that the ranging error has minimal effect on the localization results of the network, the least-squares principle is used to minimize the square of the random error vectors in E mode.

$$J={‖b-\mathbf{A}X‖}_2^2=b^{T}b+X^T{\mathbf{A}}^T\mathbf{A}X-2X^T{\mathbf{A}}^{T}b$$

(9)

where $J$ is the square of the modulus of the random error variable E. After deriving it and setting the derivative equal to zero, we obtain Equation (10). It is also known as the canonical equation for the linear least-squares problem [43].

$${\mathbf{A}}^T\mathbf{A}X-{\mathbf{A}}^{T}b=0$$

(10)

When rank (A) = n − 1, the unique solution of (10) can be expressed as:

$$X={({\mathbf{A}}^T\mathbf{A})}^{-1}{\mathbf{A}}^{T}b$$

(11)

3.2. Error Analysis

From the above analysis, it can be concluded that there is an extensive range of deviation from DV-Hop utilizing hop distance instead of straight-line distance [37]. Figure 3 is formed by eight unknown nodes and four anchor nodes randomly deployed; the radio communication radius of nodes in this network is 50 m. The average hop distance of anchor nodes A1 and A2 is 29.73 m and 26.60 m, respectively, according to Equation (1) for the following calculation.

As can be seen from the figure, anchor node A2 can communicate with nodes 3 and 6, and the hop counts are both recorded as 1. However, the difference between $l_{A2,3}$ and $l_{A2,6}$ has a large impact on the calculation of the average hop distance of anchor node A2. This mechanism of calculating the minimum hop count between nodes also leads to errors in calculating the minimum distance between unknown nodes and anchor nodes, and further errors accumulate during the localization process. Therefore, this paper considers the graded refinement of the hop count between the anchor node and the neighboring nodes, and the specific scheme is shown in Section 4.

During the traditional second phase, the unknown node receives the average hop distance estimated by the closest anchor node. For example, unknown node 1 calculates its location based on the anchor node A1. Nevertheless, anchor node A1 does not have a complete description of the full array of nodes deployed.

The least-squares method used in the third stage is computationally intensive. Moreover, if the quantity of nodes is substantial to be deployed in a broad environment, the primary least-squares method seriously affects the node survival rate. In addition, the least squares do not provide a continuous and stable localization process when some nodes run out of energy or when new nodes try to join the network.

4. Proposed Algorithm: CMWN-DV-HOP

4.1. Cyclotomic Method

The cyclotomic method was first proposed by mathematician Hui Liu, which divides a circle by its inner polygon and utilizes the circumference of the polygon to approximate the circumference of the circle [44]. As shown in Figure 4, the unit circle is divided into a square, a hexagon, an octagon, and a decagon. The segmentation factors are 4, 6, 8, and 10. It can be observed that the larger the segmentation factor is, the closer the circumference of the polygon is to the circumference of the unit circle.

We use the cyclotomic method to partition the hop value in the interval [0,1] with the segmentation factor. In this way, the hop numbers between nodes are refined, and the overall accuracy is improved. Figure 5 shows the segmentation of the hop values when the segmentation factors are 4, 6, 8, and 10. For example, when the segmentation factor a is 4, if the flooding occurs in the (0,0.25) region (containing 0.25), the hops are recorded as 0.25. At that moment, the accuracy of the minimum hop value is achieved.

4.2. Weighted Recursive Least-Squares Algorithm (WRLS)

One of the main reasons that lead to ranging deviation from the DV-Hop is not enough to consider the information of all anchor nodes that can communicate with the unknown node. The present paper uses the weighted recursive least-squares algorithm to calculate the position information of each unknown node. The weighted recursive least-squares algorithm can reduce the amount of computation and provide a stable localization process in the event of node failure or the addition of new nodes in the network.

The following details the idea of the weighted recursive least-squares scheme. Due to different measurement situations, the data obtained from these measurements influence the node’s positioning differently. We assume that the data with high measurement accuracy have a higher weight than the data with low measurement accuracy. Weighting factors are multiplied on each component of the error vector E, assuming they are $w(1),w(2),\cdots ,w(n-1)$, respectively. At this point, the error function J can be formalized as follows:

$$J=b^T\mathbf{W}b-2X^T{\mathbf{A}}^T\mathbf{W}b+X^T{\mathbf{A}}^T\mathbf{W}\mathbf{A}X$$

(12)

In Equation (12), $\mathbf{W}={[w(1),w(2),\cdots ,w(n-1)]}^T$. It should be noted that for all measurement cases, W is a matrix of positive numbers. We take the derivative of the error function J to get the unknown node coordinates X.

$$X={({\mathbf{A}}^T\mathbf{W}\mathbf{A})}^{-1}{\mathbf{A}}^T\mathbf{W}b$$

(13)

When the m-th and m+1-th measurements are:

$${\stackrel{\wedge }{X}}_m={({\mathbf{A}}_m^T{\mathbf{W}}_m{\mathbf{A}}_m)}^{-1}{\mathbf{A}}_m^T{\mathbf{W}}_m{b}_m$$

(14)

$${\stackrel{\wedge }{X}}_{m+1}={({\mathbf{A}}_{m+1}^T{\mathbf{W}}_{m+1}{\mathbf{A}}_{m+1})}^{-1}{\mathbf{A}}_{m+1}^T{\mathbf{W}}_{m+1}{b}_{m+1}$$

(15)

In Equation (15), ${\mathbf{P}}_m={[{\mathbf{A}}_m^T{\mathbf{W}}_m{\mathbf{A}}_m]}^{-1}$, and ${\mathbf{P}}_{m+1}={[{\mathbf{A}}_{m+1}^T{\mathbf{W}}_{m+1}{\mathbf{A}}_{m+1}]}^{-1}$.

$${\stackrel{\wedge }{X}}_{m+1}={\mathbf{P}}_{m+1}{\mathbf{A}}_m^T{\mathbf{W}}_m{b}_m+{\mathbf{P}}_{m+1}{a}^T(m+1)w(m+1)b(m+1)$$

(16)

$${\mathbf{P}}_m^{-1}{\stackrel{\wedge }{X}}_m={\mathbf{A}}_m^T{\mathbf{W}}_m{b}_m$$

(17)

$${\stackrel{\wedge }{X}}_{m+1}={\mathbf{P}}_{m+1}{\mathbf{P}}_m^{-1}{\stackrel{\wedge }{X}}_m+{\mathbf{P}}_{m+1}{a}^T(m+1)w(m+1)b(m+1)$$

(18)

$${\mathbf{P}}_{m+1}={[{\mathbf{P}}_m^{-1}+a^T(m+1)w(m+1)a(m+1)]}^{-1}$$

(19)

This equation ${\mathbf{P}}_m^{-1}={\mathbf{P}}_{m+1}^{-1}-a^T(m+1)w(m+1)a(m+1)$ based on the equation

$${(\mathbf{A}+\mathbf{B}\mathbf{C}\mathbf{D})}^{-1}={\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}\mathbf{B}{({\mathbf{C}}^{-1}+\mathbf{D}{\mathbf{A}}^{-1}\mathbf{B})}^{-1}\mathbf{D}{\mathbf{A}}^{-1}.$$

$${\mathbf{P}}_{m+1}={\mathbf{P}}_m-{\mathbf{P}}_m{a}^T(m+1){[w^{-1}(m+1)+a(m+1){\mathbf{P}}_m{a}^T(m+1)]}^{-1}a(m+1){\mathbf{P}}_m$$

(20)

$${\stackrel{\wedge }{X}}_{m+1}={\stackrel{\wedge }{X}}_m+{\mathbf{P}}_{m+1}{a}^T(m+1)w(m+1)[b(m+1)-a(m+1)\stackrel{\wedge }{X_m}]$$

(21)

In Equation (21), ${\mathbf{K}}_{m+1}={\mathbf{P}}_{m+1}{a}^T(m+1)w(m+1)$. The weighted recursive least-squares algorithm (WRLS) can be obtained as:

$${\stackrel{\wedge }{X}}_{m+1}={\stackrel{\wedge }{X}}_m+{\mathbf{K}}_{m+1}[b(m+1)-a(m+1)\stackrel{\wedge }{X_m}]$$

(22)

where ${\stackrel{\wedge }{X}}_m$ is the approximate coordinate of the unknown node during the previous moment, $b(m+1)$ is the currently measured value, and the corrected gain matrix is graphically represented by the letter ${\mathbf{K}}_{m+1}$.

According to the above derivation, the recursive algorithm based on the weighted least squares needs to compute the initial parameters ${\stackrel{\wedge }{X}}_0$ and ${\mathbf{P}}_0$ in Figure 6. One is to compute ${\stackrel{\wedge }{X}}_m$ and ${\mathbf{P}}_m$ utilizing a one-time algorithm based on a set of anchor node information and then ${\stackrel{\wedge }{X}}_m={\stackrel{\wedge }{X}}_0$ and ${\mathbf{P}}_0={\mathbf{P}}_m$; the other way is to assign a specific value directly and make $\stackrel{\wedge }{X_0}=\epsilon$ ($\epsilon$ is an exceedingly small real number) and ${\mathbf{P}}_0=\alpha \mathbf{I}$ ($\alpha$ is a very large real number). The cycle terminates in accordance with the following:

$$\underset{\forall i}{\max }\left|\frac{\stackrel{\wedge }{X_i}(m+1)-\stackrel{\wedge }{X_i}(m)}{\stackrel{\wedge }{X_i}(m)}\right|<\epsilon$$

(23)

4.3. CMWN-DV-Hop

4.3.1. Calculation of the Hop Value Based on the Circle Cutting Technique

Each anchor node floods its grouping information to neighboring nodes, including the initial value of 0 hops and anchor node position coordinates. The segmentation factor is introduced based on the cyclotomic method, and diverse segmentation factors divide the unknown nodes into different groups. For example, when the segmentation factor is 8, the communication area in Figure 7 is divided into A to H. Area A is the anchor node that can receive within a communication radius of 0.125R. The main idea is to estimate the distance from a transmitter to a receiver using the power of the received signal, the knowledge of the transmitted power, and the path loss model [45]. The hop count of this area node is recorded as 0.125 when a transmission is made. Area B refers to the area where the anchor node can receive packet information when it broadcasts within a communication radius of 0.25R. The hop value of unknown nodes in this geographic region is recorded as 0.25, from C to H, and so on. Based on the round-cutting technique to divide the hop count, recording the minimum number of hops will make the subsequent calculation of the actual hop distance more accurate, and the positioning accuracy will be improved. For example, for the network shown in Figure 3, if the segmentation factor is 4, the average distance per hop is calculated to be 65.40 m. The denominator has been further refined. This is closer to the actual value and makes the subsequent positioning more accurate.

$${\mathrm{Hopsize}}_1(\mathrm{a}=4)=\frac{d_{A1,A2}+d_{A1,A3}+d_{A1,A4}}{(0.25+0.75+0.5)+(0.5+0.75+1)+(0.5+0.75)}=\frac{100+154+73}{5}=65.40$$

(24)

4.3.2. Introduce the Weighting Factor ${\beta }_{\lambda }$

The weighting factor ${\beta }_{\lambda }$ is then implemented to calculate the actual distance, considering all the anchor node locations to be informed. The weighting factor is:

$${\beta }_{\lambda }=\frac{\frac{1}{{\mathrm{dis}}_{\lambda }}}{{\displaystyle \sum _{\lambda =1}^n\frac{1}{{\mathrm{dis}}_{\lambda }}}}$$

(25)

$${\mathrm{dis}}_{\lambda }={\mathrm{hop}}_{{\mathrm{UN}}_k,{\mathrm{AN}}_{\lambda }}\times {\mathrm{Hopsize}}_{\lambda }$$

(26)

n is the total number of anchor nodes (AN) that can communicate with the unknown node (UN), and the unknown node obtains the average hop distance weighted value of n anchor nodes. For an unknown node, all weighting factors are satisfied:

$${\displaystyle \sum _{\lambda =1}^n{\beta }_{\lambda }}=1$$

(27)

The average hop distance between ${\mathrm{NHopsize}}_i$ and ${\mathrm{Nd}}_{i,k}$ is:

$${\mathrm{NHopsize}}_i={\mathrm{Hopsize}}_i·{\displaystyle \sum _{i=1}^n{\beta }_{\lambda }}$$

(28)

$${\mathrm{Nd}}_{i,k}=h_{i,k}·{\mathrm{NHopsize}}_i=h_{i,k}·{\mathrm{Hopsize}}_i·{\displaystyle \sum _{i=1}^n{\beta }_{\lambda }}$$

(29)

4.3.3. WRLS Is Applied to Calculate the Unknown Node Coordinates

Based on the minimum number of hops and the hop distance ${\beta }_{\lambda }$, the weighted recursive least-squares (WRLS) algorithm is used to determine the geographic and spatial coordinates of each node. The measurements of the currently unknown nodes in the network are corrected according to the recursive algorithm based on the results of the previous parameter estimation utilizing observed data. In this way, the amount of memory can be significantly reduced, especially for nodes where the unknown nodes are close to each other, and the amount of computation can also be reduced.

5. Simulation Outcomes and Analysis

5.1. Virtual Setting and Appraisal Criteria

We selected the square two-dimensional area as the node deployment environment in this section and conducted simulation experiments on MATLAB R2020b. In order to have the relevant performance tested, the following algorithms have been compared: DV-Hop [9], CS-DV-Hop [29], NSGA-II-DV-Hop [35], BFO-GSO-DV-Hop [30], and CMWN-DV-Hop.

Experiments were carried out with three factors as variables considering the different deployment environments of wireless sensor networks. Nodes were randomized within a square area, so each set of simulation experiments was run 100 times. The parameter settings for the simulation experiment are given in

Table 1. Specification of parameters.

Parameter	Value
Node distribution	Random
Anchor node ratio	20% (5–30%)
Communication radius (R)	25 (15–40)
Number of nodes	100 (50–100)
Segmentation factor (a)	2, 4, 6, 10
Deployed environment	100 m × 100 m
Transmission signal power Ps	20 dBm
Frequency of signal f	5 GHz
Path loss exponent η	4
Reference distance d0	1 m
Transmitting energy consumption	1.50 mJ
Computational energy consumption	0.20 mJ
Receiving energy consumption	1.15 mJ

. Figure 8 shows the network deployment diagram with 100 nodes randomly distributed.

The simulation experiments were conducted based on two evaluation metrics. The first is the localization error, and the other is the average localization error. The specific calculation equations are shown below.

$$\mathrm{Localization} \mathrm{error}=\frac{{\displaystyle \sum _{i=1}^{N_u}{\mathrm{error}}_i}}{N_u}$$

(30)

$$\mathrm{Average} \mathrm{localization} \mathrm{error}=\frac{{\displaystyle \sum _{i=1}^{N_u}{\mathrm{error}}_i}}{N_u·R}\times 100\%$$

(31)

In Equation (30), ${\mathrm{error}}_i={‖X_i-X_{i0}‖}_2$, and R indicates the radius of communication. The total count of unknown nodes is represented visually by $N_u$. The estimated coordinates of node i are $X_i={[x_i,y_i]}^T$. The virtual position of unknown node i is denoted by $X_{i0}={[x_{i0},y_{i0}]}^T$.

5.2. Influence of the Number of Nodes

We test the five algorithms as mentioned above in the case of varying the nodes. R is 25 m, and the ratio of anchor nodes is 20%. The number of network nodes varies from 50 to 100. A comparison of the localization errors of five algorithms is shown in Figure 9.

It can be revealed from Figure 9 that the localization error shows an overall decreasing curve when the nodes are increased. Along with the increase in segmentation factor, the localization error of CMWN-DV-Hop progressively decreases. When the nodes are the same, CMWN-DV-Hop (a = 2) is slightly inferior to the three algorithms, including CS-DV-Hop, but outmatches DV-Hop. The two types of errors of CMWN-DV-Hop always reach their lowest when the segmentation factor is between 6 and 10, regardless of the number of nodes. When the segmentation factor is more prominent than or equal to 4, CMWN-DV-Hop delivers the best positioning property compared with the other three algorithms. Regardless of the number of nodes, CMWN-DV-Hop (a = 10) has the lowest average localization error, with a maximum reduction of 54.5890%, 53.0569%, and 54.4498% when in comparison with the other three proposed novel algorithms.

5.3. Influence of Anchor Node Ratio

We test the above localization algorithm with varying ratios of anchor nodes. R is 25 m, and the number of nodes is 100. It varies between 5% and 30% for anchor nodes. The curve variations of localization error for each algorithm are plotted in Figure 10.

The localization error for each algorithm, as a whole, tends to decrease as the proportion of anchor nodes increases. The localization error decreases significantly and gradually with increasing segmentation factor when CMWN-DV-Hop is utilized. When the anchor node ratios are the same, the performance of CMWN-DV-Hop (a = 2) is comparable to that of CS-DV-Hop and BFO-GSO-DV-Hop, while significantly superior to DV-Hop. CMWN-DV-Hop (a = 2, 4, 6, 10) has a maximum reduction in localization error of 38.7579%, 57.3950%, 67.1837%, and 70.1403%, respectively, in comparison with DV-Hop, when the anchor node ratio varies. When the segmentation factor is greater or equal to 6, CMWN-DV-Hop features the lowest positioning error compared to the three algorithms, including CS-DV-Hop. Among them, the maximum reductions are 53.0907%, 46.2620%, and 52.2841%, respectively, when the segmentation factor is 6. CMWN-DV-Hop (a = 10) has the lowest average localization error, whatever the anchor node ratio, with a maximum reduction of 59.2644%, 55.0847%, and 59.1379% when in comparison with the other three proposed novel algorithms.

5.4. Influence of Communication Radius

The five algorithms mentioned above test in the presence of varying communication radii. The total number of nodes is 100, with the proportion of anchor nodes being 20%. R varies in the range of 15 to 40 m. In Figure 11, two categories of errors for each algorithm are compared.

Inside the communication radius of these five groups, we can analyze them as follows: CMWN-DV-Hop (a = 2) and DV-Hop take the same communication radius of 25 m for minimum localization error. The three algorithms consisting of BFO-GSO-DV-Hop, which achieve the same minimum positioning error of R, are all 20 m. Meanwhile, with the increase in segmentation factor, the localization error of the CMWN-DV-Hop algorithm shows an overall decreasing trend. In particular, among these five communication radii, CMWN-DV-Hop always has the minimum localization error value when the communication radius is greater than or equal to 30 m, regardless of the segmentation factor. By observing Figure 11b, the vertical coordinates of the five algorithms tested decrease as R increases. Overall, the average orientation error does not differ significantly for CMWN-DV-Hop (a = 2) compared to the other three. Alternatively, there is a significant decrease compared to DV-Hop. Furthermore, the average orientation error of CMWN-DV-Hop (a = 4, 6, 10) is the least when R is greater than or equal to 25 m. When the communication radius is varied, CMWN-DV-Hop (a = 2, 4, 6, 10) has the maximum reduction in the average localization error by 36.6216%, 64.5825%, 74.9304%, and 81.8575%, respectively, compared with DV-Hop. Meanwhile, CMWN-DV-Hop demonstrates the best locating capabilities when a ≥ 4 and R ≥ 25 m. In particular, CMWN-DV-Hop (a = 10) is reduced by a maximum of 75.2018%, 74.2070%, and 73.5611% compared to the other three proposed novel algorithms.

5.5. Energy Consumption Analysis

Figure 12 shows the comparison of elapsed time and energy consumption for each localization algorithm. At this point, there are 25 anchor nodes and 150 unknown nodes in the network evenly distributed in a 100 × 100 area. Elapsed time is simply the amount of time that passes from the beginning of a localization process to its end. Energy consumption is the amount of energy consumed when 150 unknown nodes are located. As can be seen from the figure, the localization scheme in this paper has a shorter elapsed time and less energy consumption compared to the improved algorithm of DV-Hop.

6. Conclusions

We propose a novel localization algorithm CMWN-DV-Hop based on the cyclotomic method and weighted normalization. This paper introduces a segmentation factor based on the cyclotomic method to achieve more accurate hop values by analyzing the error sources in each stage of the original DV-Hop algorithm. In order to comprehensively consider the position information of all anchor nodes in the network, a weighting factor is introduced in calculating the hop distance calculation. A log-normal shadowing path loss model is incorporated to simulate a more realistic environment. We investigate the effects of the number of nodes, anchor node ratio, and communication radius on each localization algorithm. The simulation data show that the CMWN-DV-Hop algorithm has the smallest average localization error compared to the DV-Hop, CS-DV-Hop, NSGA-II-DV-Hop, and BFO-GSO-DV-Hop algorithms. The average localization error of the CMWN-DV-Hop algorithm decreases as the segmentation factor increases. Since the CMWN-DV-Hop (a = 10) algorithm is less sensitive to the number of nodes, it can be applied to large-scale wireless sensor networks.