Open AccessArticle

Improving Localization Accuracy Through Optimal Selection Strategy

School of Internet of Things, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

School of Information Science and Engineering, Harbin Institute of Technology, Weihai 264209, China

Author to whom correspondence should be addressed.

Electronics 2025, 14(1), 172; https://doi.org/10.3390/electronics14010172

Submission received: 6 November 2024 / Revised: 27 December 2024 / Accepted: 31 December 2024 / Published: 3 January 2025

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Abstract

A localization system is essential for providing crucial position information in various applications, such as three-dimensional (3D) warehousing, smart cities, uncrewed aerial vehicle (UAV) control, and other services that heavily rely on accurate localization. However, the transmission of wireless signals can be impacted by diverse environmental factors, leading to decreased accuracy in determining localization in scenarios involving multiple signal paths, None Line of Sight (NLOS) situations, and different types of interference. In some cases, this may render the localization system unsuitable for subsequent applications. To enhance the localization accuracy, we propose a 3D localization method using an optimization selection strategy. With this method, we make the following innovations: (1) We utilize an evaluation of feature points to minimize the negative impact of NLOS. (2) Through the backward assessment and the optimal selection of distance estimations, we obtain a more accurate localization result. In more detail, our approach implements a specific strategy for distance estimation, followed by defining the feature points within the localization field and selecting the most optimized one. Subsequently, using the chosen feature points, we evaluate the quality of the distances in reverse. We then select suitable distance estimation outcomes for further localization calculations. Ultimately, by employing the proposed 3D localization technique, we achieve a highly precise localization result. We perform simulations and experiments to assess the presented localization system. More specifically, compared with certain strategies, we improve the localization accuracy by 58.33% and 43.83% using the selection strategy. Compared with the other methods, we enhance the localization accuracy from 17.94% to 32.54%. The results from these evaluations demonstrate that our method significantly enhances 3D localization accuracy.

Keywords:

wireless localization; feature point; distance estimation; optimal selection

1. Introduction

An accurate localization system offers critical position information for countless application systems providing location-based services, including the intelligent city, the formation control of uncrewed aerial vehicles (UAVs), emergency localization, rescue and relief, and many other application systems [1,2,3,4,5,6]. However, the localization outcomes exhibit varying levels of noise due to factors such as environmental noise, wireless interference, measurement inaccuracies, the propagation of radio signals through multiple paths, and None Line of Sight (NLOS); the localization results possess distinct noise levels [7,8,9]. Alternatively, and even worse, it may cause a bad effect or a wrong action for later deep processing methods or decision-making systems.

Owing to its significance in numerous applications, localization has remained a prominent research field for several years. However, in the face of a complex localization environment, the performance of localization fails to fulfill the requirements of applications. The improvement of localization performance, particularly in terms of accuracy, has been the focus of extensive research and scholarly endeavors. Numerous scholars and researchers have dedicated their efforts to exploring and presenting diverse methodologies, strategies, and systems for localization. In the domain of WSN systems, an enhanced three-dimensional (3D) localization algorithm considering the variability in Received Signal Strength Indicator (RSSI) values was presented by Zhang et al. [10]. Previous studies [11,12] also conducted investigations on the uncertainty associated with RSSI values. In addition, a clustering-based distance estimation method was presented by Luo et al. [11]. The enhancement of the localization accuracy was observed [13]. Another effective approach to improve the precision of localization involves utilizing mapping-based or fingerprint methods. Nevertheless, the degree of flexibility was extremely elevated [14,15]: that is, it was not appropriate for an unfamiliar environment. A comprehensive uncertainty analysis of the localization procedure was conducted by Yan et al. [16]. This study provides a critical reference for enhancing accuracy. Yan et al. proposed an improved trilateration-based localization method that focused on minimizing uncertainty propagation and optimizing anchor nodes [17]. The method needs a large amount of sample data for statistical calculation.

The proposed 3D localization system incorporates the consideration of the NLOS propagation of wireless signals and the influence of each distance estimation result on localization accuracy, with the objective of optimizing distance estimation outcomes through the meticulous selection of feature points. In this method, we initially delineate the feature points. The feature points are then evaluated, and the most advantageous ones are identified. Subsequently, based on the optimal feature points, a retrospective evaluation of the accuracy of distance estimation outcomes is conducted, and the optimized results are chosen for 3D localization computation. Ultimately, a more precise localization outcome can be achieved.

During this research, we make the following contributions:

(1) With the evaluation of feature points, we can minimize the negative impact of the NLOS propagation of the wireless signal and environmental noise on the localization results.

(2) Through backward assessment and the optimal selection of distance estimation outcomes, we can obtain more accurate localization results.

(3) The proposed optimal selection strategy can be applied to the range-based localization method, especially in distribution localization.

This paper is structured as follows: Section 2 reviews and analyzes related research work. Section 3 illustrates the proposed three-dimensional localization system and its implementation details. It includes the localization system framework, the definition of the feature points, the optimal selecting of the feature points, the backward evaluation, and the optimal selection of distance estimation results. In Section 4, we perform simulations and experiments to evaluate our proposed methods. Lastly, we conclude this paper.

2. Related Works

Environmental noise and interference are the major negative factors leading to a low localization accuracy. For these problems, many improved algorithms, methods, schedules, and systems are presented [10,11,12,13,14,15,16]. They focus on improving the accuracy of distance estimation and localization accuracy.

2.1. The Distance Estimation Improvement Methods

The distance estimation accuracy determines the localization accuracy at a large scale. Luo et al. [11] considered the RSSI uncertainty in a complex environment and proposed an uncertain data clustering-based distance estimation method. Moravek et al. [12] investigated the radio channel uncertainty during distance estimation. The authors considered and estimated the measurement uncertainty and distance estimation error [15,16], but they did not conduct a further uncertainty analysis. The Monte Carlo was adopted to evaluate the uncertainties during the distance measurement [17], but it needs sufficient sample data. A comprehensive uncertainty propagation analysis provided a reference to improve the distance estimation accuracy [16]. Luo et al. [18] suggested a mapping-based or a fingerprint-based strategy to improve the distance estimation accuracy. Luo et al. [19] applied a mapping strategy to dynamic distance estimation. The result is promising. However, a novel environment was not applied, so its flexibility is limited.

2.2. The Localization Accuracy Improvement Methods

A method for 3D localization in a WSN that integrates RSSI-TOA and LSSVR techniques was proposed by Zhang et al. [10]. In order to address the challenges posed by uncertainties in the RSSI values, Luo et al. [14] introduced a localization strategy founded on interval data clustering, which significantly enhanced the accuracy. An approach utilizing uncertainty data mapping was proposed to improve the localization precision in WSNs [15]; however, this technique is restricted to environments that have been previously measured. In addition, Lohrasbipeydeh et al. [20] utilized a machine learning approach to perform uncertainty predictions associated with fingerprinting. W. Njima et al. [21] used a deep neural network as a basis for indoor localization under missing fingerprints. Kupershtein et al. [22] studied the issue of single-site emitter localization under the circumstance of multi-path fingerprints. The uncertainty of three-dimensional localization was analyzed comprehensively [16,23]. Mahdi et al. [24] considered the coordinate uncertainty of the anchor node and proposed an unknown transmission power source localization method. Mahdi et al. implemented Gaussian process regression to cope with the localization uncertainties in sensor networks [25]. Luo et al. [26] accounted for NLOS circumstances, anchor uncertainties, and clock synchronization problems when conducting localization to boost accuracy. Recognizing the propagation of uncertainties throughout the localization procedure [16,23], Yan et al. proposed an ameliorated method that minimized the propagation of uncertainties [27]. This approach efficaciously enhanced the accuracy. The cooperative localization method also considered the location uncertainty and its propagation problem [15,28]. The approach described in the literature aimed to enhance the accuracy of the distributed localization; however, it failed to take into account NLOS conditions.

3. The Localization Method Through Optimal Selection

We describe the implementation of our proposed localization system in this part. We first demonstrate the framework of this system. Then, the principle and implementation details of these sub-components will be illustrated.

3.1. The Architecture of the 3D Localization System

Figure 1 illustrates the architecture of the 3D localization system, which consists of three main components: distance estimations, the assessment and choice of the outcome for the distance estimation, and 3D localization computation. The distance estimation module outputs the distance estimation results between the anchor node and the unknown node

{d_{1}, d_{2}, d_{3}, \dots, d_{10}}

. Then the assessment and choice of outcome for distance output the high quality results

{d_{a}, d_{b}, d_{c}, \dots, d_{f}}

. Finally, the 3D localization computation perform 3D localization computing and output the 3D localization result.

Distance estimations: In this section, various techniques can be employed to precisely measure distances, including Time of Arrival (TOA), Time Difference of Arrival (TDOA), RSSI, and Symmetric Double-Sided Two-Way Ranging (SDS-TWR). The selection of an appropriate technique typically depends on the specific application scenario and practical conditions. Given the objective of enhancing the positioning accuracy in this project, we have chosen the NanoLoc-based SDS-TWR method to ensure the accurate estimation of the distance between the anchor node and the unknown node.

Assessment and choice of outcome for distance estimation: To enhance the precision of the localization, we evaluate and select the outcomes of the distance estimation. This process comprises three components: the identification of feature points, the optimization selection of the feature points, and the retrospective evaluation and the optimal selection of the distance estimation results. The initial definition is assigned to feature points, followed by the selection of optimal feature points. Finally, a backward evaluation is conducted using these selected optimal feature points to ensure high-quality and optimal distance estimation outcomes.

3D localization computation: We propose a three-dimensional localization framework so that we can apply any three-dimensional localization computation method to the framework. In this paper, we utilize four localization strategies to evaluate the proposed method. The four strategies encompass the quadrilateral method (QM), the least squares method (LSM), the maximum likelihood method (MLM), and Newton’s iteration localization method (NILM).

3.2. Distance Estimation

Assuming the presence of anchor nodes and an unidentified node in the localization system, here

m

(

m \geq 4

) denotes the number of anchor nodes. The coordinates of these anchor nodes are

(X_{1}, Y_{1}, Z_{1})

(X_{2}, Y_{2}, Z_{2})

, …,

(X_{j}, Y_{j}, Z_{j})

, …,

(X_{m}, Y_{m}, Z_{m})

, and

1 \leq j \leq m

. All the nodes are equipped with the NanoLoc RF transceiver, as depicted in Figure 2. The anchor nodes remain stationary while the unknown node is relocated within the designated localization space. When the need for localization arises, we employ the SDS-TRW strategy to accurately estimate the distances between the anchor nodes and the unknown node. Thereby enabling precise distance estimation outcomes as

D

= {

D_{1}, D_{2}, D_{3}, ......., D_{j}, ......., D_{m}

1 \leq j \leq m

. Supposing there are n unknown nodes in the system, the complexity of the distance estimation is O(mn).

3.3. Assessment and Choice of Outcome for Distance Estimation

3.3.1. Definition of Feature Points

The field of localization involves dividing the length into

l

equal segments, uniformly dissecting the width into

w

sections, and precisely partitioning the height into

h

portions,

2 \leq l

2 \leq w

2 \leq h

. Consequently, we break down the 3D localization space into

n

cubes that have identical dimensions. The calculation of

n

is executed based on Equation (1).

n = l \times w \times h

(1)

The feature points are designated as the geometric center of each cube, resulting in

n

feature points. These points are represented by

F = {F_{1}, F_{2}, F_{3}, \dots, F_{i}, \dots, F_{n}}, 1 \leq i \leq n

, and their coordinates are denoted as

(x_{1}, y_{1}, z_{1}),

(x_{2}, y_{2}, z_{2}), \dots,

(x_{i}, y_{i}, z_{i}), \dots, (x_{n}, y_{n}, z_{n})

, and

1 \leq i \leq n

3.3.2. Optimal Selection of Feature Points

The SDS-TWR approach is employed to evaluate the spatial separation between each anchor node and all the feature points,

F_{i}

. The outcome

d_{i} = {d_{i 1}, d_{i 2}, \dots, d_{i j}, \dots, d_{i n}}

is obtained, and

d_{i j}

is calculated using Equation (2). Subsequently, we ascertain the absolute error

e_{i}

between

d_{i}

and

D_{j}

. The result is expressed as

e_{i} = {e_{i 1}, e_{i 2}, \dots, e_{i j}, \dots, e_{i n}}

and

e_{i j} = |d_{i j} - D_{j}|

d_{i j} = \sqrt{{(x_{i} - X_{j})}^{2} + {(y_{i} - Y_{j})}^{2} + {(z_{i} - Z_{j})}^{2}}

(2)

We are capable of attaining the series of absolute error corrections designated as

E_{i} = {E_{i 1}, E_{i 2}, \dots, E_{i j}, \dots, E_{i n}}

E_{i j}

is determined using Equation (3).

\{\begin{array}{l} E_{i j} = 1 & (e_{i j} < 1) \\ E_{i j} = e_{i j} & (e_{i j} \geq 1) \end{array}

(3)

The assessment factor

λ = {λ_{1}, λ_{2}, \dots, λ_{i}, \dots, λ_{n}}

of the feature point

F

is subsequently determined by employing Equation (4). The higher the value is, the more approximately it is to the unknown node.

λ_{i} = \sum_{j = 1}^{m} (\frac{1}{E_{i j}})

(4)

The maximum optimized value

λ_{o}

λ

is selected. The

λ_{o}

corresponds to the optimal feature point. Here

o

corresponds to the optimal value

λ_{o}

λ

and

1 \leq o \leq n

. Therefore, the associated feature point, denoted as

(x_{o}, y_{o}, z_{o})

, is thus deemed the most suitable and optimal choice. The complexity of the optimal selection of the feature points is O(m).

We utilize the feature point to divide the localization space into different sections, and there is no iteration during the optimal selection of the feature points. It may lead to a high computing effort. But it will not disturb the convergence of the proposed 3D localization method.

3.3.3. Retrospective Evaluation and Optimal Selection of Distance Estimation Results

The distance estimation outcomes are retrospectively evaluated using the most appropriate feature points, and the relative error

ε_{j}

is computed based on Equation (5). Consequently, a sequence of relative errors

{ε_{1}, ε_{2}, \dots, ε_{m}}

can be obtained. In this context,

d_{o j}

signifies the difference between the optimal feature points and the jth anchor node. A smaller

ε_{j}

indicates a higher similarity between them.

ε_{j} = \frac{|d_{o j} - D_{j}|}{d_{o j}}

(5)

The

k

members with the smallest values were selected from the relative error sequence

{ε_{1}, ε_{2}, \dots, ε_{m}}

. The localization process utilizes the distance estimation results and anchor node coordinates, and the other

m - k

distance estimation results are discarded. The variable

k

denotes the number of selected outcomes used for the distance estimation. It is also determined according to the three-dimensional localization method,

4 \leq k \leq m

. The most optimized outcomes are assessed and selected in this manner for the estimation of distances, aiming to localize objects in three dimensions. The complexity of the retrospective evaluation and the optimal selection of the distance estimation results is O(m).

3.3.4. 3D Localization Computation

The 3D localization computation is another critical part of the system. Based on the distance evaluation results and optimal selection, we carry out a 3D localization computation. Any sort of range-based 3D localization computation approaches can be employed here, encompassing QM, LSM, MLM, and NILM. Since the former three methods are rather straightforward, we concentrate on elucidating NILM.

NILM transforms a 3D localization problem into an unconstrained minimal sum of a square problem, as shown in Equation (6).

\{\begin{array}{l} F (x, y, z) = \sum_{i = 1}^{k} {(\sqrt{{(x - x_{i})}^{2} + {(y - y_{i})}^{2} + {(z - z_{i})}^{2}} - D_{i})}^{2} \\ \min (F (x, y, z)) \end{array}

(6)

The variable

k

in this context represents the count of anchor nodes, while

D_{j}

indicates the estimated distance between the anchor nodes and the unknown node. We illustrate the processing flow of NILM in Figure 3.

The process of NILM can be summarized as follows:

(1) The initialization of the iteration. The initial position

X = X_{0}

is determined, and the Hessian matrix is initialized with an approximate identity matrix

B_{0} = I^{m \times m}

, ensuring that the iteration precision

ε

satisfies

0 < ε < 1

(2) The determination of the Newton’s search direction. The direction of Newton’s search

d_{k} = - B_{k} g_{k}

is determined. Where

g_{k}

demotes the gradient matrix in the kth direction, and

B_{k}

represents an approximate Hessian matrix in the kth direction.

(3) The determination of the factor

α_{k}

. The factor

α_{k}

is meticulously searched along the Newton’s search direction

d_{k}

, and its appropriate value is determined to ensure compliance with

X_{k + 1} = X_{k} + α_{k} d_{k}

in the (k + 1)th iteration.

(4) The rRectification of the approximate Hessian matrix

B_{k}

. The estimated Hessian matrix

B_{k}

is adjusted by

B_{k + 1} = B_{k} + Δ B_{k}

, thus satisfying the Quasi-Newtonian condition.

(5) Judging whether the iteration condition is satisfied or not. The assessment is conducted to determine whether the condition of iteration precision is met or not,

‖\nabla F (X_{k})‖ \leq ε

. If the condition is satisfied, the iteration concludes; otherwise, it will revert to step (2).

The complexity of 3D localization computation is up to the specific 3D localization method. For the specific NILM method, the complexity is O(n). As we know, the complexity of the distance estimation is O(mn). The complexity of the optimal selection of the feature points is O(m). The complexity of the retrospective evaluation and the optimal selection of the distance estimation results is O(m). Therefore, the complexity of the whole 3D localization method is O(mn) + O(m) + O(m) + O(n) = O(mn).

4. Simulation

We conduct simulations to validate the feasibility of the presented three-dimensional localization method. The effectiveness of the proposed approach is further assessed in comparison to alternative methodologies.

4.1. Simulation Arrangement

4.1.1. The Establishment of the Localization Environment

The simulation environment is a cubic area measuring 10 m on each side. We deploy 20 anchor nodes at the peak or the mid-point of each edge, as illustrated in Figure 4. The anchor nodes at the peak are denoted as red points, while, the anchor nodes at the mid-point of each edge are denoted blue triangle points.

Furthermore, each edge is evenly divided into six parts, yielding 216 cues of the same magnitude. The center point of each cube is considered as a distinct feature, resulting in a total of 216 such feature points, they are denoted as red points in Figure 5. We illustrate the feature points in Figure 5. With them, we perform the retrospective evaluation and the optimal selection of the distance estimation results and get more accurate distance estimation results.

4.1.2. Simulation Data

In the 3D space, an object coordinate is randomly selected. The distances between this coordinate and each of the twenty anchor nodes are computed. The distance results are subjected to the introduction of additive white Gaussian noise in order to simulate its impact. We also consider the NLOS noise by adding a constant value. We show the data model in Figure 6. The distance measurements are simulated by incorporating additive white Gaussian noise and NLOS noise.

4.1.3. Simulation Design

The ideal choice for distance estimation is influenced by multiple variables, including the signal-to-noise ratio (SNR), the interference from NLOS conditions, the abundance of feature points, the density of anchor nodes, and the occurrence rate of NLOS. Here, the NLOS rate represents the rate of NLOS within the total distance estimation error. We divide the simulation into five groups for the purpose of evaluating the influence of these factors, respectively. The parameters settings of the five groups are provided in Table 1. We adopt the least squares method as the localization method.

4.2. Simulation Results

4.2.1. The Impact Evaluation of the Strength of Noise

In group 1, we assess the impact of the strength of the noise. We adjust the SNR from 15 dB to −10 dB and evaluate the mean localization error. We present the simulation result in Figure 7. Based on the findings depicted in Figure 7, it is apparent that the LSM with distance selection outperforms its counterpart without distance selection when the SNR exceeds zero. This is because the principal factor influencing accuracy is NLOS. The NLOS error can be mitigated by implementing the distance selection strategy. The stability of the localization accuracy is compromised when the SNR drops below zero, leading to a shift in the primary contributing factor from NLOS to the environmental error. Consequently, the reliability of the distance selection becomes questionable.

4.2.2. The Impact Evaluation of the Strength of None Line of Sight (NLOS)

The impact of NLOS intensity on localization accuracy is being evaluated, i.e., we adjust the strength value of NLOS and evaluate the mean localization error. We show the simulation result in Figure 8. It is similar to that of group 1. When the NLOS error is below 2.5 m and comparable to the environmental error, distinguishing the distance optimization selection becomes challenging, thereby leading to a low localization accuracy. Nevertheless, the magnitude of the NLOS error is observable. When the NLOS is higher than 2.5 m, we can delete it during the distance optimization selection. The inclusion of a module for selecting optimal distances can significantly enhance the precision of the localization compared to scenarios where such a module is not present.

4.2.3. The Impact Evaluation of the Density of Feature Points

We increased the feature point density from 13 to 83 for group 3 and evaluated the localization error, as illustrated in Figure 9. It is demonstrated that when the quantity of feature points is greater than 64, the method incorporating selection demonstrates a significantly diminished localization error in comparison to the one without selection. Nevertheless, the increase in the number of feature points does not result in any additional reduction in the localization error. That indicates that the density of feature points has a practical impact on the performance of the distance selection. When the density is not big enough, mis-selection may happen, and the localization error is higher than that without selection. Conversely, when the density of feature points increases to a specific value, the selection strategy can pick out the high-quality distances, and the localization error is tiny. Thereafter, with the increment of the density of the feature points, the localization error does not decrease anymore.

4.2.4. The Impact Evaluation of the Density of Anchor Nodes

For group 3, we adjust the density of feature points from 13 to 83. The evaluation of the localization error is performed as illustrated in Figure 10. The method of feature point selection has been demonstrated to exhibit a significantly lower localization error compared to the method without selection, provided that the number of feature points exceeds 64. However, with the increment of feature points, the localization error does not decrease. That indicates that the density of feature points has a practical impact on the performance of the distance selection. When the density is not big enough, mis-selection may happen, and the localization error is higher than that without selection. Conversely, when the density of feature points increases to a specific value, the selection strategy can pick out the high-quality distances, and the localization error is tiny. Thereafter, with the increment of the density of feature points, the localization error does not decrease anymore.

4.2.5. The Impact Evaluation of the Rate of None Line of Sight (NLOS)

Figure 11 illustrates the simulation outcome of group 5. We alter the rate of none of line of sight (NLOS) in the total noise while keeping the other factors invariant. We evaluate the mean localization error of the two strategies, i.e., with optimized selection and without optimized selection.

It can be observed from Figure 11 that the method incorporating distance optimized selection has superior localization accuracy compared to the method without distance selection, particularly when the NLOS rate is below 40%. Nevertheless, when the rate exceeds 40%, the localization error escalates rapidly or even exceeds that of the technique without distance selection. That is mainly because when the NLOS rate increases to a specific level, the effect of distance selection will decrease, or, even, a low level of NLOS is deleted. Still, a high level of NLOS may be retained to perform the localization. From the analysis above, we can conclude that the proposed distance selection strategy with feature points can improve the localization error effectively.

5. Experiments

5.1. The Establishment of the Experimental Environment

In a spacious hall, we implemented the 3D-localization system, as depicted in Figure 12. As shown in Figure 12, the dimensions of the 3D space are measured to be 23.6 × 20.9 × 6.43 m³. There exist four substantial pillars, each with a perimeter of 5.3 m, positioned at approximately 9.36 m intervals from one another. Furthermore, the hall encompasses not only pedestrians but also walls and fences, thereby creating a highly intricate localization environment. The NanoLoc-based wireless localization system is illustrated in Figure 13. It consists of ten anchor nodes, an unidentified node, and a gateway node that establishes a connection with a PC via a serial communication cable. These nodes {A₁,A_2, A₃, …, A₁₀} are strategically placed at various heights throughout the hall for optimal coverage and accuracy as shown in Figure 14. Specifically illustrated in Figure 14 are six anchor nodes situated on trays, which can be adjusted between heights of 0.5 m to 1.5 m, whereas the remaining four anchor nodes are positioned at an elevation of precisely 6.43 m. For reference purposes, Table 2 presents the coordinates assigned to each individual anchor node. We illustrate the distances between the anchor nodes in Table 3.

5.2. Experiment Conduct

We conduct the localization experiment in the following steps.

(a) The anchor nodes, the unknown node, and communication node establish a wireless network after being powered on and initialized.

(b) We set the unknown node at a specific localization point.

(d) After receiving the distance estimation message, the unidentified node disseminates it to nearby anchor nodes and awaits their response messages. By leveraging the SDS-TWR approach, the unidentified node obtains the distance estimation outcomes and transmits them to a PC.

(e) The unknown node repeats the distance estimation 200 times on this localization point.

(f) The PC is responsible for executing the process of distance estimation and the computation of the 3D localization.

(g) We judge whether the localization experiment is finished or not. If it is, the investigation is over. Or else, we move the unknown node to the next localization point and go to step (c).

During the localization experiment, we set 24 localization points. We illustrate the coordinate of these points in Table 4.

5.3. Experimental Data Processing

For the treatment of the distance estimation results, we resort to three process strategies, as shown in Figure 15.

One is to utilize the three-dimensional localization methods to conduct localization directly. The other follows the “delete maximum” strategy, which considers that the NLOS probability of the maximum distance estimation results is high. It deletes the maximum distance estimation results. Then, it performs a localization computation. Another process strategy pertains to distance selection; namely, we initially employ the feature points to select the distance estimation results and subsequently undertake the localization computation. We assess the accuracy with the three strategies and the four typical localization methods.

5.4. Experimental Result

5.4.1. Processing Strategy Comparison

Employing the three strategies: direct localization, deletion maximum, and distance selection, we explore the QM, the LSM, the ML, and the NILM for conducting localization, respectively. Considering the 3D localization environment, we divide the 3D environment in to 10 × 10 × 5 parts. Therefore, the density of the feature points is 10 × 10 × 5. We present the localization results in Figure 16 and Figure 17 and in Table 5.

As illustrated in Figure 16, all four localization methods consistently produce results, thereby affirming the feasibility and validity of our suggested method. We can observe from Figure 16 and Figure 17 that the localization error of the approach with the distance selection strategy is the lowest among the three processing strategies. Specifically, based on the findings presented in Table 5, it can be observed that among the four examined localization methods, the distance selection strategy demonstrates a significant average increase in localization accuracy of 44.26%, 50.95%, 51.42%, and 60.36%, when compared to both the direct localization and the “delete maximum” strategies. Moreover, the implementation of the “delete maximum” strategy effectively mitigates any adverse impact caused by NLOS conditions. The distance selection strategy can effectively eliminate NLOS. The proposed approach holds the potential to enhance the localization accuracy in intricate environments while minimizing errors.

5.4.2. Localization Accuracy Comparison

The accuracy of the localization was assessed by utilizing the distance selection strategy for the four methods, i.e., quadrilateral, least squares, maximum likelihood, Newton’s iteration. The results are presented in Figure 18 and Figure 19. We show the average localization error of the four localization methods with the 24 localization points in Figure 18. We illustrate the average localization error of the four methods in Figure 19. We show the localization error of the four methods with the four strategies in Table 5.

The findings depicted in Figure 18 and Figure 19 and in Table 5 indicate that the quadrilateral method exhibits a higher degree of localization error compared to the alternative approaches. In contrast, both the least squares and the maximum likelihood methods demonstrate comparable levels of localization error. They are higher. Furthermore, the Newton’s iteration method demonstrates the lowest localization error among the four methods. Specifically, as shown in Figure 18, compared with the other three methods, it achieves an average enhancement of the localization accuracy of approximately 32.54%, 18.28%, and 17.94%. Consequently, the proposed localization system possesses the capability to significantly improve the precision in complex environments. That is mainly due to the iteration strategy the Newton’s iteration method. We can also see that the experimental result has a similar conclusion to the simulation results.

5.4.3. Localization Efficiency Comparison

The effectiveness of the localization efficiency is evaluated by assessing the processing time of the distances’ optimized selection algorithm and the four localization methods. From the statistics point of view, we repeat the distances’ optimized selection and our localization methods 1000 times and measure the whole processing time of the five methods, as shown in Table 6.

Table 6 illustrates that the efficiency of the distance selection with feature points is lightweight. More specifically, the processing of the distance selection is 0.024654 s, which is 87.19%, 65.33%, 62.82%, and 5.45% of the localization processing time of the four localization methods. Therefore, we adopt the proposed distance selection strategy in most real-time application systems.

6. Discussion

We consider the generality and scalability in this paper, and we apply the distance selection with feature points.

6.1. The Generality of the Optimization Selection Strategy

The optimization selection approach is implemented on four localization techniques in this study. The effectiveness and feasibility of the proposed distance selection strategy have been demonstrated through both the simulation and the experimental results. The positioning of multiple anchor nodes around an unidentified node is a common practice in localization systems. Therefore, the optimization of selecting distances between these anchor nodes and unknown nodes can be achieved. We can utilize the strategy in various range-based localization methods, including the improved version of these localization methods and other localizations. Therefore, the generality of our proposed method is extremely good.

6.2. The Scalability of the Distance Selection Strategy

In many localization application systems, we need distributed localization methods in large fields. For the proposed distance selection strategy, as it can be conducted on each unknown node, it is suitable for distribution localization. Therefore, the scalability of the distance selection strategy is fit for distribution localization. In our future work, we will conduct the proposed-method-based distribution localization and cooperated localization system.

7. Conclusions

To enhance the localization precision in a complex 3D environment, while considering the adverse effects of NLOS circumstances, we present the proposed solution, which involves the implementation of a 3D positioning system that integrates the utilization of feature points for accurate distance determination. The proposed method leverages characteristic points and employs the reverse assessment of distance estimation results to mitigate the adverse effects caused by NLOS conditions. Simulations and experiments are conducted to evaluate the efficacy of our suggested approach, demonstrating the effectiveness of our localization system.

Author Contributions

Conceptualization, N.W. and X.Y.; methodology, N.W. and Q.L.; software, Q.L.; validation, N.W., Q.L. and Y.X.; data curation, Y.X.; writing—original draft preparation, N.W. and Q.L.; writing—review and editing, N.W., X.Y., Q.L. and Y.X.; supervision, X.Y.; project administration, N.W.; funding acquisition, X.Y., Q.L. and Y.X. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (62271164), The Natural Science Foundation of Nanjing University of Posts and Telecommunications (NY224163), The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (23KJD510010), the China Postdoctoral Science Foundation (2023M741473), the Major Scientific and technological innovation project of Shandong Province of China (2020CXGC010705, 2021ZLGX-05 and 2022ZLGX04), and Shandong Provincial Natural Science Foundation (ZR2020MF017, ZR2022MF255).

Data Availability Statement

All the data encompassed in this study are obtainable from the corresponding author upon a reasonable request.

Acknowledgments

The authors are indebted to the anonymous reviewers for their insightful comments and helpful suggestions.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The architecture of the 3D localization system.

Figure 2. Localization node.

Figure 3. The processing flow of Newton’s iteration localization method (NILM).

Figure 4. The three-dimensional (3D) localization environment.

Figure 5. The deployment of the feature points.

Figure 6. The simulation data model.

Figure 7. The localization error under different signal-to-noise ratio (SNR) values.

Figure 8. The localization error with different strengths of None Line of Sight (NLOS).

Figure 9. The localization error with different densities of feature points.

Figure 10. The localization error with different densities of anchor nodes.

Figure 11. The simulation result of group 5.

Figure 12. The 3D localization environment.

Figure 13. The NanoLoc-based wireless localization system.

Figure 14. The deployment configuration of the anchor nodes.

Figure 15. The framework of the experiment’s data process.

Figure 16. The localization error observed in the four methods utilizing three strategies. (a) The localization error of the quadrilateral method. (b) The localization error of the least squares method; (c) The localization error of maximum likelihood method. (d) The localization error of the Newton’s iteration method.

Figure 17. The localization error of the three strategies: localization directly, delete maximum, and distance selection.

Figure 18. The average localization error of the four approaches: quadrilateral, least squares, maximum likelihood, and Newton’s iteration.

Figure 19. The localization error of the four localization approaches: quadrilateral, least squares, maximum likelihood, and Newton’s iteration.

Table 1. The simulation setting.

Factors	SNR	Strength of NLOS	Density of Feature Points	Density of Anchor Nodes	NLOS Rate
Group 1	15 dB~−10 dB	3 m	6³	8	25%
Group 2	10 dB	1~5 m	6³	8	25%
Group 3	10 dB	3 m	2³~8³	8	25%
Group 4	10 dB	3 m	6³	8~20	25%
Group 5	10 dB	3 m	6³	20	10~50%

Table 2. The anchor nodes’ numbers and coordinates.

No. of the Anchor Node	1 5 9	2 6 10	3 7 \	4 8 \
Coordinate	(1, 0.85, 1)	(11.8, 0.85, 1)	(22.6, 0.85, 1)	(22.6, 20.05, 1)
Coordinate	(11.8, 20.05, 1) (11.8, 19.7, 6.43)	(1, 20.05, 1) (0, 10.45, 6.43)	(11.8, 1.2, 6.43) \	(23.6, 10.45, 6.43) \

Table 3. The distances between the anchor nodes.

No. of the Anchor Node	1	2	3	4	5	6	7	8	9	10
1	0 m	10.80 m	21.60 m	28.90 m	22.03 m	22.39 m	19.20 m	11.07 m	12.09 m	25.15 m
2	10.8 m	0 m	10.80 m	22.03 m	19.20 m	19.62 m	22.03 m	16.15 m	5.44 m	16.15 m
3	21.60 m	10.80 m	0 m	19.20 m	22.03 m	22.39 m	28.90 m	25.15 m	12.09 m	11.07 m
4	28.90 m	22.03 m	19.20 m	0 m	10.80 m	12.09 m	21.60 m	25.15 m	22.39 m	11.07 m
5	22.03 m	19.20 m	22.03 m	10.80 m	0 m	5.44 m	10.80 m	16.15 m	19.62 m	16.15 m
6	22.39 m	19.62 m	22.39 m	12.09 m	5.44 m	0 m	12.09 m	14.99 m	18.50 m	14.99 m
7	19.20 m	22.03 m	28.90 m	21.60 m	10.80 m	12.09 m	0 m	11.07 m	22.39 m	25.15 m
8	11.07 m	16.15	25.15 m	25.15 m	16.15 m	14.99 m	11.07 m	0 m	14.99 m	23.60 m
9	12.09 m	5.44 m	12.09 m	22.39 m	19.62 m	18.50 m	22.39 m	14.99 m	0 m	14.99 m
10	25.15 m	16.15 m	11.07 m	11.07 m	16.15 m	14.99 m	25.15 m	23.60 m	14.99 m	0 m

Table 4. The localization points number and coordinate.

No. of Unknown Node	1 5 9 13 17 21	2 6 10 14 18 22	3 7 11 15 19 23	4 8 12 16 20 24
Coordinate	(3.4, 3.4, 1.0)	(3.4, 5.8, 1.0)	(3.4, 9.4, 1.0)	(3.4, 11.8, 1.0)
Coordinate	(3.4, 15.4, 1.0) (5.8, 11.8, 1.5) (9.4, 9.4, 1.0) (11.8, 5.8, 1.5) (15.4, 3.4, 1.0)	(5.8, 3.4, 1.5) (5.8, 15.4, 1.5) (9.4, 11.8, 1.0) (11.8, 9.4, 1.5) (15.4, 5.8, 1.0)	(5.8, 5.8, 1.5) (9.4, 3.4, 1.0) (9.4, 15.4, 1.0) (11.8, 11.8, 1.5) (15.4, 9.4, 1.0)	(5.8, 9.4, 1.5) (9.4, 5.8, 1.0) (11.8, 3.4, 1.5) (11.8, 15.4, 1.5) (15.4, 11.8, 1.0)

Table 5. The localization error of the four approaches: quadrilateral, least squares, maximum likelihood, and Newton’s iteration.

Strategies	Quadrilateral	Least Squares	Maximum Likelihood	Newton’s Iteration	Improvement
Directly	7.957 m	6.861 m	6.860 m	5.912 m	17.78%
Delete Maximum	6.115 m	5.075 m	5.050 m	4.227 m	21.29%
Distance Selection	3.855 m	2.862 m	2.826 m	1.954 m	37.30%
Improvement	44.26%	50.95%	51.42%	60.36%	\

Table 6. The process time of the five localization methods: distances’ optimized selection, quadrilateral, least squares, maximum likelihood, and Newton’s iteration.

	Distances’ Optimized Selection	Quadrilateral	Least Squares	Maximum Likelihood	Newton’s Iteration
Time (s)	24.654	28.275	37.736	39.246	452.284

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Wu, N.; Yan, X.; Luo, Q.; Xing, Y. Improving Localization Accuracy Through Optimal Selection Strategy. Electronics 2025, 14, 172. https://doi.org/10.3390/electronics14010172

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Wu N, Yan X, Luo Q, Xing Y. Improving Localization Accuracy Through Optimal Selection Strategy. Electronics. 2025; 14(1):172. https://doi.org/10.3390/electronics14010172

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Wu, Na, Xiaozhen Yan, Qinghua Luo, and Yuexiu Xing. 2025. "Improving Localization Accuracy Through Optimal Selection Strategy" Electronics 14, no. 1: 172. https://doi.org/10.3390/electronics14010172

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Wu, N., Yan, X., Luo, Q., & Xing, Y. (2025). Improving Localization Accuracy Through Optimal Selection Strategy. Electronics, 14(1), 172. https://doi.org/10.3390/electronics14010172

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