UAV-Assisted Localization of Ground Nodes in Urban Environments Using Path Loss Measurements

Abstract

This paper proposes a distance estimation error reduction framework to improve ground node localization accuracy in urban environments using an unmanned aerial vehicle (UAV) and path loss measurements. The primary goal of the framework is to bound distance estimation errors arising from inherent inaccuracies in path loss measurements. A k-means clustering algorithm is first applied to identify the region in which the ground node is located. Then, an analytical approach is used to select UAV waypoints. Moreover, a mean-based exponential smoothing approach is employed to refine the path loss measurements of the selected waypoints to mitigate the effects of multipath components that introduce significant errors in distance estimation. Finally, two estimators, maximum likelihood (ML)-based and semidefinite programming (SDP)-based relaxation, are employed to estimate the ground node’s location, validating the effectiveness of the proposed framework. Evaluations using ray tracing simulation data demonstrate a notable improvement in localization accuracy. The proposed framework effectively bounds the distance estimation errors and significantly reduces overall localization errors compared to conventional unbounded methods. Moreover, both estimators with the proposed framework achieve comparable localization accuracy, highlighting the framework’s capability to address key challenges in ML-based localization.

1. Introduction

Ground node localization in urban environments is essential for a wide range of applications, including search and rescue missions and smart city applications [1,2]. However, conventional localization technologies, such as the Global Positioning System (GPS), often exhibit limitations in these complex settings due to signal blockage caused by dense urban infrastructure [3]. Unmanned aerial vehicles (UAVs), with their maneuverability and advanced communication capabilities, offer a promising alternative for localizing ground nodes in such challenging environments [4].

Localization techniques based on wireless measurements can be categorized into range-based and range-free approaches [5]. Range-based techniques are generally more applicable and sustainable for ground node localization in outdoor scenarios [6,7]. In contrast, range-free localization techniques require the collection of extremely large datasets, making them costly and challenging to implement in outdoor scenarios. Essentially, range-based localization techniques rely on direct measurements, such as distance or angle, combined with geometric relationships to estimate the target’s location. In fact, achieving high localization accuracy in range-based techniques depends on two key factors: (1) the ranging model used to estimate the distance between reference nodes and the target and (2) the estimator employed to solve the formulated localization problem [8]. There are two widely used ranging models: Time of Arrival (TOA)-based and Received Signal Strength (RSS)-based models [9]. RSS-based models are particularly attractive compared to TOA models owing to their simplicity and lower implementation cost, primarily because they do not require additional hardware. These advantages have motivated researchers to employ RSS-based models in UAV localization [10,11,12,13]. However, the accuracy of RSS-based models is highly dependent on the path loss model [14], prompting various studies to utilize path loss models for distance estimation [15,16,17,18].

Typically, the objective functions of localization problems derived from the RSS model or path loss model are nonlinear and non-convex; therefore, a powerful estimator is required to solve them. Several estimators have been proposed in the literature to address these types of localization problems. Linearized least squares (LLS) estimators have been used to solve localization problems in [10,15,18]. While this method is simple, it provides poor performance. To improve the accuracy of LLS, a localization method based on the Weighted Least Squares Estimator (WLSE) was proposed in [19]. Although RSS measurement errors are independent, they are not identical, resulting in an unknown covariance matrix weighting. Moreover, the general assumption that noise in RSS measurements follows a normal distribution is often unrealistic in practice, making this method impractical. To overcome these limitations, the maximum likelihood (ML) estimator has been widely used in several research studies. In [11,12,20], the Particle Swarm Optimization (PSO) algorithm is employed to optimize the ML problem and estimate the location of ground nodes. In the same manner, the well-known Levenberg–Marquardt (LM) algorithm is used in [21]. Although ML estimators can be implemented without deriving a closed-form solution, their performance may suffer if a poor initial guess is selected, causing the optimization algorithm to converge to a local maximum instead of the global optimum. Recently, relaxation techniques have been leveraged to transform nonlinear and non-convex localization problems into linear and convex formulations. In [17], the nonlinear and non-convex problem, obtained from an unknown Path Loss Exponent (PLE), is first converted into a convex form using piecewise approximation, and then further linearized using Taylor’s series expansion. Additionally, semidefinite programming (SDP)-based relaxation has been proposed in [22,23]. Despite these efforts, the accuracy of these estimators remains unsatisfactory due to the approximations involved.

Although various effective estimators have been proposed in the literature to improve localization accuracy, their performance deteriorates as distance estimation errors increase. Path loss measurement error is the primary source of inaccuracy in distance estimation. In urban environments, path loss measurements are significantly influenced by multipath components due to the high number of signal reflections. Additionally, environmental and weather conditions introduce further uncertainties in path loss measurements. Unfortunately, these errors do not follow a specific statistical distribution, posing substantial challenges in achieving accurate localization.

As part of this trend, researchers have sought to improve localization accuracy either by optimizing the parameters of the path loss model or by selecting waypoints that reduce the impact of distance estimation errors in the localization process. Empirical optimization techniques have been employed to tune path loss model parameters in order to reduce the impact of measurement errors [24]. Such techniques, unfortunately, are scenario-dependent (e.g., line of sight vs. non-line of sight) and susceptible to noise. The authors in [17,25,26] treat the model parameters as unknowns that need to be estimated. However, this approach introduces increased complexity.

Regarding waypoint selection, the authors in [27] conducted an analytical investigation to bound the impact of distance estimation errors on localization accuracy in UAV-to-ground node communication scenarios. They suggested that these errors can be mitigated by selecting waypoints that satisfy specific constraints. While their recommendations showed notable improvements in localization accuracy, implementing these constraints becomes challenging in scenarios where the ground node location is unknown.

To facilitate the implementation of the recommendations from the investigations in reference [27], this paper proposes a distance estimation error reduction framework to improve ground node localization accuracy in urban environments composed of three stages. A clustering algorithm is first applied to roughly identify the region where the ground node is located within the deployment area. Based on the detected region, an analytical approach is used to determine both the boundaries of the selected waypoints and the minimum ground distance recommended in [27]. Using this information, candidate waypoints for localization are selected. Next, path loss measurements obtained at the selected waypoints are refined using a mean-based exponential smoothing approach to mitigate the path loss measurement errors caused by MPCs. The refined path loss measurements are then used to estimate the distances between the selected waypoints and the ground node. Finally, two estimators are employed to estimate the locations of the ground nodes, thereby validating the effectiveness of the proposed framework. Simulation results show that the proposed method significantly improves localization accuracy by effectively bounding distance estimation errors and reducing overall localization errors compared to the conventional (without distance estimation error reduction) method.

In summary, this paper provides the following contributions:

• We study and analyze the influence of the path loss measurement on ranging error and explore methods to mitigate the impact of this error in localization accuracy.
• We propose a distance estimation error reduction framework to reduce the effects of ranging error on localization accuracy. The framework utilizes the unsupervised learning algorithm to detect the region of the ground node and introduce a multilateration method with waypoint selection.
• We evaluate the performance of the proposed framework using realistic path loss measurements obtained through ray tracing techniques. The localization results demonstrate that the framework effectively bounds estimation errors and reduces overall localization errors compared to conventional unbounded methods. Moreover, both the proposed estimators achieve comparable localization accuracy, highlighting the framework’s ability to address key challenges in ML-based localization.

The rest of this paper is organized as follows: Section 2 describes the adopted system model for this research. Section 3 presents the analysis of impact of path loss measurement errors in distance estimation in UAV–ground node scenarios and proposes solutions to reduce it. Section 4 elaborates on the proposed distance estimation error reduction framework. The simulation configuration is presented in Section 5. Section 6 presents the results and discussions. Finally, Section 7 provides the conclusion of this work.

2. System Model

The operational setup for ground node localization is illustrated in Figure 1. We consider N ground nodes (represented as red circles) that are randomly distributed within an urban deployment area, with their locations unknown and denoted as G_n = [x_n,y_n], where n = 1,2, 3, …, N. These ground nodes periodically transmit communication signals at a frequency of 5.9 GHz. To localize the ground nodes, a UAV performs an initial scan of the deployment area, following a predefined trajectory determined by the SCAN algorithm, which has demonstrated the lowest localization error [28]. The UAV flies at a fixed predefined altitude h and navigates through M waypoints with known coordinates, denoted as W_m = [x_m,y_m,z_m], where m = 1,2, 3, …, M. The UAV altitude is selected to maximize the probability of line-of-sight connection, $P_{LOS}$, based on the derived expression in [10,29]

$${\mathrm{P}}_{\mathrm{LOS}}={\displaystyle \frac{\mathrm{a}}{1+\mathrm{a}{\mathrm{e}}^{-\mathrm{b}\varnothing }}}$$

(1)

where ∅ is the elevation angle in radians, and the coefficients a and b depend on the urban density. During flight, the UAV follows the predefined trajectory and stops at each waypoint to collect path loss measurements from the ground nodes within its communication range. As illustrated in Figure 1, $d_{\mathrm{g}}$ represents the ground distance between a ground node and the waypoint projection, while $d_s$ represents the slant distance, which can be estimated using the measured path loss. Several distance-based path loss models have been proposed for UAV-to-ground communication [30,31]. In this study, we adopt the well-known Free-Space Path Loss (FSPL) model to compute $d_s$. The measured path loss at the m-th waypoint for the transmitted signal from the n-th ground node is expressed as follows:

$${\mathrm{PL}}_{\mathrm{mn}}=20{\log }_{10}{d}_{mn}+ 47.87+{ϵ}_{\mathrm{mn}}$$

(2)

where $d_{mn}=\Vert W_m-G_n\Vert$ represents the slant distance, and ${ϵ}_{mn}$ is the path loss measurement error caused by shadowing and multipath components.

3. Analysis of Path Loss Measurement Error

In this work, path loss measurement is used for distance estimation as a primary step in localization. As mentioned above, path loss measurements are significantly affected by various factors, especially in urban environments, leading to potential errors. The major challenge is the nonlinearity of these errors. Moreover, these errors cannot be assumed to be consistent. In this section, an analytical approach is presented to investigate the impact of path loss measurement errors on distance estimation and how this impact can be bounded.

Figure 2 shows a scatter plot of path loss measurements as a function of the true distance between the UAV waypoints and the ground node. The path loss measurements are obtained using a ray-tracing technique, incorporating dependencies on shadowing and multipath components. As expected, the path loss generally increases with distance. However, the trend is not strictly linear, indicating significant variability in the measurements due to environmental factors. Notably, for shorter distances (e.g., 110–135 m), the path loss measurements are more tightly clustered, suggesting less variation in measurement errors. This is likely because, at shorter distances, the path loss measurements are taken from waypoints that maintain an LOS connection to the ground node. In contrast, waypoints that are perpendicular to the ground node exhibit greater variability in path loss measurements. In summary, the error in distance estimation using path loss measurements can be preliminarily bounded by selecting waypoints with a clear LOS connection.

However, it is well known that the impact of multipath components on the received signal can be either constructive or destructive, leading to positive or negative errors in path loss measurements. Therefore, distance estimation can result in either underestimation or overestimation. Figure 3 illustrates a scenario where distance overestimation occurs due to a positive error in path loss. The distance estimation error, $E_{ds}$, can be calculated as follows:

$$E_{ds}=d_s-{d^{\prime }}_s$$

(3)

where $d_s$ and ${d^{\prime }}_s$ are the true and measured slant distance, respectively. Based on the analysis presented in [27], the ground distance estimation error, $E_{dg}$, can be expressed as follows:

$${\mathrm{E}}_{\mathrm{dg}}={\mathrm{E}}_{\mathrm{ds}}·{\displaystyle \frac{1}{\cos \varnothing }}={\mathrm{E}}_{\mathrm{ds}}· \sqrt{1+{\displaystyle \frac{{\mathrm{h}}^2}{{{\mathrm{d}}_{\mathrm{g}}}^2}}}$$

(4)

where $d_{\mathrm{g}}$ is the ground distance, ∅ is the elevation angle, and h is the UAV altitude.

From Equation (4), it is obvious that when the UAV altitude is fixed, $E_{\mathrm{d}\mathrm{g}}$ varies with $d_{\mathrm{g}}$, which represents the distance between the true location of the ground node, G_n, and the projection of the waypoint, W′. Specifically, $E_{\mathrm{d}\mathrm{g}}$ increases as $d_{\mathrm{g}}$ decreases. This leads to the conclusion that to bound $E_{\mathrm{d}\mathrm{g}}$, waypoints located in areas perpendicular to terrestrial objects should be avoided. In other words, $E_{\mathrm{d}\mathrm{g}}$ can be bounded by introducing a minimum ground distance constraint, ${\mathrm{d}}_{\mathrm{g}\_\min }$, when selecting waypoints for localization. The minimum ground distance, ${\mathrm{d}}_{\mathrm{g}\_\min }$, can be defined as follows:

$$d_{\mathrm{g}\_\min }={\displaystyle \frac{\mathrm{h}}{\tan {\varnothing }_{\mathrm{LOS}}}}$$

(5)

where ${\varnothing }_{\mathrm{LOS}}$ is the elevation angle with a high probability of LOS waypoints.

Based on the presented analysis, we conclude that to further mitigate the impact of path loss measurement errors on distance estimation, the waypoints must maintain an LOS connection with the ground node and remain outside the minimum distance threshold.

4. Proposed Framework

Based on the analysis presented in the previous section, this section proposes a distance estimation error reduction framework to improve the localization accuracy of ground nodes in urban environments. The framework utilizes path loss measurements obtained from an initial scan of the deployment area to identify the region of the ground node using a clustering algorithm. Based on the detected region, the boundary of this area is determined based on maximizing the probability that the waypoints within it maintain an LOS connection with the ground node. Following this, the minimum distance constraint, as recommended in Section 2, is calculated to guide the selection of waypoints for localization. Since the selected waypoints are likely to be approximately the same distance from the ground node, a mean-based exponential smoothing approach is applied to refine their path loss measurements by converging them toward their mean value, thereby mitigating noise. Finally, two estimators, namely ML and SDP, are employed to estimate the location of the ground nodes and thereby validate the proposed framework. Figure 4 shows the workflow of the proposed framework.

4.1. Ground Node Region Detection

This section elaborates on the application of a clustering algorithm to detect the region of the ground node. It further explains the process of determining the boundaries of this region and details the calculation of $d_{\mathrm{g},\mathrm{m}\mathrm{i}\mathrm{n}}$, as recommended in the analysis presented in Section 3. Based on the constructed model in this work, the path loss measurements in the deployment area exhibit aggregation characteristics, where the path loss values tend to reach their lowest levels in the region of the ground node. Conventionally, the moving average window approach can be used to detect the region of the ground node by selecting the window where adjacent waypoints exhibit the lowest mean path loss. However, this approach can be unreliable, as the window selection can be skewed due to the variability in path loss measurement in some regions. However, clustering algorithms are well-suited for processing this parameter due to their ability to group data effectively. Clustering algorithms are a type of unsupervised machine learning that classify data without the need for predefined labels [32,33]. Among the various clustering algorithms, k-means clustering is recognized for its accuracy and simplicity [34]. Given a dataset D = { (W₁, PL₁), (W₂, PL₂), …, (W_m, PL_m) }, where Wm represents the 2D coordinates of the m-th waypoint and PL_m represents its associated path loss measurements, the goal is to partition the data into K clusters, C₁, C₂, …, C_K, by maximizing similarities through distance minimization. Integrating path loss into the distance function will influence cluster centroid placement. Therefore, each cluster is defined to contain spatially adjacent waypoints with similar path loss measurements. The clustering process can be formulated as minimizing the following objective function:

$$\underset{C_1,C_2,\dots ,C_K}{minimize}\sum _{k=1}^K \sum _{\left({\mathbf{W}}_m,{PL}_m\right)\in C_k} \left(\alpha {‖{PL}_m-{\mu }_k^{PL}‖}^2.\gamma {‖{\mathbf{w}}_m-{\mathbf{\mu }}_k^W‖}^2\right)$$

(6)

where ${\mu }_k^{PL}$ is the mean path loss of cluster ${\mathcal{C}}_K$, ${\mathit{\mu }}_k^w$ is the centroid of the waypoints in ${\mathcal{C}}_K$, and α and γ are weighting factors that balance the importance of path loss similarity and spatial adjacency, respectively. Typically, the region of the ground node corresponds to the cluster with the lowest mean values. The average path loss for each cluster can be calculated as follows:

$${\mathsf{\mu }}_{\mathrm{k}}^{\mathrm{PL}}={\displaystyle \frac{1}{\left|{\mathrm{C}}_{\mathrm{k}}\right|}}\sum _{{\mathrm{PL}}_{\mathrm{mn}}\in {\mathrm{C}}_{\mathrm{k}}} {\mathrm{PL}}_{\mathrm{mn}}$$

(7)

where $\left|{\mathcal{C}}_k\right|$ is the size of the cluster.

4.2. Waypoint Selection

The selection of waypoints is guided by the recommendations presented in Section 3. Specifically, the selected waypoint area is defined by a circle whose center corresponds to the projection of the cluster centroid, and whose radius is denoted by R. The radius is calculated based on the UAV altitude that achieves a satisfactory probability of LOS, $P_{LOS}$, greater than 95% and the maximum communication range, $d_{s,max}$, such that

$$\begin{array}{ll}\underset{d_s,h}{R=maximize}& \sqrt{{d_s}^2-h^2}\\ s.t.& P_{LOS} \ge 95\% \\ & d_s \le d_{s,max}\end{array}$$

(8)

where $d_{s,max}$ is the maximum communication distance that maintains the minimum RSS, typically defined by the receiver sensitivity limit. Essentially, $d_{s,max}$ is a scale-dependent value influenced by the transmitted power, antenna gains of the transmitter and receiver, and operating frequency. Analyzing Equation (8), increasing the UAV altitude reduces the size of the detected region (see Figure 5), resulting in fewer waypoints within this area. This is because increasing the UAV altitude enhances the probability of LOS (see (1)). Therefore, a tradeoff between the UAV altitude and the size of the communication area is essential.

Finally, after determining the boundaries of the ground node region, $d_{\mathrm{g},min}$, as recommended in Section 3, can be calculated using Equation (5) along with the provided values in Table 2 in reference [24]. The waypoints located within these boundaries, denoted as W_s = [x_s,y_s,z_s], s = 1,2,3, …, S, where S < M, along with their associated path loss measurements, are selected for localization.

4.3. Refine Path Loss Measurements

Since the ground node is highly likely to be close to the center of the detected region, the selected waypoints typically maintain similar distances to the ground node location. Consequently, their associated path loss measurement should be quite similar. Therefore, it is reasonable to refine these measurements toward their mean value to mitigate noise. Exponential smoothing is the common approach for refining time series observations [35]. However, since the path loss measurements of the selected waypoints are not time series observations, the exponential smoothing method is modified to perform a bias correction using a mean-based exponential smoothing approach to smooth out fluctuations in the path loss measurements for all waypoints simultaneously, as follows. Given PL_s, the path loss measurement obtained at the selected waypoint W_s, the refined path loss measurement can be represented as follows:

$${PL}_{sr}=\left(1-\rho \right){PL}_s+\rho {\overline{PL}}_s$$

(9)

where ${\overline{PL}}_s$ is the mean of path loss measurements of all selected waypoints, and $\rho$ is a weighting factor between 0 and 1. By applying this approach, each ${PL}_s$ is adjusted towards the mean, ${\overline{PL}}_s$, with strength controlled by $\rho$.

The refined path loss measurements are then used to estimate the distances between the waypoints and the ground node based on Equation (2) as follows:

$${\tilde{d}}_s={10}^{{\displaystyle \frac{{PL}_{sr}-47.87}{20}}}$$

(10)

4.4. Validation of the Proposed Framework

Various estimators have been proposed to solve localization problems formulated using RSS, which is inherently non-convex. In this work, the maximum likelihood (ML) and semidefinite programming-based Least Squares Ranging Error (SDP-LSRE) estimators [36] are selected to validate the efficiency of the proposed framework. ML is selected to assess how well the framework addresses the non-convex optimization challenge, leveraging the strategy of initializing the ML solver with an initial guess sufficiently close to the true location to facilitate convergence to an accurate solution. Conversely, SDP-LSRE is chosen for its robustness and convex relaxation capability, as demonstrated in [36]. By comparing localization performance with and without the proposed distance estimation correction framework using both estimators, we demonstrate the framework’s generalizability and its consistent improvement in localization accuracy.

The ML estimator for estimating the ground node location G_n using Equation (2) and the measured path loss at the selected waypoints W_s can be expressed as follows [36]:

$$\widehat{\mathrm{G}}=\underset{\mathrm{x},\mathrm{y}}{\mathrm{argmin}} \sum _{\mathrm{s}=1}^{\mathrm{S}}{\left[{\mathrm{PL}}_{\mathrm{sr}}-47.87-20{\log }_{10}\left(\sqrt{{\left(\right({\mathrm{x}-{\mathrm{x}}_{\mathrm{s}})}^2+{\left(\mathrm{y}-{\mathrm{y}}_{\mathrm{s}}\right)}^2)}^2+{\mathrm{h}}^2}\right)\right]}^2$$

(11)

where $\mathrm{S}$ is the index of the selected waypoint, ${PL}_{sr}$ is the redefined path loss for the selected s-th waypoint, and $x_s$ and $y_s$ are the coordinates of the selected s-th waypoint.

The SDP-LSRE estimator, obtained by utilizing the relative error estimation and semidefinite relaxation, can be represented as follows [33]:

$$\begin{array}{ll}\underset{\mathit{Z},u_s,v_s}{\widehat{G}=\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}}& {\displaystyle \sum _{s\in S}} \left({\displaystyle \frac{u_s}{{\tilde{d}}_s^2}}+{\tilde{d}}_s^2{v}_s\right)\\ \mathrm{s}.\mathrm{t}.& u_s={\left[\begin{array}{c}{\mathit{w}}_s\\ -1\end{array}\right]}^T\mathit{Z}\left[\begin{array}{c}{\mathit{w}}_s\\ -1\end{array}\right],s\in S\\ & \left[\begin{array}{cc}{v}_s& 1\\ 1& u_s\end{array}\right]\succcurlyeq {\mathbf{0}}_2,s\in S\\ & {\mathit{Z}}_{1:\mathrm{2,1}:2}={\mathit{I}}_2, \mathit{Z}⪰{\mathbf{0}}_3\\ \end{array}$$

(12)

where

$$v_s={\displaystyle \frac{1}{u_s}}$$

(13)

where ${\mathit{w}}_s={[{\mathrm{x}}_{\mathrm{s}} {\mathrm{y}}_{\mathrm{s}}]}^T$ represents the coordinates of the selected s-th waypoint, and $u_s$, $v_s$, and $\mathit{Z}$ are auxiliary variables introduced to relax the non-convexity of the problem.

5. Simulation Environment Configuration

This section describes the configuration of the simulation environment used to evaluate the performance of the proposed framework. The Wireless InSite (WI) simulator is utilized to construct a realistic UAV-to-ground node communication scenario in an urban environment, as shown in Figure 1. Based on the presented system model in Section 2, we assume that 10 ground nodes are randomly distributed within a 450 × 350 m² deployment area in an urban setting. Kuala Lumpur City Centre (KLCC) is chosen to represent the urban environment due to its diverse building layouts, which create a complex urban environment rich in multipath effects. The communication parameters for the UAV and ground nodes are listed in

Table 1. Communication parameters for UAV and ground node.

Parameter	Details
UAV and GN Antenna	Isotropic with gain 0 dB
Carrier Frequency	5.9 GHz
Bandwidth	500 MHz
Transmitted power	0 dBm
Raytracing	SBR
Propagation model	X3D

. The UAV follows a predefined trajectory to perform an initial scan using the SCAN algorithm with a resolution of 20 m, resulting in a total of 456 waypoints along the generated trajectory. Path loss predictions are generated using WI’s ray-tracing calculation engine. The UAV’s altitude is fixed, determined based on a trade-off between maximizing the line-of-sight probability $P_{LOS}$ and increasing the coverage area. To achieve this, the receiver’s sensitivity is used as a threshold for communication quality to determine the maximum communication range. According to the datasheet of DWM1001C [37], the receiver’s sensitivity in our simulation is set to −93 dBm/500 MHz. Using these communication parameters along with the Free-Space Path Loss (FSPL) model, we calculate the maximum communication range between the UAV and ground nodes. As shown in Figure 6, the communication range for the −93 dBm threshold is about 180 m. Consequently, the $P_{LOS}$ and the communication coverage radius are calculated based on Equations (1) and (8), respectively, as illustrated in Figure 7. The figure shows that the UAV altitude that obtained a $P_{LOS}$ more than 95% and an airspace coverage radius of 150 m falls between 100 m and 120 m. Therefore, we set the UAV altitude to 110 m.

6. Results and Discussion

This section presents the performance evaluation of the proposed framework. We begin by discussing the results of the k-means-based clustering algorithm, which is applied to detect the regions of ground nodes. We then analyze the localization results in terms of Root Mean Square Error (RMSE).

6.1. Detected GN Regions Using K-Means-Based Clustering

As mentioned earlier, the k-means-based clustering algorithm is used to identify the regions of ground nodes within the deployment area based on path loss measurements. To achieve this, the path loss values are grouped into k clusters based on their proximity to the cluster centroids. The cluster with the lowest mean of path loss measurements is then selected to represent the regions of the ground nodes. However, the choice of k (the number of clusters) significantly impacts the clustering result, as it directly affects how the 456 samples (corresponding to the number of UAV waypoints) are grouped. It is important to note that while each cluster may contain a different number of samples, the overall distribution of samples across clusters depends on the chosen k value. To determine the optimal number of clusters, we implemented the k-means algorithm with k values ranging from 5 to 45. For each clustering configuration, we computed the Euclidean distance error between the centroid of the cluster with the lowest path loss measurements and the actual locations of ground nodes. Figure 8 shows the average errors for ten ground nodes as a function of the number of clusters. The results show that the average error decreases significantly as the number of clusters increases, stabilizing around 15 to 20 clusters. This indicates that increasing the number of clusters improves data grouping and reduces error; however, beyond a certain point, further increases in k lead to a rise in error. The lowest error is observed when the number of clusters is set to 15.

Figure 9 illustrates the detected ground node regions using k-means clustering (k = 15) and a conventional 7 × 7 moving average window. This figure shows a radio map of the path loss, highlighting the detected regions for two ground nodes. These regions are represented by clusters of waypoints (red crosses) characterized by minimum mean path loss. Overall, k-means outperformed the moving average window approach. The errors for k-means, calculated between cluster centroids and actual ground node locations, ranged from 4.52 m to 47.63 m, with a mean error of 16.72 m. Conversely, the moving average approach yielded errors between 8.635 m and 46.6 m, with a mean error of 28.7 m. This indicates that the k-means-based clustering algorithm offers improved ground node region detection when path loss measurements are used as input features. However, for the results of the k-means-based clustering algorithm, it was observed that while the path loss measurements for the two ground nodes were partitioned into the same number of clusters (15 clusters), the sizes of the selected clusters that have the minimum average path loss (i.e., the number of waypoints per cluster) exhibited variability.

The boundaries of the detected region are determined as follows: the center of the detected region is considered as the center of the ground node detected region. Then the radius, R (i.e., $d_{\mathrm{g}\_max}$), of the detected region is determined using Equation (8). The value of $d_s$ is set to 155 m to ensure a high probability of LOS, aligning with the values specified for urban areas in Table 2 of reference [27]. As a result, the calculated $d_{\mathrm{g}\_max}$ is 110 m. However, according to Table 2 in reference [27] and Equation (8), the minimum radius $d_{\mathrm{g}\_min}$ is set to 40 m. Finally, the waypoints located between the $d_{\mathrm{g}\_max}$ and $d_{\mathrm{g}\_min}$ are selected, and their associated path loss measurements are utilized for localization.

To refine the path loss measurements of the selected waypoints, a mean-based exponential smoothing approach is applied to adjust them around their average value. The weighting factor ρ is empirically set to 0.6 using a trial-and-error method to minimize the average distance estimation error. Figure 10 shows the refined path loss measurements for sensors #2 and #4.

6.2. Localization of Ground Nodes

The refined path loss measurements are used to estimate the ground node locations. The MATLAB routine lsqnonlin is used to solve the ML problem, while the CVX MATLAB toolbox is employed to solve the SDP problem. For the ML method, the refined path loss measurements are directly applied in Equation (11) and the center of the detected region is considered as an initial guess. In contrast, for the SDP-LSRE method, the refined path loss measurements are first used to calculate the distance based on Equation (10), which is then used in the SDP-LSRE estimator as defined in Equation (12). The performance of the proposed framework is evaluated using Root Mean Square Error (RMSE), which is calculated based on the Euclidean distance between the true and the estimated locations. Moreover, the localization results of the proposed framework using k-means are compared with those obtained using moving average window and the unbounded path loss error method. The localization results for ten ground nodes are summarized in Table 2. Overall, the results indicate that the proposed framework with k-means significantly improves distance estimation accuracy and reduces localization errors compared to the other methods. The proposed framework with moving average window comes in second. The localization RMSE decreases from 45.87 m and 41.99 m for without distance estimation reduction (unbounded) to 13.25 m and 10.70 m when using the proposed framework for ML and SDP-LSRE, respectively. Moreover, the SDP-LSRE estimator outperforms the ML estimator in localization accuracy. Notably, the ML-based method exhibits a maximum error of 22.81 m, which is substantially lower than the 73.35 m observed in the unbounded ML method. Similarly, the proposed SDP-LSER-based method achieves a maximum error of 16.53 m, significantly lower than the 68.57 m recorded for the unbounded SDP-LSRE method. These results highlight the effectiveness of the proposed methods in minimizing localization errors.

Table 2. Summary of localization results.

Framework Description	Min Error	Max Error	RMSE
Bounded using K-means with ML	5.1920	22.8110	13.2586
Bounded using K-means with SDP-LSRE	5.1470	16.5350	10.7012
Bounded using moving average with ML	7.9630	50.8910	34.8019
Bounded using moving average with SDP-LSRE	7.0580	41.0960	29.2911
Unbounded with ML [36]	13.791	73.3540	45.8752
Unbounded with SDP-LSRE [36]	8.4080	68.5700	41.9959

Table 2. Summary of localization results.

Framework Description	Min Error	Max Error	RMSE
Bounded using K-means with ML	5.1920	22.8110	13.2586
Bounded using K-means with SDP-LSRE	5.1470	16.5350	10.7012
Bounded using moving average with ML	7.9630	50.8910	34.8019
Bounded using moving average with SDP-LSRE	7.0580	41.0960	29.2911
Unbounded with ML [36]	13.791	73.3540	45.8752
Unbounded with SDP-LSRE [36]	8.4080	68.5700	41.9959

Figure 11 visualizes the sorted localization errors for the ten ground nodes obtained using the proposed framework and the unbounded methods, with both estimators. It is evident that the proposed framework achieves comparable localization errors with both estimators when compared to the unbounded methods. These results highlight the framework’s effectiveness in addressing a primary challenge in ML-based localization by selecting appropriate initial guesses. It is worth noting that the ML estimator can achieve high localization accuracy when the initial guesses are carefully selected. Guesses closer to the true location yield more accurate results, as they are less prone to convergence to local optima. In the proposed framework, the initial guesses for ML are determined by the cluster centroids, which are positioned near the actual ground node locations, as described in Section 6.1.

7. Conclusions

This paper aims to enhance the accuracy of range-based localization for UAV-to-ground node scenarios in urban environments by bounding the distance estimation error derived from path loss measurements errors. We analyzed the impact of path loss measurement errors on distance estimation and explored methods to mitigate these errors. As a solution, we proposed a distance estimation error reduction framework comprising the following stages: k-means clustering is applied to identify the ground node region. Waypoints are selected to perform localization using a multilateration-based technique. A mean-based exponential smoothing approach is employed to refine the path loss measurements of the selected waypoints. ML (maximum likelihood) and SDP-LSRE (semidefinite programming-based Least Absolute Relative Error) estimators are used to compute the ground node locations. The evaluation results demonstrate a substantial reduction in localization error when implementing the proposed framework. These findings suggest that accurate localization for ground nodes can be achieved using the proposed framework. Additionally, the results provide new insights into range-based localization methods for UAV-to-ground node scenarios, which could be valuable for future research and development.