Metaheuristic Optimization for Robust RSSD-Based UAV Localization with Position Uncertainty

Abstract

Unmanned aerial vehicles (UAVs) have garnered significant research interest across various fields due to their excellent maneuverability, scalability, and flexibility. However, potential collisions and other issues can disrupt communication and hinder functionality in real-world applications. Therefore, accurate localization of UAVs is crucial. Nonetheless, environmental factors and inherent stability issues can lead to node positional errors in UAV networks, compounded by inaccuracies in transmit power estimation, complicating the effectiveness of signal strength-based localization methods in achieving high accuracy. To mitigate the adverse effects of these issues, a novel received signal strength difference (RSSD)-based localization scheme based on a robust enhanced salp swarm algorithm (RESSA) is presented. In this algorithm, an elitism strategy based on tent opposition-based learning (TOL) is proposed to promote the leader to move around the food source. Differential evolution (DE) is then used to enhance the exploration ability of each agent and improve global search. In addition, a dynamic movement mechanism for followers is designed, enabling the swarm to swiftly converge towards the food source, thereby accelerating the overall convergence process. The RSSD-based Cramér–Rao lower bound (CRLB) with position uncertainty is derived to evaluate the performance. Experimental results are presented, which show that the proposed RESSA provides better localization performance than related methods in the literature.

1. Introduction

In recent years, unmanned aerial vehicles (UAVs) have become a research hotspot in academia and industry, and are widely used in emergency communications, infrastructure inspection, military or traffic monitoring, and other aspects [1,2,3,4,5]. In these applications, the geometric position information of UAVs (including micro UAVs [6]) is critical to their flight trajectory planning or obtaining and processing the specialized information collected by drones [7,8]. To acquire accurate location information of UAVs, various positioning technologies have been developed, with the Global Positioning System (GPS) being the most widely utilized. However, in numerous real-world situations, GPS signals may be weak or completely absent, posing challenges for practical applications in environments such as densely populated areas, urban landscapes, thick vegetation, shadowed tunnel regions, disaster-stricken areas, and combat zones [7,9,10,11]. As a result, the utilization of signals transmitted by base stations (BSs) has become a favored alternative due to its reliability. This transition has led to increased interest in these geolocation technologies, which primarily use measurements (obtained by equipped devices) such as angle of arrival (AOA), time of arrival (TOA), received signal strength (RSS), time difference of arrival (TDOA), and received signal strength difference (RSSD) [12,13,14,15,16,17,18,19,20] to locate a target UAV. Among these approaches, energy-based (RSS-based and RSSD-based) localization has garnered significant research interest due to the advantages compared to other methods, namely, low cost, simple implementation, and eliminating the need for precise clock synchronization and antenna arrays [21].

Despite the considerable research devoted to this type of localization technology, many approaches assume that the transmit power (TP) is known, which is often unrealistic. In real-world applications, the instantaneous TP fluctuates due to factors like battery power, making it difficult to maintain stability [22]. An inaccurate estimation of TP can lead to additional positioning errors. Thus, suboptimal estimators were devised for unknown transmit power (UTP) conditions [12,23,24,25]. A matrix factorization-based scheme was developed in [23] to improve positioning performance. An effective robust recursive least squares approach was proposed in [25] to realize localization in non-Gaussian environments. However, most of these approaches assume the target (such as UAV) positions are stable or fixed, which is often not the case due to external factors, and this can affect the localization accuracy. The nominal and actual positions in a communication network are actually inconsistent. External factors such as wind and rain can disturb the UAV nodes and cause small position errors. Further, factors such as malicious attacks or heavy rainfall can cause even greater position uncertainty. The inherent mobility of UAV nodes can also lead to position errors [26], which can greatly degrade the localization performance. Although some techniques have taken these factors into account, the localization performance is typically guaranteed based on certain mathematical approximations and/or reasonable initial estimates [27,28]. However, these approximations may introduce extra errors, and securing a reliable initial solution can be challenging.

Without these restrictions, population-based metaheuristic algorithms have been widely employed to solve non-convex optimization problems, including particle swarm optimization (PSO) [29], the salp swarm algorithm (SSA) [30,31,32], differential evolution (DE) [33,34], simulated annealing (SA) [35], artificial bee colony (ABC) [36], and the genetic algorithm (GA) [37]. Among these methods, DE and SSA have attracted the most attention [38,39]. DE has been shown to have good convergence and be effective in applications such as minimizing energy consumption [40] and detecting unknown cyberattacks [41]. SSA is an emerging algorithm [42] that has been successfully applied in feature selection [43], parameter estimation [44,45], underwater localization [46], etc. Despite their notable success across various fields, few works to date have considered the aforementioned UAV localization problem, particularly in the context of position uncertainty, using the improved SSA combined with DE. To fill this gap, this paper considers UAV localization using RSSD measurements with node position uncertainty (NPU) and UTP based on an enhanced SSA and DE. The resulting nonconvex UAV localization problem is first formulated as a maximum likelihood (ML) problem. Then, a robust enhanced salp swarm algorithm (RESSA) is proposed to solve it. In RESSA, an elitism strategy is presented to obtain a robust initial population, and an adaptive control strategy is designed to improve the search and convergence. Furthermore, differential evolution (DE) is employed to optimize agents during the search to accelerate convergence.

The main contributions of this paper are shortened as follows:

(1)

A novel RESSA method is proposed to solve the RSSD-based ML UAV localization problem with NPU and UTP in a wireless network.

(2)

An elitism strategy using tent opposition-based learning (TOL) is presented to obtain a diverse population. An adaptive control strategy is designed to avoid local optima. In addition, a DE mechanism is employed to improve global search and accelerate convergence.

(3)

An RSSD-based CRLB considering the position errors of UAV nodes and UTP is derived as a performance benchmark.

(4)

Simulation experiments are conducted to demonstrate the RESSA performance in UAV localization and compare it with several state-of-the-art schemes.

The rest of this paper is organized as follows: Section 2 presents the related work. The RSSD-based UAV localization problem with NPU and UTP is formulated in Section 3. The RESSA framework using SSA is given in Section 4. Further, the CRLB with UAV node position errors is derived and the complexity of related methods is analyzed. Section 5 presents the performance of the proposed RESSA and compares it with state-of-the-art methods. Finally, some concluding remarks are given in Section 6.

2. Related Work

There has been significant research to achieve accurate localization [4,7,47,48,49,50]. A distributed algorithm for the relative positioning of UAVs was presented in [47], which first derives the local geometry using the weighted semi-definite programming (SDP) algorithm and then integrates it into the global geometry utilizing statistical information. However, this takes enough iterations to achieve good accuracy ignoring the external interference. To cope with these interference, a UAV-assisted anti-interference system was developed in [48], which can not only realize the self-positioning and synchronization of UAVs but also realize the positioning of ground users based on TDOA information. To reduce the need for precise synchronization, a novel TDOA-based UAV localization algorithm was proposed in [4] that only requires basic UAV-BS synchronization to solve the reformulated Least Square (LS) problem, where fine search and coarse search are used to obtain the optimal estimate. And some other attention has been paid to RSS information. A new localization framework was designed in [49] using the drone as a mobile node for wireless sensor networks (WSNs) to obtain high localization accuracy. A joint UAV localization approach was designed in [7] to improve accuracy using multiple points of trajectory and multiple RSS measurements from different BSs. In addition, some hybrid schemes were explored. A novel hybrid TOA/AOA localization approach, named the azimuth TOA/1AOA algorithm, was proposed for UAV-assisted WSNs to provide good accuracy with low computational complexity [50].

Among these works, signal strength-based solutions typically employ either RSS measurements or RSSD measurements. Further, most of them only consider the case when TP is known. In fact, the unknown case is more common. This has been considered in [12,15,21,22,51,52,53,54,55,56]. A unified SDP approach was proposed in [55] to deal with unknown parameters. A two-stage RSS-based scheme was developed in [52] to mitigate the effects of signal attenuation and absorption in ocean sensor networks. The RSS-based cooperative localization problem was reformulated to a mixed SDP-SOCP problem in [53], which was solved using the epigraph method, Taylor expansion, and semi-definite relaxation (SDR). In [12], opposition-based learning (OBL) and adaptive redirection were used with DE for localization. For RSSD-based schemes, the advantages of them over RSS-based schemes include not requiring the TP and coordination between anchor nodes and the target [24]. An IDE-based method for RSSD localization considering Gaussian mixture noise was presented in [21]. Based on this, robust fault-tolerant localization was proposed in [15] to deal with faulty nodes. A bounded RLBM-SDP algorithm was presented in [56] to obtain the optimal position considering linear bias.

Most estimators in the above literature are based on approximations or good initial solutions. However, approximations often introduce errors, and a good initial solution can be difficult to obtain, so the positioning performance using RSS and RSSD measurements is limited. To this end, scholars have explored some alternative approaches to enhance the accuracy of drone self-positioning by utilizing machine learning techniques or heuristic optimization methods. For instance, using machine learning, a real-time positioning method for the target UAV based on Taylor series linearization is proposed in [57] to obtain a near-optimal solution. However, this approach necessitates extensive data collection to ensure high accuracy in this indoor environment. In addition, some research focused on heuristic optimization algorithms. An efficient PSO method was proposed in [9] to achieve two-dimensional (2D) UAV positioning, which utilizes hierarchical PSO (HPSO) and a new particle update scheme based on particle number to reduce the computational complexity and improve the accuracy. However, in real-world scenarios, non-line-of-sight (NLOS) situations often occur. Then, an efficient UAV-assisted localization strategy was proposed in [10] to adapt to NLOS environments, which plans appropriate search trajectories to ensure that there are at least three anchor nodes around each target, and then use the PSO algorithm for position estimation.

Although they have achieved some success in these scenarios, most approaches operate under the assumption that node positions are both known and constant, which is difficult to achieve in practice due to external and/or internal factors [16]. Several estimators have been developed that consider node position errors. A robust weighted LS (RWLS) estimator was proposed in [28] to reduce estimation errors. The S lemma and SDR were used to reformulate the problem. RCRM-SDP was developed in [27] to obtain a robust solution for large signal to noise ratios (SNRs). However, these still rely on some approximations or good initial solutions to guarantee the localization performance. Recently, SSA-based methods have been considered for target localization [58,59,60,61,62,63] in various scenarios. An SSA-based approach was developed in [58] for passive localization using TDOA measurements. A range-free SSA-based multi-objective localization scheme was designed in [60] to reduce positioning error using distance vector hop (DV-Hop). QSSA was combined with reliable anchor pair selection in [63] to obtain node positions in anisotropic WSNs. Underwater SSA (USSA) was presented in [61] for underwater node positioning. These studies all attained significant success independently of the aforementioned conditions. Combined with our previous work, the superior positioning performance of the OLAM-IDE algorithm [21] for non-Gaussian noise environments offers us fresh perspectives. To this end, considering both UTP and NPU, this paper proposes a novel UAV localization technique—i.e., RESSA—to improve localization accuracy, which does not depend on the previous two.

3. Problem Formulation

Considers a two-dimensional (2D) wireless network having N UAVs as anchor nodes with known coordinates and an unknown target UAV to be located. Suppose the unknown target UAV location is $\mathit{x}={\left[x_1,x_2\right]}^T$ and the actual UAV anchor node positions are ${\mathit{a}}_i={\left[a_{i1},a_{i2}\right]}^T$, $i=1,2,\dots ,N,N>3,$ where T denotes matrix transpose. The extension to three dimensions can easily be obtained. It is assumed that the coordinates of UAVs as anchor nodes are not in a straight line to avoid ambiguity, which is reasonable. The distance between the target UAV and ith anchor UAV node is

$$v_i=∥\mathit{x}-{\mathit{a}}_{\mathit{i}}∥,i=1,2,...,N.$$

(1)

The received signal power $P_i$ at the ith anchor based on a log-distance path loss model with no noise can be expressed as [15]

$${\displaystyle \frac{P_i}{P_t}}=Q_0{\left(v_i\right)}^{-\xi },i=1,2,\dots ,N,$$

(2)

where $Q_0$ is the corresponding path loss for the reference distance $v_0=1$ m; $P_t$ is the transmit power; and $\xi$ is the path loss exponent, which is typically between 1 and 6. The value of $\xi$ depends on the environment; in free space, $\xi =2$, so this is considered the a priori value.

A log-normal model is typically considered for dynamic wireless signals. In this case, the received power follows a log-normal distribution, so (2) can be reformulated as

$${\stackrel{⌣}{P}}_i\left(dB\right)=P_t\left(dB\right)-10\xi {\log }_{10}\left(v_i\right)+n_i,$$

(3)

where ${\stackrel{⌣}{P}}_i\left(dB\right)=P_i\left(dB\right)-Q_0\left(dB\right)$ and $n_i$ is measurement noise, which is assumed to be Gaussian with $n_i\sim N\left(0,{\sigma }_i^2\right)$.

Without loss of generality, let node 1 be the reference node. The RSSD between the first and ith nodes is ${\stackrel{⌣}{P}}_{i1}={\stackrel{⌣}{P}}_i-{\stackrel{⌣}{P}}_1$, which can be expressed as

$${\stackrel{⌣}{P}}_{i1}=-10\xi {log}_{10}\left({\displaystyle \frac{∥\mathit{x}-{\mathit{a}}_i∥}{∥\mathit{x}-{\mathit{a}}_1∥}}\right)+n_{i1}.$$

(4)

where $n_{i1}\sim \left(0,{\sigma }_i^2+{\sigma }_1^2\right)$. For simplicity, we assume the noise variances are identical, i.e., ${\sigma }_i^2={\sigma }_1^2={\sigma }^2$, so $n_{i1}\sim \left(0,2{\sigma }^2\right)$. Note that the RSSD source localization problem can be solved using (4) without knowledge of the transmit power $P_t$. This avoids the introduction of additional errors due to inaccurate power estimation.

Let $\Delta {\mathit{a}}_i={\left[\Delta a_{i1},\Delta a_{i2}\right]}^T$ be the ith node position error, $\Delta a_i\sim N\left(0,{\sigma }_a^2\right)$, due to interference and other errors. Then, the assumed position is

$$\begin{array}{c}\hfill {\tilde{\mathit{a}}}_i={\mathit{a}}_i+\Delta {\mathit{a}}_i\end{array}$$

(5)

Considering ${\tilde{\mathit{a}}}_i$ instead of ${\mathit{a}}_i$, the conditional probability density function (PDF) of $n_{1i}$ for the data matrix $\stackrel{⌣}{\mathit{P}}={[{\stackrel{⌣}{P}}_{12},{\stackrel{⌣}{P}}_{13},\dots {\stackrel{⌣}{P}}_{1N}]}^T$ can be expressed as

$$p\left(\stackrel{⌣}{\mathit{P}}\left|\mathit{x}\right.\right)=\underset{i=1}{\overset{N}{\Pi }}{\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{i1}}}exp\left(-{\displaystyle \frac{{\left({\stackrel{⌣}{P}}_{1i}-10\xi {log}_{10}\left(\frac{∥\mathit{x}-{\mathit{a}}_i-\Delta {\mathit{a}}_i∥}{∥\mathit{x}-{\mathit{a}}_1-\Delta {\mathit{a}}_1∥}\right)\right)}^2}{2{\sigma }_{i1}^2}}\right),$$

(6)

where ${\stackrel{⌣}{P}}_{1i}=-{\stackrel{⌣}{P}}_{i1},{\sigma }_{1i}^2={\sigma }_1^2+{\sigma }_i^2$, i = 2, …, N.

The maximum likelihood (ML) estimate $\widehat{\mathit{x}}$ of $\mathit{x}$ can then be obtained by maximizing the PDF $lnp\left(\stackrel{⌣}{\mathit{P}}\left|\mathit{x}\right.\right)$ in (6), which can be converted to a minimization problem, denoted as

$$\begin{array}{c}\hfill \widehat{\mathit{x}}=\underset{\mathit{x}}{argmin}f\left(\mathit{x}\right),\end{array}$$

(7)

where

$$\begin{array}{c}\hfill f\left(\mathit{x}\right)=\sum _{i=2}^N{\displaystyle \frac{1}{{\sigma }_{1i}^2}}{\left({\stackrel{⌣}{P}}_{1i}-10\xi {log}_{10}\left({\displaystyle \frac{∥\mathit{x}-{\mathit{a}}_i-\Delta {\mathit{a}}_i∥}{∥\mathit{x}-{\mathit{a}}_1-\Delta {\mathit{a}}_1∥}}\right)\right)}^2,\end{array}$$

(8)

To solve the optimization problem in (7), for convenience, we define the search space of the solution as

$$\begin{array}{c}\hfill \mathit{K}=\left\{\left(K_1,K_2\right):\left\{l{b}_m\le K_m\le u{b}_m\right\},m=1,2\right\},\end{array}$$

(9)

where $l{b}_m$ and $u{b}_m$ are lower and upper bounds, respectively. The notation employed in this paper is given in

Table 1. Notation.

Variable	Description
$\mathit{x}$	target location
${\mathit{a}}_i$	actual location of the ith anchor
${\tilde{\mathit{a}}}_i$	assumed location of the ith anchor
$\Delta {\mathit{a}}_i$	position error of the ith anchor
$v_i$	distance between the target and ith anchor
$v_0$	reference distance
$Q_0$	path loss for $v_0=1$ m
$P_t$	transmit power
$\xi$	path loss exponent
N	number of anchors
$P_i$	noise-free received signal power at node i
${\stackrel{⌣}{P}}_i$	noisy received signal power at node i
${\stackrel{⌣}{P}}_{i1}$	RSSD between the first and ith nodes
$\mathit{K}$	search space
$l{b}_m$, $u{b}_m$	lower and upper bounds of the search space
$l{b}_m^{\prime }$, $u{b}_m^{\prime }$	lower and upper bounds for the current individuals
$X_m^1$	coordinate of the leader in the mth dimension
$X_m^{\tau }$, ${\left(X_m^{\tau }\right)}^{\prime }$	coordinate of the th follower in the mth dimension
$X_m^p$, $X_m^q$	two individuals selected from the current population
$F_m$	food source in the mth dimension
$c_1$, $c_2$, $c_3$	control parameters
$c^{\prime }$	convergence parameter
${\mathit{t}}^{\tau }$	chaotic tent map vector for individual $\tau$
${\mathit{\nu }}_m^{\tau ,+}$	random vector generated using ${\mathit{t}}^{\tau }$
${\mathit{\nu }}_m^{\tau ,-}$	opposite vector of ${\mathit{\nu }}_m^{\tau ,+}$
${\mathit{Z}}^{\tau }$	$\tau$th initial individual
${\tilde{\mathit{Z}}}^{\tau }$	$\tau$th mutated individual
${\stackrel{⌣}{\mathit{Z}}}^{\tau }$	$\tau$th individual after crossover
${\stackrel{⌢}{\mathit{Z}}}^{\tau }$	$\tau$th updated individual
L	population size
$it$	current iteration
$i{t}_{max}$	maximum number of iterations
$\vartheta$	scaling factor
$\varphi$	crossover probability
$MC$	number of Monte Carlo trials

.

4. Robust Localization Using RESSA

In this section, we first present the SSA and then describe the proposed RESSA framework. The chaotic tent map, adaptive control strategy, and DE are explained in detail. The RESSA steps are also given for clarity. Then, the computational complexity of the proposed RESSA is analyzed and compared with other algorithms. In addition, the CRLB for the RSSD model with node position errors is derived as a benchmark, as shown in Appendix A.

4.1. Salp Swarm Algorithm

SSA is a swarm optimizer based on the behavior of ocean salps [42]. The initial salps are generated randomly. Each salp represents a candidate solution to the problem to be solved. A population of salps can be divided into a leader, which is the first salp in the group, and the followers. The leader determines the direction of movement of the swarm so the followers move with the leader.

The coordinates of the leader $X_m^1$ are updated as follows [43,64]:

$$\begin{array}{c}\hfill X_m^1=\left\{\begin{array}{cc}{F}_m+c_1\times \left(c_2\times \left(u{b}_m-l{b}_m\right)+l{b}_m\right)& c_3\ge 0.5\\ F_m-c_1\times \left(c_2\times \left(u{b}_m-l{b}_m\right)+l{b}_m\right)& c_3<0.5\end{array}\right.,\end{array}$$

(10)

where $F_m$ are the coordinates of the food source (target) in the mth dimension; $u{b}_m$ and $l{b}_m$ are the upper and lower bounds, respectively; and $c_1$, $c_2$, and $c_3$ are control parameters. $c_2$ and $c_3$ are uniformly distributed in [0,1], and $c_1$ decreases iteratively according to

$$\begin{array}{c}\hfill c_1=2\times e^{-{\left({\displaystyle \frac{\delta \times it}{i{t}_{max}}}\right)}^2},\end{array}$$

(11)

where $\delta$ is a constant and $it$ and $i{t}_{max}$ are the current and maximum numbers of iterations, respectively. The relationship between $c_1$ and $it$ is shown in Figure 1 for three values of $\delta$. Typically, $\delta =4$ is employed.

The coordinates of the $\tau$th follower in the mth dimension are updated as

$$\begin{array}{c}\hfill X_m^{\tau }={\displaystyle \frac{1}{2}}\times \left(X_m^{\tau }+X_m^{\tau -1}\right)\end{array}$$

(12)

4.2. RESSA Framework

(1)

Elitism initialization

Chaotic maps have been widely used in many fields due to their ergodic and semi-random properties [65,66]. The tent map [32,67,68] is used here for initialization and is defined as

$$\begin{array}{c}\hfill t^{\tau +1}=\left\{\begin{array}{cc}{t}^{\tau }/0.7& t^{\tau }<0.7\\ {\displaystyle \frac{10}{3}}\left(1-t^{\tau }\right)& t^{\tau }\ge 0.7\end{array}\right.,\end{array}$$

(13)

where $\tau$ is the iteration number.

Figure 2 gives the tent map versus the iteration number $\tau$ for two initial values. This shows that different initial values $t^1$ produce very different sequences. The tent map values in dimension m for a population of size L, ${\left\{{\mathit{t}}_m^{\tau }=\left[t_1^{\tau },...,t_m^{\tau }\right]\right\}}_{\tau =1}^L$, are

$$\begin{array}{c}\hfill t_m^{\tau +1}=\left\{\begin{array}{cc}{t}_m^{\tau }/0.7& \mathrm{if}{t}_m^{\tau }<0.7\\ {\displaystyle \frac{10}{3}}\left(1-t_m^{\tau }\right)& \mathrm{if}{t}_m^{\tau }\ge 0.7\end{array}\right.,\end{array}$$

(14)

Then, the population vectors ${\left\{{\mathit{\nu }}^{\tau ,+}=\left[{\nu }_1^{\tau ,+},...,{\nu }_m^{\tau ,+}\right]\right\}}_{\tau =1}^L$ are calculated using

$$\begin{array}{c}\hfill {\mathit{\nu }}_m^{\tau ,+}=l{b}_m+\left(u{b}_m-l{b}_m\right)t_m^{\tau }.\end{array}$$

(15)

According to OBL, the opposite vectors can help accelerate convergence and improve exploration. These vectors are created using

$$\begin{array}{c}\hfill {\mathit{\nu }}_m^{\tau ,-}=l{b}_m+u{b}_m-{\mathit{\nu }}_m^{\tau ,+}.\end{array}$$

(16)

The initial population ${\left\{{\mathit{Z}}^{\tau ,1}\right\}}_{\tau =1}^L$ is obtained from the $2L$ vectors ${\left\{{\mathit{\nu }}_m^{\tau ,+}\cup {\mathit{\nu }}_m^{\tau ,-}\right\}}_{\tau =1}^L$ as the L vectors with the lowest fitness.

(2)

Adaptive control

According to (12), the coordinates of the followers only depend on their coordinates and those of adjacent individuals, so the behavior is simple. Thus, if the leader falls into a local optimum, so will the followers, which can result in poor estimation. To avoid early convergence, a parameter $c^{\prime }$ is introduced, which is based on the control parameter $c_1$ and the tent map, so followers can have the same adaptive update ability as the leader to avoid falling into the local optimum and accelerate convergence. The new follower is given by

$$\begin{array}{c}\hfill {\left(X_m^{\tau }\right)}^{\prime }=X_m^{\tau }+{\displaystyle \frac{c^{\prime }}{2}}\left(X_m^p+X_m^q\right),\end{array}$$

(17)

where $X_m^p$ and $X_m^q$, $p\ne q$, are two salps selected from the current population to improve the exploration, and 

$$\begin{array}{c}\hfill c^{\prime }=t^{\tau }{c}_1,\end{array}$$

(18)

where $t^{\tau }$ is the tent map. Figure 3 shows that regardless of the value of $t^1$, $c^{\prime }$ decreases and approaches 0. Over 200 iterations, the range of values decreases from $\left[0,2\right]$ to $\left[0,0.5\right]$.

(3)

Differential evolution

DE is employed to further prevent early SSA convergence. This algorithm has been successfully employed in many optimization frameworks. The main DE steps are mutation, crossover, and selection. Here, DE is combined with SSA to generate better candidate solutions.

(a)

Mutation

For a population ${\mathit{Z}}^{\tau },\tau =1,2,\dots ,L$, a mutant population is obtained as

$$\begin{array}{c}\hfill {\tilde{\mathit{Z}}}^{\tau }={\mathit{Z}}^{{{\rm Y}}_1}+\vartheta \left({\mathit{Z}}^{{{\rm Y}}_2}-{\mathit{Z}}^{{{\rm Y}}_3}\right),\end{array}$$

(19)

where ${\mathit{Z}}^{{{\rm Y}}_1}$, ${\mathit{Z}}^{{{\rm Y}}_2}$, and ${\mathit{Z}}^{{{\rm Y}}_3}$ are three individuals in the current population; ${{\rm Y}}_1,{{\rm Y}}_2$, and ${{\rm Y}}_3$ are three distinct indices in $\left[1,L\right]$; and $\vartheta =U\left({\vartheta }_{min},{\vartheta }_{max},\left[\begin{array}{cc}1& D\end{array}\right]\right)$ is the scale factor with ${\vartheta }_{min}=0.2$ and ${\vartheta }_{max}=0.3$.

(b)

Crossover

From the current and mutant individuals, a crossover probability $\varphi$ is employed to produce a trial population ${\left\{{\stackrel{⌣}{\mathit{Z}}}^{\tau }=\left[{\stackrel{⌣}{\mathit{Z}}}_1^{\tau },\dots ,{\stackrel{⌣}{\mathit{Z}}}_m^{\tau }\right]\right\}}_{\tau =1}^L$, which is denoted as

$$\begin{array}{c}\hfill {\stackrel{⌣}{\mathit{Z}}}_m^{\tau }=\left\{\begin{array}{c}\begin{array}{cc}{\tilde{\mathit{Z}}}_m^{\tau },& \mathrm{if}\begin{array}{c}c\le \varphi \end{array}\end{array}\hfill \\ \begin{array}{cc}{\mathit{Z}}_m^{\tau },& \mathrm{if}\begin{array}{c}c>\varphi \end{array}\end{array}\hfill \end{array}\right.,\end{array}$$

(20)

where c is a random number in [0,1].

(c)

Bounding

To avoid errors caused by abnormal individuals, a boundary processing method is used, where these individuals outside the search space $\mathit{K}$ are modified as follows:

$$\begin{array}{c}\hfill {\stackrel{⌢}{\mathit{Z}}}_m^{\tau }=\left\{\begin{array}{cc}l{b}_m^{\prime }+\phi \left(u{b}_m^{\prime }-l{b}_m^{\prime }\right),& \mathrm{if}\begin{array}{c}\begin{array}{c}{\stackrel{⌣}{\mathit{Z}}}_m^{\tau }<l{b}_m^{\prime },{\stackrel{⌣}{\mathit{Z}}}_m^{\tau }>u{b}_m^{\prime }\end{array}\end{array}\hfill \\ {\stackrel{⌣}{\mathit{Z}}}_m^{\tau },& \mathrm{if}\begin{array}{c}\begin{array}{c}l{b}_m^{\prime }\le {\stackrel{⌣}{\mathit{Z}}}_m^{\tau }\le u{b}_m^{\prime }\end{array}\end{array}\hfill \end{array}\right.,\end{array}$$

(21)

where $\phi$ is uniformly distributed in [0,1], and $l{b}_m^{\prime }$ and $u{b}_m^{\prime }$ are the lower and upper bounds of the current L individuals, respectively, given by

$$\begin{array}{c}\hfill l{b}_m^{\prime }=\underset{{\mathit{Z}}_m^{\tau }}{argmin}{\left\{{\mathit{Z}}_m^{\tau }\right\}}_{\tau =1}^L,\end{array}$$

(22)

$$\begin{array}{c}\hfill u{b}_m^{\prime }=\underset{{\mathit{Z}}_m^{\tau }}{argmax}{\left\{{\mathit{Z}}_m^{\tau }\right\}}_{\tau =1}^L,\end{array}$$

(23)

(d)

Selection

The best individuals are selected from the current ${\left\{{\mathit{Z}}^{\tau }\right\}}_{\tau =1}^L$ and obtained ${\left\{{\stackrel{⌢}{\mathit{Z}}}^{\tau }\right\}}_{\tau =1}^L$ by evaluating their fitness, which is

$$\begin{array}{c}\hfill {\mathit{Z}}^{\tau }=\left\{\begin{array}{c}\begin{array}{cc}{\stackrel{⌢}{\mathit{Z}}}^{\tau },& \mathrm{if}\begin{array}{c}f\left({\stackrel{⌢}{\mathit{Z}}}^{\tau }\right)<f\left({\mathit{Z}}^{\tau }\right)\end{array}\end{array}\hfill \\ \begin{array}{cc}{\mathit{Z}}^{\tau },& \mathrm{otherwise}\end{array}\hfill \end{array}\right.,\tau =1,\dots ,L\end{array}$$

(24)

(4)

RESSA

The RESSA is proposed to solve the problem of robust localization in UAV networks when there is uncertainty in the anchor UAV node position coordinates. The RESSA flowchart is presented in Figure 4. It employs the SSA shown in Algorithm 1. In this algorithm, the followers move to their next positions based on the leader coordinates. However, it is difficult to ensure that a valid solution is obtained in each iteration, and this can cause convergence to a poor solution. Thus, direct application of the SSA does not guarantee a good solution.

A robust SSA, denoted RESSA, is proposed that combines chaos mechanism, OBL theory, an adaptive control (AC) strategy, and the DE mechanism. The incorporation of chaos mechanism and OBL theory aims to provide a robust elitism population to avoid falling into a local optimum and, thus, improve the convergence speed. The AC strategy employs a flexible control parameter to ensure appropriate follower movement. Furthermore, the DE steps such as mutation and crossover significantly bolster both exploration and exploitation capabilities, yielding a greater variety of potential solutions, and further improves convergence speed and estimation accuracy.

Algorithm 1 Salp swarm algorithm (SSA).

1 Input: lower and upper bounds $l{b}_m$, $u{b}_m$; number of dimensions m; population size L; maximum number of iterations $i{t}_{max}$; constants $\delta$, $c_1$, $c_2$, and $c_3$ 2 Construct a random initial population 3 Calculate the fitness of the L individuals 4 Find the best solution ${\mathit{Z}}_{best}$ 5 Set $it=1$ 6 While ($it\le i{t}_{max}$) do 7    Update $c_1$ using (11) 8    While ($\tau \le L$ ) do 9        If ($\tau ==1$) then 10        Update the coordinates of the leader using (10) and (11) 11      Else 12        Update the coordinates of the followers using (12) 13      End If 14   Amend outliers outside $\mathit{K}$ 15   Update the food source by calculating and comparing the fitness of current slaps 16    $\tau =\tau +1$ 17   End while 18   Select the best slap as ${\mathit{Z}}_{new}$ 19   $it=it+1$ 20 End while 21 Output: ${\mathit{Z}}_{best}$ as the last ${\mathit{Z}}_{new}$

The RESSA is given in Algorithm 2 and has the following steps:

• Step 1: Establish the cost function using (7) and (8).
• Step 2: Initialize all parameters such as the lower bound $l{b}_m$; upper bound $u{b}_m$; minimum scale factor ${\vartheta }_{min}$; maximum scale factor ${\vartheta }_{max}$; crossover probability $\varphi$; and control parameters $c_1$, $c_2$, and $c_3$.
• Step 3: Generate the initial population using (14)–(16).
• Step 4: Evaluate the fitness of the individuals and choose the L with the lowest fitness and update $c_1$ and $c^{\prime }$.
• Step 5: If $\tau ==1$, update the coordinates of the leader using (10) and (11).
• Step 6: If $\tau \ge 2$, update the coordinates of the followers using (17) and (18).
• Step 7: Obtain mutation population using (19).
• Step 8: Conduct crossover using (20) to update the population.
• Step 9: Check and modify outliers using (21)–(23).
• Step 10: Select the best individuals by calculating their fitness.
• Step 11: Repeat Steps 5 to 10 until $it>i{t}_{max}$.
• Step 12: Obtain the optimal UAV position estimate (the best slap) and its fitness.

Algorithm 2 Proposed RESSA.

1 Input: $l{b}_m$, $u{b}_m$, $m=1,2$; L; $i{t}_{max}$; $\delta$; $c_1$, $c_2$, and $c_3$; scale factor $\vartheta$; crossover probability $\varphi$ 2 Set cost function using (7) and (8) 3 Obtain the initial population using (14)–(16) 4 Choose the L individuals with the lowest fitness 5 Choose the best solution as ${\mathit{Z}}_{best}$ 6 Set $it=1$ 7 While ($it\le i{t}_{max}$) do 8    Update $c_1$ using (11) and obtain the tent map value ${\mathit{t}}^{\tau }$ 9    Update $c^{\prime }$ using (18) 10    While ($\tau \le L$ ) do 11      If ($\tau ==1$) then 12         Update the coordinates of the leader using (10) and (11) 13      Else 14         Update the coordinates of the followers using (17) and (18) 15     End If 16   Conduct mutation using (19) 17   Perform crossover using (20) 18   Adjust outliers using (21)-(23) 19   Implement selection using (24) 20    $\tau =\tau +1$ 21   End while 22   Select the best slap as ${\mathit{Z}}_{new}$ 23   If ($f\left({\mathit{Z}}_{new}\right)\le f\left({\mathit{Z}}_{best}\right)$) then 24     ${\mathit{Z}}_{best}$= ${\mathit{Z}}_{new}$ 25   End If 26   $it=it+1$ 27 End while 28 Output: ${\mathit{Z}}_{best}$

4.3. Complexity Analysis

The algorithms considered in this paper are PSO, DE, and TN localization using a faster algorithm (TLFA) [12], OLAM-IDE [21], CSSA [32], DESSA [39], and the proposed RESSA. Their computational complexity depends on the maximum number of iterations $\epsilon$ or generations G, cost of the objective function $\beta$, population size L, and number of anchor nodes N. The complexity of these methods is given in

Table 2. Computational complexity.

Method	Complexity
PSO	$O\left(\epsilon LN\right)$
DE	$O\left(GLN\right)$
TLFA	$O\left(GLN+Llog\left(2L\right)\right)$
OLAM-IDE	$O\left(G^{\prime }LN+Llog\left(2L\right)\right)$
CSSA	$O\left(L\left(L+\epsilon +\beta \right)N\right)$
DESSA	$O\left(2LN+L\right)\left(\left(\epsilon -L^2\right)/2+L^{2}N\right)$
RESSA	$O\left(L\left(L+2\epsilon +\beta \right)N\right)$

. The individual complexity of the leader and follower salps is $O\left(\epsilon L\right)$. Considering initialization including the tent map, their complexity is $O\left(L\right)$ and $O\left(L^2-L\right)$, and their sum has complexity $O\left(L^2\right)$. The objective function is $O\left(\beta L\right)$, and updating the optimal solution is $O\left(\epsilon L\right)$, so the complexity of CSSA is $O\left(L\left(L+\epsilon +\beta \right)N\right)$. The complexity of DE is $O\left(\epsilon L\right)$. Adding these, the computational complexity of RESSA is $O\left(L\left(L+2\epsilon +\beta \right)N\right)$. For OLMA-IDE, $G^{\prime }$ is the number of generations when the termination condition is satisfied.

5. Performance Evaluation

In this section, the performance of the proposed RESSA algorithm for UAV localization is evaluated via simulation and compared with the related algorithms PSO, DE, TLFA [12], OLAM-IDE [21], CSSA [32], and DESSA [39]. The RSSD-based CRLB is considered for comparison purposes. All simulations were conducted on a laptop computer equipped with an Intel(R) Core(TM) i5-3500U CPU@2.10 GHz and 12 GB RAM. The algorithms were implemented using Matlab R2019a and the Windows 10 operating system. The simulation parameters are given in

Table 3. Simulation parameters For 7 methods.

Method	Parameters
PSO	$w\in \left[0.2,0.9\right];c=2$
DE	$F=0.9;CR=0.9$
TLFA	$F=0.9;CR=0.9$
OLAM-IDE	$F\in \left[0.4,0.5\right];CR\in \left[0.6,0.7\right]$
CSSA	$\begin{array}{c}{c}_1=2\times exp\left(-{\left(4\times it/i{t}_{max}\right)}^2\right);c_2=\left[0,1\right];\hfill \\ c_3=\left[0,1\right];c_t=t\times 2\times exp\left(-{\left(0.8\times it/i{t}_{max}\right)}^2\right)\hfill \end{array}$
DESSA	$\begin{array}{c}{F}_{min}=0.2;F_{max}=0.8;CR=0.1;\hfill \\ c_1=2\times exp\left(-{\left(4\times it/i{t}_{max}\right)}^2\right);\hfill \\ c_2=\left[0,1\right];c_3=\left[0,1\right]\hfill \end{array}$
RESSA	$\begin{array}{c}F\in \left[0.2,0.3\right];CR=0.1;\hfill \\ c_1=2\times exp\left(-{\left(4\times it/i{t}_{max}\right)}^2\right);\hfill \\ c_2=\left[0,1\right];c_3=\left[0,1\right],c^{\prime }=t{c}_1\hfill \end{array}$

.

To obtain a dynamic communication environment, the positions of the anchor UAV nodes and target UAV were changed randomly each Monte Carlo trial. The UAV nodes were deployed in a square area with sides $length$, so the area is $lengt{h}^2$ m² (This can also be extended to three dimensions with $lengt{h}^3$ m³.). The lower and upper bounds are $l{b}_1,l{b}_2=0$ and $u{b}_1,u{b}_2=50$, respectively. The other parameters are $L=12$, $i{t}_{max}=50$, $t^1=0.6$, $\delta =4$, $\vartheta \in \left[0.2,0.3\right]$, and $\varphi =0.1$. The root mean square error (RMSE) is used to evaluate the estimation performance, as described in [15], and is defined as

$$\begin{array}{c}\hfill \mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^{\mathrm{MC}}{\left({\widehat{\mathit{x}}}_i-{\mathit{x}}_i\right)}^2}{\mathrm{MC}}}}\end{array}$$

(25)

where ${\widehat{\mathit{x}}}_i$ is the estimated position of target node ${\mathit{x}}_i$ at the ith trial and $\mathrm{MC}=1000$ is the number of trials.

5.1. Performance in Noise

The RMSE of the target UAV position estimates for seven methods and the CRLB versus the noise variance ${\sigma }^2$ is presented in Figure 5 with ${\sigma }_a=1$, $\xi =3$, $length=50$, and $N=7$ and $N=9$. As expected, an increase in the noise variance increases the RMSE of the seven methods and, thus, the localization accuracy. For example, with ${\sigma }^2=5$ and 9 dB, for $N=7$ in Figure 5a, the differences in RMSE between RESSA and the CRLB are 0.10 m and 0.51 m, respectively, whereas they are 2.79 m and 2.63 m for PSO, 2.23 m and 1.54 m for CSSA, 1.67 m and 1.68 m for DESSA, 1.26 m and 1.62 m for DE, 0.68 m and 2.00 m for TLFA, and 0.38 m and 1.02 m for OLAM-IDE. The RMSE results in Figure 5b for $N=9$ show that the RMSE is lower than in Figure 5a, which indicates that increasing the number of anchor nodes improves the positioning accuracy. This is because more information is available for the estimation. For example, when ${\sigma }^2=7$ dB, the improvement from $N=7$ to $N=9$ is 38.2% for RESSA, 43.8% for PSO, 34.1% for CSSA, 37.7% for DESSA, 29.6% for DE, 35.8% for TLFA, and 33.8% for OLAM-IDE.

Figure 5 indicates that although the performance of some algorithms such as DE, TLFA, and DESSA is relatively close when the noise variance is small, as the noise level increases, the proposed algorithm provides the best estimation performance. In addition, it is obvious that PSO and CSSA are more susceptible to noise than the other algorithms. Moreover, the number of anchor nodes has a significant effect on performance. It is also observed that with more information, the performance of the DE-based methods—namely, TLFA and OLAM-IDE—is relatively close and both are better than the original DE algorithm.

5.2. Performance with Different Path Loss Exponents

Figure 6 presents the RMSE of the target UAV position estimates for the seven algorithms and the CRLB versus the path loss exponent $\xi$ with $\sigma =3$, ${\sigma }_a=1$, $length=50$, and $N=7$ and $N=9$. This shows that the RMSE of all methods decreases as $\xi$ increases from 2 to 6. For example, the RMSE decreases for RESSA from 8.21 m to 5.87 m, PSO from 5.89 m to 3.88 m, CSSA from 4.99 m to 4.86 m, DESSA from 8.75 m to 6.39 m, DE from 8.07 m to 6.11 m, TLFA from 7.92 m to 6.76 m, and OLAM-IDE from 8.71 m to 5.89 m. Comparing Figure 6a,b indicates that, as before, an increase in the number of anchor nodes improves the estimation accuracy. However, in all cases, RESSA provides the best performance, which is closest to the CRLB. Conversely, PSO and CSSA have the worst performance, while the performance of the other algorithms is similar. For example, with $\xi =4$ and $N=7$, from Figure 6a, the improvement in RMSE with RESSA over PSO is 26.4%, over CSSA is 13.5%, DE is 12.8%, TLFA is 11.8%, DESSA is 9.4%, and OLAM-IDE is 4.2%. The corresponding improvement with $\xi =4$ and $N=7$ from Figure 6b is 43.4% over PSO, 40.7% over CSSA, 15.2% over OLAM-IDE, 11.8% over DE, 10.9% over TLFA, and 3.9% over DESSA.

5.3. Performance with Different Node Position Errors

Figure 7 presents the RMSE versus node position error standard deviation ${\sigma }_a$ with $\sigma =4$, $\xi =3$, $length=30$, and $N=7$ and $N=9$. This shows that an increase in ${\sigma }_a$ results in an increase in the RMSE, and the RMSE is lower in Figure 7b as there are more anchor nodes. PSO and DE have significant variations in RMSE with respect to ${\sigma }_a$. When ${\sigma }_a\le 11$ dB, PSO is worse than DE, but the opposite occurs when ${\sigma }_a>11$ dB. However, RESSA provides the best estimation performance, which is closest to the CRLB. For example, with $N=7$ and ${\sigma }_a=6$ dB, there is a 0.33 m difference between RESSA and the CRLB, whereas the difference with the other methods is higher, i.e., 0.72 m for CSSA, 0.85 m for OLAM-IDE, 0.89 m for TLFA, 0.97 m for DE and DESSA, and 1.48 m for PSO, respectively. With $N=9$, the corresponding results from Figure 7b are 0.35 m, 0.63 m, 0.69 m, 0.97 m, 0.98 m, 1.05 m, and 1.59 m for RESSA, TLFA, OLAM-IDE, DE, DESSA, CSSA, and PSO, respectively. Furthermore, when ${\sigma }_a=16$ dB, although the performance of RESSA is close to CSSA in Figure 7a, there is a significant difference in Figure 7b. In Figure 7a, OLAM-IDE has the second-best performance while CSSA, DESSA, and TLFA are similar, but the performance of OLAM-IDE and TLFA, CSSA, and DE is similar in Figure 7b.

5.4. Performance with Different Side Lengths

Figure 8 gives the cumulative distribution function (CDF) of the estimation error for the 7 methods with $\sigma =3$, ${\sigma }_a=1$, $\xi =3$, $N=7$, and $length=20$ m, 40 m, 60 m, and 80 m. This shows that an increase in the area of the region increases the RMSE primarily due to greater signal attenuation.

Table 4. CDF for 7 methods with different area side lengths.

Method	$\mathit{length}=\mathbf{20}$ m		$\mathit{length}=\mathbf{80}$ m
	Probability (75%)	Probability (95%)	Probability (75%)	Probability (95%)
PSO	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 2.39$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 6.88$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 9.91$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 29.52$ m
DE	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 2.20$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 5.23$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 9.64$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 23.21$ m
TLFA	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 2.18$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 6.39$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 8.72$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 20.14$ m
OLAM-IDE	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 2.10$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 5.27$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 8.80$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 21.99$ m
CSSA	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 2.30$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 4.84$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 8.55$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 21.68$ m
DESSA	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 2.24$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 6.55$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 9.44$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 27.82$ m
RESSA	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 2.04$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 4.51$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 8.06$ m	$∥\widehat{\mathit{x}}-\mathit{x}∥\le 19.79$ m

gives the corresponding errors for different probabilities. This shows that RESSA has $∥\widehat{\mathit{x}}-\mathit{x}∥\le 2.04$ m and $∥\widehat{\mathit{x}}-\mathit{x}∥\le 8.06$ m for a 75% probability with $length=20$ m and 80 m, and $∥\widehat{\mathit{x}}-\mathit{x}∥\le 4.51$ m and $∥\widehat{\mathit{x}}-\mathit{x}∥\le 19.79$ m for a 95% probability, respectively. These results are much better than those of the other algorithms, which confirms the superiority of the proposed approach.

6. Conclusions and Future Works

In this paper, a novel received signal strength difference (RSSD)-based scheme, RESSA, was proposed for UAV localization considering unknown transmit power and node position errors. The UAV localization problem was first solved using an ML framework after conversion. Then, an elitism strategy based on TOL was designed to facilitate the movement of the leader around the target UAV. DE is employed to enhance the search capability of each agent and improve the balance with global search. Moreover, a dynamic motion mechanism was developed to enable the UAV swarm to move faster towards the target. Experimental results were presented showing that the proposed UAV localization scheme provides better estimation accuracy than related methods in the literature.

Although we achieved an effective closed-form solution under node position uncertainty, the simulation environment is idealized. For instance, noise generated from communications between drones tends to exhibit non-Gaussian characteristics, which should be further considered in future research to better reflect the real-world communication dynamics among drones. Furthermore, some other parameters like the path loss exponent show significant variability depending on environmental conditions, leading to uncertainties in their actual values. Incorporating these uncertainties will also be of interest and crucial in future studies to enhance drone positioning accuracy. Moreover, signal propagation may be disrupted by external attacks, potentially causing failures in some UAV nodes and significantly degrading positioning performance. Regrettably, this paper does not address this specific scenario. Therefore, future research should focus on developing secure drone positioning systems capable of withstanding external attacks and deriving a closed-form solution for such scenarios, which would be highly beneficial and meaningful.