An Enhanced MOPSO Method for Distributed Radar Topology Optimization

Abstract

Time difference of arrival (TDOA) localization enables high-accuracy positioning by analyzing arrival-time differences of target signals at distributed radar nodes, whose performance strongly depends on radar node topology. However, existing studies tend to focus more on improving localization accuracy, while overlooking the impact of radar geometric layout and surveillance coverage on localization performance. To this end, this paper proposes a topology optimization method for a distributed radar system based on an improved non-dominated sorting multi-objective particle swarm optimization (NS-MOPSO) algorithm. A geometric localization model is developed for a distributed TDOA radar system. Based on this model, three optimization objectives are formulated, including minimizing geometric dilution of precision (GDOP), maximizing target coverage, and improving the geometric balance of node placement. These three objective functions are incorporated into the NS-MOPSO framework to achieve a more reasonable radar geometric distribution. To enhance the optimization performance, a series of strategies are adopted, such as non-dominated sorting for Pareto-based solution selection, an improved crowding-distance scheme to encourage balanced multi-objective optimization, and Gaussian mutation to increase solution diversity and reduce the risk of premature convergence. To validate the proposed method, both simulation studies and real-world experiments were conducted under different node deployment scenarios. The results show that the optimized topology achieves a 6.4% reduction in RMSPE and a 4.3% increase in the proportion of high-quality localization regions compared with the best-performing comparative method, while also demonstrating faster convergence and improved stability. These findings confirm the effectiveness and robustness of the proposed approach in enhancing localization accuracy, expanding effective coverage, and improving overall system performance.

1. Introduction

Passive emitter localization is a fundamental capability in electronic countermeasures and intelligent sensing systems. Among existing techniques, TDOA has been widely adopted due to its independence from emitter priors and its relatively high accuracy and multipath resilience [1]. Nevertheless, complex noise conditions and inconsistent bias characteristics across sensors motivate numerous studies on modeling, compensation, and robust estimation. Jeon refined the TDOA–TOA model to mitigate relay-induced delays [2], while Bahrampour reduced noise-dominated errors using an iterative mean-based strategy [3]. Cui restored multi-station associations under low co-visibility by incorporating coarse positional constraints [4]. Under large-scale passive satellite scenarios, Shu revealed that Earth Constraint dominates the positioning geometry, thereby simplifying the TDOA model while preserving essential geometric relationships [5]. For asynchronous or time-varying environments, Tu’s parameterized TDOA formulation enables the reconstruction of quasi-synchronous delay measurements [6], whereas Li’s Bayesian fusion method enhances robustness under NLOS conditions by accounting for correlated errors [7]. Zuo employed an LPNN-based maximum-likelihood framework to achieve solutions closer to theoretical limits under strong noise [8]. Under more challenging noise conditions, Zhang’s ICWLS further suppresses ML bias through explicit bias modeling and linearized constraints [9]. In addition, TDOA can be combined with DOA to improve emitter discrimination under complex electromagnetic environments [10]. In scenario-specific applications, Bhandari utilized a response-surface-based wave-speed correction approach to enhance AE source localization in anisotropic materials [11], while Liu integrated MEMS sensors with SWD–STA/LTA–AIC and an improved WLS algorithm to achieve centimeter-level accuracy in outdoor ground-vibration localization [12]. For multi-source and asynchronous conditions, Li adopted a sparse-reconstruction framework to enable automatic clustering and matching of multiple emitters, ensuring stable localization performance even without synchronization [13].

Beyond advances in error modeling and algorithm design, the geometric topology of a sensing system has a more fundamental impact on localization accuracy. Extensive studies have shown that different array configurations—such as T-shaped, Y-shaped, and rhombus layouts—exhibit pronounced GDOP variations across directions, and that baseline length and orientation directly influence the regional error distribution [14]. For MIMO radar systems, both the CRLB of position estimation and the CRB of target-velocity estimation demonstrate that symmetric or uniformly expanded deployments provide superior observability [15,16]. Under deployment constraints, Liang employed ROI–minimax CRLB combined with ADMM to obtain near-optimal sensor placements [17]. For communication-range-limited scenarios, Sadeghi proposed a tangent-based layout strategy to improve the FIM structure [18]. Yang enhanced regional surveillance coverage using a PSO-based optimization framework [19], while other studies formulated a dual-objective problem involving K-coverage and RMSE to achieve a balanced solution through multi-neighborhood search [20]. In three-dimensional configurations, Bellabas analyzed and verified that center-oriented layouts with approximately 120° baseline angles possess clear advantages in geometric observability [21].

At the system level, multi-performance trade-offs, node reliability, and computational cost have also become important research directions. Cao’s COM-W algorithm performs CRLB-based weighted fusion over multiple three-node combinations and demonstrates more robust estimation performance than traditional WLS under various geometric and noise conditions [22]. Giunta developed a statistical warning mechanism based on second- and fourth-order cross-correlation residuals, enabling automatic identification and exclusion of faulty sensors and thereby improving system reliability under non-ideal conditions [23]. Li reformulated sensor-subset selection as a combinatorial optimization problem, significantly reducing computational complexity while maintaining accuracy close to exhaustive search, thus providing practical real-time capability [24]. In addition, for multi-task and resource-constrained scenarios, several studies have attempted to build multi-objective optimization frameworks (e.g., MOPSO-NRCD and SC-MOPSO) to improve the quality of the Pareto front, although these works primarily focus on algorithmic evolutionary mechanisms and pay limited attention to geometric factors driving localization performance [25,26]. Zhang W addressed energy-constrained deployments by coupling CRLB-based sensor and resource co-optimization, which strengthens the trade-off between performance and energy expenditure [27]. Overall, existing studies on bias suppression, asynchronous compensation, multi-source separation, material modeling, geometric optimization, coverage enhancement, node reliability, and online selection have collectively strengthened the accuracy and robustness of TDOA systems under complex environments, providing a solid foundation for subsequent research on distributed radar topology optimization.

Despite the progress achieved in distributed radar geometry optimization, existing studies still exhibit two fundamental limitations. First, most frameworks treat geometric performance as an aggregated indicator embedded within localization accuracy or coverage metrics, rather than modeling spatial geometric consistency as an explicit structural objective. As a result, optimization processes tend to improve performance at discrete evaluation points while overlooking global spatial stability across the surveillance region. Second, in high-dimensional or deployment-constrained scenarios, heterogeneous objective scales often lead to dominance bias in evolutionary search, causing uneven population distributions and limited Pareto diversity. This issue becomes more pronounced in large TDOA-sensitive regions and multi-target configurations, where maintaining both localization precision and geometric stability remains challenging. These observations indicate that an effective topology optimization framework should not only coordinate accuracy and coverage objectives, but also explicitly regulate spatial geometric consistency under realistic deployment constraints. Motivated by this perspective, this work develops a geometry-aware multi-objective topology optimization framework for distributed TDOA localization. The main contributions are summarized as follows:

• A unified three-objective topology optimization formulation is developed for distributed TDOA radar systems. The model jointly considers localization accuracy, coverage capability, and geometric balance, and explicitly introduces spatial geometric consistency as an independent regulation dimension under practical deployment constraints.
• A radar-oriented refinement of multi-objective PSO is proposed. An objective-aware crowding strategy is introduced to alleviate dominance bias in heterogeneous objective spaces, while mutation and reinitialization operators are restructured by embedding geometric feasibility constraints and GDOP-related indicators.
• A comprehensive validation framework integrating simulation and field experiments is established. Monte Carlo simulations and real-world outdoor tests jointly verify localization performance, coverage effectiveness, and geometric stability, demonstrating consistent improvements over conventional deployment strategies.

The remainder of this paper is organized as follows. Section 2 reviews related work on distributed TDOA localization, with particular emphasis on the impact of node topology on geometric performance. Section 3 presents the system modeling and formulates the proposed multi-objective topology optimization problem, followed by a detailed description of the improved multi-objective particle swarm optimization algorithm. Section 4 evaluates the proposed method through comprehensive simulation and experimental studies under various deployment scenarios. Finally, Section 5 concludes the paper and discusses directions for future research.

2. Related Work

The localization accuracy of distributed TDOA radar systems is fundamentally determined by the geometric relationship between radar nodes and the target [14,28]. Although extensive studies have focused on signal processing techniques, synchronization strategies, and robust estimation algorithms for TDOA localization [2,3,4,5,6,7,8,9,29], it is widely recognized that the spatial configuration of radar nodes has a direct impact on error propagation characteristics and localization performance [14,30].

Even when measurement accuracy and noise conditions remain unchanged, different deployment geometries can lead to significantly different localization performance, such as spatially non-uniform accuracy distribution and the emergence of localization blind regions [28,31,32]. To quantitatively characterize these geometric effects, performance metrics including GDOP and the CRLB are commonly adopted to evaluate how node placement influences localization accuracy and robustness [15,30,31,32].

The following subsections, therefore, focus on reviewing topology-related geometric performance in distributed TDOA localization systems, with particular emphasis on the role of node deployment in shaping localization accuracy, coverage completeness, and geometric stability.

2.1. Impact of Node Topology on Geometric Performance

It is well recognized that localization accuracy in TDOA systems is strongly affected by the geometric relationship between the target and the distributed radar nodes [28]. From a geometric perspective, the relative positions of radar nodes determine the intersection angles of the corresponding hyperbolic loci, which directly influence the sensitivity of localization results to measurement noise. When the node geometry is unfavorable, small perturbations in time-difference measurements may be significantly amplified, resulting in large localization errors even under moderate noise conditions [14,30].

To quantitatively describe this geometric effect, metrics such as GDOP and CRLB are widely adopted in the literature [15,30,31,32]. These metrics characterize how measurement errors propagate through the localization model under a given node configuration, thereby providing an intrinsic evaluation of geometric observability independent of specific signal processing algorithms. Extensive studies have demonstrated that different deployment patterns—such as uniform linear arrays, T-shaped arrays, Y-shaped arrays, and rhombus configurations—exhibit markedly different GDOP distributions across the surveillance region [28,33,34,35]. In particular, certain symmetric or compact layouts may achieve favorable performance along specific directions, while suffering from rapid degradation of localization accuracy in other regions. As illustrated in Figure 1, the GDOP exhibits a pronounced spatially non-uniform distribution over the monitored area. Favorable localization performance is achieved only in limited regions, whereas significant performance degradation occurs near the boundaries of the area, potentially leading to localization blind zones. Moreover, small perturbations in target position or measurement noise can cause large variations in localization error when the node geometry is unfavorable, indicating poor geometric stability [35].

A prominent characteristic observed in many conventional deployments is the strong spatial non-uniformity of localization performance. Even when the average GDOP over the region appears acceptable, localized areas with extremely high GDOP values may still exist, leading to practical localization blind zones [31,32]. These regions are especially problematic in applications requiring stable and reliable positioning over wide areas, as they introduce significant performance uncertainty and limit the effective coverage of the system. Moreover, the anisotropic nature of GDOP distributions implies that localization accuracy may vary drastically with target position, which is undesirable for multi-target or dynamic tracking scenarios.

In addition to accuracy degradation, geometric instability also affects the robustness of distributed TDOA systems. When node placements lack sufficient angular diversity or exhibit poor baseline orientation, the localization solution becomes highly sensitive to both measurement noise and modeling errors, resulting in fluctuating estimation performance [14,28]. This issue is further exacerbated near the boundaries of the monitored region, where geometric constraints tend to weaken and error amplification becomes more pronounced. As a result, topology design that focuses solely on minimizing average error metrics may fail to ensure consistent performance across the entire surveillance area.

These observations indicate that effective topology design for distributed TDOA localization should not only aim at reducing overall localization error, but also emphasize geometric balance and spatial performance stability. A comprehensive evaluation of node topology, therefore, requires simultaneous consideration of average accuracy, regional coverage, and the uniformity of geometric performance, which motivates the adoption of geometry-aware optimization frameworks in subsequent studies.

2.2. Limitations of Existing Topology Design Approaches

Although prior research has clearly demonstrated the importance of node topology in distributed TDOA localization, several limitations remain in existing topology design approaches. Firstly, many deployment strategies are developed under simplified assumptions or rely on fixed and predefined layouts, which may not be well-suited to complex or constrained real-world environments [36]. In practical scenarios, radar deployment is often subject to terrain restrictions, infrastructure constraints, and limited feasible regions, under which idealized geometric configurations cannot be directly realized. As a result, the effectiveness of these methods may degrade significantly when applied outside their assumed conditions.

Secondly, a considerable portion of optimization-based approaches formulate topology design as a single-objective problem, typically focusing on minimizing average localization error or a single geometric metric [37]. While such formulations can yield performance improvements with respect to a specific criterion, they often fail to adequately balance competing requirements, including localization accuracy, spatial coverage, and geometric stability. In distributed TDOA systems, improving one aspect of performance may lead to degradation in another, making single-objective optimization insufficient for addressing practical deployment needs [19,20].

Moreover, the robustness and stability of optimized topologies under varying target positions and realistic deployment conditions are not always thoroughly evaluated. Many existing studies emphasize performance gains under specific test scenarios or limited regions, without systematically examining how localization accuracy and geometric performance vary across the entire surveillance area [28,31]. This limitation is particularly critical for distributed systems operating in dynamic or multi-target environments, where consistent performance over wide spatial regions is essential.

These limitations indicate the need for more flexible and comprehensive topology optimization frameworks that explicitly incorporate multiple geometric performance criteria. In particular, multi-objective optimization methods provide a promising direction for addressing the inherent trade-offs in distributed TDOA node deployment [25,26]. By enabling the simultaneous consideration of localization accuracy, coverage completeness, and geometric balance, such frameworks are better suited to maintaining robustness and adaptability under realistic deployment constraints, thereby motivating the multi-objective, geometry-aware topology optimization approach adopted in this work.

3. Proposed Method

3.1. Problem Description and System Overview

In this section, the proposed distributed radar topology optimization method is introduced. To provide an intuitive overview of the considered localization scenario and problem setting, Figure 2 illustrates the overall positioning scene and system framework adopted in this work. The detailed formulation and methodology of each component are presented in the following subsections.

Based on the TDOA, emitter localization refers to estimating the position of an unknown emitter in a distributed radar network by exploiting the time differences of target signals received at multiple radar nodes. This technique is widely used in electronic countermeasures, battlefield surveillance, and related scenarios. The localization scene consists of a Radar Surveillance Area (RSA) and a Radar Deployment Area (RDA), as illustrated in Figure 3. The RSA represents the potential region where the emitter may be located, whereas the RDA defines the area where distributed radar nodes can be deployed. During localization, each radar node receives the target’s transmitted signal, and the time differences of arrival among nodes are utilized to estimate the emitter’s position based on the known coordinates of the radar nodes.

From the relationship between the signal and the geometric configuration, let the emitter position be denoted as p, and the true coordinate of the i-th radar node be s_i. The time difference of arrival between the signal arriving at the i-th node and the reference node 0 satisfies the following:

$${\tau }_{i0}={\displaystyle \frac{\parallel \mathit{p}-{\mathit{s}}_i\parallel -\parallel \mathit{p}-{\mathit{s}}_0\parallel }{c}}+n_{i0}$$

(1)

where c denotes the speed of light, and $n_{i0}$ represents the noise associated with the time-difference measurement. The TDOA measurement ${\tau }_{i0}$ represents the difference in the propagation distances between the emitter and two radar nodes. Geometrically, each TDOA value defines a branch of a hyperbola, where the two radar nodes act as the foci. Therefore, the true emitter position must lie on the corresponding hyperbolic curve. By combining multiple TDOA measurements with respect to the reference node, the emitter location is estimated as the intersection of several such hyperbolic loci, which form the basis of TDOA-based passive localization.

In TDOA localization accuracy analysis, RMSPE is a key metric for evaluating system performance, as it comprehensively reflects the influence of measurement noise and radar node deployment on localization accuracy. The measured range-difference vector between the target and the radar nodes is expressed as $\mathbf{\Delta }\mathbf{r}$. The localization model can thus be formulated as follows:

$$\Delta \mathbf{r} = \mathbf{h}\left(\mathbf{x}\right) + \mathbf{n}$$

(2)

where $\mathbf{h}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ represents the theoretical range-difference vector determined by the target position and the node coordinates, and $\mathbf{n}$ is the TDOA measurement noise vector (with covariance matrix ${\mathit{Q}}_{\mathit{n}}$). By linearizing $\mathbf{h}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ around the true target position ${\mathrm{x}}_0$, the linearized observation model is given by the following:

$$\Delta \mathbf{y} = \mathbf{H}\Delta \mathrm{x} + \mathbf{n}$$

(3)

where $\Delta \mathbf{y} = \Delta \mathbf{r} - \mathbf{h}\left({\mathrm{x}}_0\right)$ denotes the range-difference residual, and $\Delta \mathrm{x} = \mathrm{x} - {\mathrm{x}}_0$ represents the localization estimation error. H is the Jacobian matrix, whose elements describe the geometric relationship between node coordinates and the target position, directly reflecting the geometric constraints imposed by the radar topology. According to the least-squares estimation theory, the covariance matrix of the localization error ${\mathrm{P}}_{\mathrm{err}}$ satisfies the following:

$${\mathbf{P}}_{\mathrm{e}\mathrm{r}\mathrm{r}}=({\mathbf{H}}^{\mathrm{T}}{\mathbf{Q}}_n^{-1}\mathbf{H}{)}^{-1}$$

(4)

Based on the derived error covariance, localization performance can be quantitatively evaluated using both RMSPE and GDOP, which characterize the combined effects of measurement noise and node geometry, respectively.

The RMSPE, which characterizes the square root of the trace of the localization error covariance matrix, is defined as follows:

$$\mathrm{RMSPE} = \sqrt{\mathrm{tr}\left({\mathbf{P}}_{\mathbf{err}}\right) }= \sqrt{\mathrm{tr}\left[{\left({\mathbf{H}}^{\mathrm{T}}{{\mathit{Q}}_{\mathit{n}}}^{-1}\mathbf{H}\right)}^{-1}\right]}$$

(5)

where $\mathrm{tr}\left(·\right)$ denotes the trace of a matrix. This expression clearly reveals the decisive role of the radar topology in determining the localization accuracy. Through the Jacobian matrix H, the geometric configuration of radar nodes directly influences the propagation characteristics of localization errors. When the nodes are arranged properly and the geometric constraint capability is strong, the system becomes significantly less sensitive to measurement noise, thereby yielding a smaller RMSPE and achieving higher localization accuracy.

To further characterize the influence of node geometry on the amplification of localization errors, the GDOP is introduced as a geometry-based performance metric by neglecting the heterogeneity of measurement noise weights. Under the assumption that all TDOA measurements share identical noise variance, GDOP can be expressed as follows:

$$\mathrm{GDOP} = \sqrt{\mathrm{tr}\left[{\left({\mathbf{H}}^{\mathrm{T}}\mathbf{H}\right)}^{-1}\right]}$$

(6)

This metric is solely determined by the Jacobian matrix H and directly reflects the geometric constraint capability of the node topology. A smaller GDOP value indicates a more favorable geometric configuration, under which the amplification effect of measurement noise on localization errors is significantly reduced. In practical systems, the RMSPE reflects the overall localization accuracy resulting from the combined effects of node geometry and measurement noise, whereas GDOP is used to characterize the intrinsic quality of the node geometry itself. Therefore, the joint use of RMSPE and GDOP enables a more comprehensive evaluation of the impact of node topology on localization performance.

3.2. Multi-Objective Optimization Framework

3.2.1. Deployment Constraints and Feasible Regions

In practical distributed TDOA deployment, node placement must satisfy not only optimization objectives but also geometric feasibility and engineering constraints. Therefore, this subsection formulates the deployment constraints and feasible region modeling required for topology optimization. These constraints ensure that radar nodes can be physically deployed within allowable regions, maintain valid geometric visibility with respect to the targets, and remain compatible with practical sensing and scanning limitations. The geometric constraint feasibility in practical deployment must consider algorithmic solvability and engineering limitations, which are mainly addressed from the following three perspectives.

(1)

Geometric Deployment Constraints

Radar node positions $\widehat{\mathrm{s}}\left(\mathrm{t}\right)$ must lie within the RDA. For each node, its sensing field of view for emitter detection is denoted as ${\widehat{\mathrm{s}}}_{\mathrm{i}}^{\mathrm{bd}}$. To ensure geometric visibility, the included angle between any pair of node–target bearing vectors must not exceed the maximum allowable scanning angle ${\mathsf{\alpha }}_{\mathrm{i}}^{\max }$:

$$\mathrm{\angle }({\omega }_{b_d,i},{\omega }_{b_b,i})<{\alpha }_i^{max}$$

(7)

This condition guarantees that the target remains within the coverage sector and that geometric constraints remain valid.

(2)

Feasible Steering-Direction Constraints

In practical distributed radar systems, each sensing node is equipped with a directional antenna whose effective field of view (FoV) is inherently limited. As a result, the antenna steering direction cannot arbitrarily cover the entire angular space and must be properly aligned with the target-bearing directions to ensure effective signal reception. To account for this practical limitation, the antenna steering angle of each node is constrained such that the target of interest always lies within the antenna’s effective scanning sector. Accordingly, to guarantee target visibility under the limited antenna field of view, the steering angle of node i at time t must satisfy the following:

$$\left|\begin{array}{c}{\alpha }_i\left(t\right)-{\theta }_{i,k}\left(t\right)\end{array}\right|\le {\displaystyle \frac{\Delta {\alpha }_i}{2}},\forall k\in \mathcal{T}$$

(8)

where ${\alpha }_i\left(t\right)$ denotes the steering angle of the antenna at node i at time t, ${\theta }_{i,k}\left(t\right)$ represents the bearing angle from node i to target k, and $\Delta {\alpha }_i$ corresponds to the effective field of view of the antenna. The set $\mathcal{T}$ denotes the collection of targets within the region of interest. This constraint ensures that the antenna main lobe of each node is properly oriented toward the target coverage region, thereby guaranteeing target visibility and reliable signal acquisition. At the same time, by limiting the steering range of each node individually, sufficient angular diversity among different sensing nodes is preserved, which contributes to enhanced geometric robustness and improved localization performance.

(3)

Algorithmic Resource Constraints

During multi-objective optimization, the velocity ${\mathrm{v}}_{\mathrm{m}}\left(\mathrm{t}\right)$ of each particle in the swarm must remain within predefined bounds: ${\mathrm{v}}_{\mathrm{m}}\left(\mathrm{t}\right)\in \left[{\mathrm{v}}_{\min },{\mathrm{v}}_{\max }\right]$, to avoid excessive oscillation or divergence during the search process. Since nodes may collaborate to detect multiple emitters, computational and communication loads must remain feasible. Based on the modified MOPSO algorithm, the search over the geometric-constraint space should balance the three key objectives—localization accuracy, coverage completeness, and geometric uniformity—thereby ensuring both stability and efficiency in the final optimized topology.

3.2.2. Objective Functions

Building upon the above three-objective formulation, the resulting optimization problem involves inherently conflicting objectives. Unlike conventional multi-objective deployment formulations that treat geometric performance as an aggregated indicator within localization accuracy, this work explicitly elevates geometric stability to an independent optimization dimension. This reformulation shifts the optimization objective from performance aggregation to structural consistency regulation across the surveillance region.

Based on the above performance metrics and deployment constraints, the distributed radar topology design problem can be formulated as a constrained multi-objective optimization problem. The objective is to determine the spatial configuration of radar nodes that achieves an optimal balance among localization accuracy, coverage capability, and geometric stability under practical deployment conditions. To achieve a balanced consideration of localization accuracy, coverage completeness, and geometric uniformity, a modified multi-objective metric vector is constructed as follows:

$$\mathbf{F}\left(\mathbf{X}\right)=\left[\begin{array}{ccc}{\mathsf{\Lambda }}_1(\mathbf{\Phi },{\dot{\mathbf{s}}}^{ks}),& 1-{\mathsf{\Gamma }}_2(\mathbf{\Phi },{\dot{\mathbf{s}}}^{rsa}),& 1-{\mathsf{\Psi }}_3(\mathbf{\Phi },{\dot{\mathbf{s}}}^{geo})\end{array}\right]+\lambda V\left(\mathbf{X}\right)$$

(9)

where X denotes the deployment positions and pointing angles of all radar nodes in the distributed system, and Φ represents the geometric configuration of the system, the sets ${\dot{\mathbf{s}}}^{ks}$, ${\dot{\mathbf{s}}}^{rsa}$, and ${\dot{\mathbf{s}}}^{geo}$ correspond to the sampling points within the key-task region, the surveillance–coverage region, and the geometric–balance analysis region, respectively, the term $\lambda V\left(\mathbf{X}\right)$ represents the penalty imposed on solutions that violate engineering or coverage constraints, where the penalty coefficient $\lambda$ controls the penalty strength, ensuring that infeasible solutions can be effectively excluded without compromising the convergence stability of the optimization algorithm, the indicator ${\mathsf{\Lambda }}_1(\mathbf{\Phi },{\dot{\mathbf{s}}}^{ks})$ is constructed to evaluate the localization accuracy of the system over the key-task region, based on the root-mean-square position error at each sampling point, and is defined as follows:

$${\mathsf{\Lambda }}_1(\mathbf{\Phi },{\mathbf{s}}^{ks})={\displaystyle \frac{1}{\left|{\dot{\mathbf{s}}}^{ks}\right|}}\sum \mathrm{RMSPE}(p)$$

(10)

where $\left|{\dot{\mathbf{s}}}^{ks}\right|$ denotes the number of sampling points within the key-task region. A smaller value of ${\mathsf{\Lambda }}_1$ corresponds to better localization accuracy, the indicator ${\mathsf{\Gamma }}_2(\mathbf{\Phi },{\dot{\mathbf{s}}}^{rsa})$ reflects the spatial surveillance–coverage capability of the system. Specifically, the subset of sampling points satisfying the coverage criterion is denoted as ${\mathsf{\Omega }}_{\mathrm{cov}}$, and the corresponding coverage efficiency is given by the following:

$${\mathsf{\Gamma }}_2(\mathbf{\Phi },{\dot{\mathbf{s}}}^{rsa})={\displaystyle \frac{\left|{\mathsf{\Omega }}_{\mathrm{cov}}\right|}{\left|{\dot{\mathbf{s}}}^{rsa}\right|}}$$

(11)

The indicator ${\mathsf{\Psi }}_3(\mathbf{\Phi },{\dot{\mathbf{s}}}^{geo})$ characterizes the geometric balance of the spatial configuration and primarily reflects the stability of the GDOP distribution. To eliminate the influence of absolute GDOP magnitude, this study adopts a normalized dispersion measure, expressed as follows:

$${\mathsf{\Psi }}_3\left(\mathbf{\Phi },{\mathbf{s}}^{geo}\right)=1-{\displaystyle \frac{\sigma \left(\mathrm{GDOP}\right)}{max\left(\mathrm{GDOP}\right)-min\left(\mathrm{GDOP}\right)}}$$

(12)

where $\sigma \left(\mathrm{GDOP}\right)$ denotes the standard deviation of GDOP values over the specified region, describing the degree of fluctuation. A smaller standard deviation corresponds to a ${\mathsf{\Psi }}_3$ value approaching 1, indicating that the system exhibits better geometric balance across the spatial region.

The topology optimization problem can be expressed as follows:

$$\left\{\begin{array}{c}\underset{\mathbf{S}}{min} \mathbf{F}\left(\mathbf{X}\right)\\ s.t.{\mathbf{s}}_i\in RDA,i=1,\dots ,N,\\ {\mathit{p}}_j\in RSA,j=1,\dots ,N,\\ \angle \left({\mathit{\omega }}_{b_i},{\mathit{\omega }}_{b_j}\right)\le {\alpha }_i^{max}.\end{array}\right.$$

(13)

To clarify the distinction from existing multi-objective radar deployment strategies such as Yang et al. [25] and Wang et al. [26], it should be noted that prior studies have explored multi-objective formulations centered on performance-oriented metrics (e.g., GDOP, coverage, positioning accuracy), where geometric characteristics are typically reflected through aggregated performance indicators. Building upon this foundation, the present work reformulates geometric stability as an explicit optimization objective within the deployment model. Instead of relying solely on average GDOP or coverage enhancement, spatial geometric balance is quantified through the normalized standard deviation of GDOP across the surveillance region. This modeling perspective shifts the optimization emphasis from improving performance at sampled locations to regulating the global consistency of geometric conditions. In this sense, existing works primarily enhance system performance at discrete evaluation points, whereas the proposed formulation introduces a dispersion-control mechanism to constrain spatial imbalance over the entire region. Practically, minimizing GDOP dispersion limits performance fluctuation across space, suppresses extreme geometric disparities between subregions, and improves robustness against localized degradation. As a result, the optimized topologies exhibit enhanced structural stability, particularly in large-scale surveillance scenarios.

3.3. Proposed NS-MOPSO-Based Topology Optimization Algorithm

To address the multiple constraints, multi-objective coupling, and non-convex characteristics inherent in the distributed radar topology optimization problem, this paper develops an improved multi-objective particle swarm optimization algorithm (Non-dominated Sorting Multi-Objective Particle Swarm Optimization, NS-MOPSO). By integrating the non-dominated sorting mechanism with global search capabilities of particle swarm optimization, the proposed method achieves efficient exploration of the high-dimensional solution space, enabling effective optimization of radar deployment topologies. Building upon this foundation, a Gaussian mutation–inspired reinitialization strategy is incorporated. This strategy introduces lightweight perturbations to a subset of particles when stagnation is detected during the evolution process, allowing poorly performing particles to escape local optima, thereby enhancing population diversity and improving the global stability of convergence under complex constraints [38,39]. Based on the above algorithmic framework, the overall optimization procedure of the proposed NS-MOPSO algorithm is summarized in Figure 4.

3.3.1. Particle Encoding and Constraint Handling

In the algorithmic encoding scheme, the position coordinates and antenna pointing angles of radar nodes are unified into a single particle position vector. For a system containing N radar nodes, the particle position is expressed as follows:

$$\mathbf{X} = \left[{\mathrm{x}}_1,{\mathrm{y}}_1,{\mathsf{\theta }}_1,{\mathrm{x}}_2,{\mathrm{y}}_2,{\mathsf{\theta }}_2,\cdots ,{\mathrm{x}}_{\mathrm{N}},{\mathrm{y}}_{\mathrm{N}},{\mathsf{\theta }}_{\mathrm{N}}\right]$$

(14)

where $\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$ denotes the coordinate of the i-th radar node and ${\mathsf{\theta }}_{\mathrm{i}}$ denotes its antenna pointing angle. The variables satisfy the following:

$$\left\{\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}\in \left[{\mathrm{x}}_{\min },{\mathrm{x}}_{\max }\right]\\ {\mathrm{y}}_{\mathrm{i}}\in \left[{\mathrm{y}}_{\min },{\mathrm{y}}_{\max }\right]\\ {\mathsf{\theta }}_{\mathrm{i}}\in \left[0,2\mathsf{\pi }\right)\end{array}\right.$$

(15)

Random initialization within the deployment region ensures diversity and prevents premature convergence, providing a solid basis for subsequent multi-objective search [14].

To determine whether radar nodes satisfy coverage constraints, a feasibility indicator is defined as follows:

$$\mathrm{c}\left(\mathsf{\Phi },{\mathsf{\omega }}_{\mathrm{n},\mathrm{k}}^{\mathrm{bd}}\right) = 1$$

(16)

where $\mathsf{\Phi }$ represents the particle’s location parameters, and ${\mathsf{\omega }}_{\mathrm{n},\mathrm{k}}^{\mathrm{bd}}$ corresponds to the k-th boundary sample point of the n-th constraint class. To quantify the overall constraint violation, a penalty function is adopted as follows:

$$\mathrm{V}\left(\mathbf{X}\right) = \sum _{\mathrm{k} = 1}^{\mathrm{K}}\sum _{\mathrm{n} = 1}^{{\mathrm{N}}_{\mathrm{k}}^{\mathrm{bd}}}\max \left\{0,1-\mathrm{c}\left(\mathbf{X},{\mathsf{\omega }}_{\mathrm{n},\mathrm{k}}^{\mathrm{bd}}\right)\right\}$$

(17)

where V(X) denotes the accumulated violation of geometric and coverage constraints under the solution X, $\mathrm{c}\left(\mathbf{X},{\mathsf{\omega }}_{\mathrm{n},\mathrm{k}}^{\mathrm{bd}}\right)$ is a feasibility indicator used to evaluate whether the constraint at sample point k of class n is satisfied. When a constraint is fully satisfied, c ≥ 1, and the penalty is 0. When this is violated, a penalty term 1 − c(·) is generated, and the total penalty is accumulated to form V(X).

3.3.2. Non-Dominated Sorting with Objective-Aware Crowding Queue Strategy

In multi-objective radar topology optimization, the optimization objectives often exhibit strong trade-offs, making it difficult to identify a single dominant solution. To effectively balance localization accuracy, coverage capability, and geometric balance, a non-dominated sorting strategy is adopted to rank candidate solutions based on Pareto dominance. Furthermore, to alleviate solution clustering and preserve population diversity during the evolutionary process, a crowding queue strategy is incorporated into the sorting and selection procedure.

To reflect the priority of different emitter task regions in localization accuracy, task weights are incorporated into the localization performance term:

$$\mathsf{\Lambda }(\mathsf{\Phi },{\dot{\mathrm{S}}}^{\mathrm{ks}})={\sum }_{r=1}^R w_r\cdot {\mathsf{\Lambda }}_r(\mathsf{\Phi },{\dot{\mathrm{S}}}^{\mathrm{ks}})$$

(18)

where R is the number of emitter task regions, and $w_r$ is the weight coefficient of the r-th task region, satisfying the following:

$${\sum }_{r=1}^R w_r=1$$

(19)

For each particle in the swarm, a non-dominated sorting procedure is conducted based on the multi-objective vector $\mathbf{F}\left(\mathbf{X}\right)=[f_1,f_2,f_3]$ to determine the dominance rank of each particle [40]. $f_1,f_2,f_3$ correspond to the localization accuracy, coverage capability, and geometric balance objectives, respectively, and are defined as follows:

$$\left\{\begin{array}{c}{\mathrm{f}}_1=\mathsf{\Lambda }\left(\mathbf{X},{\dot{\mathrm{S}}}^{\mathrm{ks}}\right)+{\lambda }_1\mathrm{V}\left(\mathbf{X}\right)\\ {\mathrm{f}}_2=1-\mathsf{\Gamma }\left(\mathbf{X},{\dot{\mathrm{S}}}^{\mathrm{rsa}}\right)+{\lambda }_2\mathrm{V}\left(\mathbf{X}\right)\\ {\mathrm{f}}_3=1-{\mathsf{\Psi }}_3(\mathbf{\Phi },{\dot{\mathbf{s}}}^{geo})+{\lambda }_3\mathrm{V}\left(\mathbf{X}\right)\end{array}\right.$$

(20)

For particles belonging to the same rank, an objective-aware crowding distance is calculated to evaluate the diversity of particle distribution in the objective space while emphasizing critical performance metrics, which is defined as follows:

$$d_i=\sum _{m=1}^3{\alpha }_m{\displaystyle \frac{\left|f_m\right(i+1)-f_m(i-1\left)\right|}{max{f}_m-min{f}_m}}$$

(21)

where ${\alpha }_m$ denotes the weighting coefficient associated with the m-th objective, reflecting its relative importance in the radar topology optimization problem. Specifically, ${\alpha }_m$ is defined as the inverse of the standard deviation of the objective values within the current population:

$${\tilde{\alpha }}_m={\displaystyle \frac{1}{\sigma \left(f_m\right)}}$$

(22)

where $\sigma \left(f_m\right)$ denotes the standard deviation of the m-th objective across all particles. To ensure a balanced contribution of all objectives, the weighting coefficients are further normalized such that their sum equals one as follows:

$${\alpha }_m={\displaystyle \frac{{\tilde{\alpha }}_m}{{\sum }_{k=1}^M {\tilde{\alpha }}_k}},\sum _{m=1}^M {\alpha }_m=1$$

(23)

This adaptive weighting strategy assigns lower weights to objectives with large fluctuations and higher weights to more stable objectives, thereby mitigating single-objective dominance in crowding-distance computation. Figure 5 illustrates a comparison between the conventional and weighted crowding mechanisms. Under the traditional scheme (Figure 5a), differences in objective scales cause certain objectives to dominate the crowding contribution, resulting in clustered selections near extreme regions. In contrast, the weighted strategy in Figure 5b yields a more uniformly distributed Pareto set, improving diversity and structural stability.

To place the proposed objective-aware strategy in the context of existing crowding-distance refinements, the relative crowding distance (RCD) method in [25] is briefly discussed. RCD mitigates dominance caused by heterogeneous objective magnitudes through prior normalization, thereby reducing scale imbalance during Pareto selection. The proposed method addresses a similar heterogeneity issue but from a different regulation perspective. Instead of relying solely on magnitude normalization, it incorporates the fluctuation characteristics of each objective within the evolving population. By embedding adaptive weights derived from the inverse standard deviation of objective values into the crowding-distance formulation, the selection process is adjusted according to inter-objective dispersion rather than numerical scale alone. Such fluctuation-aware regulation enhances the representation of objectives with relatively small variation ranges. In radar topology optimization, this is particularly beneficial for geometric consistency metrics, whose fluctuation amplitude is typically lower than that of localization error.

To further illustrate the differences among crowding mechanisms under the proposed three-objective model, Figure 6 presents the selection outcome of a representative generation under identical problem settings. All gray points correspond to candidate solutions from the same population at that generation, while the colored points indicate the subset of individuals selected for survival according to each crowding mechanism. The three objectives $f_1$, $f_2$ and $f_3$ correspond to localization error, coverage performance, and geometric consistency, respectively. Due to their modeling characteristics, the objectives exhibit significantly different variation ranges. Specifically, the localization error objective $f_1$ shows the largest fluctuation, whereas the geometric consistency objective $f_3$ varies within a relatively smaller range. As illustrated in Figure 6a, when variation magnitudes differ substantially, the standard crowding-distance mechanism tends to concentrate selections along the objective with larger fluctuation, resulting in uneven distribution in the multi-objective space. The normalization-based mechanism in Figure 6b partially alleviates scale dominance; however, the representation of low-variation objectives remains limited. In contrast, Figure 6c demonstrates that the proposed objective-aware strategy achieves a more balanced distribution across the three objectives, particularly enhancing the representation of $f_3$. In this context, a desirable selection distribution refers to a well-spread set of selected solutions across the three-objective space, ensuring balanced representation of localization accuracy, coverage performance, and geometric consistency. Excessive concentration along a single objective dimension may lead to biased exploration and reduced diversity. Therefore, a more uniformly distributed set of colored points indicates improved diversity preservation and better structural balance among the objectives. As observed in Figure 6c, the proposed objective-aware crowding strategy achieves such a distribution, which is beneficial for maintaining balanced multi-objective trade-offs under heterogeneous objective scales.

To provide a quantitative comparison of the selection characteristics, four evaluation metrics are defined at the selection level. These metrics capture objective bias, coverage capability, structural uniformity, and spatial dispersion of the selected solutions. Specifically, let $S=\left\{\mathbf{f}\right(k){\}}_{k=1}^{N_s}$ denote the set of selected individuals in a representative generation:

$${\mathbf{f}}^{\left(k\right)}=[f_1^{\left(k\right)},f_2^{\left(k\right)},f_3^{\left(k\right)}]$$

(24)

This corresponds to the localization error, coverage performance, and geometric consistency, respectively.

To ensure comparability among heterogeneous objectives, each objective is normalized within S as follows:

$${\tilde{f}}_j^{\left(k\right)}={\displaystyle \frac{f_j^{\left(k\right)}-\underset{S}{min} f_j}{\underset{S}{max} f_j-\underset{S}{min} f_j+\epsilon }}$$

(25)

where ε is a small positive constant introduced to avoid division by zero.

Based on the normalized objective vectors, the following metrics are defined as follows:

(1)

${\mathrm{Bias}}_{f1}$

$${\mathrm{Bias}}_{f1}=1-{\displaystyle \frac{\mathrm{std}\left({\tilde{f}}_1\right)}{\mathrm{std}\left({\tilde{f}}_1\right)+\mathrm{std}\left({\tilde{f}}_2\right)+\mathrm{std}\left({\tilde{f}}_3\right)}}$$

(26)

This metric quantifies the selection preference toward the high-variation objective $f_1$. Smaller values indicate a stronger bias toward $f_1$-dominated solutions.

(2)

${\mathrm{Cov}}_{f3}$

$${\mathrm{Cov}}_{f3}=\underset{S}{max} {\tilde{f}}_3-\underset{S}{min} {\tilde{f}}_3$$

(27)

This metric measures the directional coverage along the geometric consistency objective. Larger values indicate broader coverage in the $f_3$ dimension.

(3)

Uniformity

$$\mathrm{Uniformity}=\mathrm{std}(\{\parallel {\tilde{\mathbf{f}}}^{\left(k\right)}-{\tilde{\mathbf{f}}}^{\left(n\right(k\left)\right)}{\parallel }_2\})$$

(28)

where n(k) denotes the nearest neighbor of individual k in the normalized objective space. Smaller values indicate more regular spacing among selected individuals.

(4)

Spread-3D

$$\mathrm{Spread}-3\mathrm{D}=\sqrt{\mathrm{var}({\tilde{f}}_1)+\mathrm{var}({\tilde{f}}_2)+\mathrm{var}({\tilde{f}}_3)}$$

(29)

This metric reflects the overall dispersion of selected solutions in the three-objective space. Larger values indicate greater global coverage.

The comparison under the formulated three-objective model is presented in

Table 1. Quantitative comparison of selection behavior under three crowding mechanisms.

Method	${\mathbf{Bias}}_{\mathit{f}\mathbf{1}}$	${\mathrm{Cov}}_{\mathit{f}\mathbf{3}}$	Uniformity	Spread-3D
Standard	0.3231	0.5744	0.6559	0.5305
RCD	0.2269	0.5394	0.3132	0.3923
objective-aware crowding	0.2075	0.7311	0.2456	0.4415

.

As shown in Table 1, the normalization-based RCD mechanism alleviates the dominance of the localization error objective compared with the standard crowding strategy. The proposed objective-aware crowding further enhances this regulation effect while achieving the highest coverage along the geometric consistency objective. In terms of structural distribution, it exhibits more regular spacing among selected individuals compared with the RCD method. Overall, the objective-aware mechanism improves geometric coverage and structural coordination simultaneously, leading to a more balanced and task-aligned representation across objectives under the formulated multi-objective model.

3.3.3. Stagnation-Aware Mutation and Genetic Reheating Strategy

To enhance the global search capability and convergence stability of the multi-objective particle swarm in complex topology spaces, it is necessary to alleviate the diversity degradation and premature convergence commonly observed in standard non-dominated sorting algorithms. Therefore, when population stagnation is detected over several consecutive generations, a lightweight random-perturbation mechanism is introduced. Specifically, the Gaussian update rules are restructured by embedding geometric feasibility constraints and GDOP-related indicators, where the position and angle perturbations are restricted within the radar deployment area (RDA) and surveillance region limits to maintain geometric feasibility during perturbation. The updated position and steering angle are given by the following:

$$\left\{\begin{array}{c}{\mathrm{p}}^{\prime }=\mathrm{p}+\mathsf{\xi },\mathsf{\xi }\sim \mathcal{N}\left(0,{\mathsf{\sigma }}_{\mathrm{p}\mathrm{o}\mathrm{s}}^2\mathrm{I}\right)\\ {\mathsf{\theta }}^{\prime }=\mathrm{wrap}(\mathsf{\theta }+\mathrm{\eta }),\mathrm{\eta }\sim \mathcal{N}(0,{\mathsf{\sigma }}_{\mathrm{d}\mathrm{i}\mathrm{r}}^2\mathrm{I})\end{array}\right.$$

(30)

where $\mathrm{p}$ denotes the current positions of the radar nodes, $\mathsf{\theta }$ represents the corresponding antenna steering angles, ${\mathrm{p}}^{\prime }$ and ${\mathsf{\theta }}^{\prime }$ are the mutated node positions and steering angles, respectively, and $\mathsf{\xi }$ and $\mathrm{\eta }$ are zero-mean Gaussian perturbations with standard deviations, ${\sigma }_{\mathrm{p}\mathrm{o}\mathrm{s}}$ and ${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{r}}$, used to regulate the perturbation intensity of node positions and angular directions. The operator wrap(·) performs angular normalization, constraining the angles within [−π, π] to prevent angular discontinuity.

To further prevent premature convergence in geometry-constrained deployment spaces while maintaining population diversity, a reinitialization mechanism is incorporated into the disturbance framework [41]. When the algorithm fails to improve the objective value (e.g., composite GDOP) over several consecutive generations, the worst-performing individuals in the current swarm are selected and reinitialized, thereby enhancing the global exploration capability. This process can be expressed as follows:

$${\mathbf{x}}_i\leftarrow {\mathrm{\Pi }}_{\mathcal{F}}\left(\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{F}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}\right(\left)\right),{\mathbf{v}}_i\leftarrow \mathbf{0},i\in {\mathcal{I}}_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}$$

(31)

where ${\mathcal{I}}_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}$ denotes the indices of particles with the largest GDOP values in the current population. The operator RandomFeasible() randomly generates feasible samples within the deployment region to satisfy boundary constraints, and ${\mathrm{\Pi }}_{\mathcal{F}}$ denotes the feasibility projection operator used to ensure physically valid positions. Through such periodic reinitialization, the algorithm avoids premature convergence and maintains adequate global search ability while ensuring stable local refinement.

Although mutation and partial reinitialization have been widely adopted in general multi-objective PSO frameworks [42], their direct application may not fully capture the geometric coupling characteristics and practical deployment constraints inherent in distributed radar topology optimization. In the proposed framework, these operators are adapted in a problem-oriented manner. Specifically, Gaussian mutation is restricted within the radar deployment area (RDA) and surveillance region limits to maintain geometric feasibility during perturbation. In addition, reinitialization is triggered based on the stagnation of a composite GDOP-related indicator rather than a general objective stagnation criterion, facilitating targeted escape from locally imbalanced geometric configurations. Through this constraint-aware mutation and GDOP-guided reinitialization strategy, local geometric refinement and global exploration are coordinated under radar-specific structural considerations. Accordingly, the proposed diversity regulation mechanism can be viewed as a scenario-adaptive modification of conventional operators, enhancing their suitability for distributed radar topology optimization.

3.3.4. Adaptive Learning and Search Regulation Mechanism

To further strengthen the balance between global exploration and local exploitation at different evolutionary stages and to maintain the overall convergence stability of the algorithm, an adaptive learning mechanism is introduced. The inertia weight is updated linearly with the iteration index according to the following:

$$\mathsf{\omega }\left(\mathrm{t}\right) = {\mathsf{\omega }}_{\max }-{\displaystyle \frac{\mathrm{t}}{{\mathrm{T}}_{\max }}}\left({\mathsf{\omega }}_{\max }-{\mathsf{\omega }}_{\min }\right)$$

(32)

Meanwhile, the cognitive and social learning factors are also adjusted adaptively as follows:

$${\mathrm{c}}_1\left(\mathrm{t}\right) ={ \mathrm{c}}_{1,\min }+{\displaystyle \frac{\mathrm{t}}{{\mathrm{T}}_{\max }}}\left({\mathrm{c}}_{1,\max }-{\mathrm{c}}_{1,\min }\right)$$

(33)

$${\mathrm{c}}_2\left(\mathrm{t}\right)={\mathrm{c}}_{2,\max }-{\displaystyle \frac{\mathrm{t}}{{\mathrm{T}}_{\max }}}\left({\mathrm{c}}_{2,\max }-{\mathrm{c}}_{2,\min }\right)$$

(34)

where ${\mathsf{\omega }}_{\max }$ and ${\mathsf{\omega }}_{\min }$ denote the upper and lower bounds of the inertia weight. ${\mathrm{c}}_{1,\max }$, ${\mathrm{c}}_{1,\min }$, ${\mathrm{c}}_{2,\max }$, ${\mathrm{c}}_{2,\min }$ represent the upper and lower bounds of the cognitive and social learning factors. ${\mathrm{T}}_{\max }$ is the maximum number of iterations. In the early stage of the algorithm, a larger inertia weight combined with a higher cognitive factor ${\mathrm{c}}_1$ helps particles perform global exploration and expand the search space. As the iteration progresses, the inertia weight gradually decreases while the social learning factor ${\mathrm{c}}_2$ increases, which enhances the local exploitation ability and improves the stability of convergence accuracy. This strategy accelerates the particles’ convergence toward the Pareto-optimal front [42].

Compared with existing multi-objective evolutionary methods for radar topology optimization, the proposed NS-MOPSO emphasizes problem-oriented mechanism restructuring and scenario adaptability. First, diversity enhancement operators are redesigned under radar-specific geometric and deployment constraints, embedding RDA/RSA feasibility regulation into mutation and employing GDOP-aware stagnation detection for targeted reinitialization. Second, the integration of objective-aware crowding and adaptive learning facilitates a balanced trade-off between convergence stability and Pareto diversity in constrained deployment spaces. Third, by considering the coupling among localization accuracy, coverage capability, and geometric consistency, the framework supports coordinated performance improvement across heterogeneous objectives. Overall, the proposed method constitutes a structurally embedded refinement of multi-objective PSO, in which conventional operators are systematically reconfigured within a radar-aware optimization architecture. This design enables geometry-consistent and deployment-constrained topology evolution tailored to distributed radar systems.

The detailed procedure of the proposed NS-MOPSO-based topology optimization is presented in Algorithm 1.

Algorithm 1 NS-MOPSO-based Topology Optimization

Require: Particle swarm size N, maximum iteration number ${\mathrm{T}}_{\max }$, objective evaluation function F(X), constraint violation function c(X), initialization ranges for particle position X = {xi, yi, θi} and velocity vi, Parameters ${\mathsf{\omega }}_{\max }{,\mathsf{\omega }}_{\min },{\mathrm{c}}_{1,\max },{\mathrm{c}}_{1,\min },{\mathrm{c}}_{2,\max },{\mathrm{c}}_{2,\min }$. Ensure: Final Pareto-optimal solution set $P_{\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{o}}$ and optimal topology structure ${\mathrm{X}}^{*}$. 1: Randomly initialize particle swarm: $X=\left\{\right(x_i,y_i,{\theta }_i){\}}_{i=1}^N$, and compute fitness F(X). 2: Perform non-dominated sorting and obtain initial Pareto set ${\mathrm{P}}_{\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{o}}$, compute crowding distance $d_i$ 3: Assign initial inertia weight ${\omega }_0$ and learning factors ${\mathrm{c}}_1$, ${\mathrm{c}}_2$ 4: while $t<T_{max}$ do 5:  Update inertia weight according to (32) 6:  Update learning factors according to (33) and (34) 7:  For each particle, evaluate objective fitness $F\left(X_i\right)$ 8:  Perform non-dominated sorting to update Pareto set $P_{\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{o}}$ 9:  Update velocity and position:     $v_i\leftarrow \omega v_i+c_1{r}_1(p_i^{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}-x_i)+c_2{r}_2(g^{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}-x_i)$, $x_i\leftarrow x_i+v_i$ 10:  Apply self-adaptive mutation to $(p_i,{\theta }_i)$ to enhance diversity 11:  If $\underset{j}{max}||F_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}\left(t\right)<F_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}(t-1)||<{\epsilon }_F$ or $|HV(P^t)-HV(P^{t-1})|<{\epsilon }_{HV}$ 12:  then    Select the worst $J_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}$ particles 13:    Reinitialize these particles using RandomFeasible() 14:    Recompute fitness and update Pareto set 15:  End if 16:  Increment iteration count $t\leftarrow t+1$ 17: end while 18: return Pareto and ${\mathrm{X}}^{*}$

4. Performance Evaluation and Analysis

4.1. Simulation Experiments and Analysis

4.1.1. Simulation Environment Setup

To reflect practical application scenarios of distributed TDOA-based radar localization, this study considers a representative urban sensing environment and constructs a simulation framework for topology optimization. Under fixed deployment conditions, the focus is placed on evaluating the effectiveness of the proposed topology optimization method in improving geometric conditions and localization performance in complex environments.

To verify the effectiveness of the proposed topology optimization approach, a typical TDOA localization scenario is constructed as follows. The radar deployment area is set to [−40 km, 40 km] × [0 km, 80 km], and the emitter monitoring area is defined as [−200 km, 200 km] × [0 km, 380 km]. Within the RSA, two key emitting sources (KS) are placed, with their parameters listed in

Table 2. KS Region Parameters.

KS	Coordinate Range (km)	Localization Accuracy (m)	Weight
1	[−154, −119]/[176, 223]	330	0.4
2	[114, 156]/[176, 223]	330	0.6

. Among the two signal sources, KS2 corresponds to a higher-priority task that requires more stringent localization accuracy. Therefore, a larger weighting factor (0.6) is assigned to KS2 to reflect its greater importance in the overall system performance evaluation. This setting follows common practice in multi-objective modeling, where relative weights are used to encode task priorities under practical deployment considerations. The radar system contains five nodes, whose key parameters are shown in

Table 3. Radar System Parameters.

Parameter	Value
Signal Frequency	3 GHz
Bandwidth	5 MHz
Noise Figure	10 dB
Antenna Gain	10 dB

, including a signal frequency of 3 GHz and a bandwidth of 5 MHz. The proposed NS-MOPSO is then applied, with a swarm size of 100 and a maximum iteration number of 150. All simulation experiments and algorithmic evaluations were implemented using MATLAB R2024b.

4.1.2. Simulation-Based Experimental Setup

To validate the effectiveness of the distributed radar deployment topology optimization method proposed in this paper, eight comparison experiments are designed: random deployment, uniform array deployment, tangent-based layout [18], single-objective PSO, NSGA-II, standard MOPSO, MOPSO-NRCD, and the proposed NS-MOPSO. The experiments focus on analyzing the influence of different optimization strategies on localization accuracy (RMSPE), monitoring coverage rate, and geometric balance. This allows us to comprehensively evaluate the improvements in localization performance and topological stability achieved by the algorithm.

In the topology optimization experiments, the random deployment group serves as a naive baseline for performance comparison. The tangent-based layout is introduced as a non-evolutionary, deterministic geometric baseline, which directly calculates node positions via tangent equations without iterative optimization. The uniform array group is used to evaluate the degradation of geometric constraints under traditional deployment conditions. The single-objective PSO group adopts the traditional particle swarm optimization algorithm, which optimizes for a single objective. Building on this, the standard MOPSO group incorporates a multi-objective balancing mechanism to simultaneously optimize localization accuracy and coverage performance. The NSGA-II and MOPSO-NRCD groups represent State-of-the-Art multi-objective evolutionary strategies, providing a competitive benchmark for performance comparison. The improved NS-MOPSO group proposed in this paper further integrates non-dominated sorting, adaptive mutation, and GDOP-guided reinitialization strategies to enhance global search capability and convergence stability, thereby achieving coordinated optimization of localization accuracy, coverage rate, and geometric uniformity across the three objectives.

4.1.3. Simulation Results and Analysis

First, RMSPE is adopted to evaluate localization accuracy under different radar deployment strategies, and the results are shown in Figure 7. Each curve is obtained from 100 independent Monte Carlo trials under identical deployment constraints with measurement noise and initialization perturbations. The random deployment scheme exhibits the largest error (mean 445.3 m), followed by the tangent-based layout (393.7 m) and the uniform array (428.1 m). Evolutionary optimization methods significantly reduce RMSPE: single-objective PSO achieves 356.8 m, NSGA-II and standard MOPSO reduce it to 319.1 m and 312.7 m, respectively, and MOPSO-NRCD further lowers it to 302.2 m. The proposed NS-MOPSO achieves the lowest average RMSPE of 287.5 m, corresponding to a reduction of approximately 35% compared with random deployment and about 5% relative to MOPSO-NRCD. These results indicate that integrating geometric balance regulation and scenario-aware diversity control leads to more stable spatial configurations and improved TDOA localization accuracy.

The monitoring coverage rate reflects the proportion of the target region effectively observed by distributed radar nodes, and the comparative results are presented in Figure 8. Each curve is obtained from 100 independent Monte Carlo trials under identical deployment constraints. The random deployment and uniform array schemes achieve average coverage rates of 58.5% and 59.6%, respectively, while the tangent-based layout yields a moderate coverage rate of 62.0%, indicating limited adaptability in complex surveillance environments for these non-evolutionary strategies. Evolutionary optimization methods substantially improve coverage performance, raising it above 68% for multi-objective approaches. In particular, MOPSO-NRCD attains 72.9%, while the proposed NS-MOPSO further increases the average coverage to 77.6%, corresponding to an absolute improvement of approximately 19% over random deployment and about 4.7% relative to MOPSO-NRCD. These results indicate that integrating geometric balance regulation and scenario-aware diversity control leads to a more effective spatial configuration of radar nodes, thereby enhancing surveillance completeness and overall network robustness.

Using the multi-objective normalized indicator as the evaluation metric, the convergence behaviors of different algorithms are compared in Figure 9. The proposed NS-MOPSO approaches the Pareto-optimal region more rapidly during the early stage of evolution, exhibiting a faster convergence rate than GA, standard MOPSO, and MOPSO-NRCD. In the later stage, the fluctuation amplitude of its objective indicator remains comparatively smaller, indicating improved convergence stability. After 150 iterations, NS-MOPSO achieves the lowest normalized fitness value, further confirming its superior convergence performance. The enhanced performance can be attributed to the joint effect of geometry-constrained mutation, GDOP-aware reinitialization, and objective-aware crowding regulation, which collectively maintain population diversity while preventing premature convergence. As a result, the proposed framework achieves a favorable balance between convergence speed and solution stability, demonstrating its suitability for distributed radar topology optimization under practical deployment constraints.

GDOP is a core metric that quantifies the explanatory effectiveness of localization geometry. A smaller GDOP value indicates stronger geometric constraints and higher localization accuracy. Figure 10 compares the differences in geometric performance between the uniformly distributed array and the optimized deployment. Simulation results show that the GDOP distribution of the optimized deployment is significantly better than that of the traditional uniform array: the average GDOP decreases from 28.91 to 18.63, representing a reduction of 35.6%, the minimum GDOP remains at 3.00, while the maximum GDOP is effectively controlled within 100.00. In addition, the proportion of regions where the GDOP of the optimized deployment is lower than 20 reaches 85.1%, which is 37.2% higher than that of the uniform array (47.9%). This indicates that the optimized deployment significantly enhances the overall geometric stability of the system and improves global localization accuracy.

From the perspective of spatial distribution, the GDOP of the uniform array exhibits noticeable anisotropy, with good performance along the array baseline direction but rapid degradation toward both sides. In contrast, the optimized deployment achieves a more homogeneous GDOP distribution by reasonably adjusting node positions, and it maintains better localization stability, particularly in boundary regions (as shown in Figure 11). Further 3D surface analysis demonstrates that the optimized deployment effectively eliminates the high-GDOP “blind zones” that exist in the uniform array, resulting in a flatter and more balanced accuracy distribution surface.

4.1.4. Computational Cost and Scalability Analysis

To further evaluate the computational burden and scalability of the proposed method in practical deployment scenarios, both the theoretical complexity structure and empirical runtime are analyzed. All algorithms were implemented in MATLAB and executed on an AMD Ryzen 7 5800H @ 3.20 GHz with 16 GB RAM, consistent with the experimental environment used throughout this study.

From a theoretical perspective, the computational cost of multi-objective radar topology optimization mainly consists of three components: objective function evaluation, non-dominated sorting, and crowding-distance computation. Let $N_{\mathrm{p}\mathrm{o}\mathrm{p}}$ denote the population size, G the number of iterations, M the number of objectives, K the number of radar nodes, and S the number of spatial sampling points. During each generation, every individual must evaluate localization error, coverage performance, and geometric consistency over all sampling points. The dominant computational term can therefore be expressed as $O(G\cdot N_{\mathrm{p}\mathrm{o}\mathrm{p}}\cdot S\cdot K)$, which typically governs the overall runtime in distributed TDOA topology optimization. For algorithms involving non-dominated sorting, additional computational overhead arises from sorting and crowding operations. The sorting cost increases with population size, and crowding-distance updates require objective-wise ordering. Although these operations introduce extra computations, they remain secondary compared with objective evaluation in typical grid-based deployment optimization.

Among the compared methods, uniform array, random deployment, and the tangent-based layout do not involve iterative optimization and therefore incur negligible computational cost. The compared methods were selected according to their representativeness and practical relevance in distributed radar topology optimization. Intermediate-complexity variants generally share similar computational structures and do not introduce fundamentally different optimization mechanisms beyond the selected representatives. Standard MOPSO is mainly dominated by objective evaluation and archive updating. MOPSO-NRCD introduces normalization-based crowding and neighborhood constraint checks; however, these additional operations do not alter the fundamental computational structure. The proposed NS-MOPSO incorporates an objective-aware crowding strategy, where additional computations are limited to population-level fluctuation statistics and adaptive weight updating. Since these operations are significantly less expensive than objective evaluation, the proposed strategy does not introduce a substantial increase in overall computational cost. It should be noted that all evolutionary optimization processes are performed offline prior to system deployment and do not affect real-time localization performance. Therefore, this analysis focuses solely on the deployment optimization stage. Under the main experimental configuration (K = 5, $N_{\mathrm{p}\mathrm{o}\mathrm{p}}$ = 100, G = 150), each method was independently executed 30 times, and the average runtime is reported in

Table 4. Average runtime under five-node configuration.

Method	Average Runtime (ms)
Uniform Array	0.11
Random Deployment	0.19
Tangent-based Layout	0.24
MOPSO	92.4
NSGA-II	186.7
MOPSO-NRCD	108.3
NS-MOPSO	97.6

.

The results indicate that evolutionary algorithms incur higher offline computational cost than fixed deployment strategies. However, the runtime of the proposed method is comparable to standard MOPSO and lower than MOPSO-NRCD and NSGA-II, demonstrating that the introduced strategy does not significantly increase computational burden.

To further investigate scalability, the number of radar nodes was increased to 5, 10, 20, and 50 while maintaining other parameters. The results are summarized in

Table 5. Average runtime under different node scales (ms).

Method	5 Nodes	10 Nodes	20 Nodes	50 Nodes
Uniform Array	0.11	0.13	0.15	0.19
Random Deployment	0.19	0.22	0.26	0.33
Tangent-based Layout	0.24	0.28	0.34	0.45
MOPSO	92.4	181.5	362.8	901.3
NSGA-II	186.7	369.2	738.5	1789.4
MOPSO-NRCD	108.3	213.7	425.9	1062.8
NS-MOPSO	97.6	191.2	382.4	892.1

.

As summarized in Table 5, the low-complexity baseline methods exhibit only marginal runtime growth with increasing node scale. In contrast, PSO-based methods increase approximately linearly with the number of nodes, while NSGA-II shows a more rapid growth trend. Specifically, when scaling from 5 to 50 nodes, NS-MOPSO increases from 97.6 ms to 892.1 ms, maintaining a runtime within one second even under the 50-node configuration, which demonstrates favorable scalability and practical feasibility. Although evolutionary algorithms inevitably incur higher offline computational cost than fixed deployment strategies, the runtime growth remains within the same order of magnitude across PSO-based methods. Importantly, the proposed NS-MOPSO does not introduce additional complexity escalation compared with standard MOPSO or MOPSO-NRCD. The near-linear runtime growth indicates that the performance improvements are not achieved at the expense of exponential or quadratic computational expansion, thereby ensuring scalability for large-scale distributed radar networks.

4.1.5. Ablation Study on Key Components

To further quantify the contribution of each designed module, an ablation study is conducted under the identical simulation settings described in Section 4.1.1. Each variant is generated by removing a single key component while keeping all other configurations unchanged. The quantitative results of these component-wise variants are summarized in

Table 6. Ablation Results of Key Components in NS-MOPSO.

Variant	RMSPE (m)	Coverage (%)	GDOP < 20 (%)
Full NS-MOPSO	287.5	77.6	89.2
w/o objective-aware crowding	301.9	73.5	78.4
w/o GDOP-guided reinitialization	305.2	74.1	73.7
w/o adaptive mutation	298.7	72.3	81.5

Note: GDOP < 20 (%) represents the proportion of the target area where the geometric dilution of precision is below 20, indicating a more stable localization geometry. All results are averaged over 100 Monte Carlo trials.

.

As shown in Table 6, removing the objective-aware crowding strategy leads to a noticeable degradation in both localization accuracy and coverage consistency, indicating its important role in maintaining diversity across objectives during multi-objective evolution. Similarly, excluding the GDOP-guided reinitialization mechanism results in a significant increase in RMSPE and a marked reduction in the GDOP< 20 ratio (from 89.2% to 73.7%), suggesting that this mechanism effectively mitigates stagnation in suboptimal geometric configurations. Although the adaptive mutation operator provides a comparatively moderate improvement, it consistently enhances global exploration capability without introducing excessive computational burden.

Collectively, these results demonstrate that the observed performance gains are not solely attributable to parameter tuning but arise from the coordinated design and structured integration of multiple complementary mechanisms. Therefore, the proposed framework represents a structured integration of complementary mechanisms, in which each component contributes measurably to the overall improvement in localization accuracy, coverage quality, and geometric stability.

4.2. Experimental Results and Analysis

To further validate the effectiveness of the proposed distributed TDOA-based radar topology optimization method in practical application scenarios, this section constructs an experimental platform based on a real outdoor environment. It conducts multiple node deployment and localization tests in an open-field setting. Compared with the aforementioned simulation scenarios, real-world environments involve more complex non-ideal factors, including target scattering characteristics, background clutter, environmental noise, and clock offsets among distributed radar nodes, all of which may significantly affect localization accuracy. Therefore, real-scene experimental evaluation enables a more comprehensive assessment of the performance advantages and robustness of the optimized topology under practical deployment conditions.

4.2.1. Experimental Scenario and Equipment

The experimental scenario is selected in an open outdoor area with no significant obstructions. The test site has an approximate length of 102 m and a width of 68 m, exhibiting typical characteristics of open-area outdoor localization environments and representing application scenarios such as intelligent traffic monitoring and outdoor target tracking. Considering the site scale and the sensing characteristics of the radar system, the radar deployment area is defined as 102 m × 20 m, while the radiation source (target activity) region is set to 20 m × 20 m, as illustrated in Figure 12. This configuration enables effective coverage of multi-point localization tasks within a typical range of 40–100 m. In subsequent experimental groups, four radar nodes are deployed at different positions within the designated deployment area. By adjusting the node coordinates, multiple deployment configurations—including uniform array layouts and random layouts—are constructed to evaluate the influence of radar geometry on localization performance.

The real-world experiments were conducted using four 80 GHz millimeter-wave radars designed for traffic monitoring, as shown in Figure 13. This radar system is specifically developed for outdoor traffic surveillance applications and features strong ranging capability and reliable multi-target detection performance. The typical detection range is 200–250 m, exceeding the spatial extent of the experimental area and providing sufficient signal quality and multipath diversity for TDOA-based ranging. All radar nodes were installed at a uniform height of approximately 1.5 m above ground to mitigate multipath effects from ground reflections. The radars were connected to an edge computing platform via network interfaces, enabling all devices to share a unified time synchronization signal. This configuration ensures the consistency and comparability of time-of-arrival measurements across different radar nodes, thereby supporting the subsequent TDOA-based localization process.

4.2.2. Experimental Design

In the selected test area, six representative radar topology configurations are sequentially constructed in the experiments, including a uniform array, a random deployment, MOPSO, NSGA-II, MOPSO-NRCD, and NS-MOPSO. For each deployment configuration, multiple target locations are sampled within the predefined radiation source region, as illustrated in Figure 14, and the time-of-arrival information recorded by each radar node is collected. Meanwhile, the corresponding ground-truth target positions are measured and used as references for performance evaluation. Under a unified data acquisition and processing framework, the localization results obtained from different topology configurations are systematically compared. The evaluation focuses on localization error performance, GDOP distribution characteristics, radiation area coverage capability, and stability differences induced by geometric configurations, thereby validating the performance improvements achieved by the optimized topology in real-world experimental scenarios.

After completing data acquisition across multiple real-world experimental scenarios, the localization results obtained under different radar topologies are uniformly aggregated and statistically analyzed to establish an error- and coverage-based performance evaluation framework. Specifically, for each estimated position, the experimental system’s target position estimates are matched and compared with the corresponding ground-truth reference location to obtain the localization error at each sampling point. On this basis, statistical analysis is conducted on the error data to compute the RMSPE, effective coverage rate, and the cumulative distribution function (CDF) of the localization error. Through comparative analysis of these metrics across different topology configurations, the localization accuracy, coverage capability, and error distribution characteristics of various radar deployment schemes in real-world scenarios are systematically evaluated.

4.2.3. Experimental Results Analysis

Based on the above experimental evaluation and data analysis, the comparative performance of different radar topology configurations under practical deployment conditions is summarized in Figure 15, Figure 16 and Figure 17, which present localization accuracy, surveillance coverage, and convergence characteristics, respectively. In terms of localization accuracy, both uniform array and random deployment are inherently limited by fixed or unstructured geometric layouts, resulting in relatively high average positioning errors and more pronounced fluctuations. Multi-objective optimization methods provide measurable improvements; however, their performance remains sensitive to local geometric imbalance. By contrast, the proposed NS-MOPSO consistently achieves the lowest average RMSPE and the smallest error variance across different scenarios. Specifically, compared with random deployment (2.32 m), the proposed method reduces RMSPE to 1.60 m, corresponding to an approximate 31% improvement. Even relative to the best-performing comparative method, MOPSO-NRCD (1.71 m), it achieves an additional improvement of about 6.4%. These gains stem from the coordinated effects of non-dominated sorting, objective-aware diversity regulation, and geometry-constrained mutation and reinitialization strategies, which jointly promote a more balanced solution distribution and enhance robustness under practical deployment constraints.

With respect to effective surveillance coverage, Figure 16b presents a clear performance hierarchy among the deployment strategies. The baseline schemes, namely random deployment and uniform array, achieve average coverage rates of 37.5% and 43.6%, respectively, reflecting limited adaptability under fixed or unstructured geometric layouts. In contrast, topology optimization substantially enhances coverage performance: MOPSO and NSGA-II improve the coverage to 60.8% and 61.3%, while MOPSO-NRCD further increases it to 62.6%. The proposed NS-MOPSO achieves the highest coverage of 66.9%, corresponding to an absolute improvement of 4.3 percentage points over MOPSO-NRCD and 6.1 points over standard MOPSO, and a significantly larger gain compared with the baseline deployments. These results indicate that the integration of geometry-aware diversity regulation and scenario-constrained operators enables a more effective spatial distribution of radar nodes, thereby reducing coverage gaps and improving overall surveillance completeness under practical deployment constraints.

Furthermore, the CDF analysis of positioning errors, shown in Figure 17, provides additional insight into the reliability of different deployment strategies. The uniform array and random deployment exhibit more right-shifted and flatter CDF curves, indicating larger error dispersion and heavier tail behavior. In contrast, the optimized topology methods produce more left-concentrated distributions, reflecting improved positioning consistency. Among them, the proposed NS-MOPSO achieves the leftmost and steepest CDF curve, meaning that a higher proportion of sampling points fall within the lower error intervals. For a given cumulative probability level, NS-MOPSO consistently attains smaller positioning errors than the comparative approaches. These results demonstrate that the proposed topology optimization not only reduces the average error but also effectively suppresses large-error occurrences, thereby enhancing overall localization reliability under practical deployment conditions.

5. Conclusions

This paper presents a topology optimization framework for distributed radar TDOA localization systems. To address the limitations of conventional uniform deployment strategies, an improved multi-objective particle swarm optimization approach, termed NS-MOPSO, is developed. A geometric TDOA localization model is established to characterize the influence of the spatial configuration on localization performance, and a unified multi-objective formulation is constructed to jointly optimize localization accuracy, coverage capability, and geometric balance. Experiments validate the effectiveness of the proposed approach. The proposed NS-MOPSO achieves a 6.4% reduction in RMSPE and a 4.3% increase in the proportion of regions with favorable geometric conditions compared with the best-performing comparative method (MOPSO-NRCD), while maintaining stable convergence behavior. The optimized topology provides improved localization reliability, enhanced coverage efficiency, and reduced spatial error variability under practical deployment constraints. Overall, the proposed framework represents a problem-oriented refinement of multi-objective PSO for radar topology optimization. By embedding radar-specific geometric constraints into diversity regulation and operator design, it enhances structural stability in heterogeneous objective spaces and offers insights for multi-objective optimization in other geometrically coupled sensing systems.