Dual-Trail Stigmergic Coordination Enables Robust Three-Dimensional Underwater Swarm Coverage

Abstract

Swarm coverage by unmanned underwater vehicles (UUVs) is essential for inspection, environmental monitoring, and search operations, but remains challenging in three-dimensional domains under limited sensing and communication. Pheromone-based stigmergic coordination provides a low-bandwidth alternative to explicit communication, yet conventional single-field models are susceptible to depth-dependent sensing inconsistencies and multi-source signal interference. This paper introduces a dual-trail stigmergic coordination framework in which a virtual pheromone field encodes short-term motion cues while an auxiliary coverage trail records the accumulated exploration effort. UUV motion is guided by the combined gradients of these two fields, enabling more consistent behavior across depth layers and mitigating ambiguities caused by overlapping pheromone sources. At the macroscopic level, swarm evolution is modeled by a coupled system of partial differential equations (PDEs) describing vehicle density, pheromone concentration, and coverage trail. A Lyapunov functional is constructed to derive sufficient conditions under which perturbations around the uniform coverage equilibrium decay exponentially. Numerical simulations in three-dimensional underwater domains demonstrate that the proposed framework reduces coverage holes, limits redundant overlap, and improves robustness with respect to a single-pheromone baseline and a potential-field-based controller. These results indicate that dual-field stigmergic control is a promising and scalable approach for UUV coverage in constrained underwater environments.

1. Introduction

The ocean covers more than two thirds of the Earth’s surface and supports critical activities related to climate regulation, food resources, and subsea infrastructure [1]. As offshore wind farms, pipelines, communication cables, and marine monitoring networks expand, the demand for reliable underwater inspection, environmental observation, and search operations continues to grow [2]. Unmanned underwater vehicles (UUVs) have therefore become indispensable tools, especially as missions extend over larger areas where single-vehicle operations often fall short in terms of endurance, coverage, and reliability [3,4]. Swarm robotics offers a promising paradigm in this context. Deploying multiple simple robots rather than a single complex platform provides benefits such as scalability, redundancy, and adaptability [5]. Recent reviews highlight underwater swarm robotics as an emerging approach with strong potential for large area exploration, cooperative sensing, and distributed environmental monitoring [6,7]. However, underwater swarms face significant constraints: communication bandwidth is extremely limited, global positioning systems are unavailable, and sensing remains noisy and range restricted [8,9]. These factors challenge reliable coordination, consistent coverage, and collective decision-making in three-dimensional underwater environments. Complementary progress has been made in multi-vehicle coverage and path planning. Studies have explored multi-AUV coverage planning in 3D spaces, hybrid optimization approaches for cooperative navigation, and aerial multi-UAV coverage algorithms that address environmental complexity [10]. Surveys on area coverage further consolidate techniques such as grid-based decomposition, frontier exploration, and learning-based coverage [11]. Yet these methods often assume accurate localization, abundant communication, or predominantly planar environments, which limit their applicability to realistic underwater conditions. Biological systems offer an alternative perspective through stigmergy—indirect coordination mediated by modifications in the environment. Examples include pheromone trails in insects, chemical markers in microbes, and hydrodynamic or chemical cues used by marine organisms [12,13]. Such mechanisms inspire artificial pheromone and stigmergic coordination methods in swarm robotics [14,15]. Physical or virtual pheromone systems based on visual markers, magnetic materials, or projected fields have demonstrated effective coordination in various robotic platforms [16]. Pheromone-inspired strategies have also been explored for exploration, coverage, and target search, showing advantages in distributed environments that lack high-bandwidth communication [17,18]. Despite their promise, conventional single-pheromone-field approaches face limitations in underwater coverage tasks. A single field must simultaneously encode local motion cues and global coverage status; consequently, robots may cluster in already explored zones or overlook coverage gaps, particularly when many agents contribute to the same field [19,20]. Depth-dependent sensing further leads to inconsistent perceptions across layers, and most stigmergic methods lack formal convergence guarantees for uniform coverage [21,22]. Recent work on dispersion and multi-layer coordination suggests that combining multiple fields or biological metaphors can improve distribution performance, but such methods have rarely been adapted specifically for underwater swarm coverage [23,24]. This study focuses on a central question: How can a swarm of UUVs achieve robust, near-uniform coverage of a three-dimensional underwater region using only local sensing and minimal communication, while overcoming the intrinsic limitations of single-field pheromone models. To address this problem, we propose a dual-trail stigmergic coordination framework. In this framework, each UUV senses two distinct scalar fields: a virtual pheromone field representing recent activity and motion cues, and a coverage trail field encoding the spatial distribution of exploration effort. Robot motion is guided by the combined gradients of these fields, enabling the swarm to avoid overcrowded regions and prioritize under-explored areas even under limited sensing conditions. At the macroscopic scale, a coupled system of partial differential equations (PDEs) is used to describe swarm evolution, including vehicle density, pheromone concentration, and the coverage trail. Drawing on advances in stigmergy modeling and swarm dispersion theory, a Lyapunov functional is constructed to derive sufficient conditions for convergence toward a uniform coverage state. Numerical simulations in three-dimensional underwater environments demonstrate that the proposed framework reduces coverage holes, limits redundant overlap, and improves robustness relative to single-pheromone and potential-field baselines. Integrating insights from ocean engineering, swarm robotics, and dynamical systems, this work aims to provide a scalable and theoretically grounded coordination mechanism suitable for deployment in real underwater missions, including environmental monitoring, infrastructure inspection, and search-and-rescue operations.

The remainder of this paper is organized as follows: Section 2 formulates the mathematical model governing the evolution of agent density, pheromone concentration, and coverage memory. Section 3 presents a Lyapunov-based stability analysis to demonstrate convergence toward uniform coverage, providing a theoretical guarantee of system performance. Section 4 validates the model through numerical simulations and evaluates its resilience to environmental perturbations. Finally, Section 5 summarizes the findings and discusses implications for practical deployment in resource-constrained underwater environments.

2. Problem Formulation

This section develops the mathematical description of the dual-trail stigmergic coverage model for an underwater swarm. The objective is to characterize the macroscopic evolution of vehicle density and the two environmental traces that govern local motion and long-term coverage balance.

Let $\Omega \subset {\mathbb{R}}^3$ denote a bounded operation domain with a piecewise smooth boundary $\partial \Omega$. The spatial variable $x\in \Omega$ represents a fixed Eulerian coordinate, independent of time t. The collective state of the swarm is modeled by a scalar density field $\rho (x,t):\Omega \times [0,\infty )\to {\mathbb{R}}_{\ge 0}$. Regarding the stability analysis, while logarithmic entropies of the form $V\sim \int \rho ln\rho$ are often cited for naturally enforcing density positivity, we employ a quadratic Lyapunov functional $V\sim \int {(\rho -{\rho }^{\ast })}^{2}dx$. This quadratic form characterizes the $L^2$ norm of the perturbation, thereby providing a physically rigorous measure of the spatial variance or the energy of inhomogeneity relative to the equilibrium state ${\rho }^{\ast }$.

Two environmental fields mediate stigmergic coordination:

• Virtual pheromone field $p(x,t)$, describing short-term activity cues deposited during motion.
• Coverage trail $c(x,t)$, representing accumulated exploration over longer horizons.

Both traces are continuously updated by the swarm and influence the drift component of the density dynamics.

2.1. Macroscopic Density Dynamics

Following standard continuum-swarm modeling, the density field satisfies the conservation law

$${\partial }_t\rho (x,t)+\nabla ·\left(\rho (x,t)\mathbf{u}(x,t)\right)=\nabla ·(D_{\rho }\nabla \rho (x,t)),$$

(1)

where $D_{\rho }>0$ denotes effective diffusion arising from sensing uncertainty and local disturbances.

The drift velocity is determined by the gradients of the two trails:

$$\mathbf{u}(x,t)=-{\chi }_p\nabla p(x,t)-{\chi }_c\nabla c(x,t),$$

(2)

where ${\chi }_p>0$ and ${\chi }_c>0$ quantify the sensitivity to virtual pheromone and coverage trail, respectively.

2.2. Virtual Pheromone Model

The virtual pheromone is generated proportionally to local swarm density and decays at a fast rate. Its evolution is described by the reaction-diffusion equation

$${\partial }_{t}p(x,t)=D_p\Delta p(x,t)-{\lambda }_{p}p(x,t)+{\alpha }_p\rho (x,t),$$

(3)

where $D_p>0$ is the diffusion coefficient, ${\lambda }_p>0$ the evaporation rate, and ${\alpha }_p>0$ the deposition gain.

2.3. Coverage-Trail Model

The long-term coverage trail evolves according to

$${\partial }_{t}c(x,t)=D_c\Delta c(x,t)-{\lambda }_{c}c(x,t)+{\alpha }_c\rho (x,t),$$

(4)

where $D_c\ll D_p$ and ${\lambda }_c\ll {\lambda }_p$, reflecting the slower time-scale associated with accumulated spatial usage.

2.4. Coupled Dual-Trail System

Equations (1), (3) and (4) yield the complete dual-trail stigmergic PDE model:

$$\begin{array}{cc}\hfill {\partial }_t\rho & =\nabla ·(D_{\rho }\nabla \rho +{\chi }_p\rho \nabla p+{\chi }_c\rho \nabla c),\hfill \\ \hfill {\partial }_{t}p& =D_p\Delta p-{\lambda }_{p}p+{\alpha }_p\rho ,\hfill \\ \hfill {\partial }_{t}c& =D_c\Delta c-{\lambda }_{c}c+{\alpha }_c\rho .\hfill \end{array}$$

(5)

No-flux boundary conditions

$$\nabla \rho ·\mathbf{n}=\nabla p·\mathbf{n}=\nabla c·\mathbf{n}=0\mathrm{on}\partial \Omega$$

(6)

ensure mass conservation and prevent artificial inflow or outflow of the two traces. Initial conditions are given by

$$\rho (x,0)={\rho }_0\left(x\right),p(x,0)=p_0\left(x\right),c(x,0)=c_0\left(x\right).$$

(7)

This coupled structure provides the basis for analyzing convergence toward uniform coverage. In particular, it allows us to study how perturbations of the density and trail fields around a spatially uniform equilibrium evolve in time, which is formalized by a Lyapunov-based stability analysis in the next section.

3. Lyapunov Stability Analysis

The macroscopic formulation adopted in this work follows a standard continuum approximation of large-scale swarms. Under this representation, individual UUV motions are abstracted into density fluxes driven by local sensory cues. While this abstraction omits discrete vehicle dynamics, it allows us to analyze global spatial redistribution patterns through partial differential equations (PDEs), which are more amenable to stability analysis. Similar continuum models have proven effective in capturing collective behaviors in chemotaxis, aggregation, and agent-based swarm systems.

The dual-trail system introduces two distinct environmental traces with different time scales. The virtual pheromone field p(x,t) represents immediate activity and decays rapidly, while the coverage trail c(x,t) evolves slowly to reflect accumulated spatial usage. The separation of time scales is not merely a modeling convenience; it is intended to mirror operational constraints of UUV sensing, in which short-term signals can be estimated more accurately than their long-term counterparts. These differing decay and diffusion parameters play a critical role in shaping the qualitative behavior of the swarm.

3.1. Uniform Equilibrium and Perturbation Variables

We are interested in spatially uniform coverage states. A time-independent, spatially constant triple $({\rho }^{\ast },p^{\ast },c^{\ast })$ is an equilibrium if

$$0=0,0=-{\lambda }_p{p}^{\ast }+{\alpha }_p{\rho }^{\ast },0=-{\lambda }_c{c}^{\ast }+{\alpha }_c{\rho }^{\ast }.$$

(8)

The first equation is trivially satisfied by any constant ${\rho }^{\ast }$; consistency with total mass conservation fixes ${\rho }^{\ast }=M/|\Omega |$. The remaining two equations then give $p^{\ast }={\displaystyle \frac{{\alpha }_p}{{\lambda }_p}}{\rho }^{\ast }$ and $c^{\ast }={\displaystyle \frac{{\alpha }_c}{{\lambda }_c}}{\rho }^{\ast }$.

To study stability we introduce deviations

$$r(x,t)=\rho (x,t)-{\rho }^{\ast },s(x,t)=p(x,t)-p^{\ast },q(x,t)=c(x,t)-c^{\ast }.$$

(9)

Mass conservation implies

$${\int }_{\Omega }r(x,t)dx={\int }_{\Omega }\rho (x,t)dx-{\int }_{\Omega }{\rho }^{\ast }dx=M-M=0,\forall t\ge 0,$$

(10)

so the density deviation has zero spatial mean at all times.

Substituting $\rho ={\rho }^{\ast }+r$, $p=p^{\ast }+s$, $c=c^{\ast }+q$ into the governing equations, we obtain the perturbation system:

$$\begin{array}{cc}\hfill {\partial }_{t}r& =\nabla ·(D_{\rho }\nabla r+{\chi }_p({\rho }^{\ast }+r)\nabla s+{\chi }_c({\rho }^{\ast }+r)\nabla q),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\partial }_{t}s& =D_p\Delta s-{\lambda }_{p}s+{\alpha }_{p}r,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\partial }_{t}q& =D_c\Delta q-{\lambda }_{c}q+{\alpha }_{c}r,\hfill \end{array}$$

(13)

with boundary conditions

$$(D_{\rho }\nabla r+{\chi }_p({\rho }^{\ast }+r)\nabla s+{\chi }_c({\rho }^{\ast }+r)\nabla q)·\mathbf{n}=0,\nabla s·\mathbf{n}=0,\nabla q·\mathbf{n}=0.$$

(14)

We work in the standard Hilbert spaces $L^2(\Omega )$ and $H^1(\Omega )$, with ${\parallel u\parallel }_2^2={\int }_{\Omega }{\left|u\right|}^{2}dx$ and ${\parallel \nabla u\parallel }_2^2={\int }_{\Omega }{|\nabla u|}^{2}dx$.

3.2. Lyapunov Functional and Energy Identity

We introduce a quadratic Lyapunov functional

$$V\left(t\right)={\displaystyle \frac{1}{2}}{\int }_{\Omega }(a_r{r}^2+a_s{s}^2+a_q{q}^2)dx,$$

(15)

Remark 1

(Justification of the Quadratic Form). The Lyapunov functional $V\left(t\right)$ defined in Equation (15) is constructed as a weighted $L^2(\Omega )$ norm of the perturbation state vector $(r,s,q)$. In the context of coverage control, the term ${\int }_{\Omega }{r}^{2}dx$ physically represents the spatial variance of the swarm density; minimizing this quantity is equivalent to maximizing the uniformity of coverage near the equilibrium. While logarithmic functionals are often employed in biological chemotaxis models to ensure global positivity, the quadratic form is rigorously sufficient for analyzing local exponential stability. It effectively linearizes the “energy” of the error dynamics, allowing us to quantify the competition between the stabilizing influence of diffusion $(D_{\rho },D_p,D_c)$ and the destabilizing influence of stigmergic advection (${\chi }_p,{\chi }_c$). The weights $a_r,a_s,a_q$ act as scaling factors to symmetrize the coupling contributions, ensuring that the energy dissipation rate is strictly positive definite. with fixed weights $a_r,a_s,a_q>0$. This functional is strictly positive for any non-trivial perturbation and vanishes only at the equilibrium. Differentiating V along solutions yields:

$$\dot{V}\left(t\right)={\int }_{\Omega }(a_{r}r{\partial }_{t}r+a_{s}s{\partial }_{t}s+a_{q}q{\partial }_{t}q)dx.$$

(16)

3.3. Density Contribution

Using the perturbation equation for r:

$${\int }_{\Omega }{a}_{r}r{\partial }_{t}rdx=a_r{\int }_{\Omega }r\nabla ·(D_{\rho }\nabla r+{\chi }_p({\rho }^{\ast }+r)\nabla s+{\chi }_c({\rho }^{\ast }+r)\nabla q)dx.$$

(17)

Integrating by parts and using Neumann boundary conditions, the diffusion term becomes:

$$-a_r{D}_{\rho }{\int }_{\Omega }{|\nabla r|}^{2}dx=-a_r{D}_{\rho }{\parallel \nabla r\parallel }^2.$$

(18)

The cross-gradient terms are:

$$-a_r{\chi }_p{\int }_{\Omega }({\rho }^{\ast }+r)\nabla r·\nabla sdx,-a_r{\chi }_c{\int }_{\Omega }({\rho }^{\ast }+r)\nabla r·\nabla qdx.$$

(19)

3.4. Pheromone and Coverage Contributions

From the equation for s:

$${\int }_{\Omega }{a}_{s}s{\partial }_{t}sdx=a_s{\int }_{\Omega }s(D_p\Delta s-{\lambda }_{p}s+{\alpha }_{p}r)dx.$$

(20)

Integration by parts gives $-a_s{D}_p{\parallel \nabla s\parallel }^2-a_s{\lambda }_p{\parallel s\parallel }^2+a_s{\alpha }_p{\int }_{\Omega }srdx$. Similarly, for q:

$${\int }_{\Omega }{a}_{q}q{\partial }_{t}qdx=-a_q{D}_c{\parallel \nabla q\parallel }^2-a_q{\lambda }_c{\parallel q\parallel }^2+a_q{\alpha }_c{\int }_{\Omega }qrdx.$$

(21)

3.5. Combined Energy Balance

Collecting terms, we obtain

$$\dot{V}=-a_r{D}_{\rho }{\parallel \nabla r\parallel }^2-a_s{D}_p{\parallel \nabla s\parallel }^2-a_q{D}_c{\parallel \nabla q\parallel }^2-a_s{\lambda }_p{\parallel s\parallel }^2-a_q{\lambda }_c{\parallel q\parallel }^2+C,$$

(22)

where the coupling contribution is

$$C=-a_r{\chi }_p{\int }_{\Omega }({\rho }^{\ast }+r)\nabla r·\nabla sdx-a_r{\chi }_c{\int }_{\Omega }({\rho }^{\ast }+r)\nabla r·\nabla qdx+a_s{\alpha }_p{\int }_{\Omega }srdx+a_q{\alpha }_c{\int }_{\Omega }qrdx.$$

(23)

3.6. Control of Nonlinear Couplings

We assume that the perturbation remains small. Specifically, suppose there exists $\delta >0$ such that if initial deviations are small, ${\parallel r(·,t)\parallel }_{L^{\infty }}\le {\rho }^{\ast }/2$ for all $t\ge 0$. Under this assumption, ${\rho }^{\ast }+r(x,t)\le {\displaystyle \frac{3}{2}}{\rho }^{\ast }:=C_{\rho }$.

The stability analysis presumes that the initial density deviation $r(\mathbf{x},0)$ satisfies ${\parallel r(·,0)\parallel }_{L^{\infty }}<ϵ{\rho }^{\ast }$, where $ϵ\in (0,1)$ is a bound ensuring that the linearization of the advective coupling terms remains valid. Physically, this implies that the initial swarm deployment must possess a non-zero spatial spread rather than a singularity. This condition prevents the chemotactic collapse phenomena often observed in classical Keller-Segel models and ensures that the repulsive coverage force is sufficient to disperse the agents before irreversible aggregation occurs.

3.7. Gradient–Gradient Couplings

Using Cauchy–Schwarz and Young’s inequality (for any ${\epsilon }_1>0$):

$$a_r{\chi }_p{\int }_{\Omega }({\rho }^{\ast }+r)\nabla r·\nabla sdx\le a_r{\chi }_p{C}_{\rho }\left({\displaystyle \frac{{\epsilon }_1}{2}}{\parallel \nabla r\parallel }^2+{\displaystyle \frac{1}{2{\epsilon }_1}}{\parallel \nabla s\parallel }^2\right).$$

(24)

Similarly for the term involving $\nabla r·\nabla q$ with ${\epsilon }_2>0$:

$$a_r{\chi }_c{\int }_{\Omega }({\rho }^{\ast }+r)\nabla r·\nabla qdx\le a_r{\chi }_c{C}_{\rho }\left({\displaystyle \frac{{\epsilon }_2}{2}}{\parallel \nabla r\parallel }^2+{\displaystyle \frac{1}{2{\epsilon }_2}}{\parallel \nabla q\parallel }^2\right).$$

(25)

3.8. Zero-Order Couplings

For the mixed terms, using Young’s inequality with ${\epsilon }_3,{\epsilon }_4>0$:

$$\begin{array}{cc}\hfill a_s{\alpha }_p{\int }_{\Omega }srdx& \le a_s{\alpha }_p\left({\displaystyle \frac{{\epsilon }_3}{2}}{\parallel r\parallel }^2+{\displaystyle \frac{1}{2{\epsilon }_3}}{\parallel s\parallel }^2\right),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill a_q{\alpha }_c{\int }_{\Omega }qrdx& \le a_q{\alpha }_c\left({\displaystyle \frac{{\epsilon }_4}{2}}{\parallel r\parallel }^2+{\displaystyle \frac{1}{2{\epsilon }_4}}{\parallel q\parallel }^2\right).\hfill \end{array}$$

(27)

Combining these bounds into C, we find constants ${\theta }_i,\beta$ such that:

$$C\le {\theta }_1{\parallel \nabla r\parallel }^2+{\theta }_2{\parallel \nabla s\parallel }^2+{\theta }_3{\parallel \nabla q\parallel }^2+{\theta }_4{\parallel s\parallel }^2+{\theta }_5{\parallel q\parallel }^2+\beta {\parallel r\parallel }^2.$$

(28)

Substituting back into $\dot{V}$:

$$\dot{V}\le -(a_r{D}_{\rho }-{\theta }_1){\parallel \nabla r\parallel }^2-(a_s{D}_p-{\theta }_2){\parallel \nabla s\parallel }^2-\cdots +\beta {\parallel r\parallel }^2.$$

(29)

By selecting weights and ${\epsilon }_i$ appropriately, we define positive constants ${\gamma }_i$ such that:

$$\dot{V}\le -{\gamma }_1{\parallel \nabla r\parallel }^2-{\gamma }_2{\parallel \nabla s\parallel }^2-{\gamma }_3{\parallel \nabla q\parallel }^2-{\gamma }_4{\parallel s\parallel }^2-{\gamma }_5{\parallel q\parallel }^2+\beta {\parallel r\parallel }^2.$$

(30)

3.9. Poincaré Inequality and Exponential Decay

Since ${\int }_{\Omega }rdx=0$, we use the Poincaré inequality ${\parallel r\parallel }^2\le {\displaystyle \frac{1}{{\lambda }_1}}{\parallel \nabla r\parallel }^2$, where ${\lambda }_1>0$ is the first Neumann eigenvalue. Substituting this gives:

$$\dot{V}\le -\left({\gamma }_1-{\displaystyle \frac{\beta }{{\lambda }_1}}\right){\parallel \nabla r\parallel }^2-{\gamma }_2{\parallel \nabla s\parallel }^2-{\gamma }_3{\parallel \nabla q\parallel }^2-{\gamma }_4{\parallel s\parallel }^2-{\gamma }_5{\parallel q\parallel }^2.$$

(31)

If parameters satisfy ${\gamma }_1>\beta /{\lambda }_1$, there exists $\kappa >0$ such that $\dot{V}\left(t\right)\le -{\kappa (\parallel \nabla r\parallel }^2+{\parallel \nabla s\parallel }^2+{\parallel \nabla q\parallel }^2+{\parallel s\parallel }^2+{\parallel q\parallel }^2)$.

Using the norm equivalence $m_1{(\parallel r\parallel }^2+{\parallel s\parallel }^2+{\parallel q\parallel }^2)\le V\left(t\right)\le m_2{(\parallel r\parallel }^2+{\parallel s\parallel }^2+{\parallel q\parallel }^2)$ and Poincaré’s inequality again, there exists $\mu >0$ such that

$$\dot{V}\left(t\right)\le -\mu V\left(t\right).$$

(32)

Integrating yields $V\left(t\right)\le V\left(0\right)e^{-\mu t}$. Thus, the coupled dual-trail system converges exponentially to the uniform coverage equilibrium under small perturbations.

4. Numerical Simulations and Analysis

To evaluate the proposed dual-trail stigmergic framework under realistic underwater conditions, we performed numerical simulations of the coupled PDE system in a three-dimensional domain. The setup mirrors typical inspection scenarios, probing recovery from biased initial conditions, resilience to unsteady flows, and dynamic obstacle handling compared to a single-pheromone baseline where the coverage trail is deactivated ($D_c={\chi }_c={\alpha }_c=0$). A comprehensive list of model parameters, chosen to satisfy theoretical stability constraints, is provided in

Table 1. Simulation Parameters and Physical Interpretation.

Parameter	Symbol	Value	Unit	Physical Interpretation
Domain Size	$\Omega$	$40\times 40\times 20$	m	Operational volume for underwater inspection
Grid Resolution	$\Delta h$	0.5	m	Spatial granularity of the virtual fields
Time Step	$\Delta t$	0.02	s	Integration step (satisfying CFL condition)
Diffusion (Density)	$D_{\rho }$	0.12	${\mathrm{m}}^2/\mathrm{s}$	Effective swarm diffusivity / sensor noise
Diffusion (Pheromone)	$D_p$	0.20	${\mathrm{m}}^2/\mathrm{s}$	Virtual diffusion rate of pheromone packets
Diffusion (Coverage)	$D_c$	0.10	${\mathrm{m}}^2/\mathrm{s}$	Virtual diffusion of coverage memory
Decay Rate (Pheromone)	${\lambda }_p$	0.6	${\mathrm{s}}^{-1}$	Evaporation rate of short-term cues
Decay Rate (Coverage)	${\lambda }_c$	0.25	${\mathrm{s}}^{-1}$	Fading rate of long-term coverage memory
Sensitivity (Pheromone)	${\chi }_p$	0.30	${\mathrm{m}}^2/\mathrm{s}$	Response strength to aggregation cues
Sensitivity (Coverage)	${\chi }_c$	0.45	${\mathrm{m}}^2/\mathrm{s}$	Repulsion strength from visited areas

. The computational domain $\Omega =[0,40]\times [0,40]\times [0,20]$ m is discretized with a spatial step $h=0.5$ m. The system is evolved for $T_{max}=200$ s with a time step $\Delta t=0.02$ s, satisfying CFL conditions for the diffusion-taxis dynamics.

4.1. Initial Condition Relaxation Dynamics

To solve the coupled system of nonlinear PDEs (Equation (5)), we employ a finite difference method on a uniform staggered Cartesian grid. Time integration is performed using a first-order explicit Euler scheme, while spatial derivatives are discretized as follows:

• Diffusion Terms ($D\Delta ·$): Approximated using second-order central differences.
• Advection Terms ($\nabla ·\left(\rho \mathbf{u}\right)$): Discretized using a first-order Upwind Scheme to ensure numerical stability and preserve the non-negativity of the density function $\rho$. The upwind direction is determined dynamically based on the sign of the local drift velocity vector $u=-{\chi }_p\nabla p-{\chi }_c\nabla c$.

The discrete update rule for the density at grid point $(i,j,k)$ at time step n is:

$${\displaystyle \frac{{\rho }_{i,j,k}^{n+1}-{\rho }_{i,j,k}^n}{\Delta t}}=D_{\rho }\mathcal{L}{\left({\rho }^n\right)}_{i,j,k}-\left({\displaystyle \frac{{\mathcal{F}}_{i+1/2}^x-{\mathcal{F}}_{i-1/2}^x}{h}}+{\displaystyle \frac{{\mathcal{F}}_{j+1/2}^y-{\mathcal{F}}_{j-1/2}^y}{h}}+{\displaystyle \frac{{\mathcal{F}}_{k+1/2}^z-{\mathcal{F}}_{k-1/2}^z}{h}}\right)$$

(33)

here $\mathcal{L}\left({\rho }^n\right)$ denotes the discrete Laplacian operator, and ${\mathcal{F}}^{x,y,z}$ represent the numerical fluxes in the $x,y,z$ directions calculated via the upwind rule (e.g., ${\mathcal{F}}_{i+1/2}^x={\rho }_i{u}_{i+1/2}$ if $u_{i+1/2}>0$, else ${\rho }_{i+1}{u}_{i+1/2}$).

All simulations were conducted using a fixed pseudo-random number generator seed to ensure reproducibility. The specific model parameters are detailed in Table 1.

Table 1 lists the parameters used in the simulation. Although the governing equations can be non-dimensionalized, we retain physical units (SI) to provide operational context. The diffusion coefficients ($D_{\rho },D_p,D_c$) quantify the “randomness” or exploration tendency of the swarm and the spread of information fields. The decay rates (${\lambda }_p,{\lambda }_c$) effectively set the “memory horizon” of the system; a higher ${\lambda }_p$ implies that pheromone cues vanish quickly, enforcing immediate responsiveness, while a lower ${\lambda }_c$ allows the coverage trail to persist, preventing redundant revisits over long durations. The sensitivity parameters (${\chi }_p,{\chi }_c$) act as control gains, determining the speed at which UUVs react to environmental gradients.

We consider a rectangular underwater region

$$\Omega =[0,L_x]\times [0,L_y]\times [0,L_z]$$

(34)

with sizes $L_x=L_y=40$ m and $L_z=20$ m. The domain is discretized into a uniform grid of $N_x\times N_y\times N_z=80\times 80\times 40$ cubic cells with spacing $h=0.5$ m.

The macroscopic swarm state is described by the UUV density $\rho (x,t)$, virtual pheromone field $p(x,t)$, and coverage trail $c(x,t)$, governed by

$$\begin{array}{cc}\hfill {\partial }_t\rho & =D_{\rho }\Delta \rho +{\chi }_p\nabla ·(\rho \nabla p)+{\chi }_c\nabla ·(\rho \nabla c),\hfill \\ \hfill {\partial }_{t}p& =D_p\Delta p-{\lambda }_{p}p+{\alpha }_p\rho ,\hfill \\ \hfill {\partial }_{t}c& =D_c\Delta c-{\lambda }_{c}c+{\alpha }_c\rho ,\hfill \end{array}$$

(35)

with homogeneous Neumann boundary conditions on all fields.

Unless stated otherwise, we use the following non-dimensionalized parameters, chosen to satisfy the Lyapunov stability conditions derived in Section 3:

$$\begin{array}{cc}\hfill & D_{\rho }=0.12,D_p=0.20,D_c=0.10,\hfill \\ \hfill & {\lambda }_p=0.6,{\lambda }_c=0.25,{\alpha }_p=1.0,{\alpha }_c=0.8,\hfill \\ \hfill & {\chi }_p=0.30,{\chi }_c=0.45.\hfill \end{array}$$

(36)

The time step is $\Delta t=0.02$ s, which satisfies the CFL constraint for the combined diffusion-taxis dynamics. Simulations run for $T_{max}=200$ s, corresponding to ${10}^4$ time steps. The uniform coverage equilibrium is given by

$${\rho }^{\ast }={\displaystyle \frac{M}{|\Omega |}},p^{\ast }={\displaystyle \frac{{\alpha }_p}{{\lambda }_p}}{\rho }^{\ast },c^{\ast }={\displaystyle \frac{{\alpha }_c}{{\lambda }_c}}{\rho }^{\ast },$$

(37)

where M is the total swarm mass. To test convergence, the initial density $\rho (x,0)$ is biased toward one corner of the domain, mimicking the deployment of a UUV group from a single launch point. Initial fields $p(x,0)$ and $c(x,0)$ are set to zero. For comparison, we implement a single-trail baseline by disabling the coverage trail,

$${\alpha }_c=0,{\chi }_c=0,D_c=0,$$

(38)

so that the swarm relies solely on a virtual pheromone field p for motion guidance.

While the stability analysis utilizes a continuum PDE formulation, the practical deployment involves discrete UUVs. The continuum variable $\rho (x,t)$ corresponds to the probability density function (PDF) of agent locations in a stochastic control framework. In a physical implementation, the drift velocity $\mathbf{u}(x,t)=-{\chi }_p\nabla p-{\chi }_c\nabla c$ serves as the reference velocity vector for the discrete agents. The “virtual pheromone” and “coverage trail” are not physical substances but digital information maps updated essentially via acoustic modems or relative sensing. The discrete agents estimate the local gradients $\nabla p$ and $\nabla c$ using finite differences of sensor measurements or data shared with neighbors within communication range $R_{comm}$. The robustness predicted by the PDE analysis manifests in practice as the swarm’s ability to “self-heal” coverage gaps despite individual agent failures or sensor noise.

4.2. Recovery from Biased Initial Deployment

We first consider a static environment without flow or obstacles. The initial density is concentrated in the top-right, shallow region of the domain, with a Gaussian profile in $(x,y)$ and a depth bias toward small z. In the single-trail baseline, the swarm initially spreads along the pheromone gradients, but after a transient phase the density profile develops persistent clusters near the initial launch region. The coverage trail is absent, so regions that are repeatedly visited keep reinforcing the virtual pheromone field, leading to self-induced attraction.

As a result, $\mathcal{I}\left(t\right)$ decreases slowly and saturates at a non-zero level, indicating incomplete homogenization. The redundancy index $\mathcal{R}\left(t\right)$ approaches values close to one, reflecting substantial re-visiting of the same volume elements.

In contrast, the dual-trail system quickly suppresses this bias. The coverage trail $c(x,t)$ accumulates in over-visited areas, generating repulsive gradients that counteract the attractive effect of the virtual pheromone field. As time evolves, the density spreads throughout the full volume, and both horizontal and vertical slices show a nearly uniform swarm distribution.

Quantitatively, $\mathcal{I}\left(t\right)$ exhibits a clear exponential decay over several decades, with an observed decay rate closely matching the theoretical Lyapunov exponent $\mu$. This confirms that the PDE-level stability analysis accurately predicts the macroscopic relaxation behavior. At the same time, the redundancy index $\mathcal{R}\left(t\right)$ stabilizes at values approximately 40–50% lower than in the single-trail baseline, demonstrating that the dual-trail mechanism not only equalizes coverage but also uses fewer repeated passes to do so.

To quantify the macroscopic behavior, we monitor four metrics:

• Coverage uniformity:

  $$\mathcal{I}\left(t\right)={\displaystyle \frac{1}{|\Omega |}}{\int }_{\Omega }{(\rho (x,t)-{\rho }^{\ast })}^{2}dx,$$

  (39)

  which measures deviations from the uniform density.
• Redundancy index:

  $$\mathcal{R}\left(t\right)={\displaystyle \frac{1}{|\Omega |}}{\int }_{\Omega }{\displaystyle \frac{c(x,t)}{{max}_{x}c(x,t)+\epsilon }}dx,$$

  (40)

  with $\epsilon$ a small regularization constant. Larger $\mathcal{R}$ indicates more repeated traversal of already explored regions.
• Coverage completeness:

  $$\mathsf{\Gamma }\left(t\right)={\displaystyle \frac{\left|\right\{x\in \Omega :c(x,t)<c_{th}\left\}\right|}{|\Omega |}},$$

  (41)

  where $c_{th}$ is a minimal coverage threshold. Smaller $\Gamma$ corresponds to fewer coverage holes.
• Relaxation of the Lyapunov energy:

  $$V\left(t\right)={\displaystyle \frac{1}{2}}{\int }_{\Omega }\left(a_r{(\rho -{\rho }^{\ast })}^2+a_s{(p-p^{\ast })}^2+a_q{(c-c^{\ast })}^2\right)dx,$$

  (42)

  which should decay exponentially if the PDE system operates in the theoretically stable regime.

All metrics are evaluated at every time step using discrete sums over the grid.

4.3. Robustness to Hydrodynamic Perturbations

To test robustness, we augment the density equation with a divergence-free advection field representing unsteady underwater current,

$${\partial }_t\rho +\nabla ·\left(\rho \mathbf{u}\right)=D_{\rho }\Delta \rho +{\chi }_p\nabla ·(\rho \nabla p)+{\chi }_c\nabla ·(\rho \nabla c),$$

(43)

where $\mathbf{u}(x,t)$ is a synthetic time-varying flow composed of a slow background drift and local vortical pulses. In addition, we add small-amplitude spatio-temporal noise to the source terms of p and c to mimic sensing and deposition uncertainties.

In the single-trail system, the combined effect of biased pheromone reinforcement and flow perturbations produces large-scale clustering: density accumulates in regions where vortices trap particles, and the lack of a coverage-aware field allows these clusters to persist. The coverage completeness $\mathsf{\Gamma }\left(t\right)$ remains above 10%, indicating substantial persistent holes, and the Lyapunov energy $V\left(t\right)$ no longer decays monotonically, with intermittent re-growth during strong perturbation events.

The dual-trail framework behaves qualitatively differently. Flow perturbations temporarily distort the density field, but the coverage trail accumulates in regions where trajectories linger, making those regions less attractive over time. Once local vortices decay, UUVs are pushed toward under-covered volumes. In all tested parameter sets, $\mathsf{\Gamma }\left(t\right)$ decreases below 2% and remains bounded, even under repeated perturbation bursts. The Lyapunov energy retains an overall exponential trend with small, bounded oscillations, confirming that the system remains within the basin of attraction predicted by the theoretical stability conditions.

4.4. Spatio-Temporal Evolution and Swarm Dynamics

We first examine the macroscopic evolution of the swarm starting from an initially aggregated configuration (Figure 1a and Figure 2a,d). The two control strategies lead to qualitatively different behaviors. Under the single-trail protocol, existing pheromone traces reinforce subsequent agent trajectories, creating a positive feedback loop that amplifies initial density variations. Within 100 s, the swarm develops several high-density clusters (Figure 1b). In the three-dimensional case, these clusters become long-lived and localized by t = 200 s (Figure 2b,c). The resulting distribution exhibits pronounced coverage gaps, indicating that the single-trail approach has limited ability to escape local minima generated by self-attraction. Under the dual-trail framework, the coverage field c(x,t) gradually accumulates in over-visited regions, generating repulsive gradients that counterbalance attraction to the virtual pheromone field. As a result, the swarm disperses throughout the domain and fills previously under-explored regions. By t = 100 s, the density distribution on the horizontal slice is close to uniform (Figure 1c). In the three-dimensional simulations, the dual-trail strategy produces a nearly homogeneous density profile by t = 200 s (Figure 2e,f). These observations are consistent with the intended role of the coverage trail as a long-term balancing mechanism.

4.5. Quantitative Performance and Lyapunov Energy Decay

To quantify performance, we monitor the uniformity index $\mathcal{I}\left(t\right)$, redundancy index $R\left(t\right)$, void fraction $\mathsf{\Gamma }\left(t\right)$, and Lyapunov energy $V\left(t\right)$ defined in Equations (38)–(41).

Figure 3a shows that, for the single-trail approach, the uniformity index $\mathcal{I}\left(t\right)$ decreases initially but then saturates at a relatively high value, indicating incomplete homogenization. In contrast, $\mathcal{I}\left(t\right)$ under the dual-trail framework decays by several orders of magnitude and reaches values on the order of ${10}^{-4}$ by $t=200\mathrm{s}$, corresponding to a substantially more uniform swarm distribution.

The redundancy index $R\left(t\right)$ in Figure 3b highlights a similar trend. The single-trail system develops high redundancy due to repeated traversal of already covered regions. With the dual-trail mechanism, $R\left(t\right)$ converges to a lower steady-state level, approximately $45\%$ smaller than that of the baseline within the tested parameter range.

Figure 3c reports the void fraction $\mathsf{\Gamma }\left(t\right)$. The single-trail system maintains a non-negligible fraction of uncovered volume, whereas $\mathsf{\Gamma }\left(t\right)$ under the dual-trail framework rapidly decreases to values close to zero and remains small thereafter, indicating that coverage holes are substantially reduced.

Finally, Figure 3d compares the evolution of the Lyapunov energy $V\left(t\right)$. In semi-logarithmic coordinates, the curve corresponding to the dual-trail system aligns well with a straight line after an initial transient, consistent with the exponential decay predicted by the Lyapunov analysis. This agreement suggests that the PDE-level stability result provides a useful approximation of the macroscopic relaxation behavior observed in the simulations.

4.6. Recovery from Localized Disturbances

We next evaluate robustness to hydrodynamic disturbances by adding a time-varying, divergence-free flow field $u(x,t)$ to the density equation and injecting small-amplitude noise into the source terms of p and c, as described in Equation (43).

In the single-trail system, the combined effect of biased pheromone reinforcement and flow structures can lead to vortex trapping (Figure 4a): density accumulates in regions where vortices retain agents for extended periods, while other parts of the domain remain under-visited. In these simulations, the void fraction $\mathsf{\Gamma }\left(t\right)$ exhibits substantial fluctuations and does not return to the low values observed in the static-flow case.

The dual-trail framework responds differently. Regions where trajectories linger accumulate a higher coverage-trail level, which makes those regions progressively less attractive. Once local vortices weaken, agents are driven toward under-covered volumes (Figure 4b). Across the tested parameter sets, $\mathsf{\Gamma }\left(t\right)$ remains below approximately $2\%$ after transients, even under repeated perturbation events (Figure 4c). The Lyapunov energy $V\left(t\right)$ continues to exhibit an overall decaying trend with bounded oscillations, suggesting that the system remains within the basin of attraction characterized by the theoretical stability conditions.

We also probe recovery from a localized external “blow” perturbation that temporarily removes agents from part of the domain (Figure 4c). The dual-trail system restores a nearly uniform configuration and returns the void fraction to below $2\%$ within a moderate time interval, illustrating the self-healing capability provided by the coverage-trail memory.

4.7. Gradient Compensation Mechanism for Shadow-Zone Mitigation

To clarify the mechanisms underlying the observed differences, we examine the coverage fields in the vicinity of obstacles. In single-trail models, an obstacle or local repulsive region can create a persistent low-signal “shadow” zone downstream (Figure 5). Because the pheromone field remains weak in this region, subsequent agents receive little guidance to enter it, and coverage voids may persist.

The dual-trail framework is designed to mitigate this effect by exploiting the slower coverage-trail dynamics. As shown in Figure 5b, the coverage field $c(x,t)$ increases in both the vicinity of obstacles and the downstream region where agents repeatedly pass. In the dual-trail controller, this accumulated coverage acts as a repulsive contribution, while neighboring under-covered regions remain relatively more attractive. The resulting balance between repulsion and attraction promotes trajectories that enter and fill the previous shadow zone, leading to a more continuous coverage field at the macroscopic scale.

Three-dimensional simulations indicate that this mechanism reduces long-lived aggregation and coverage voids that arise in the single-trail system, while also lowering steady-state redundancy and improving robustness to flow-induced trapping.

Three-dimensional numerical simulations substantiate that the dual-trail framework fundamentally resolves the irreversible aggregation and coverage voids associated with the shadowing effect in conventional single-pheromone models by introducing an antagonistic active wake-filling mechanism. Quantitative results indicate that the proposed method achieves globally uniform coverage while reducing steady-state redundancy by approximately $45\%$ , exhibiting superior resilience against vortex trapping and robust self-healing capabilities under strong advective flows and sudden external perturbations. Crucially, the linear decay profile of the Lyapunov energy on a semi-logarithmic scale provides rigorous numerical validation of the system’s exponential convergence toward equilibrium, thereby guaranteeing the determinism and robustness of the swarm control strategy in both theoretical and practical contexts.

5. Conclusions

The dual-trail stigmergic framework presented in this study addresses several limitations of single-field coordination strategies for underwater swarm coverage. By distinguishing short-term and long-term components of environmental feedback, the framework reduces self-reinforcing aggregation and promotes balanced spatial redistribution. The accompanying PDE model and Lyapunov analysis provide an analytical viewpoint that complements empirical observations and clarifies the conditions under which uniform coverage can be expected.

The numerical simulations demonstrate that, across the tested scenarios, the dual-trail mechanism improves coverage uniformity, reduces redundant revisits, and maintains performance under moderate hydrodynamic disturbances. While the continuum model abstracts away individual UUV dynamics and sensing uncertainty, its ability to reproduce key qualitative features of the swarm suggests that it may serve as a useful foundation for further algorithmic development.

A critical advantage of the proposed stigmergic framework is its distributed nature. Although the macroscopic behavior is described by global PDEs, individual UUVs do not solve these equations globally. Instead, each agent performs local operations: sensing field values, updating local virtual maps, and computing gradients via neighbor consensus. The computational complexity for each agent is $O\left(M\right)$, where M is the number of local grid cells or neighbors in communication range, independent of the global domain size. For a typical local map resolution of ${10}^3$ cells, the required operations are in the range of mega-FLOPS (floating-point operations per second), which is well within the capability of standard low-power embedded processors found on UUVs (e.g., ARM Cortex-M or Nvidia Jetson series), which typically offer performance in the range of 10-500 GFLOPS.

Several directions remain open. First, bridging the gap between the continuum model and discrete multi-UUV implementations would provide a clearer path toward field deployment. Second, testing the framework under more realistic ocean conditions (e.g., anisotropic diffusion, intermittent communication, or depth-varying currents) would help evaluate its operational robustness. Third, integrating additional mission objectives such as risk-aware navigation or multi-scale adaptive coverage may extend the applicability of the dual-trail approach.

Overall, the results indicate that dual-field stigmergic coordination is a promising direction for underwater swarm coverage, offering both conceptual simplicity and analytical tractability. Future work will focus on “hardware-in-the-loop” simulations and small-scale tank tests with swarm agents to validate the local gradient estimation and obstacle avoidance protocols derived from this continuum model.