Distributed cooperative simultaneous localization and mapping for dense micro-robot swarms: A stigmergic approach with hardware-constrained sensor fusion

Abstract

To address the challenge of deploying dense micro-robot swarms where classical simultaneous localization and mapping (SLAM) methods are computationally infeasible, we propose a hardware-constrained, stigmergic cooperative SLAM framework. Our system enables swarms to map unknown environments in real time, without a central coordinator or high-bandwidth links. Our method introduces five novel components: (i) Stigmergic Counter-Consensus—a bounded, monotone, and bandwidth-frugal consensus rule over occupancy counters; (ii) ATOP-Raycast—an Adaptive Thin-Obstacle-Preserving Bresenham variant with probabilistic endpoint diffusion; (iii) Proximal Delta Encoding of map updates using tilewise run-length and majority masks; (iv) a Budget-Aware extended Kalman filter that codesigns fusion rate and numerical precision with MCU limits; and (v) a Tri-Force Frontier-Cohesion controller yielding emergent exploration while maintaining communication neighborhoods. In real-world validation with 40 robots, the framework achieves a thin-feature retention rate of 92.4% and a final map Intersection-over-Union (IoU) of 0.89. This performance is sustained with a minimal communication overhead of ∼110 bytes per packet, demonstrating near-linear scalability on ESP32-class hardware while preserving critical geometry. We provide algorithmic details, complexity bounds, convergence guarantees, and validate our approach through a comprehensive suite of simulations. Together, these yield near-linear scalability to 40 + robots at 20 Hz on ESP32-class hardware, preserve thin obstacles, and achieve low collision rates with modest communication. We provide algorithmic details, complexity bounds, convergence guarantees, and validate our approach through a comprehensive suite of simulations.

Introduction

Dense cooperative simultaneous localization and mapping (SLAM) becomes most attractive precisely where resources are tight: Dozens of micro-robots with short-range ranging, low-power radios, and MCU-class processors deployed in cluttered interiors with narrow passages and thin obstacles. In this regime, mainstream multi-robot pipelines that rely on camera features and global pose-graph optimization tend to exceed available compute, memory, and bandwidth, while uniform-resolution occupancy grids can erode the very thin structures that determine safe navigation.^1–3 The consequence is a persistent gap between what distributed SLAM achieves on powerful platforms and what is feasible on truly minimal agents typical of swarm robotics. ⁴

Occupancy grids remain a robust representational choice for embedded mapping, but classical realizations accumulate floating-point log-odds through probabilistic ray updates that assume ample arithmetic precision and memory.^3,5,6 Moreover, when free space is repeatedly “carved” along rays, uniform grids tend to thin—and eventually erase—slender objects unless additional modeling is introduced. On MCU-class hardware, the heavy use of floating point and global high-resolution maps is especially problematic, creating tension between geometric fidelity and resource limits. At the systems level, state-of-the-art multi-robot visual SLAM frameworks, such as COVINS and Kimera-Multi, demonstrate impressive accuracy by optimizing large, shared pose graphs,^7,8 and more recent decentralized approaches reduce bandwidth by exchanging compressed features and selected constraints. ⁹ However, these methods still inherit the computational weight of feature extraction, data association, and nonlinear optimization (often powered by engines such as iSAM2 or g2o), making them ill-suited to swarms built from minimal processors and short-range radios.^10,11 A different operating point is therefore needed—one that treats arithmetic, memory, and communication as first-class constraints rather than afterthoughts.

Recent literature from 2023 to 2024 has further crystallized the divide between high-fidelity visual reconstruction and resource-constrained swarm mapping. Lajoie and Beltrame ¹² introduced Swarm-SLAM, a decentralized framework that prioritizes sparse inter-robot loop closures to reduce bandwidth, yet it still assumes floating-point optimization back-ends that may tax micro-controllers. Conversely, the push for “ultra-lightweight” architectures has gained momentum ¹³ ; recent implementations on centimeter-scale platforms demonstrate that discarding non-linear optimization for direct filter-based fusion is essential for scaling to dense clusters. Contemporary surveys confirm that while deep-learning-enhanced pipelines (e.g., neural implicit mapping) are defining the upper bound of accuracy, they remain computationally prohibitive for the milliwatt-class edge devices targeted in this study. ¹⁴

The present study adopts a hardware-first, stigmergic perspective: Cooperation emerges from local, monotone edits to a shared map rather than from explicit global optimization. Concretely, bounded integer occupancy counters are updated via Kümmerle-style discrete raycasting to fit fixed-point arithmetic. ¹⁵ Thin-structure loss is countered by diffusing endpoint evidence to immediate neighbors and refining resolution locally only where edge energy persists. Map changes are encoded as small, proximal deltas and exchanged opportunistically with nearby robots. A merge rule that is idempotent and monotone yields order-independent convergence under standard assumptions for time-varying interaction graphs, aligning with consensus theory in networked multi-agent systems.^16–18 For exploration, a lightweight frontier-seeking policy blended with avoidance and cohesion preserves radio neighborhoods while expanding coverage. ¹⁹

This framing leads to the following research questions. First, can a bounded, integer-only occupancy update—augmented with local endpoint diffusion and adaptive refinement—preserve thin obstacles under realistic noise and limited sensing range? Second, can proximal exchanges of versioned map deltas, merged with a conservative idempotent rule, deliver consistent cooperative maps without centralized coordination? We posit this is achievable provided the proximity graph satisfies B-connectivity—a condition of “connectivity over time.” In layman's terms, this does not require the swarm to form a complete, simultaneous network like a conference call. Instead, it functions like a digital relay race: As long as robots meet and exchange data frequently enough within a bounded time window $T_B$ , information successfully percolates from one end of the swarm to the other, bridging gaps in the instantaneous network topology? Third, can the entire pipeline operate within the compute, memory, and bandwidth budgets of MCU-class hardware while sustaining real-time control frequencies? To investigate these questions, this work introduces an integrated framework for distributed cooperative SLAM tailored to dense micro-robot swarms. The system comprises: A thin-obstacle-preserving raycast update on bounded counters; a stigmergic, idempotent tile-merge with convergence guarantees under time-varying connectivity; a proximal delta encoder that transmits only changed runs; a budget-aware extended Kalman filter (BA-EKF) that codesigns fusion rate and fixed-point precision; and a tri-objective exploration controller that balances frontier pursuit, obstacle avoidance, and cohesion. In simulation and hardware demonstrations, the stack maintains real-time operation on MCU-class processors while retaining thin geometry and achieving seam-free merges, positioning stigmergic, hardware-constrained SLAM as a practical pathway for dense swarms.^1,7,18

Related work

To clarify the contributions of this study and strictly delineate our approach from high-resource paradigms, we divide the related work into foundational occupancy mapping techniques and the emerging body of research on hardware-constrained swarm coordination.

Foundations of occupancy mapping and consensus

Classical range-based mapping relies on occupancy grids as a compact, robust representation for fusing uncertain measurements.^3,5 In their standard form, grids are updated by floating-point log-odds integration with Bayesian inverse sensor models, a design that suits desktop-class processors but stresses microcontrollers due to precision, memory, and latency demands. A second drawback, independent of hardware, is geometric: Repeated free-space “carving” along rays tends to erode thin structures when resolution is uniform and endpoints are noisy. To reconcile fidelity with embedded constraints, the present work returns to discrete, integer-arithmetic raycasting ¹⁶ and redesigns the update around bounded integer counters. This preserves real-time execution on MCU-class hardware while curbing unbounded evidence growth that can destabilize fusion and degrade map usability. ⁵ In parallel, local endpoint diffusion and on-demand refinement address the fragility of slender obstacles without resorting to globally finer grids. In multi-robot SLAM, visual–inertial systems have set the benchmark for accuracy by exchanging inter-robot constraints and optimizing a collaborative pose graph. Centralized or server-assisted frameworks such as COVINS aggregate contributions and perform global optimization on a powerful back-end. Fully distributed systems such as Kimera-Multi perform inter-robot place recognition and decentralized pose-graph optimization on board, with robust outlier rejection and dense metric-semantic reconstructions. ²⁰ Bandwidth-aware pipelines move in a complementary direction: Data-efficient decentralized visual SLAM transmits compact descriptors and selected constraints to reduce communication, ⁹ and DOOR-SLAM adds distributed outlier rejection to stabilize peer-to-peer operation. ¹⁰ Despite their strengths, these methods inherit the computational footprint of feature extraction, place recognition, and nonlinear optimization—assumptions that clash with the kilobytes-and-milliwatts envelope typical of dense micro-robot swarms. The operating point pursued here is therefore different: Agents carry only short-range ranging and low-power radios; cooperation is built around compact occupancy deltas and conservative merges, not global optimization.

Hardware-constrained and stigmergic swarm SLAM

The communication philosophy behind this design is stigmergic. Work on virtual/digital pheromones showed that swarms can coordinate by emitting and sensing locally diffused “marks,” achieving group-level structure without centralized arbitration. ²⁰ Broader theories of stigmergic self-organization in swarm intelligence suggest that such local, monotone edits to a shared medium can yield robust, scalable behavior.^21,22 We transpose this logic to SLAM: Occupancy updates are the marks; transmissions are proximal (neighbor only); and merges are idempotent and monotone, resisting oscillations under lossy, asynchronous contact. Exploration policy must reinforce, not undermine, agreement. Frontier-based exploration remains an effective way to convert partial grids into motion by steering toward the boundary between known-free and unknown space, ²³ but in dense teams pure frontier pursuit can fragment the communication graph and elevate collision risk. Blending frontier seeking with obstacle avoidance and cohesion keeps loose neighborhoods intact, improving information mixing while maintaining coverage efficiency. ²⁴ The convergence lens comes from distributed consensus on time-varying graphs. Foundational results show that local, order-independent updates converge when the interaction graph is repeatedly connected (B-connectivity) and disturbances are bounded.^17,18 By designing the merge as a bounded, monotone operation over integer counters, the proposed method inherits these guarantees while staying within MCU limits. Even bandwidth-aware variants like DOOR-SLAM or Swarm-SLAM, ¹² which filter redundant inter-robot constraints, typically rely on floating-point back-ends that exceed MCU capabilities. Recent surveys and analyses on switching-graph consensus further contextualize these assumptions and their robustness to link variability and delays.^25–27

Addressing strict hardware constraints requires a fundamental shift away from global optimization. Pires et al. ²⁸ demonstrated cooperative localization for resource-constrained swarms by prioritizing relative range measurements, though their focus was primarily on state estimation rather than dense environmental mapping. Chaves et al. ²⁹ introduced SwarMap, a crowd-sourced approach that aggregates partial occupancy maps from low-cost agents. However, SwarMap relies on an efficient but centralized server structure to fuse contributions, whereas our approach is fully decentralized. Most recently, “ultra-lightweight” paradigms have further validated that discarding non-linear optimization in favor of direct filter-based fusion is essential for scaling to dense clusters ( $N>40$ ). ³⁰

Consider a swarm of N micro-robots indexed by $i\in \{1,\dots ,N\}$ operating in a planar environment. Each robot maintains a pose estimate $x_i(t)\in SE(2)$ via a fixed-point BA EKF and a local occupancy grid ${\mathcal{M}}_i(t)$ consisting of bounded integer counters $c_{i,k}\in [-C_{max},C_{max}]$ . Sensing integrates B short-range IR beams using the Adaptive Thin-Obstacle-Preservation (ATOP) discrete ray-march, where endpoints diffuse quantized support to 8-neighbors to preserve thin structures, and grid resolution adapts via local $2\times$ refinement. Communication relies on opportunistic broadcasts forming a time-varying proximity graph $\mathcal{G}(t)$ , assumed to be B-connected over a horizon $T_B$ . Messages carry proximal delta encoding (PDE) packets with versioned, run-length encoded deltas, which are fused by the stigmergic counter-consensus (SCC) using a monotone, idempotent merge rule to ensure consistency. Exploration is governed by a Tri-Force Frontier-Cohesion (TFC) controller that blends frontier attraction, obstacle repulsion, and cohesion. The system design maximizes map coverage (IoU) and inter-robot agreement while minimizing communication cost and collisions, subject to a strict $\le 50$  ms per-frame compute budget on ESP32-class hardware.

To clearly situate the proposed framework within the broader landscape of multi-robot mapping, Table 1 provides a comparative analysis of our method against established centralized and distributed SLAM paradigms. While visual-inertial frameworks such as COVINS and Kimera-Multi set the benchmark for metric accuracy, they fundamentally rely on floating-point non-linear optimization (e.g., pose graphs) and high-bandwidth feature exchange, requirements that exceed the computational and energy budgets of dense micro-robot swarms. Conversely, classical occupancy grid approaches are computationally lighter but prone to eroding thin geometric structures due to aliasing and repeated free-space carving. The proposed method occupies a distinct operating point: By enforcing fixed-point integer arithmetic, employing stigmergic, proximal communication, and integrating the ATOP thin-obstacle preservation mechanism, it achieves scalable, consistent mapping on MCU-class hardware where traditional pipelines are infeasible.

Table 1.

Comparison of related work vs. proposed framework.

Dimension	Centralized Visual SLAM	Distributed Visual Simultaneous Localization and Mapping (SLAM)	Classical Occupancy Grids	Proposed Method
Compute paradigm	Global non-linear optimization (Server-based)	Distributed pose-graph optimization (High-end onboard)	Floating-point log-odds integration	Fixed-point integer arithmetic
Communication	High bandwidth (Feature clouds/Constraints)	Medium bandwidth (Compressed descriptors)	N/A	Minimal bandwidth (Proximal, versioned tile deltas)
Connectivity	Star topology (Vulnerable to node failure)	Mesh (Requires reliable links for outliers)	N/A	Ad hoc/DTN (Requires only B-connectivity)
Map representation	Sparse feature map/Global pose graph	Metric-semantic mesh/Pose graph	Uniform high-res grid (Float)	Adaptive thin-obstacle preservation (ATOP) integer grid with local refinement
Thin obstacle handling	Dependent on dense reconstruction layer	Dependent on mesh generation resolution	Prone to erosion/aliasing	Explicit preservation via digital diffusion

To clearly position our five primary contributions, Table 2 provides a comparative analysis against the closest contemporary swarm mapping frameworks—namely DOOR-SLAM, Swarm-SLAM, SwarMap, and recent ultra-lightweight filter-based methods. Although these state-of-the-art approaches have successfully mitigated bandwidth constraints through compressed feature exchange or prioritized loop closures, they typically remain reliant on floating-point optimization back-ends or centralized map aggregation that exceed the strict power and memory budgets of MCU-class micro-robots. As detailed in Table 2, the proposed framework is uniquely distinguished by its strict adherence to bounded integer arithmetic and stigmergic coordination across all five core modules. By codesigning the entire pipeline—from the ATOP thin-obstacle preservation to the TFC cohesive exploration policy—for hardware-constrained edge devices, this approach enables dense swarms to achieve global map consistency without any off-board processing or non-linear pose-graph optimization.

Table 2.

Mapping of claimed architectural components against state-of-the-art decentralized and swarm SLAM frameworks.

Related work	Architecture	Validation	ATOP	BA-EKF	PDE	SCC	TFC
DOOR-SLAM 10	Distributed	Sim & real	Not addressed (Relies on float/mesh)	Not addressed (Non-linear back-end)	Partially addressed (Data-efficient constraints)	Not addressed (Pose-graph rejection)	Not addressed
Swarm-SLAM 12	Decentralized	Sim & real	Not addressed	Not addressed (Floating-point optimization)	Partially addressed (Sparse loop closures)	Not addressed	Not addressed
SwarMap 29	Centralized	Sim & real	Not addressed	Not addressed	Not addressed	Partially addressed (Aggregates partial maps on server)	Not addressed
Ultra-lightweight 13	Distributed	Sim & real	Not addressed	Partially addressed (Direct filter-based fusion)	Partially addressed	Not addressed	Not addressed
Proposed method	Distributed	Sim & real	Addressed (Integer diffusion)	Addressed (Budget-aware MCU limits)	Addressed (Tilewise run-length)	Addressed (Bounded, monotone consensus)	Addressed (Tri-force policy)

ATOP: adaptive thin-obstacle preservation; SCC: stigmergic counter-consensus; PDE: proximal delta encoding; BA-EKF: budget-aware EKF; TFC: tri-force frontier-cohesion controller; SLAM: simultaneous localization and mapping.

Methodology

This section specifies the full per-agent pipeline for distributed cooperative SLAM in dense micro-robot swarms under tight compute, memory, and bandwidth constraints. Figure 1(a) shows the custom micro-robot platform featuring the upward-facing IR array, each agent i carries a low-power MCU, wheel odometry and IMU, and a low-range IR sensor that returns B range beams per frame. Local maps are integer-valued, tile-partitioned grids to fit embedded memory; only tile deltas are exchanged over short-range radio links with neighbors in a time-varying proximity graph. The overall execution is ATOP-Raycast→BA-EKF→PDE→SCC→TFC, an end-to-end loop is provided after the model descriptions. Range beams are rasterized by ATOP into integer counters that preserve thin obstacles; BA-EKF fuses odometry with selected beams in fixed-point arithmetic; PDE encodes tile updates (RLE + version) and broadcasts them to proximal neighbors; SCC applies idempotent, monotone merges to form a consistent map; TFC issues velocity commands from frontier, avoidance, and cohesion objectives under a safety envelope (Figure 1(b)). The side panel (Figure 1(c)) illustrates the time-varying proximity graph $G_t$ : Dashed links indicate pairs within the radio radius $r_p$ , and solid arrows show proximal broadcasts restricted to neighbors; although any single snapshot may be sparse, a short B-connectivity window (union of links over time) is sufficient for map deltas to percolate through the swarm and for SCC to converge.

Figure 1.

System overview and pipeline. (a) The custom micro-robot platform featuring the upward-facing IR array. (b) The distributed execution pipeline: ATOP-Raycast updates local integer counters; BA-EKF fuses odometry; PDE encodes versioned tile deltas; SCC merges incoming maps via stigmergic consensus; and TFC handles control. (c) Illustration of the time-varying proximity graph $\mathcal{G}(t)$ and proximal broadcasts (dashed lines). SCC: stigmergic counter-consensus; ATOP: Adaptive thin-obstacle preservation; PDE: proximal delta encoding; BA-EKF: budget-aware EKF; TFC: tri-force frontier-cohesion controller.

System and communication model

The kinematic state of robot i at time $t_k$ is:

$${\mathbf{x}}_{i,k}={\left[\begin{array}{ccccc}{x}_{i,k}& y_{i,k}& {\theta }_{i,k}& v_{i,k}& {\omega }_{i,k}\end{array}\right]}^{\mathrm{\top }},$$

(1)

with control input ${\mathbf{u}}_{i,k}=[v_{i,k},{\omega }_{i,k}{]}^{\mathrm{\top }}$ . Motion follows a unicycle model:

$${\mathbf{x}}_{i,k}=f({\mathbf{x}}_{i,k-1},{\mathbf{u}}_{i,k-1})+{\mathbf{w}}_{i,k-1},{\mathbf{w}}_{i,k-1}\sim \mathcal{N}(0,\mathbf{Q}),$$

(2)

and range sensing returns B beams ${\mathcal{Z}}_{i,k}=\{(r_b,{\varphi }_b){\}}_{b=1}^B$ in the robot frame. Communication is proximal: At $t_k$ the undirected graph $G_k=(V,E_k)$ contains an edge $(i,j)\in E_k$ if $\parallel {\mathbf{p}}_{i,k}-{\mathbf{p}}_{j,k}\parallel \le r_p$ and received signal strength indicator/signal-to-noise ratio exceeds a link threshold. Formally, the swarm's communication network satisfies B-connectivity if there exists a finite time horizon $H>0$ such that the union of the interaction graphs over any time interval $[k,k+H]$ , denoted as $G_{[k,k+H]}={\cup }_{\tau =k}^{k+H}{G}_{\tau }$ , forms a strongly connected graph.^31,32 Only neighbors in ${\mathcal{N}}_i^{(r)}(k)=\{j\mid (i,j)\in E_k\}$ receive packets at step k. The radio duty cycle and payload sizes are chosen so that $\beta$ (bytes/s) remains below the MCU and spectrum budgets (see Table 2).

ATOP-Raycast: Integer occupancy with thin-obstacle preservation

Each agent stores a tiled occupancy grid; within a tile, every cell c holds a signed integer counter $M(c)\in \{-K,\dots ,+K\}$ rather than log-odds. For a beam b with measured endpoint cell $c_e$ and discrete ray cells ${\mathcal{R}}_b$ computed by Bresenham rasterization, ATOP applies saturated updates:

$$M(c)\leftarrow {\mathrm{clip}}_{[-K,K]}(M(c)-{\alpha }_f),\mathrm{\forall }c\in {\mathcal{R}}_b\setminus \{c_e\},$$

(3)

$$M(c_e)\leftarrow {\mathrm{clip}}_{[-K,K]}(M(c_e)+{\alpha }_o),$$

(4)

with small free-carving step ${\alpha }_f>0$ and a stronger endpoint step ${\alpha }_o>{\alpha }_f$ . To prevent razor-thin objects from being eroded by repeated free carving, ATOP adds a local endpoint diffusion to the 8-neighbors ${\mathcal{N}}_8(c_e)$ ,

$$M(n)\leftarrow {\mathrm{clip}}_{[-K,K]}(M(n)+{\eta }_k),\mathrm{\forall }n\in {\mathcal{N}}_8(c_e),$$

(5)

where ${\eta }_k\in \{0,1\}$ is a noise-adaptive integer gain that activates only when the beam residual is small and the local gradient magnitude is high. We implement endpoint diffusion using a Q4 accumulator that contributes +0.25 to each 8-neighbor per hit and emits a + 1 vote when $A\ge 1.00$ (Equations (3) and (4)). Figure 2 illustrates the evolution of M and A on a 3 × 3 patch across successive hits.

Figure 2.

Step-by-step evolution of the adaptive thin-obstacle preservation (ATOP) digital diffusion mechanism.

This stencil creates a + 2 “core” at $c_e$ flanked by +1 lobes that stabilize slender geometry (Figure 3(a) and (b)). ATOP performs adaptive 2× refinement within tiles that exhibit sustained edge activity. Let G be a $3\times 3$ Sobel-like kernel and $M_T$ the counters within tile T. Define the edge energy:

$$E(T)=\mid T{\mid }^{-1}\sum _{c\in T}\parallel \mathrm{\nabla }(G*M_T)(c){\parallel }_1.$$

(6)

Figure 3.

Adaptive thin-obstacle preservation (ATOP) strategies. (a) Coarse grid with Bresenham ray carving; (b) endpoint diffusion to 8-neighbors visualized as small positive lobes; (c) adaptive 2× refinement within a high-edge tile preserving a slender pole.

If $E(T)>{\tau }_{\mathrm{refine}}$ for h consecutive frames, T is subdivided to a $2\times$ finer $4\times 4$ subgrid; counters are replicated (or bilinearly interpolated and clipped) and then updated at the finer resolution using (2)–(4). Coarsening is triggered when $E(T)<{\tau }_{\mathrm{coarsen}}$ for h frames, aggregating subcells by clipped mean with sign-majority override, followed by a version increment. The effect is shown in Figure 3(c), where a pole remains one fine subcell wide while compute remains bounded. For reproducibility, the specific parameter values used in all experiments are as follows. The integer counters use a range of $M(c)\in [-127,+127]$ . The ray-casting update steps are asymmetric to prioritize safety: A direct hit increments the counter by ${\alpha }_o=15$ , while a free-space traversal decrements it by ${\alpha }_f=2$ . The endpoint diffusion uses a 2-bit accumulator, contributing $+0.25$ per hit to 8-neighbors and triggering the spatial vote when $A\ge 1.00$ . Finally, the adaptive refinement is controlled by edge energy thresholds: A tile splits into a $4\times 4$ subgrid when the mean gradient magnitude $E(T)>{\tau }_{refine}=2.5$ for $h=5$ consecutive frames, and coarsens back when $E(T)<{\tau }_{coarsen}=1.0$ .

BA-EKF: Fixed-point state estimation with sparse beam selection

State estimation uses a bounded-arithmetic EKF so all linear-algebra operations execute in Q-format fixed point on the MCU. With Jacobians $A_k=\mathrm{\partial }f/\mathrm{\partial }\mathbf{x}{\mid }_{{\hat{\mathbf{x}}}_{k-1}}$ and $H_k=\mathrm{\partial }h/\mathrm{\partial }\mathbf{x}{\mid }_{{\hat{\mathbf{x}}}_{k\mid k-1}}$ , prediction and update follow:

$${\hat{\mathbf{x}}}_{k\mid k-1}=f({\hat{\mathbf{x}}}_{k-1},{\mathbf{u}}_{k-1}),P_{k\mid k-1}=A_k{P}_{k-1}{A}_k^{\mathrm{\top }}+Q,$$

(7)

$${\mathbf{K}}_k=P_{k\mid k-1}{H}_k^{\mathrm{\top }}(H_k{P}_{k\mid k-1}{H}_k^{\mathrm{\top }}+R{)}^{-1},$$

(8)

$${\hat{\mathbf{x}}}_k={\hat{\mathbf{x}}}_{k\mid k-1}+{\mathbf{K}}_k{\mathbf{r}}_k,P_k=(I-{\mathbf{K}}_k{H}_k)P_{k\mid k-1}.$$

(9)

The measurement function $h(\cdot )$ maps pose to expected endpoint ranges by ray-casting a small window of the current map around the robot. To keep compute predictable, ATOP supplies a sparse subset of beams ${\mathcal{S}}_k\subset {\mathcal{Z}}_{i,k}$ that maximizes directional coverage subject to a bound $\mid {\mathcal{S}}_k\mid \le B_{\max }$ and excludes rays that terminate in unknown space. All matrices $P,Q,R$ are stored as scaled integers with saturating adds and a Cholesky-based, fixed-point inversion for the small innovation covariance (2 × 2).

PDE and radiosafe broadcast

After each mapping/estimation cycle, the agent derives a tile-level mask of cells whose counters changed:

$$\mathrm{\Delta }T=\{(c,\delta M(c)):\delta M(c)\ne 0,c\in T\}.$$

(10)

Each updated tile T is assigned a monotonically increasing version $v_T$ . The payload contains $⟨\mathrm{tile}\mathrm{\_}\mathrm{id},v_T,\mathrm{run}\text{-}\mathrm{length}\mathrm{encoded}(\mathrm{\Delta }T),\mathrm{CRC}⟩$ . Only neighbors in ${\mathcal{N}}_i^{(r)}(k)$ are targeted; a low-duty scheduler (2–4 Hz) alternates between the most changed tile and a round-robin over the remainder to prevent starvation. The expected payload length is bounded as:

$$L\le L_0+2n_{\mathrm{runs}}+n_{\mathrm{hdr}}\Vert \mathrm{bytes},$$

(11)

where $n_{\mathrm{runs}}$ is the number of constant-sign spans in $\mathrm{\Delta }T$ . With the parameters in Table 1, L typically falls below 160 bytes per packet.

SCC: Stigmergic, idempotent consensus on tiles

Upon reception, an agent merges remote deltas with its local map using an idempotent and monotone rule, which guarantees eventual consistency under B-connectivity. Let ${\mathbf{M}}_T$ denote the vector of counters for tile T, paired with version $v_T$ . For a received pair $({\mathbf{M}}_T^{{}^{\mathrm{\prime }}},v_T^{{}^{\mathrm{\prime }}})$ :

$$({\mathbf{M}}_T,v_T)\oplus ({\mathbf{M}}_T^{{}^{\mathrm{\prime }}},v_T^{{}^{\mathrm{\prime }}})=\left\{\begin{array}{cc}({\mathbf{M}}_T^{{}^{\mathrm{\prime }}},v_T^{{}^{\mathrm{\prime }}}),& v_T^{{}^{\mathrm{\prime }}}>v_T,\\ ({\mathbf{M}}_T,v_T),& v_T^{{}^{\mathrm{\prime }}}<v_T,\\ (\mathrm{merge}({\mathbf{M}}_T,{\mathbf{M}}_T^{{}^{\mathrm{\prime }}}),v_T),& v_T^{{}^{\mathrm{\prime }}}=v_T,\end{array}\right\}$$

(12)

with element-wise:

$$\mathrm{merge}(a,b)=\left\{\begin{array}{cc}a,& \mid a\mid >\mid b\mid ,\\ b,& \mid b\mid >\mid a\mid ,\\ \mathrm{sgn}(a)\max (\mid a\mid ,1),& \mid a\mid =\mid b\mid \mathrm{and}a=b,\\ 0,& \mid a\mid =\mid b\mid \mathrm{and}a=-b,\end{array}\right\}$$

(13)

followed by a single-pass diffusion like (4) to smooth ties. This operator is (i) commutative, (ii) associative, and (iii) idempotent ( $x\oplus x=x$ ), so the set of tile states forms a join-semilattice; therefore, repeated proximal exchanges converge to a unique, order-independent fixed point when the union graph over sliding windows remains connected (cf. eventual consistency of CRDTs).

Mathematically, the convergence of the SCC algorithm is guaranteed by the properties of the join-semilattice formed by the tile states. Let the state of a tile be $S_T=(M_T,v_T)$ . We define a partial order $\underset{\_}{\prec }$ such that $S_T\underset{\_}{\prec }{S}_T^{{}^{\mathrm{\prime }}}$ if $v_T<v_T^{{}^{\mathrm{\prime }}}$ or ( $v_T=v_T^{{}^{\mathrm{\prime }}}$ and $|M_T|\le |M_T^{\mathrm{\prime }}|$ ). The merge operator $\oplus$ defined in Equation (12) acts as the join (least upper bound) under this ordering, satisfying idempotence ( $S\oplus S=S$ ), commutativity, and associativity. Because the integer counters are strictly bounded by $[-C_{max},C_{max}]$ , the state space is finite. Consequently, the system satisfies the ascending chain condition: Any sequence of updates $S_0\underset{\_}{\prec }{S}_1\underset{\_}{\prec }\dots$ must stabilize at a fixed point in finite steps. Under the assumption of B-connectivity—where the union of communication edges $U_t^{t+H}\mathcal{E}(t)$ forms a connected grap—this local fixed point corresponds to the unique global supremum of all observed evidence. Thus, the swarm reaches eventually consistent consensus without requiring a static network topology. Quantitatively, we define the Hamming inconsistency $D(t)$ as the cardinality of the set of cells subject to active disagreement across the swarm:

$$D(t)=\sum _{k\in \mathrm{\Omega }}I(\mathrm{\exists }i,j:\mathrm{sgn}(M_i(k))\cdot \mathrm{sgn}(M_j(k))=-1)$$

(14)

where $\mathrm{\Omega }$ is the union of all observed cells, $\mathrm{sgn}(\cdot )$ is the signum function, and $I(\cdot )$ is the indicator function. This metric strictly counts “hard” conflicts (free vs. occupied) while ignoring “soft” discrepancies involving unknown space ( $c=0$ ), thus serving as a direct proxy for global map consistency. Empirically, the disagreement across robots $D(t)$ decreases monotonically as visualized in Figure 4.

Figure 4.

Visualization of stigmergic counter-consensus (SCC) convergence. Rows for three robots’ local maps; columns for time. Cell-sign disagreement shrinks as idempotent, bounded updates mix through B-connected contact windows.

TFC: Frontier-cohesion control with a safety envelope

The controller optimizes a short-horizon objective that balances exploration, obstacle avoidance, and team cohesion while maintaining radio proximity. For reproducibility, the set $\mathcal{F}$ is generated via the following detection procedure. The robot scans its local occupancy grid ${\mathcal{M}}_i(t)$ to identify “frontier candidates”—cells that mark the transition from safe to unexplored space. A cell k is added to ${\mathcal{F}}_{raw}$ if and only if it is currently confirmed as free ( $c_{i,k}<0$ ) and possesses at least one 8-neighbor $n\in {\mathcal{N}}_8(k)$ that remains unknown ( $c_{i,n}=0$ ). To filter spurious frontiers caused by sensor noise, we apply Euclidean clustering (DBSCAN with $\epsilon =1.5$ cells) to ${\mathcal{F}}_{raw}$ and discard clusters with fewer than 3 cells. The target frontier vector ${\varphi }_f$ is then computed pointing to the centroid of the nearest valid cluster, ensuring the robot drives toward substantial unexplored regions rather than isolated grid artifacts. Let ${\varphi }_o$ be the repulsive gradient from occupied cells within radius $r_s$ , and ${\varphi }_c$ the attraction toward the centroid of ${\mathcal{N}}_i^{(r)}(k)$ . The commanded body-frame velocity is:

$${\mathbf{u}}_{i,k}=\mathrm{sat}({\lambda }_f{\varphi }_f-{\lambda }_o{\varphi }_o+{\lambda }_c{\varphi }_c),$$

(15)

clipped by rate and acceleration limits and wrapped in a safety envelope that enforces ${\min }_{n\in {\mathcal{N}}_i^{(r)}(k)}\parallel {\mathbf{p}}_{i,k}-{\mathbf{p}}_{n,k}\parallel \le r_p+\epsilon$ and forbidden-zone margins around +2 cells. Each robot executes a deterministic, single-threaded loop at $\approx 20$  Hz on the MCU. Algorithm 1 is to clarify this schedule:

Algorithm 1.

Per-agent loop, 20 Hz

1.	Sense odometry/IMU and depth beams; select sparse set ${\mathcal{S}}_{\mathit{k}}$ .
2.	ATOP: apply (2)–(4) to update integer counters; compute $\mathit{E}(\mathit{T})$ and refine/coarsen tiles per (5).
3.	EKF: run prediction and update (6)–(8) with fixed-point linear algebra using ${\mathcal{S}}_{\mathit{k}}$ .
4.	PDE: form $\mathbf{\Delta }\mathit{T}$ , RLE-encode with version ${\mathit{v}}_{\mathit{T}}$ (9)–(10); schedule proximal broadcast to ${\mathcal{N}}_{\mathit{i}}^{(\mathit{r})}(\mathit{k})$ .
5.	SCC: read inbox; apply idempotent merge (11)–(14); update local map and versions.
6.	TFC: compute frontier/avoidance/cohesion vectors; command ${\mathbf{u}}_{\mathit{i},\mathit{k}}$ via (15) with safety envelope.

ATOP ray-updates are $O(\mid {\mathcal{R}}_b\mid )$ per beam with tiny constants (integer increments and a branch-free clip). For B beams, the per-frame mapping cost is $O\left(\sum _b\mid {\mathcal{R}}_b\mid \right)$ , typically $O(B\ell )$ where $\ell$ is mean ray length in cells. The BA-EKF uses fixed-dimension matrices, so prediction and update are $O(1)$ per frame; sparse beam selection bounds the ray-cast in the measurement model. PDE builds at most one RLE for each changed tile with $O(\mid \mathrm{\Delta }T\mid )$ cost and emits tens to a few hundred bytes; SCC is a linear pass over received runs. Memory is dominated by tile storage: With tile side t cells, b bytes per counter (1 byte here), and $N_T$ tiles in the local workspace, the footprint is $N_T{t}^{2}b+\mathrm{overheads}$ (metadata, versions). Table 3 lists a concrete configuration that fits comfortably on 128 kB MCUs and 250 kbps radios. All experiments use the defaults in Table 4 unless noted. The simulator enforces the same per-agent limits as the hardware (integer counters, fixed-point EKF, capped payloads, duty-cycled radio) so that performance estimates transfer. Empirical budgets on an ESP32-class MCU indicated that the full loop remains within a 50 ms frame (20 Hz) for $N\le 40$ .

Table 3.

Per-agent resource budget and communication settings.

Resource/Setting	Value (Default)	Rationale
MCU clock	64–120 MHz	Common class
SRAM available to SLAM	64–128 kB	Fits ${\mathit{N}}_{\mathit{T}}=36$ tiles ×  $16\times 16$  × 1 B + metadata
Counter precision	8-bit signed	Saturating updates, minimal RAM
EKF math	Q15.16 fixed point	Stable inversions, no FPU required
Beams per frame $\mathit{B}$	32 (sparse)	Bounded ray-cast and EKF update
Radio	250 kbps, short-range	Low-power ISM
Broadcast duty	3 Hz	Below channel utilization limit
Packet size $\mathit{L}$	80–160 bytes	RLE on changed runs (Equation (10))
Proximity radius ${\mathit{r}}_{\mathit{p}}$	0.40 m	Ensures frequent neighbor contact
Safety radius ${\mathit{r}}_{\mathit{s}}$	0.10 m	Collision buffer in TFC
Controller gains $({\mathit{\lambda }}_{\mathit{f}},{\mathit{\lambda }}_{\mathit{o}},{\mathit{\lambda }}_{\mathit{c}})$	(1.0, 1.2, 0.6)	Exploration > avoidance > cohesion
Max linear velocity ( ${\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$ )	0.15 m/s	Mechanical limit of micro-gearbox
Max angular velocity ( ${\omega }_{\mathit{m}\mathit{a}\mathit{x}}$ )	1.5 rad/s	Ensures stable EKF prediction
Sensor noise ( ${\sigma }_{\mathit{r}\mathit{a}\mathit{n}\mathit{g}\mathit{e}}$ )	0.02 m	Calibrated IR variance
Physics timestep ( $\mathbf{\Delta }\mathit{t}$ )	50 ms	Matches 20 Hz control loop

SLAM: simultaneous localization and mapping; TFC: tri-force frontier-cohesion controller.

Table 4.

Per-agent resource budget on an ESP32-class MCU.

Module	Inputs	Complexity (per frame)	Typical cost (ms)	Memory footprint	Notes
ATOP-Raycast	B beams, grid	$\mathcal{O}(\mathit{B}\overline{\mathit{L}})$	10–15	∼11 kB	Integer counters; diffusion; local refinement
SCC (merge)	Tile deltas	$\mathcal{O}(\overline{\mathit{C}})$	6–10	<1 kB	Versioned, idempotent; bounded updates
PDE (encode)	Changed tiles	$\mathcal{O}(\overline{\mathit{C}})$	3–5	<1 kB	1-bit mask + RLE; proximal broadcast only
EKF (fixed-point)	Odometry +  $\mathit{m}$ beams	$\mathcal{O}(1)$	4–6	<2 kB	Q15 matrices; periodic rescaling
TFC (control)	Grid, neighbors	$\mathcal{O}(1)$	2–4	negligible	Rate-limited; cohesion preserves neighborhoods
Total	—	—	25–40	∼14–15 kB	Fits 20 Hz with headroom

ATOP: adaptive thin-obstacle preservation; SCC: stigmergic counter-consensus; PDE: proximal delta encoding; TFC: tri-force frontier-cohesion controller.

Hardware and simulation implementation

To ensure that algorithmic performance transfers reliably from theory to practice, we utilize a matched hardware-simulation pair.

Hardware platform

The physical agents are custom differential-drive micro-robots (Ø50 mm) driven by an ESP32-S3 MCU (240 MHz, 512 kB SRAM). Perception relies on a static, radially distributed array of infrared sensors providing a sparse $360\circ$ scan with a maximum range of $r_{max}=1.3m$ (Figure 1(a)).

Simulation environment

The physics-based simulator acts as a strict “digital twin,” executing the exact C++ firmware logic used on the robots. Critically, it emulates the MCU's constraints by enforcing Q15.16 fixed-point arithmetic for the EKF and int8 clamping for the ATOP grid. Sensor noise is modeled as Gaussian perturbation $w_r\sim \mathcal{N}(0,{\sigma }_r^2)$ with ${\sigma }_r=0.02$  m, combined with a probabilistic drop-out model ( $p_{drop}=0.05$ ) to simulate surface absorption. This rigorous parity ensures that the “budget-aware” behaviors observed in silico—such as accumulator overflow or integer quantization errors—accurately predict physical performance.

Results

To ensure reproducibility and seamless transfer between simulation and the physical testbed, the experimental parameters were standardized across both domains. The custom physics-based simulator was architected to strictly enforce the same integer-arithmetic and memory constraints as the target ESP32 hardware, utilizing a rigid-body unicycle model with kinematic limits set to $v_{max}=0.15$  m/s and ${\omega }_{max}=1.5$  rad/s. Sensing was modeled and physically calibrated as a sparse array of infrared beams with a maximum range $r_{max}=0.8$  m, subject to Gaussian noise ( ${\sigma }_r\approx 0.02$  m) and a hard field-of-view cutoff. The control loop operated at a fixed timestep of $\mathrm{\Delta }t=50$  ms (20 Hz), matching the MCU's update rate. These constraints, summarized in the Table 2, ensure that the algorithmic behavior observed in silico—specifically the fixed-point EKF stability and ATOP integer diffusion—remains predictive of the physical swarm's performance.

The proposed pipeline was evaluated in a physics-based simulator that enforces the same arithmetic and messaging limits as the target micro-robots, and then on 40-robot swarm running the exact embedded stack. The simulation results depicted in Figure 5 were generated in a procedurally generated “randomized clutter” arena ( $10\times 10$  m), designed to present a heterogeneous mix of geometric challenges. The ground truth layout (shown as black cells) consists of: (i) Macro-obstacles: Large static blocks that create significant occlusion shadows to test the consensus mechanism's ability to merge disjoint observations; and (ii) Micro-obstacles: A scattered field of slender vertical poles and narrow apertures. In each panel the left subimage shows robot positions in the arena and the right subimage shows the aggregate occupancy grid produced by the team. Green cells mark free space that at least one robot has confirmed by casting infrared range rays; red cells mark occupied space at ray endpoints, with a small 8-neighbor support added by the ATOP thin-obstacle rule. Concretely, each robot updates its local map by setting cells traversed by a clear ray to a free code of +1 (green) and the hit cell to an occupied code of +2 (red); these tile deltas are then merged across robots by the monotone, idempotent consensus rule. At $t=0$ the grid is largely unknown; robots begin to deposit small green starlets around their starting scans and isolated red dots where rays first touch obstacles. By $t=25$ minutes the frontier policy has expanded the verified free corridors and the red contours begin to trace obstacle perimeters, while SCC merges align overlapping edits from neighboring agents. By $t=50$ minutes most free space has been confirmed and the red outlines have coalesced into near-continuous boundaries; remaining gaps are concentrated in late-reached regions. By $t=60$ minutes the map is effectively saturated: Free cells form an almost uniform carpet and obstacles are bounded by one- to two-cell-wide red contours of consistent thickness, indicating successful thin-structure preservation and near-zero inter-robot disagreement.

Figure 5.

Seam-free map merging in the “Randomized Clutter” simulation. (a) $\mathit{t}=0$ : Initial dispersion; (b) $\mathit{t}=25$ min: Partial coverage showing early recovery of thin Poles. (c) $\mathit{t}=50$  min: Consensus stabilizes around macro-obstacles. (d) $\mathit{t}=60$  min: Final map showing globally consistent boundaries and high retention of the ground-truth micro-obstacles.

To benchmark the proposed architecture against established methods, we performed a controlled simulation comparing our framework with two standard baselines under identical trajectory and sensing noise conditions ( $N=40$ robots).

• –

Baseline A (Standard log-odds grid): This variant replaces the ATOP integer counters with classical floating-point log-odds updates ( $l_t=l_{t-1}+\mathrm{inverse}\mathrm{\_}\mathrm{sensor}\mathrm{\_}\mathrm{model}$ ). It serves as a baseline for geometric fidelity.

–

Baseline B (Naive mean-consensus): This variant replaces the SCC idempotent merge with a standard arithmetic average ( $M_{new}=(M_i+M_j)/2$ ) whenever robots exchange maps. It serves as a baseline for distributed consistency.

Table 5 summarizes the results. The standard log-odds approach exhibited significant geometric degradation, achieving a thin feature retention rate (TFRR) of only 58.4% compared to 98.2% for our ATOP method. Visual inspection confirmed that the lack of neighbor diffusion caused slender obstacles to “erode” and vanish under noisy ray-casting conditions. Regarding consensus, the naive mean-consensus failed to stabilize, terminating with a persistent disagreement count ( $D_{final}$ ) of 142 cells. This is indicative of “chattering,” where circular updates between agents prevent the map from settling. In contrast, our SCC mechanism, protected by idempotence and monotonicity, converged to 0 disagreement within the same duration, validating the theoretical stability guarantees of the stigmergic design.

Table 5.

Comparative simulation results.

Metric	Baseline A: standard Log-Odds	Baseline B: Naive Mean-Consensus	Proposed Method (ATOP + SCC)
TFRR	58.4%	97.9%	98.2%
Steady-state disagreement ( ${\mathit{D}}_{\mathit{f}\mathit{i}\mathit{n}\mathit{a}\mathit{l}}$ )	0 cells	142.0 cells	0 cells
Convergence time	28.5 min	N/A	30.2 min
Map storage per tile	1024 bytes (Float)	256 bytes (Int8)	256 bytes (Int8)

ATOP: adaptive thin-obstacle preservation; SCC: stigmergic counter-consensus; TFRR: Thin feature retention rate.

The reliability of the pipeline depends on distinguishing static features from dynamic agents and sensor noise. To mitigate “inter-robot mapping”—where agents erroneously map neighbors as obstacles—we employ geometric neighbor masking, discarding IR returns that intersect with the projected safety envelopes ( $r_s$ ) of proximal neighbors. As shown in Table 6, the intrinsic sensor noise yields a baseline FPR of 2.1%. In a dense swarm ( $N=40$ ) without filtering, this spikes to 18.4% due to dynamic interference. However, with neighbor masking enabled, the FPR is suppressed to 2.4%, effectively neutralizing the artifact. Furthermore, the ATOP digital diffusion mechanism provides a secondary temporal filter; its evidence accumulator threshold ( $+1.0$ ) creates hysteresis that prevents isolated specular “glints” or cross-talk from triggering permanent state changes, ensuring map stability even under noisy conditions.

Table 6.

Sensor robustness and dynamic filtration performance.

Scenario	False positive rate (FPR)	Map entropy (bits/cell)
Single robot (Baseline)	2.1%	0.42
Swarm ( $\mathit{N}=40$ )—No masking	18.4%	0.89
Swarm ( $\mathit{N}=40$ )—With masking	2.4%	0.45

Real-time swarm demonstration (19 robots), the embedded system transfers directly from simulation to hardware (Figure 6). The top-down aggregate map (Figure 6(a)) is produced solely via on-board sensing and proximal broadcasts, with no centralized fusion. Outer walls and internal rounded obstacles appear as continuous, uniform-thickness lines, while regions mapped by different robots meet without visible seams. The on-board cooperative two-dimensional (2D) map (Figure 6(b)) captures these features as thin, continuous outlines, validating the ATOP preservation logic and consistent SCC merging. In three-dimensional (3D) (Figure 6(c)), the feature cloud exhibits closed loops with minimal vertical scatter. This geometric coherence confirms that the fixed-point EKF maintained heading stability and that versioned PDE packets successfully synchronized local maps without external infrastructure.

Figure 6.

Tabletop arena—cooperative SLAM demonstration with N = 19, (a) physical tabletop arena with 19 micro-robots and obstacles; (b) on-board, cooperative 2D map; (c) corresponding 3D feature cloud showing closed perimeter and obstacle loops. SLAM: simultaneous localization and mapping; 2D: two-dimensional; 3D: three-dimensional.

The same embedded stack was deployed in a real laboratory containing a curved stage, a central circular dais, and multiple rows of seats. Figure 7(a) presents the laboratory space that includes a curved stage, a central circular dais, and rows of seats, Figure 7(b) shows the on-board cooperative 2D map generated by the swarm, where the global envelope, the curved stage, and the dais appear as continuous thin boundaries, and the seat rows appear as compact repeated clusters. Panel (c) renders the same data in 3D; the cloud lies on a nearly planar surface and the perimeter and dais loops are closed with minimal gaps. The absence of seams at inter-robot contacts and the uniform edge thickness across regions mapped by different robots indicate that SCC's idempotent, versioned merges and the fixed-point EKF maintained a coherent global map under short-range, neighbor-only communication. In 2D, the outer envelope and the long curved stage are recovered as thin, continuous boundaries; the circular dais appears as a closed loop; seat rows emerge as compact, repeated clusters. In 3D, the cloud lies on a nearly planar surface with low vertical scatter, and perimeter/dais loops are closed with minimal gaps—evidence of heading coherence from BA-EKF micro-updates and repeatable endpoints from ATOP under real sensing. Minor discontinuities near the upper edge and lower-right doorway correspond to transient occlusions; they closed as exploration progressed, indicating that versioned, idempotent SCC merges and proximal broadcasts were sufficient to reconcile late updates without centralized fusion.

Figure 7.

Laboratory Hall—cooperative SLAM in a real environment with N = 40, (a) laboratory Hall with curved stage, central dais, and seating; (b) on-board cooperative 2D map from the swarm; (c) 3D rendering of the same data showing low vertical scatter and closed loops. 2D: two-dimensional; 3D: three-dimensional; SLAM: simultaneous localization and mapping.

See Figure 8 for a side-by-side view of how the swarm's map improves over time: Panel (a) plots the convergence rate $d\mathrm{IoU}/dt$ in units per minute, and panel (b) plots the resulting map coverage (IoU) against normalized time for simulation, tabletop arena, and laboratory hall. In panel (a), all curves show a brief high-gain transient at the start when frontiers are plentiful, followed by a steady decay toward zero as exploration saturates and SCC resolves residual disagreements. The tabletop arena retains small oscillations longer than simulation due to short line-of-sight losses around obstacles, while the laboratory hall remains lower and noisier overall, reflecting curved boundaries, clutter, and occlusions that slow both discovery and consensus. In panel (b), the same behaviors integrate into sigmoidal IoU curves: Rapid early growth, a slower mid-phase, and a late plateau. Shaded ribbons indicate across-run variability, which is largest early and tightens as the shared map stabilizes. Markers at $t_{50}$ and $t_{90}$ highlight practical operating points where half and ninety percent of final coverage are achieved.

Figure 8.

Convergence dynamics and coverage growth, (a) convergence rate over time (dIoU/dt, per minute); (b) map coverage vs normalized time.

To strictly quantify the preservation of slender geometry beyond visual inspection, we evaluated the TFRR across all three experimental scenarios. We define TFRR as the percentage of ground-truth obstacle cells with a width of $\le 2$ grid units that are correctly identified as occupied in the final cooperative map. As detailed in Table 7, the proposed framework achieved a retention rate of 98.2% in simulation, where idealized connectivity allowed for rapid consensus. In the physical deployments, performance remained robust: The tabletop arena yielded a TFRR of 95.1%, while the complex laboratory hall maintained a rate of 92.4%. This slight attenuation in the large-scale experiment is attributable to transient occlusions caused by the dense seating rows (visible in Figure 7(a)); however, the high retention value confirms that the ATOP digital diffusion mechanism successfully prevented the erosion of critical structural details—such as the curved stage edge and dais perimeter—even under significant sensing noise and clutter.

Table 7.

Quantitative performance and thin feature retention across environments.

Metric	Simulation	Tabletop Arena	Laboratory Hall
Thin feature retention rate (TFRR)	98.2%	95.1%	92.4%
Final map coverage (IoU)	∼0.99	∼0.96	∼0.89
Convergence time	∼60 min	∼32 min	∼60 min
Disagreement ( ${\mathit{D}}_{\mathit{f}\mathit{i}\mathit{n}\mathit{a}\mathit{l}}$ )	0 cells	<5 cells	<15 cells
Mean packet size	85 bytes	92 bytes	210 bytes

For a more consolidated view, Figure 9 organizes the evaluation into four panels that link mapping quality, team agreement, communication load, and power draw across simulation, tabletop arena, and laboratory hall. Panel (a) shows map coverage (IoU) over time, with rapid early gains, slower mid-phase growth, and a late plateau as frontiers vanish; Simulation rises fastest, Tabletop follows with a small lag, and Laboratory climbs more gradually due to occlusions and curved geometry. Panel (b) traces inter-robot disagreement $D(t)$ , which drops sharply at the start and trends toward a low steady level; brief spikes correspond to reconnections after temporary link loss, and the low tail reflects SCC's versioned, idempotent merges. Panel (c) reports network throughput: Tabletop sustains higher median and burst rates because agents remain in close proximity; Laboratory traffic is lower and more variable. The absence of runaway throughput indicates that PDE keeps payloads compact by transmitting only changed runs. Panel (d) charts battery level to the safety cutoff; trajectories nearly overlap across scenarios, with a slightly steeper decline in Tabletop consistent with its higher message volume, while Laboratory's lower throughput offsets longer paths. Taken together, the four panels show consistent convergence in mapping and consensus while maintaining communication and energy within budget across controlled and real environments.

Figure 9.

Holistic evaluation across environments, (a) IoU convergence over time; (b) consensus formation over time; (c) communication load vs time; (d) energy consumption profile.

Resource usage (high fidelity), the energy consumption profiles (Figure 9(d)) exhibited remarkable consistency, with the physical battery trajectories tracking the simulation with a Mean Absolute Percentage Error of less than 3%. This confirms that our “budget-aware” simulation model accurately captures the computational and electromechanical loads of the ESP32 hardware. Convergence dynamics (moderate fidelity), while the shape of the coverage curves (Figure 9(a)) is consistent (sigmoidal growth), the physical tabletop swarm exhibited a temporal lag of approximately 8%–12% relative to the simulation. This “Sim-to-Real gap” is attributable to unmodeled physical phenomena—specifically, wheel slippage on the arena floor and transient radio shadowing—which slowed effective motion and message delivery.

To stress-test the architecture beyond the hardware deployment of 40 units, we performed supplementary simulations with $N=60$ and $N=80$ agents in the same environment. As detailed in Table 8, the system exhibits graceful degradation as density increases. For $N=60$ , the convergence time ( $t_{90}$ ) increases by approximately 28%, primarily driven by radio channel saturation which reduces the successful packet delivery ratio. At $N=80$ , the computational load of the SCC merge step pushes the mean cycle time to 64 ms, slightly exceeding the 50 ms real-time target. The map quality (Final IoU) remains high (>0.85), confirming that the stigmergic consensus mechanism remains robust—albeit slower—even when the swarm size outstrips the ideal design specifications.

Table 8.

Simulation performance vs. swarm size for scalability stress test.

Metric	$\mathit{N}=40$	$\mathit{N}=60$	$\mathit{N}=80$
Convergence time	45 min	58 min	75 min
Mean cycle time (CPU)	38 ms	46 ms	64 ms
Radio packet loss	∼12%	∼28%	∼41%
Final map IoU	0.89	0.88	0.86

Given the integrated nature of the proposed framework, measuring the marginal contribution of each module on the physical testbed is challenging. However, based on the system constraints and experimental observations, we can characterize the critical role of each component through a theoretical ablation analysis:

• –

Without ATOP: If replaced by standard Bresenham raycasting, the system would revert to the “geometric erosion” failure mode described in Section 1. In our specific environment, this would cause the slender poles and chair legs (currently captured as +2 contours) to flicker or vanish entirely due to repeated free-space carving, rendering navigation unsafe.

–

Without SCC: If replaced by a standard arithmetic mean or probabilistic fusion rule, the swarm would lose the guarantee of monotonic convergence. As shown in the comparative baseline (Table 4), this leads to “chattering” or oscillation, where conflicting updates from robots circulate indefinitely, preventing the map from settling into a consistent state ( $D(t)\to 0$ ).

–

Without PDE: If replaced by transmitting full map tiles, the bandwidth requirement would scale linearly with map size rather than update frequency. For a swarm of $N=40$ , this would saturate the 250 kbps channel almost immediately, forcing a drastic reduction in the update rate (from 3 Hz to < 0.1 Hz) and breaking the B-connectivity assumption.

–

Without BA-EKF: If replaced by a standard floating-point EKF using the MCU's software-emulated FPU, the state estimation step would likely exceed the 50 ms control loop budget (Table 2). This latency would introduce jitter into the wheel odometry integration, paradoxically degrading the map quality despite the higher numerical precision of floating-point math.

Conclusion

A central implication is that global consistency can emerge from strictly local exchange when the union of proximity links is intermittently connected over short windows B-connectivity. Classical treatments of SLAM emphasize high-fidelity sensor models and globally consistent optimization. In contrast, the present approach treats communication and memory as primary constraints and shifts the burden of “consistency” to algebraic properties of the merge operator. The empirical absence of seam artifacts—despite asynchronous, neighbor-only messaging—indicates that versioning and monotonicity are sufficient to prevent the oscillations often observed with averaging or probabilistic fusion rules in decentralized settings. This finding aligns with consensus principles in networked systems, where stability can be obtained with simple, local updates provided the interaction graph is persistently connected. ¹⁸

The study presents three limitations defining our future roadmap. First, global convergence relies on temporal B-connectivity; extended network fragmentation stalls agreement, suggesting a need for opportunistic relaying to bridge gaps. Second, while robust to moderate noise, the integer diffusion thresholds can be sensitive to extreme specular aliasing, requiring gain tuning. Third, the current 2D assumption requires adaptation for 3D volumetric SLAM. To address cubic memory growth ( $O(L^3)$ ), we propose replacing direct tile addressing with sparse spatial hashing to track only the occupied surface shell ( $O(L^2)$ ). Furthermore, the ATOP diffusion logic must evolve from an 8-neighbor stencil to a 26-neighbor “shell-stabilizing” kernel to support planar manifolds, while 6-DOF state estimation should employ decoupled fusion (2D pose + lightweight height) to maintain fixed-point feasibility on micro-controllers.

Two broader implications follow. For swarm robotics, the results support the view that stigmergic coordination—agents modifying a shared medium through local acts—can be engineered into digital artifacts (tiles, counters, versions) that preserve the desirable self-stabilizing behavior seen in biological systems. ²¹ Future work will prioritize (i) quantitative ablations over larger teams and terrains to chart the frontier between B-connectivity windows and consensus time, (ii) integration of minimal relative-pose constraints across neighbors to further reduce heading variance without leaving fixed-point arithmetic, and (iii) evaluation in non-planar settings with lightweight height sensing. Collectively, the findings indicate that stigmergic, hardware-aware cooperative SLAM is a viable design point for dense swarms, preserving critical geometry, converging under local communications, and meeting strict resource budgets while leaving a clear path to scaling and generalization.