Decentralized Coverage Control

• Decentralized coverage control is a framework that uses distributed algorithms for mobile agents to achieve full or optimal sensor coverage based solely on local data.
• It leverages geometric tessellation, grid-based alignment, and gradient descent methods to ensure efficient deployment in both 2D and 3D spaces under communication constraints.
• The approach enhances robustness and scalability by integrating consensus protocols, randomized strategies, and learning-based controllers to reduce communication overhead and improve convergence.

Decentralized coverage control is the paper and synthesis of distributed algorithms enabling a population of mobile agents—typically robots or sensors constrained to use only local sensing, computation, and inter-agent communication—to collectively achieve complete or optimal coverage of a spatial domain. The objective ranges from full sensor deployment (every point contained within a sensing footprint) to optimal allocation for density-weighted sensing or service, typically under communication, actuation, and robustness constraints. Over the last two decades, decentralized coverage has unified approaches drawn from geometric partitioning, randomization, graph consensus, optimization, and learning, with rigorous analysis of convergence, communication complexity, and scalability.

1. Canonical Objectives and Problem Formulation

The general decentralized coverage problem requires a set of agents to deploy in a domain $M$ to satisfy a performance metric (e.g., complete coverage, density-weighted minimization), subject to only local information. Canonical settings include:

• Complete geometric coverage: Deploy $n$ identical mobile sensors in $M\subset\mathbb{R}^d$ so that $\forall\,p\in M$, $\exists\,i:\,\|p-p_i(\infty)\|\leq R_s$. This strict requirement is central in 3D sensor networks (e.g., underwater, aerial) (Nazarzehi, 2016).
• Locational cost minimization: Minimize

$$H(P) = \sum_{i=1}^N \int_{V_i(P)} \phi(x)\|x-p_i\|^2 dx,$$

where $V_i(P)$ is the Voronoi partition for agent $i$ and $\phi$ an importance density (Kia et al., 18 Jul 2024).

• k-order coverage: Extend coverage to require $k\leq n$ agents per point, defining $k$-order Voronoi diagrams and associated “dominant regions” $W_i$ (Tabatabai et al., 2019).
• Dynamic or multi-objective coverage: Address time-varying densities $\phi(x,t)$ (ergodic control, Nash equilibria) (Abraham et al., 2018) or multiple concurrent densities (fair/constrained coverage) (Cervino et al., 17 Sep 2024).

A fundamental constraint is decentralization: each agent’s control law is a function only of its local observations and neighbor messages. Connectivity is often encoded by a (dynamic) graph $G=(V,E)$ where $(i,j)\in E$ iff $\|p_i-p_j\| \leq R_c$.

2. Geometric Tessellations and Spatially Optimal Grids

A key mechanism in decentralized coverage—especially for complete domain coverage—is geometric tessellation. In 2D, equilateral triangular (T) grids achieve minimal vertex count for strict coverage due to centroidal worst-case point conditions (Yang, 2017). In 3D, among space-filling polyhedra, the truncated octahedron (TO) grid provides the maximal volumetric quotient $V_p/V_s\approx0.68$, with grid edge $L=2R_s/\sqrt{5}$ ensuring the circumscribed sphere equals the sensor range (thus minimizing redundant coverage) (Nazarzehi, 2016). The vertex set is parameterized as

$$v(\alpha_1,\alpha_2,\alpha_3) = \left(x_0 + (2\alpha_1 + \alpha_3)\tfrac{2R_s}{\sqrt{5}},\, y_0 + (2\alpha_2 + \alpha_3)\tfrac{2R_s}{\sqrt{5}},\, z_0 + \alpha_3\tfrac{2R_s}{\sqrt{5}}\right),\,\, \alpha_j\in\mathbb{Z}.$$

Grid choice, both in 2D and 3D, underpins almost all coverage-control formulations where pure partitioning or grid-based allocation is possible, and is essential for analytic convergence proofs and sensor-density minimization.

3. Decentralized Algorithms and Local Information Exchange

3.1 Randomized Grid-Based Approaches

To achieve complete coverage, grid-based approaches use two main steps:

1. Consensus on grid alignment: Each agent maintains a local estimate $q_i$ of a grid origin and synchronizes via consensus iterations

$$X_i(k+1) = \frac{X_i(k) + \sum_{j\in N_i(k)} X_j(k)}{1 + |N_i(k)|},$$

with similar updates for $Y_i$, $Z_i$ (Nazarzehi, 2016). Uniform joint connectivity ensures almost-sure convergence to a common origin.

1. Vertex occupation via random walk: After consensus, each robot moves to its nearest unoccupied grid vertex and randomly spreads to unoccupied neighbors, using local communication to maintain a Boolean occupancy map. With at least as many robots as grid vertices, the system is absorbed in a complete coverage state almost surely in finite time due to an absorbing Markov chain structure.

This architecture generalizes readily to obstacle-free domains and higher dimensions (hexagonal prism, rhombic dodecahedron, etc.) (Yang, 2017).

3.2 Metric Gradient-Based Decentralized Laws

For optimal coverage with respect to a density $\phi$, agents adopt a negative-gradient controller

$$\dot p_i = -k\frac{\partial H}{\partial p_i} = -2k m_i(p_i-C_{V_i}),$$

with $m_i$ the mass of $V_i$ and $C_{V_i}$ its centroid. This law is decentralized: each agent requires only its position, the positions of Voronoi neighbors, and local information about $\phi$ over its own cell. The standard Lloyd algorithm operates in this regime, with guaranteed convergence to centroidal Voronoi tessellations (Kia et al., 18 Jul 2024).

When obstacles and local information are dominant, spatial decomposition and buffered Voronoi cells—computed via point cloud spherical mirroring and local convex hulls—ensure decentralized, collision-free, and deadlock-resistant policies (Wang et al., 4 Mar 2024).

3.3 Consensus and Weighted-Averaging Methods

In contexts such as coordinated sweeping or orientation control, agents use local nearest-neighbor averaging (Vicsek-type rules) and cluster/graph-based similarity weighting. Global consensus is achieved under weak ergodicity (second eigenvalue modulus $<1$) (Ahmad, 2016), and similarity-weighted updates halve convergence iterations compared to vanilla averaging.

4. Communication Complexity, Triggering, and Robustness

Communication efficiency is critical. Self-triggered update laws, as in $k$-order coverage, allow agents to accumulate bounded uncertainty about neighbors’ positions, exchanging messages only when the uncertainty ball exceeds a descent guarantee

$$\text{if } \text{bnd}_i(t) \geq \max\{\|p_i-C_{g_i}\|, \epsilon\} \text{ then trigger communication}.$$

This achieves an $\gtrsim80\%$ reduction in messages and communication power with negligible effect on final coverage quality (Tabatabai et al., 2019).

Absorbing Markov chain models ensure robustness to message loss and agent failure: so long as neighbor communication graphs remain connected on average and agent count $n\geq |{\rm grid}|$, convergence remains almost sure (Nazarzehi, 2016). For energy-constrained agents, hybrid switching logic using local coverage gradients and “request-to-charge” policies ensures global task progress and no agent runs out of charge, with bounded price of decentralization relative to centralized schemes (Meng et al., 2020).

5. Generalizations: Learning-Based and Multi-Objective Decentralized Coverage

5.1 Imitation- and Learning-Based Decentralized Policies

Recent approaches employ neural architectures—e.g., CNN–GNN–MLP pipelines (LPAC), spatial transformers (MADP)—to enable scalable, fully distributed action selection (Agarwal et al., 10 Jan 2024, Vatnsdal et al., 21 Sep 2025). Each agent fuses a local importance map, neighbor features, and absolute/relative coordinates via learned aggregators, imitating centralized Lloyd or CVT experts. Such policies generalize to team-size and environment distribution shifts, and outperform Lloyd or classical greedy policies in real and simulated benchmarks.

5.2 Multi-Objective, Constrained, and Robust Coverage

The multi-objective setting posits $M$ heterogeneous IDFs $\phi_1,\ldots,\phi_M$ and seeks, for example, to minimize the worst-case coverage cost ("fair coverage") or constrain each field’s cost below a threshold ("constrained coverage") (Cervino et al., 17 Sep 2024). Dual decomposition reduces these to single-field coverage with dynamically reweighted combinations $\phi_\lambda=\sum \lambda_m\phi_m$, solvable by a standard decentralized controller (e.g., LPAC). Primal-dual update rules on the Lagrangian multipliers, implemented with only local communication, ensure feasibility and 30–50% cost reduction over baseline decentralized CVT.

Robustness can also be imposed structurally: bearing-rigidity constraints are encoded in decentralized model-predictive controllers, maintaining formation integrity and enabling rapid recovery from robot loss or adversarial errors (Pant et al., 23 Feb 2025).

6. Implementation, Performance, and Design Considerations

The implementation of decentralized coverage control algorithms requires careful tradeoffs between sensor/communication range, communication frequency, computation per agent, and required convergence time. Key practical guidelines include (Nazarzehi, 2016):

• Set sensing range and communication radius so that the communication graph remains connected; e.g., truncated octahedron: $R_c \geq (4/\sqrt{5})R_s \approx 1.79R_s$.
• Choose grid size or control gain based on robot hardware and domain dimensions.
• Round-synchronized operation and message exchange are assumed in most consensus and grid-based schemes.
• Communication and computation per agent scale as $O(|N_{\text{neighbors}}|)$ per round for standard algorithms; for LPAC and MADP architectures, the per-agent load is dominated by 5–8 GNN or attention layers and a shallow MLP (Agarwal et al., 10 Jan 2024, Vatnsdal et al., 21 Sep 2025).

Performance evaluation shows that optimal grid selection (e.g., truncated octahedron) yields not only theoretical but empirical improvement in number of steps to coverage (e.g., 17 for TO, 28 for cubic, 22–23 for other polyhedra, with 100 agents in a $10R_s$ cube). Limitations include fragility to obstacles unless explicitly handled, sensitivity to robot density and range, and, for learning-based controllers, degraded performance out-of-distribution.

7. Open Problems and Directions

Although decentralized coverage control is theoretically mature for strict, static, obstacle-free domains, major open problems remain:

• Real-world non-convexity is only partially addressed by conformal mapping and sectorial partitioning, which can enforce load balance and Riemannian centroid-seeking even in multiply connected domains (Feng et al., 7 Feb 2025).
• Learning-based controllers generalize better than hand-crafted rules, but still require large-scale data from (centralized) experts, and transfer guarantees remain empirical (Agarwal et al., 10 Jan 2024, Vatnsdal et al., 21 Sep 2025).
• Full integration of exploration–exploitation tradeoffs (e.g., through GP-UCB inspired cost functionals) and time-varying, partially observed environments is a current frontier (Guidone et al., 4 Nov 2025).
• Quantification of the price of decentralization and formal regret bounds for continuous, spatially distributed teams are active areas of research.

Decentralized coverage control thus operates at the interface of geometry, optimization, stochastic processes, distributed consensus, and learning, with a continuing trajectory toward robustness, scalability, and formal performance guarantees in heterogeneous and dynamic environments.