Distributed Diffeomorphic Coverage Control

• Distributed diffeomorphic coverage control is a method that transforms complex, nonconvex geometries into canonical domains using smooth conformal mappings.
• It partitions the transformed workspace into convex, collision-free sectors while dynamically balancing workloads with decentralized gradient control.
• The approach guarantees stability, rapid convergence, and topological robustness, as validated by simulations with near-perfect load balancing and obstacle avoidance.

A distributed diffeomorphic coverage control framework is a class of methodologies for deploying multi-agent systems (MASs) to achieve provably optimal and load-balanced coverage of highly non-convex, annular, or multi-holed domains through the use of global or distributed conformal mappings, sectorial partitions, pull-back Riemannian metrics, and decentralized gradient-based control. This approach rigorously transforms complex geometries into tractable canonical domains where decentralized algorithms can be formulated for scalable, collision-free, and locally computable coverage allocation, guaranteeing stability and robust convergence properties (Feng et al., 7 Feb 2025, Feng et al., 20 Jan 2026).

1. Diffeomorphic (Conformal) Mapping of the Workspace

The foundational principle of the framework is the construction of a (quasi-)conformal diffeomorphism between the original non-convex or poriferous workspace and a canonical domain—typically an annulus or a multi-holed disk. For a planar annulus-type domain $$D_\mathrm{nonconvex} = (\Omega \setminus O) \cup \partial O$$, a composite diffeomorphism $f := \varpi \circ \imath \circ \theta^*$ is constructed, with each stage $g$ being $C^\infty$ with $\det J_g > 0$, ensuring smoothness and invertibility. The mapping passes through:

• A rectangular conformal map ($\theta^*$) to $[0,L^*]\times[0,1]$,
• An exponential embedding ($\imath(z) = \exp(2\pi(z-L^*))$) to the annulus,
• A final quasi-conformal automorphism ($\varpi$) to minimize angle distortion.

For surfaces of genus $0$ with $n$ holes, the distributed poly-annulus conformal mapping $\tau: S \to \Xi$ is used, involving local disk- or annulus-type conformal maps, gluing via Möbius transformations and welding, global similarity transformations for interface alignment, geometric rectification, and solution to the Laplace-Dirichlet equation by a Beltrami solver. This produces a diffeomorphic image $\Xi$ in which the coverage problem is reformulated (Feng et al., 7 Feb 2025, Feng et al., 20 Jan 2026).

2. Sectorial Partition and Workload Allocation

In the mapped (annular or poly-annular) domain $\Xi$, the workspace is partitioned into $N$ sectorial subregions via $N$ "partition bars" parameterized by angles $\psi_i$ or $\hat\psi_i$ ($i=1,\dots,N$), which define regions $\{E_i(\psi)\}$ or $\{\hat E_i(\hat\psi)\}$ between successive bars. This partition is constructed such that:

• Each subregion is convex with respect to the Riemannian metric induced via pull-back, ensuring a unique minimal geodesic between points within each $E_i$,
• The preimage of each partition under $f^{-1}$ or $\tau^{-1}$ divides the original domain into connected, non-overlapping, collision-free subregions (Feng et al., 7 Feb 2025, Feng et al., 20 Jan 2026).

On multi-holed domains, strictly collision-free partition bars are achieved by the buffer-based sequence mechanism, which imposes ordered detours around obstacles (holes) with strictly increasing buffer distances $\hat r_{k,i}$ for bar $i$ and obstacle $k$, guaranteeing that $\hat\Gamma_i \cap \hat O_k = \emptyset$ and $\hat\Gamma_i \cap \hat\Gamma_j = \emptyset$ for $i \neq j$. The workload perturbation from these geometric detours can be made arbitrarily small with $\beta\to 0$ (Feng et al., 20 Jan 2026).

3. Global Coverage Cost and Decentralized Control Law

A global cost functional is defined to quantify the quality of coverage and agent deployment. In the mapped domain, with (possibly non-uniform) density $\rho$, the cost is: $$J(\psi, p) = \sum_{i=1}^N \int_{E_i(\psi)} d_l^2(q, p_i)\; \rho(q) \, dq,$$ where $d_l$ is the distance under the pull-back Riemannian metric $\eta$. Workload per sector is $m_i = \int_{E_i} \rho(q)\, dA(q)$. To dynamically achieve equitable workload, the partition bars evolve via

$\dot\psi_i = k_\psi (m_i - m_{i-1}),$

which is the Laplacian flow on a ring graph and yields exponential convergence to balanced workloads (Feng et al., 7 Feb 2025, Feng et al., 20 Jan 2026).

Agents apply a distributed gradient-descent control law in the Riemannian metric: $$u_i = -k_p \eta^{-1}(p_i) \nabla_{p_i} J, \quad \text{or}\quad \mathbf u_i = -k_p\, \eta_i^{-1}(p_i)\, \frac{\partial J}{\partial p_i}.$$ Each agent requires only knowledge of its local subregion and adjacent bar positions for computation; no global state is necessary, making the scheme fully distributed.

Obstacle and inter-agent avoidance are encoded in the pull-back metric. For poriferous domains, an obstacle-weighted metric

$$\hat \eta_0(\hat p) = (1 + \hat \sigma_0(\hat p))^2 \mathrm{diag}(1, \hat r^2)$$

with additional dynamic weights for agent-agent separation enforces geodesic completion, barrier separation, and strictly positive definiteness on the feasible set (Feng et al., 20 Jan 2026).

4. Collision-Free Guarantees and Topological Robustness

The geometric and topological construction ensures partition bars and agent flow remain strictly collision-free for all time. The buffer-based partition bars with buffer sequence factor $\beta > 0$ enforce a minimal clearance, preventing intersection of bars and obstacles and preserving sector order throughout the evolution. The workload error induced by geometric detours is bounded by $C_{\hat\delta}\beta$ and can be suppressed arbitrarily by decreasing $\beta$ (Feng et al., 20 Jan 2026). The topological universality result confirms that any compact genus-0, $n$-holed surface supports such a partition and associated coverage dynamics via existence of $n$ independent harmonic one-forms, guaranteeing applicability across a wide class of nonconvex domains.

5. Theoretical Properties: Stability, Convergence, and Quasi-Optimality

The framework guarantees strong analytical properties:

• Existence and reachability: Compactness and continuity of $J$ ensure the existence of a global minimizer for the agent positions and partition configuration.
• Workload-balancing and Input-to-State Stability (ISS): The partition dynamics are exponentially (ISS-)stable with respect to bounded disturbances from detours, with workload imbalance $\|\mathbf e(t)\|$ converging exponentially up to a small offset determined by $\beta$.
• Agent convergence: Agents asymptotically approach the Riemannian centroids of their subregions; local exponential stability is enforced by positive-definite Riemannian Hessians at equilibrium.
• Quasi-optimality of time-discrete implementation: Discretization introduces three error sources (time step, angular quantization, geometric detour). The global cost error is bounded as $$|J(\hat\psi^*,p^*) - J(\hat\psi^A,p^A)| \le C_1(\Delta t)^2 + C_2 (2\pi/K^*) + C_3\beta + \epsilon_{\mathrm{conv}}$$, with controllable terms vanishing as their respective parameters are refined (Feng et al., 7 Feb 2025, Feng et al., 20 Jan 2026).

6. Algorithmic Implementation and Distributed Coordination

Implementation proceeds by:

1. Computing the (distributed) conformal diffeomorphism and its inverse to generate the mesh in $\Xi$.
2. Initializing bar and agent positions, then, for each quantized bar configuration, simulating coupled bar and agent updates over a time window.
3. Recording local costs and workloads, communicating among neighbors to select the configuration with minimal cost.
4. Pulling back the optimal configuration to the original space, after which agents continue to apply their control law until convergence.

The algorithm requires only local neighbor-wise communication of bar angles and local costs; all other computations (workload, position updates) are fully decentralized (Feng et al., 7 Feb 2025, Feng et al., 20 Jan 2026).

7. Performance Characterization and Empirical Validation

Extensive simulations in both nonconvex annular domains and genus-0, $n$-holed (poriferous) surfaces confirm:

• Achieving near-perfect workload balance with workload RMSE as low as $6.75\times10^{-3}$ and imbalance $<0.002$ for $N=6$ agents and $n=3$ holes.
• Rapid convergence (within $134$ iterations or $20$ seconds) under nominal operation; rapid re-partition under agent loss.
• Robust geometric obstacle avoidance and preservation of strictly connected subregions throughout simulation.
• Superior performance relative to Voronoi and control-barrier-function-based baselines in nonconvex and obstacle-rich domains, particularly in maintaining connectivity and reachability of centroids (Feng et al., 7 Feb 2025, Feng et al., 20 Jan 2026).

In summary, the distributed diffeomorphic coverage control framework synthesizes conformal geometry, decentralized sectorial partitioning, and gradient-based Riemannian control to provide provably convergent, load-balanced, and collision-avoiding MAS coverage strategies for complex planar and surface domains. The architecture is topologically universal for genus-0, $n$-holed workspaces, robust to agent failure, and supports explicit error quantification for practical deployments (Feng et al., 7 Feb 2025, Feng et al., 20 Jan 2026).