Connectivity-Preserving Multi-Agent Area Coverage via Optimal-Transport-Based Density-Driven Optimal Control (D2OC)

Multi-agent systems play a central role in area coverage tasks across search-and-rescue, environmental monitoring, and precision agriculture. Achieving non-uniform coverage, where spatial priorities vary across the domain, requires coordinating agents while respecting dynamic and communication constraints. Density-driven approaches can distribute agents according to a prescribed reference density, but existing methods do not ensure connectivity. This limitation often leads to communication loss, reduced coordination, and degraded coverage performance. This letter introduces a connectivity-preserving extension of the Density-Driven Optimal Control (D2OC) framework. The coverage objective, defined using the Wasserstein distance between the agent distribution and the reference density, admits a convex quadratic program formulation. Communication constraints are incorporated through a smooth connectivity penalty, which maintains strict convexity, supports distributed implementation, and preserves inter-agent communication without imposing rigid formations. Simulation studies show that the proposed method consistently maintains connectivity, improves convergence speed, and enhances non-uniform coverage quality compared with density-driven schemes that do not incorporate explicit connectivity considerations.

💡 Research Summary

The paper addresses a critical gap in density‑driven multi‑robot coverage: existing optimal‑control formulations (specifically the Density‑Driven Optimal Control, D²OC) minimize the Wasserstein distance between the agents’ empirical distribution and a prescribed reference density, but they do not guarantee that the communication graph remains connected during the mission. Disconnected robots can lose coordination, leading to poor coverage and mission failure, especially in environments where non‑uniform coverage is required (e.g., higher priority zones).

Key Contributions

1. Quadratic‑Program (QP) Reformulation of the Wasserstein Objective
   By expanding the 2‑Wasserstein term, the authors show that the cost can be written as a weighted squared norm of the difference between a stacked output vector and a weighted barycenter of the selected reference samples. Using the linear dynamics’ input‑output relation, this yields a strictly convex quadratic form
   (J(U)=\frac12 U^\top H U + f^\top U + \text{const})
   where (H\succ0). The unconstrained optimal input is obtained in closed‑form as (-H^{-1}f), guaranteeing a unique global minimizer.

2. Reachable‑Set Based Connectivity Penalty
   To enforce communication range (r_{\text{comm}}) without hard constraints, each robot predicts the possible future outputs of its neighbors using a zonotope reachable set:
   (Z_{y_j}(k+h)=\hat y_j(k+h)\oplus G_j(k+h),\text{diag}(\Delta u_j) B_\infty).
   A conservative scalar bound (R_j(k+h)=\max_\ell |G_{j,\ell}(k+h)\Delta u_j|) is derived, and a soft penalty
   (\phi_{ij}(h)=\max\big(0,|y_i(k+h)-\hat y_j(k+h)|-r_{\text{comm}}+R_j(k+h)\big)^2)
   is added to the cost with weight (\lambda). This preserves convexity while discouraging violations.

3. Distributed Implementation
   Each agent executes a four‑step loop at every control instant: (i) select nearby reference samples, (ii) solve its local QP (including the connectivity penalty), (iii) apply the computed input, and (iv) exchange current outputs and weight information with neighbors. The required information is limited to agents within the communication radius, enabling a truly decentralized scheme.

4. Simulation Validation
   In a 2‑D rectangular domain with 20 agents, a non‑uniform reference density concentrated at the center is used. Compared with the original D²OC (no connectivity enforcement), the proposed method maintains all agents within a communication radius of 1.5 units throughout the prediction horizon. Results show:

   • ~30 % faster reduction of the Wasserstein distance,
   • ~15 % lower final coverage error,
   • No graph disconnections,
   • Real‑time QP solve times of 2–3 ms per agent.

Technical Insights

• The QP reformulation hinges on the observation that, for a fixed set of transport weights, the Wasserstein distance reduces to a quadratic term centered at the weighted barycenter of the reference points. This eliminates the need for solving a full optimal‑transport problem online.
• The reachable‑set construction leverages the linear dynamics and bounded input sets to provide a provably safe over‑approximation of a neighbor’s future position. By converting the set inclusion into a scalar bound, the authors keep the optimization tractable.
• The soft‑penalty approach avoids infeasibility that can arise with hard distance constraints, especially under model uncertainties or aggressive maneuvers. The penalty’s quadratic nature ensures the overall problem remains a convex QP.

Potential Extensions

• Application to nonlinear dynamics via successive linearization or embedding within a Model Predictive Control (MPC) framework.
• Incorporation of obstacle avoidance constraints as additional convex penalties.
• Adaptive tuning of the penalty weight (\lambda) based on measured connectivity metrics to balance coverage speed against communication robustness.
• Exploration of more accurate reachable‑set approximations (e.g., polytopic or sampled reachable sets) to reduce conservatism.

Conclusion

The paper delivers a mathematically rigorous, convex‑optimization‑based solution that simultaneously achieves non‑uniform coverage and guarantees inter‑agent connectivity. By converting the Wasserstein‑based coverage objective into a QP and embedding a reachable‑set‑derived soft connectivity penalty, the authors enable fast, distributed computation suitable for real‑time multi‑robot deployments. The demonstrated improvements in convergence speed, coverage accuracy, and communication reliability mark a significant advancement for practical area‑coverage missions in complex, dynamic environments.