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Safe Decentralized Density Control of Multi-Robot Systems using PDE-Constrained Optimization with State Constraints

Longchen Niu, Gennaro Notomista

TL;DR

The paper addresses safe, scalable density control for multi-robot swarms under localization and motion noise by modeling the swarm density $\rho(r,t)$ with a PDE and enforcing safety via a CLF-CBF optimization framework. It develops a decentralized OBC that distributes safety constraints over local neighborhoods, using a worst-case neighbor prediction to maintain feasibility and forward invariance of the safety set. The discretized PDE-based controller yields a strictly convex quadratic program with a unique solution, and the authors prove a sufficient local-to-global safety condition (Proposition) ensuring global safety from local constraints. Empirical validation through simulations and quadcopter experiments demonstrates real-time performance, robustness to disturbances, and practical applicability to density coverage tasks.

Abstract

In this paper, we introduce a decentralized optimization-based density controller designed to enforce set invariance constraints in multi-robot systems. By designing a decentralized control barrier function, we derived sufficient conditions under which local safety constraints guarantee global safety. We account for localization and motion noise explicitly by modeling robots as spatial probability density functions governed by the Fokker-Planck equation. Compared to traditional centralized approaches, our controller requires less computational and communication power, making it more suitable for deployment in situations where perfect communication and localization are impractical. The controller is validated through simulations and experiments with four quadcopters.

Safe Decentralized Density Control of Multi-Robot Systems using PDE-Constrained Optimization with State Constraints

TL;DR

The paper addresses safe, scalable density control for multi-robot swarms under localization and motion noise by modeling the swarm density with a PDE and enforcing safety via a CLF-CBF optimization framework. It develops a decentralized OBC that distributes safety constraints over local neighborhoods, using a worst-case neighbor prediction to maintain feasibility and forward invariance of the safety set. The discretized PDE-based controller yields a strictly convex quadratic program with a unique solution, and the authors prove a sufficient local-to-global safety condition (Proposition) ensuring global safety from local constraints. Empirical validation through simulations and quadcopter experiments demonstrates real-time performance, robustness to disturbances, and practical applicability to density coverage tasks.

Abstract

In this paper, we introduce a decentralized optimization-based density controller designed to enforce set invariance constraints in multi-robot systems. By designing a decentralized control barrier function, we derived sufficient conditions under which local safety constraints guarantee global safety. We account for localization and motion noise explicitly by modeling robots as spatial probability density functions governed by the Fokker-Planck equation. Compared to traditional centralized approaches, our controller requires less computational and communication power, making it more suitable for deployment in situations where perfect communication and localization are impractical. The controller is validated through simulations and experiments with four quadcopters.
Paper Structure (11 sections, 2 theorems, 15 equations, 5 figures)

This paper contains 11 sections, 2 theorems, 15 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work and Contributions
  3. Mathematical Background
  4. Decentralized Control
  5. Decentralization Formulation
  6. Safety Guarantee
  7. Numerical Implementation
  8. Application and Results
  9. Simulation
  10. Quadcopter Experiment
  11. Conclusion

Key Result

Proposition 1

Let each robot $i \in \{1, \dots ,N\}$ implement the proposed controller eq: opt_main. Then, if every robot satisfies the bound below (e.g., by tuning $\beta, \beta_i, D, \epsilon \dots$) then the global CBF condition, $\beta h + \dot h \geq 0,$ is guaranteed to hold.

Figures (5)

  • Figure 1: Schematic for the safe decentralized OBC. Each robot $i$ computes the worst-case safety threat, $\lambda_i$, from its neighbors, and the maximum safety improvement, $\dot h_i^\mathbf{self}$, by itself. These parameters determine if a slack $\delta_i$ is necessary for the CBF feasibility in controller \ref{['eq: opt_main']}.
  • Figure 2: Time sequence of the simulation. The star indicates the center of the target, green is the target density (darker is higher), and red is the invariance area. Blue represents robots' positions with trajectory trails, purple represents communication links, and the dashed contour is the robot's PDF.
  • Figure 3: Density matching, Safety, and Bound Plots. Lower $V$ value represents better target tracking. Higher $h$ corresponds to being further from the danger area. The right plot shows the terms in bound \ref{['eq: final bound swarm']}.
  • Figure 4: (a) Experiment setup. Quadcopters in yellow, $\mathcal{A}$ in red, green X-axis, red Y-axis, and the green star is the center of $\rho_d$. (b) Time sequence of the experiment. The star indicates the center of the target, green is the target density (darker is higher), and red is the invariance area. Blue represents robots' positions with trajectory trails, purple communication links, yellow control commands, and the dashed contour is the robot's PDF.
  • Figure 5: Density matching, Safety, and Bound Plots. Lower $V$ value represents better target tracking. Higher $h$ corresponds to being further from the danger area. The right plot shows the terms in bound \ref{['eq: final bound swarm']}.

Theorems & Definitions (8)

  • Proposition 1: Sufficient Condition for Local-to-Global Safety Guarantee
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 2: Existence, Uniqueness, and Safety of the Discretized Decentralized Controller
  • proof
  • Remark 4