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Game-Theoretic Safe Multi-Agent Motion Planning with Reachability Analysis for Dynamic and Uncertain Environments (Extended Version)

Wenbin Mai, Minghui Liwang, Xinlei Yi, Xiaoyu Xia, Seyyedali Hosseinalipour, Xianbin Wang

TL;DR

This work addresses safe, scalable multi-agent motion planning under dynamics and uncertainty by fusing reachability analysis with a dynamic potential game, yielding decentralized NE approximations via ND-iBR. It introduces MA-FRS to explicitly model uncertainty propagation and collision margins, embedding reachability-based penalties into the agents’ costs. The resulting framework achieves finite-step convergence to an $\varepsilon$-NE while maintaining safety under bounded disturbances, as validated by 2D/3D simulations and real-world 4-vehicle experiments with MPC execution. The approach offers robust, collision-free coordination in dynamic environments and provides a practical path toward decentralized, safety-guaranteed multi-agent planning in real-world systems.

Abstract

Ensuring safe, robust, and scalable motion planning for multi-agent systems in dynamic and uncertain environments is a persistent challenge, driven by complex inter-agent interactions, stochastic disturbances, and model uncertainties. To overcome these challenges, particularly the computational complexity of coupled decision-making and the need for proactive safety guarantees, we propose a Reachability-Enhanced Dynamic Potential Game (RE-DPG) framework, which integrates game-theoretic coordination into reachability analysis. This approach formulates multi-agent coordination as a dynamic potential game, where the Nash equilibrium (NE) defines optimal control strategies across agents. To enable scalability and decentralized execution, we develop a Neighborhood-Dominated iterative Best Response (ND-iBR) scheme, built upon an iterated $\varepsilon$-BR (i$\varepsilon$-BR) process that guarantees finite-step convergence to an $\varepsilon$-NE. This allows agents to compute strategies based on local interactions while ensuring theoretical convergence guarantees. Furthermore, to ensure safety under uncertainty, we integrate a Multi-Agent Forward Reachable Set (MA-FRS) mechanism into the cost function, explicitly modeling uncertainty propagation and enforcing collision avoidance constraints. Through both simulations and real-world experiments in 2D and 3D environments, we validate the effectiveness of RE-DPG across diverse operational scenarios.

Game-Theoretic Safe Multi-Agent Motion Planning with Reachability Analysis for Dynamic and Uncertain Environments (Extended Version)

TL;DR

This work addresses safe, scalable multi-agent motion planning under dynamics and uncertainty by fusing reachability analysis with a dynamic potential game, yielding decentralized NE approximations via ND-iBR. It introduces MA-FRS to explicitly model uncertainty propagation and collision margins, embedding reachability-based penalties into the agents’ costs. The resulting framework achieves finite-step convergence to an -NE while maintaining safety under bounded disturbances, as validated by 2D/3D simulations and real-world 4-vehicle experiments with MPC execution. The approach offers robust, collision-free coordination in dynamic environments and provides a practical path toward decentralized, safety-guaranteed multi-agent planning in real-world systems.

Abstract

Ensuring safe, robust, and scalable motion planning for multi-agent systems in dynamic and uncertain environments is a persistent challenge, driven by complex inter-agent interactions, stochastic disturbances, and model uncertainties. To overcome these challenges, particularly the computational complexity of coupled decision-making and the need for proactive safety guarantees, we propose a Reachability-Enhanced Dynamic Potential Game (RE-DPG) framework, which integrates game-theoretic coordination into reachability analysis. This approach formulates multi-agent coordination as a dynamic potential game, where the Nash equilibrium (NE) defines optimal control strategies across agents. To enable scalability and decentralized execution, we develop a Neighborhood-Dominated iterative Best Response (ND-iBR) scheme, built upon an iterated -BR (i-BR) process that guarantees finite-step convergence to an -NE. This allows agents to compute strategies based on local interactions while ensuring theoretical convergence guarantees. Furthermore, to ensure safety under uncertainty, we integrate a Multi-Agent Forward Reachable Set (MA-FRS) mechanism into the cost function, explicitly modeling uncertainty propagation and enforcing collision avoidance constraints. Through both simulations and real-world experiments in 2D and 3D environments, we validate the effectiveness of RE-DPG across diverse operational scenarios.

Paper Structure

This paper contains 19 sections, 2 theorems, 33 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Relevant Investigations
  3. Game-based Multi-Agent Motion Planning
  4. Safety-Aware Motion Planning under Uncertainty
  5. Novelty and Contribution
  6. Preliminaries
  7. Notations and Definitions
  8. Overview and Key Modeling Components
  9. Safe Multi-Agent (MA) Motion Planning via Integrating Forward Reachable Set (FRS) into Dynamic Potential Game (DPG)
  10. DPG Formulation for Multi-Agent Motion Planning
  11. ND-iBR-based Problem Transformation
  12. Cost Enhancement via Reachability Analysis
  13. Tracking Cost
  14. Velocity Constraint Penalty
  15. Collision Penalty with MA-FRS
  16. ...and 4 more sections

Key Result

Theorem 1

Consider the above DPG with a set of agents $\mathcal{N}$. For each player $i\in\mathcal{N}$, let their strategy set be nonempty and compact. Suppose $J_i$ admits the decomposition eq:Ji where $L_i(\cdot), L_i^\text{F}(\cdot)$ are both continuous, and eq:Li_and_Lif_decompose also holds. Fix $\vareps and set $\mathbf u_i^{k+1}$ as: $\mathbf u_i^{k+1}=\arg\min_{\mathbf u_i} J_i(\boldsymbol{x}^{(0)},

Figures (8)

  • Figure 1: System overview of RE-DPG. Initial states, goal states, and neighborhood information are processed by the MA-FRS and DPG formulation modules, where safety-aware cost functions are established. The ND-iBR module then iteratively refines the planned nominal trajectories by incorporating neighboring agents' strategies. Note that the planning and tracking of nominal trajectories are decoupled. The finalized trajectories are executed using a receding horizon MPC controller to handle disturbances, while a state estimation mechanism continuously provides feedback to ensure safe trajectory execution in dynamic environments.
  • Figure 2: Examples of 2D and 3D ellipsoidal FRSs along the trajectories of two double integrator agents. The green ellipsoids represent the Minkowski sum of the agents’ FRSs at the end of their trajectories.
  • Figure 3: Pairwise distances between agents regarding the settings of 7 agents and 12 agents related to Fig. \ref{['fig:agent7']} and Fig. \ref{['fig:agent12']}, respectively. The dashed line at the bottom of each plot corresponds to the desired distance threshold.
  • Figure 4: Comparison results across three evaluation metrics with varying numbers of agents and disturbance levels.
  • Figure 5: Scenario 1: Snapshots and visualization.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Definition 1: Minkowski Sum
  • Definition 2: Ellipsoids in $n$-Dimensional Space
  • Definition 3: Translation of an Ellipsoid
  • Definition 4: Ellipsoidal Approximation of the Minkowski Sum for Concentric Ellipsoids seo2019robust
  • Definition 5: Open-Loop Nash equilibrium (NE)
  • Definition 6: DPG for Multi-Agent Motion Planning
  • Theorem 1: Convergence of i$\varepsilon$-BR
  • Remark 1: Practical sufficiency of $\varepsilon$-NE and local minimum
  • Remark 2: Neighbor screening and computational benefits
  • Theorem 2: Ellipsoidal intersection criterion
  • ...and 1 more