MultiNash-PF: A Particle Filtering Approach for Computing Multiple Local Generalized Nash Equilibria in Trajectory Games

Abstract: Modern robotic systems frequently engage in complex multi-agent interactions, many of which are inherently multi-modal, meaning they can lead to multiple distinct outcomes. To interact effectively, robots must recognize the possible interaction modes and adapt to the one preferred by other agents. In this work, we propose an efficient algorithm for capturing the multimodality in multi-agent interactions. We leverage a game-theoretic planner to model interaction outcomes as equilibria where \emph{each equilibrium} corresponds to a distinct interaction \emph{mode}. We then develop an efficient algorithm to identify all the equilibria, allowing robots to reason about multiple interaction modes. More specifically, we formulate interactive planning as Constrained Potential Trajectory Games (CPTGs) and model interaction outcomes by local Generalized Nash equilibria (GNEs) of the game. CPTGs are a class of games for which a local GNE can be found by solving a single constrained optimal control problem where a potential function is minimized. We propose to integrate the potential game approach with implicit particle filtering, a sample-efficient method for non-convex trajectory optimization. We utilize implicit particle filtering to identify the coarse estimates of multiple local minimizers of the game's potential function. MultiNash-PF then refines these estimates with optimization solvers, obtaining different local GNEs. We show through numerical simulations that MultiNash-PF reduces computation time by up to 50\% compared to a baseline. We further demonstrate the effectiveness of our algorithm in real-world human-robot interaction scenarios, where it successfully accounts for the multi-modal nature of interactions and resolves potential conflicts in real-time.