Symmetry-Breaking in Multi-Agent Navigation: Winding Number-Aware MPC with a Learned Topological Strategy

In distributed multi-agent navigation without explicit communication, agents can fall into symmetry-induced deadlocks because each agent must autonomously decide how to pass others. To address this problem, we propose WNumMPC, a hierarchical navigation method that quantifies cooperative symmetry-breaking strategies via a topological invariant, the winding number, and learns such strategies through reinforcement learning. The learning-based Planner outputs continuous-valued signed target winding numbers and dynamic importance weights to prioritize critical interactions in dense crossings. Then, the model-based Controller generates collision-free and efficient motions based on the strategy and weights provided by the Planner. Simulation and real-world robot experiments indicate that WNumMPC effectively avoids deadlocks and collisions and achieves better performance than the baselines, particularly in dense and symmetry-prone scenarios. These experiments also suggest that explicitly leveraging winding numbers yields robust sim-to-real transfer with minimal performance degradation. The code for the experiments is available at https://github.com/omron-sinicx/WNumMPC.