Efficient Homotopy-Aware Multi-Agent Path Planning in the Plane

Core Concepts

An efficient framework using Dynnikov coordinates for homotopy-aware multi-agent path planning in the plane, which can generate homotopically distinct solutions.

Abstract

The paper proposes an efficient framework for homotopy-aware multi-agent path planning in the plane. The key ideas are:

Calculating homotopy classes as solutions of an unlabeled multi-agent path planning problem, where each goal is reached by only one agent.
Using the Dynnikov coordinates, a representation of braid group elements by tuples of integers, to efficiently maintain the homotopy-augmented graph.

By combining this framework with revised prioritized planning, the authors obtained a method to generate homotopically distinct solutions of multi-agent path planning in the plane. They proved the completeness of this method for grid cases under certain assumptions.
The authors experimentally demonstrated that the runtime of their method grows approximately quadratic with respect to the number of agents. They also showed that optimizing homotopically distinct solutions generated by their method leads to finding low-cost trajectories.

Stats

We propose an efficient framework using the Dynnikov coordinates for homotopy-aware multi-agent path planning in the plane.
Our method can generate homotopically distinct solutions of multi-agent path planning problem in the plane by combining our framework with revised prioritized planning and proved its completeness in the grid world under specific assumptions.
Experimentally, we demonstrated the scalability of our method for the number of agents.
We also confirmed the usefulness of homotopy-awareness by showing experimentally that generation of homotopically distinct solutions by our method contributes to planning low-cost trajectories for a swarm of agents.

Quotes

Homotopy is a straightforward topological feature of paths, but difficult to calculate due to its non-abelian nature.
Whereas the homotopy group for (labeled) multi-agent path planning in the plane is the pure braid group, for unlabeled multi-agent path planning, it is the (full) braid group.
The Dynnikov coordinates were introduced by Dynnikov [2002].

Key Insights Distilled From

Homotopy-Aware Multi-Agent Path Planning in Plane

by Kazumi Kasau... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2310.01945.pdf

Deeper Inquiries

How can the proposed framework be extended to handle scenarios with obstacles, where the size of agents cannot be ignored

To extend the proposed framework to handle scenarios with obstacles where the size of agents cannot be ignored, several modifications and enhancements can be implemented. One approach could involve incorporating collision avoidance algorithms to ensure that agents do not collide with obstacles or each other. This could involve adapting existing collision detection and avoidance techniques, such as potential fields or velocity obstacles, to work in conjunction with the homotopy-aware planning framework. Additionally, the size of the agents would need to be taken into account when calculating paths to navigate around obstacles. This could involve defining the size of each agent as a parameter in the planning process and adjusting the paths accordingly to avoid collisions. By integrating obstacle avoidance strategies and considering the physical dimensions of the agents, the framework can be extended to handle scenarios with obstacles effectively.

Can the homotopy-aware planning be combined with efficient optimal approaches, such as increasing-cost tree search or reduction-based methods, to achieve both optimality and homotopy-awareness

Homotopy-aware planning can be combined with efficient optimal approaches to achieve both optimality and homotopy-awareness in multi-agent path planning scenarios. One potential approach is to integrate increasing-cost tree search algorithms with the homotopy-aware framework. By incorporating the concept of homotopy classes into the cost function of the tree search algorithm, the planner can prioritize paths that not only minimize cost but also explore different homotopy classes. This integration would allow the planner to generate optimal solutions while ensuring that they are homotopically distinct. Additionally, reduction-based methods can be enhanced to incorporate homotopy-awareness by considering the topological features of paths in the reduction process. By combining these optimal approaches with homotopy-aware planning, planners can achieve a balance between optimality and topological awareness in multi-agent path planning tasks.

What decentralized control strategies can be developed by leveraging the braid group representation of agent trajectories to enable effective coordination with minimal communication

Leveraging the braid group representation of agent trajectories opens up possibilities for developing decentralized control strategies that enable effective coordination with minimal communication. One strategy could involve assigning each agent a specific braid representation that encodes its trajectory and interactions with other agents. By sharing only the braid representations with neighboring agents, each agent can autonomously navigate and coordinate its movements based on the shared braid information. This decentralized approach allows agents to synchronize their actions without the need for explicit communication of detailed trajectory information. Additionally, the braid group structure can facilitate the prediction of future agent trajectories based on their current braid representations, enabling agents to anticipate and adapt to the movements of others in a coordinated manner. By leveraging the braid group representation, decentralized control strategies can achieve efficient coordination and collaboration among multiple agents in complex environments.