Planning Optimal Paths for Multiple Robots on Graphs

Abstract: In this paper, we study the problem of optimal multi-robot path planning (MPP) on graphs. We propose two multiflow based integer linear programming (ILP) models that computes minimum last arrival time and minimum total distance solutions for our MPP formulation, respectively. The resulting algorithms from these ILP models are complete and guaranteed to yield true optimal solutions. In addition, our flexible framework can easily accommodate other variants of the MPP problem. Focusing on the time optimal algorithm, we evaluate its performance, both as a stand alone algorithm and as a generic heuristic for quickly solving large problem instances. Computational results confirm the effectiveness of our method.

• The paper presents two ILP formulations that optimally schedule multi-robot paths by minimizing either makespan or total travel distance.
• It employs a multiflow approach to convert complex, high-dimensional path planning into tractable optimization problems, outperforming traditional heuristics.
• Experimental results on puzzles and dense grid graphs demonstrate scalability and practical applicability in various robotic coordination tasks.

Planning Optimal Paths for Multiple Robots on Graphs

This paper addresses the computationally complex problem of Multi-Robot Path Planning (MPP) on graphs, a task paramount for various domains including assembly, evacuation, formation control, and search and rescue. The authors propose two Integer Linear Programming (ILP) models based on multiflow formulations to find optimal solutions: one minimizes the time until the last robot reaches its target, and the other minimizes the total distance traveled by all robots.

The study acknowledges the intrinsic difficulty of MPP, primarily due to the coupling of robots' paths, which results in high dimensionality and expanded state space. Despite the computational intensity of these problems, especially in tightly coupled scenarios, where traditional heuristics like A* tend to falter, the proposed methods offer computational feasibility and optimality assurances by employing state-of-the-art ILP solvers.

Methodology

Two distinct ILP models form the crux of this research:

1. Time Optimal Model: This model minimizes the time taken for the last robot to reach its goal. By employing a network flow perspective, it ensures complete and optimal path solutions through iterative refinement, where time expansions of the graph determine the minimal feasible time step for achieving optimal solutions.
2. Distance Optimal Model: Here, the goal is to minimize the total path length of all robots. This formulation extends upon the time expansion approach, encoding path traversal costs into the ILP to achieve a minimal distance path plan, which can be further refined using the results from the time-optimal strategies.

Results and Evaluation

Experimental validations are conducted on various scenarios including the classical $n^2$-puzzle, grid graphs of different sizes, and obstacles interspersed environments to test scalability. Notable results include:

• The ILP-based approach successfully solves the 9-puzzle optimally in just over a second on average and extends its utility to the more computationally demanding 16-puzzle within notably quicker times than traditional BFS.
• For more extensive grid graphs up to $32 \times 32$, the models efficiently handle dense obstacle scenarios, confirming the robustness of the conflated ILP models.
• The approaches demonstrate scalability, adapting to increased problem complexity with proportional computational time which hints at potential optimization and hybridization strategies for practical applications.

Implications and Future Prospects

The implications of the research extend across robotic coordination tasks—systems where autonomous multi-agent systems must exhibit high efficiency. The solutions outline a clear path towards enhancing robotics autonomy in deterministic environments, particularly with respect to improving battery life and minimizing operational time in applications requiring precise coordination.

Future advancements in this domain could explore enhancing ILP models for real-time adaptability, employing learning-based heuristics to preemptively manage encountered obstacles, and developing hybrid methods that blend ILP strengths with domain-specific heuristics, thereby potentially bypassing some of the solver limits. Moreover, employing non-linear methods might offer further insights into the inherently dynamic and stochastic environments typical in many robotics applications.

In summary, this paper presents a rigorous examination and a novel ILP approach to multi-robot path planning, contributing substantially to the theoretical and practical landscape of robotic operations in constrained and complex environments.