Scalable Asymptotically-Optimal Multi-Robot Motion Planning

Abstract: Finding asymptotically-optimal paths in multi-robot motion planning problems could be achieved, in principle, using sampling-based planners in the composite configuration space of all of the robots in the space. The dimensionality of this space increases with the number of robots, rendering this approach impractical. This work focuses on a scalable sampling-based planner for coupled multi-robot problems that provides asymptotic optimality. It extends the dRRT approach, which proposed building roadmaps for each robot and searching an implicit roadmap in the composite configuration space. This work presents a new method, dRRT* , and develops theory for scalable convergence to optimal paths in multi-robot problems. Simulated experiments indicate dRRT* converges to high-quality paths while scaling to higher numbers of robots where the naive approach fails. Furthermore, dRRT* is applicable to high-dimensional problems, such as planning for robot manipulators

• The paper introduces a novel sampling-based planner that uses implicit tensor product roadmaps to achieve asymptotically optimal multi-robot motion planning.
• The approach leverages an oracle-guided expansion with Voronoi regions to efficiently explore high-dimensional composite configuration spaces, ensuring scalability.
• Experimental results show that the planner outperforms traditional methods in path cost and computational efficiency, handling up to a dozen robots simultaneously.

Scalable Asymptotically-Optimal Multi-Robot Motion Planning

Introduction

The paper "Scalable Asymptotically-Optimal Multi-Robot Motion Planning" (1706.09932) addresses the complexities involved in multi-robot motion planning within high-dimensional spaces. Traditional approaches either focus on constructing explicit roadmaps in the composite space of all robots or tackle the problem through decoupled planning strategies. The primary challenge these methods face is scalability due to the exponential growth in computational requirements with increasing numbers of robots. The paper introduces a novel sampling-based planner that extends previous methodologies to achieve asymptotically optimal solutions while remaining scalable.

Problem Formulation

In the multi-robot planning problem, multiple robots must navigate a shared workspace while avoiding collisions with both obstacles and other robots. Each robot operates within its configuration space, and the challenge is to find a trajectory that optimizes a chosen cost function. The cost function examined in this paper primarily considers the total path length across all robots. The paper rigorously defines the conditions under which the proposed method can find optimal paths and establish the robustness of the solutions.

Proposed Methodology

The cornerstone of the proposed solution is an innovative use of implicit tensor product roadmaps to explore the composite configuration space of multiple robots efficiently. The authors build on previous work to construct individual roadmaps for each robot and perform searches within the implicit tensor product of these roadmaps, rather than explicitly constructing the entire high-dimensional composite roadmap. This implicit roadmap search enables scalability and asymptotic optimality, addressing both the memory and computational challenges faced by traditional methods.

Oracle-Guided Expansion

An important component of the proposed planner is an oracle function used during the expansion phase of the roadmap. This function leverages the notion of Voronoi regions in configuration space to guide sampling and expansion towards the most promising regions. The paper presents a detailed description of the oracle mechanism that ensures that every expansion step can potentially lead to an optimal solution.

Theoretical Guarantees

The authors provide formal proofs to establish the asymptotic optimality of their method. The main theoretical contribution is the demonstration that the implicit tensor product roadmap can converge to paths that are arbitrarily close to the optimal one, given a sufficient number of samples and iterations. Several lemmas and theorems articulate the convergence properties of the planner, with particular focus on robustness and the efficiency of the sampling process.

Experimental Validation

The paper includes extensive experimental validation to demonstrate the practical performance and scalability of the proposed planner. Tests were conducted in various environments, including a two-dimensional polygonal space and more complex scenarios involving dual-arm robot manipulators. The results indicate that the method consistently finds high-quality solutions and scales to scenarios where other methods fail, effectively handling larger numbers of robots without compromising on path quality.

Key Findings

1. Scalability: The proposed approach manages to effectively plan for up to a dozen robots simultaneously, a task that becomes infeasible with traditional explicit roadmap methods.
2. Optimality: Solutions converged quickly to high-quality paths, often outperforming benchmark methods in terms of both path cost and computational efficiency.
3. Practicality: The method effectively operates in high-dimensional spaces typical of robotic manipulations, such as those of dual-arm robots.

Implications and Future Work

This work significantly advances the field of multi-robot planning by providing a scalable, efficient, and theoretically grounded approach. While the current implementation focuses on specific cost functions and robot dynamics, the framework can be extended to encompass more complex robot models, including kinodynamic constraints. Future explorations could also incorporate learning algorithms to enhance the sampler efficiency and further reduce convergence times in dynamic environments.

Conclusion

The paper presents a rigorous and well-structured approach to addressing the inherent complexities of multi-robot motion planning. By advancing the scalability and optimality of solutions, it paves the way for more sophisticated applications of multi-agent systems in robotics, particularly in environments requiring high-dimensional planning and operations. The theoretical contributions are backed by robust experimental results, confirming the efficacy of the proposed methodology in practical settings.