[Paper Review] Path Planning in Unknown Environments Using Optimal Transport Theory

Haoyan Zhai, Magnus Egerstedt+1 more|arXiv (Cornell University)|Sep 25, 2019

Robotic Path Planning Algorithms50 references4 citations

TL;DR

This paper proposes a deterministic, graph-based path planning algorithm for unknown environments using optimal transport theory and the Fokker-Planck equation (FPE). It incrementally builds a potential-guided graph from the start toward the target, dynamically updating with new obstacle information, and guarantees completeness—finding a feasible path in finite time when one exists, even in high-dimensional spaces.

ABSTRACT

This paper introduces a graph-based, potential-guided method for path planning problems in unknown environments, where obstacles are unknown until the robots are in close proximity to the obstacle locations. Inspired by optimal transport theory, the proposed method generates a graph connecting the initial and target configurations, and then finds a path over the graph using the available environmental information. The graph and path are updated iteratively when newly encountered obstacle information becomes available. The resulting method is a deterministic procedure proven to be complete, i.e., it is guaranteed to find a feasible path, when one exists, in a finite number of iterations. The method is scalable to high-dimensional problems. In addition, our method does not search the entire domain for the path, instead, the algorithm only explores a sub-region that can be described by the evolution of the Fokker-Planck equation. We demonstrate the performance of our algorithm via several numerical examples with different environments and dimensions, including high-dimensional cases.

Motivation & Objective

• Address the challenge of path planning in unknown environments where obstacles are only detected upon proximity.
• Overcome limitations of existing methods such as livelock in re-planning, high computational cost in high dimensions, and local minima in potential fields.
• Develop a deterministic, complete algorithm that avoids exhaustive domain search by focusing on relevant sub-regions via FPE dynamics.
• Ensure scalability and efficiency by restricting exploration to regions governed by the Fokker-Planck equation’s evolution.
• Provide theoretical guarantees on convergence and path feasibility using optimal transport theory and gradient flow in Wasserstein space.

Proposed method

• Uses a potential function to guide graph generation, with nodes added based on minimizing potential energy in the configuration space.
• Employs a region-growing strategy rooted in the Fokker-Planck equation (FPE), which models the evolution of probability distributions under optimal transport.
• Constructs a tree-structured graph starting from the initial configuration, expanding toward the target using a gradient descent-like procedure on the potential function.
• Dynamically updates the graph and path when new obstacle data is detected, ensuring adaptability to changing environments.
• Limits exploration to a sub-region described by the FPE’s support, avoiding full-space search and improving computational efficiency.
• Integrates optimal transport theory to ensure escape from local minima by leveraging the gradient flow structure of the FPE in Wasserstein space.

Experimental results

Research questions

• RQ1 Can a path planning algorithm in unknown environments guarantee completeness without exhaustive domain search?
• RQ2 How can optimal transport theory be leveraged to guide graph-based pathfinding while avoiding local minima?
• RQ3 What is the role of the Fokker-Planck equation in restricting the search space to a computationally tractable sub-region?
• RQ4 How can the algorithm maintain determinism and completeness while incrementally incorporating new obstacle information?
• RQ5 Can the method scale effectively to high-dimensional configuration spaces common in multi-robot systems?

Key findings

• The algorithm is proven to be complete: it guarantees to find a feasible path in finite iterations if one exists in the known environment.
• The graph grows linearly with respect to the dimension of the configuration space, enabling scalability to high-dimensional problems.
• The search is confined to a sub-region governed by the Fokker-Planck equation’s support, significantly reducing computational burden compared to full-space exploration.
• The method avoids local minima due to the underlying gradient flow structure of the FPE in Wasserstein space, ensuring convergence to global minimizers.
• The path is optimal with respect to the currently known environment at each planning step, and the algorithm detects infeasibility when no path exists.
• Numerical experiments demonstrate robust performance across various 2D and high-dimensional environments, including narrow passages and complex obstacle layouts.