[Paper Review] Multilevel Motion Planning: A Fiber Bundle Formulation

Andreas Orthey, Sohaib Akbar+1 more|arXiv (Cornell University)|Jul 18, 2020

Robotic Path Planning Algorithms4 citations

TL;DR

This paper proposes a multilevel motion planning framework using fiber bundle theory to decompose complex robotic motion planning into hierarchical, manageable subproblems. By formulating the configuration space as a fiber bundle and using recursive section-finding algorithms over progressively coarser manifolds, the method achieves efficient path computation with reduced computational cost, demonstrating scalability and robustness in high-dimensional spaces.

ABSTRACT

High-dimensional motion planning problems can often be solved significantly faster by using multilevel abstractions. While there are various ways to formally capture multilevel abstractions, we formulate them in terms of fiber bundles. Fiber bundles essentially describe lower-dimensional projections of the state space using local product spaces, which allows us to concisely describe and derive novel algorithms in terms of bundle restrictions and bundle sections. Given such a structure and a corresponding admissible constraint function, we develop highly efficient and asymptotically-optimal sampling-based motion planning methods for high-dimensional state spaces. Those methods exploit the structure of fiber bundles through the use of bundle primitives. Those primitives are used to create novel bundle planners, the rapidly-exploring quotient-space trees (QRRT*), and the quotient-space roadmap planner (QMP*). Both planners are shown to be probabilistically complete and almost-surely asymptotically optimal. To evaluate our bundle planners, we compare them against classical sampling-based planners on benchmarks of four low-dimensional scenarios, and eight high-dimensional scenarios, ranging from 21 to 100 degrees of freedom, including multiple robots and nonholonomic constraints. Our findings show improvements up to 2 to 6 orders of magnitude and underline the efficiency of multilevel motion planners and the benefit of exploiting multilevel abstractions using the terminology of fiber bundles.

Motivation & Objective

• To address the computational intractability of high-dimensional motion planning in robotic systems.
• To develop a hierarchical decomposition strategy that reduces planning complexity by leveraging geometric structure in configuration space.
• To formalize motion planning as a section-finding problem over fiber bundles, enabling scalable and structured path computation.
• To improve efficiency and scalability of motion planning in complex environments through multilevel abstraction.
• To integrate shortest path computation and recursive section-finding to enable robust and adaptive trajectory generation.

Proposed method

• The configuration space is modeled as a fiber bundle, with base space representing coarse-level navigation and fibers representing fine-level degrees of freedom.
• A recursive algorithm, FindSection, computes motion paths by traversing hierarchical levels, starting from a coarse path in the base space.
• ShortestPath is used to compute an initial path in the previous level’s configuration space, which is then used as input to FindSection.
• The algorithm employs a depth-limited search with a maximal branching factor to control computational complexity.
• It uses a flag (FF) to determine whether to prioritize feasibility (true) or optimality (false) during section search.
• The method alternates between feasibility checks and section-finding to refine the path across levels.

Experimental results

Research questions

• RQ1 How can motion planning in high-dimensional configuration spaces be decomposed into computationally tractable subproblems?
• RQ2 Can fiber bundle theory provide a geometric framework for hierarchical motion planning with guaranteed convergence?
• RQ3 What is the impact of recursive section-finding on path quality and computational efficiency?
• RQ4 How does the integration of shortest path and section-finding improve robustness in complex environments?
• RQ5 To what extent can multilevel abstraction reduce the computational cost of motion planning?

Key findings

• The method successfully computes feasible motion paths in high-dimensional configuration spaces by leveraging hierarchical decomposition.
• The use of fiber bundle structure enables a natural abstraction of configuration space into base and fiber components, simplifying planning.
• Recursive section-finding with depth and branching constraints maintains computational tractability while ensuring path feasibility.
• The algorithm demonstrates improved scalability over monolithic planning approaches, particularly in complex, high-degree-of-freedom scenarios.
• The integration of feasibility checks (ptc) and path refinement improves convergence and robustness of the solution.
• The approach achieves a balance between path quality and computational cost through adaptive search control.