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arxiv: 2606.12070 · v1 · pith:4V52IJN2new · submitted 2026-06-10 · 💻 cs.RO

Fibration Trees: A Unified Approach to Multi-Robot Motion Planning

Andreas Orthey , Florian T. Pokorny , Lydia E. Kavraki This is my paper

Pith reviewed 2026-06-27 09:22 UTC · model grok-4.3

classification 💻 cs.RO
keywords fibration treesmulti-robot motion planningsampling-based planningRRTstate space projectionsdecompositionsprobabilistic completenessconfiguration space
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The pith

Fibration trees unify sequential prioritization, parallel decomposition, and task-space projections under one formalism for multi-robot planning.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces fibration trees as a way to model state-space projections using fibrations, which connect higher-dimensional spaces to lower-dimensional ones. This structure lets the authors combine methods that were previously handled separately, such as prioritizing robots one after another or breaking the problem into parallel subspaces. From this they derive the Fibration-RRT planner, which runs on any user-supplied fibration tree and inherits probabilistic completeness from earlier RRT variants. Experiments on teams of robots with as many as 96 degrees of freedom show the planner solving 32 different scenarios by exploiting the supplied trees. A reader would care because the approach replaces ad-hoc combinations of projection techniques with a single, coherent representation that scales to high-dimensional multi-robot problems.

Core claim

By modeling projections as fibrations, fibration trees unify sequential prioritization, parallel decomposition, and task-space projections under a single coherent formalism. The resulting Fibration-RRT planner operates on user-defined fibration trees, generalizes quotient-space RRT and discrete RRT, includes task-space projections, and is proven probabilistically complete.

What carries the argument

Fibration trees: trees whose nodes are state spaces and whose edges are fibrations, each fibration modeling a projection from a higher-dimensional space to a lower-dimensional or simplified space.

If this is right

  • Fibration-RRT handles arbitrary mixtures of prioritization and decomposition strategies without custom code for each combination.
  • The planner remains probabilistically complete while operating on any valid user-supplied fibration tree.
  • Task-space projections can be inserted directly into the planning hierarchy alongside configuration-space projections.
  • High-dimensional instances with up to 96 degrees of freedom become tractable when the supplied tree exploits the problem structure.
  • An open-source implementation allows direct reuse on new multi-robot scenarios.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same fibration-tree structure could be attached to other sampling-based planners besides RRT variants.
  • Automated discovery or verification of suitable fibration trees would remove the current dependence on manual design.
  • The formalism may transfer to single-robot high-dimensional planning or to non-robotics domains that rely on state-space projections.

Load-bearing premise

Users can supply fibration trees whose fibrations correctly capture the geometric and topological relationships between the full configuration space and the chosen subspaces or task spaces.

What would settle it

A multi-robot problem equipped with a fibration tree that correctly models all required projections, yet Fibration-RRT returns no solution after exhaustive sampling or fails to exhibit the probabilistic-completeness property.

Figures

Figures reproduced from arXiv: 2606.12070 by Andreas Orthey, Florian T. Pokorny, Lydia E. Kavraki.

Figure 1
Figure 1. Figure 1: We simplify multi-robot motion planning problems using fibration trees, which are hierarchical representations of state spaces where nodes model state space simplifications. An example is the Hopf fibration S 3 → S 2 , which can be understood as a simplification of S 3 by S 2 . A graph on S 2 is shown (bottom right), with the graph restriction on S 3 visualized as the fibers of the Hopf fibration. The colo… view at source ↗
Figure 2
Figure 2. Figure 2: Fibration trees as unification tool to represent parallel, sequential, and partial fibrations. For each fibration, we list a non-exhaustive list of methods which make use of this particular instantiation of a fibration tree together with a tailor-made solver to exploit the particular structure. Please see Sec. 2 for a more comprehensive list of methods and Sec. 4.2 for related definitions. currently not po… view at source ↗
Figure 3
Figure 3. Figure 3: Overview about planning with fibration trees. Given a state space A, a fibration tree formalizes abstractions imposed onto A. This fibration tree contains, as an example, additional state spaces B, C, D, E, F which are simplifications of the original state space A. A number of projections are attached as edges between state spaces, which describe how state spaces are related. Given such a fibration tree an… view at source ↗
Figure 4
Figure 4. Figure 4: Examples of the different fibration types we consider in this paper as visualized on the torus T 2 . Left: Sequential fibration with a projection T 2 → S 1 . The circle (green) has a path segment (darkgreen), which is lifted to a path restriction (green, on torus). Middle: A parallel fibration of T 2 to two circle spaces, one for the major axis (green), one for the minor axis (orange) of the torus. Two pat… view at source ↗
Figure 5
Figure 5. Figure 5: Overhead Benchmarks. Success graphs of the benchmarks for the Multi-Disks scenarios where we plan in component space (Fibration tree has a single root node which represents the component state space). Top row shows the scenarios, bottom row shows the success graphs, whereby the x-axis shows time in log-space, while the y-axis shows the success rate from 0 to 100 percent. Colors indicate the planner as show… view at source ↗
Figure 11
Figure 11. Figure 11: In the last scenario, Path Velocity Decomposition ( [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 6
Figure 6. Figure 6: Experiment used for benchmarking. Movable robots are depicted in light green, obstacles in gray, and movable obstacles in light red. First and second row are multi-robot navigation scenarios, while the third row uses task constraints on the endeffector and dynamic obstacles in space-time. 10 0 10 1 10 2 0 25 50 75 100 (a) Multi Disks 10 0 10 1 10 2 0 25 50 75 100 (b) Airship Coordination 10 1 10 2 0 25 50 … view at source ↗
Figure 7
Figure 7. Figure 7: Success graphs of the benchmarks for scenarios from [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Success graphs for the prioritization benchmarks. Prepared using sagej.cls [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Success graphs for the decomposition benchmarks. 10 1 10 2 0 25 50 75 100 (a) Vertical Maze 10 2 10 3 0 25 50 75 100 (b) Fixed on Wall 10 1 10 2 10 3 0 25 50 75 100 (c) Mobile Manipulators 10 1 10 2 0 25 50 75 100 (d) Path Velocity Decomposition Fibration-RRT [Ours] Task-RRT [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Success graphs of the benchmarks for scenarios from [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Fibration trees used for task-space evaluations. Colors indicate type of projection: Parallel , Sequential , and Partial . 7.1 Limitations While we showed Fibration-RRT to perform well on all scenarios, there are still several improvements possible. Handling of variable-volumetric fibers In restriction sampling, we handle all fibers as if they would have the same volume. Often, however, the volume of fibe… view at source ↗
Figure 12
Figure 12. Figure 12: Hopf fibration with associated mappings. The Hopf map f from S 3 to S 2 , the inverse hopf map h from S 2 and fiber S 1 to S 3 , and the stereographic projection g from S 3 to 3-dimensional space. be an element of S 1 . We can then lift the sphere element together with the fiber element by using h(ϕ, θ, λ) =     −t1 sin(λ) +t1 cos(λ) t2 cos(λ) + t3 sin(λ) t3 cos(λ) − t2 sin(λ)     (4) whereby t1 =… view at source ↗
read the original abstract

State space projections and decompositions have emerged as powerful tools to tackle the curse of dimensionality in high-dimensional, multi-robot motion planning problems. However, existing methods lack a unified framework which seamlessly handles combinations of projections (prioritization or task-space) and decompositions (parallel or decoupled subspaces). To fill this gap, we introduce fibration trees, which are trees consisting of state spaces as nodes and fibrations as edges, whereby a fibration models a projection from a higher-dimensional space to a lower-dimensional (or simplified) space. By modeling projections as fibrations, we unify sequential prioritization, parallel decomposition, and task-space projections under a single, coherent formalism. Building on this, we develop the rapidly-exploring random fibration trees (Fibration-RRT) planner, a sampling-based motion planner that generalizes strategies from quotient-space RRT (for sequential prioritizations) and discrete RRT (for parallel decompositions), while allowing the inclusion of task-space projections. Fibration-RRT operates on user-defined fibration trees and is proven to be probabilistically complete. To test the generality and efficiency of Fibration-RRT, we provide an open-source implementation and conduct experiments on 32 scenarios using multi robot teams with up to 96 degrees of freedom. Our results indicate that Fibration-RRT efficiently solves high-dimensional problems by exploiting user-defined fibration trees, thereby establishing fibration trees as a powerful, unified framework for multi-robot motion planning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces fibration trees—trees with configuration spaces as nodes and fibrations as edges—as a unified formalism for modeling projections (prioritization, task-space) and decompositions (parallel, decoupled) in multi-robot motion planning. It presents Fibration-RRT, a sampling-based planner that generalizes quotient-space RRT and discrete RRT, claims a probabilistic-completeness proof for any user-defined fibration tree, and reports experiments on 32 scenarios with robot teams up to 96 DOF, supported by an open-source implementation.

Significance. If the unification and completeness results hold under the stated assumptions, the framework could provide a coherent way to combine existing projection and decomposition strategies, reducing the need for ad-hoc planner variants in high-dimensional problems. The open-source code and scale of the experimental evaluation (32 scenarios, 96 DOF) are concrete strengths that would allow the community to test the approach directly.

major comments (2)
  1. [Abstract; Fibration-RRT definition and completeness proof] Abstract and Fibration-RRT definition section: the probabilistic-completeness claim is stated to apply to any user-defined fibration tree, yet the manuscript provides no explicit characterization of the validity conditions on the fibrations (specifically, the homotopy lifting property required for path lifting and recursive sampling to cover the full space). This assumption is load-bearing for both the unification and the completeness guarantee; without it, the recursive argument generalizing quotient-space RRT and discrete RRT can fail for merely surjective projections.
  2. [Experiments (32 scenarios)] Experiments section: the claim that Fibration-RRT 'efficiently solves high-dimensional problems by exploiting user-defined fibration trees' rests on results from 32 scenarios, but the manuscript does not report per-scenario success rates, failure modes when invalid fibrations are supplied, or ablation on tree validity, making it impossible to assess whether the completeness guarantee translates to the tested instances.
minor comments (2)
  1. [Abstract] Abstract: the term 'fibration' is introduced without a brief topological definition or pointer to the relevant section, which may hinder readers outside algebraic topology.
  2. [Introduction and definitions] Notation: the distinction between a fibration (as a map with the lifting property) and a general projection is not consistently emphasized in early sections, risking conflation with prior quotient-space work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract; Fibration-RRT definition and completeness proof] Abstract and Fibration-RRT definition section: the probabilistic-completeness claim is stated to apply to any user-defined fibration tree, yet the manuscript provides no explicit characterization of the validity conditions on the fibrations (specifically, the homotopy lifting property required for path lifting and recursive sampling to cover the full space). This assumption is load-bearing for both the unification and the completeness guarantee; without it, the recursive argument generalizing quotient-space RRT and discrete RRT can fail for merely surjective projections.

    Authors: We agree that the completeness result requires fibrations to satisfy the homotopy lifting property. The manuscript adopts the standard topological definition of a fibration (which includes this property by definition) when modeling projections and decompositions. However, the referee is correct that an explicit statement of these validity conditions is not provided in the Fibration-RRT definition or completeness section. We will revise the manuscript to include a clear characterization of the required fibration properties, ensuring the unification and probabilistic-completeness claims are properly conditioned. revision: yes

  2. Referee: [Experiments (32 scenarios)] Experiments section: the claim that Fibration-RRT 'efficiently solves high-dimensional problems by exploiting user-defined fibration trees' rests on results from 32 scenarios, but the manuscript does not report per-scenario success rates, failure modes when invalid fibrations are supplied, or ablation on tree validity, making it impossible to assess whether the completeness guarantee translates to the tested instances.

    Authors: We will add per-scenario success rates to the experiments section in the revision to provide more granular support for the efficiency claims. The 32 scenarios were evaluated using valid user-defined fibration trees consistent with the framework's assumptions. The completeness guarantee does not apply to invalid fibrations (i.e., those lacking the homotopy lifting property), so experiments focused on valid cases; we will add a clarifying note on this distinction and potential issues with invalid trees, though a full ablation on invalid cases is not planned as it falls outside the method's stated scope. revision: partial

Circularity Check

0 steps flagged

No circularity: new definitional framework with independent completeness argument

full rationale

The paper introduces fibration trees as a novel tree structure with fibrations as edges and develops Fibration-RRT as a planner operating on user-supplied instances of this structure. Probabilistic completeness is asserted to follow from the recursive sampling properties generalized from prior quotient and discrete RRT methods. No quoted equation or definition reduces a claimed prediction or result to a fitted parameter, self-citation chain, or input by construction. The central unification and completeness statements rest on the new formalism itself rather than on quantities defined in terms of previously fitted values or unverified self-citations, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The framework introduces fibration trees as a new organizing structure whose validity depends on the user's choice of projections; no free parameters are fitted inside the planner itself. The probabilistic-completeness proof relies on standard assumptions of sampling-based planning.

axioms (1)
  • standard math Standard assumptions of sampling-based motion planning (uniform sampling, collision checking, etc.) suffice for probabilistic completeness when the fibration tree is fixed.
    Invoked in the statement that Fibration-RRT is probabilistically complete.
invented entities (2)
  • fibration tree no independent evidence
    purpose: Unified representation of projections and decompositions
    New structure introduced to connect state spaces via fibrations; no independent evidence outside the paper is supplied.
  • Fibration-RRT no independent evidence
    purpose: Sampling-based planner operating on fibration trees
    New algorithm that generalizes prior RRT variants; correctness claimed via proof but no external verification supplied.

pith-pipeline@v0.9.1-grok · 5799 in / 1499 out tokens · 17970 ms · 2026-06-27T09:22:51.063346+00:00 · methodology

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Reference graph

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