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arxiv: 2607.00065 · v1 · pith:CFXFAYBHnew · submitted 2026-06-30 · 💻 cs.RO · cs.AI

Optimal any-angle path planning in static and dynamic environments

Yiyuan Zou , Clark Borst This is my paper

Pith reviewed 2026-07-02 19:07 UTC · model grok-4.3

classification 💻 cs.RO cs.AI
keywords any-angle path planningoptimal pathfindingdynamic obstaclesgrid-based navigationSIPPpath planning accelerationvisibility checks
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The pith

Elliptical forward expansion and field-of-view scanning yield Zeta*-SIPP, an optimal any-angle planner more than twenty times faster than prior methods in dynamic settings.

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

The paper develops two techniques to accelerate optimal any-angle path planning on grids while keeping solution guarantees intact in both static and dynamic obstacle environments. Elliptical forward expansion restricts the search space to ellipse-based neighborhoods, and field of view replaces conventional line-of-sight checks for faster visibility tests. These are combined using inverted scanning from open nodes and forward scanning from closed nodes, producing Zeta* for static cases and Zeta*-SIPP for dynamic cases. A sympathetic reader would care because the resulting speed allows repeated replanning during motion through open areas such as warehouses or oceans without accepting suboptimal routes. The central demonstration is that Zeta*-SIPP outperforms the prior state-of-the-art optimal dynamic planner by more than a factor of twenty.

Core claim

The paper shows that elliptical forward expansion and field-of-view replacements, integrated via inverted and forward scanning, produce Zeta* that matches the performance of existing static any-angle planners while extending directly to dynamic environments through Zeta*-SIPP, which is more than twenty times faster than TO-AA-SIPP yet still returns provably optimal any-angle paths.

What carries the argument

Elliptical forward expansion, which limits search to ellipse neighborhoods, combined with field-of-view visibility replacement and the choice of inverted versus forward scanning to connect nodes.

If this is right

  • Zeta* achieves performance comparable to Anya in static environments yet extends more readily to other settings.
  • Zeta*-SIPP handles both static and dynamic obstacles while preserving optimality.
  • Inverted scanning connects visual links from open nodes and forward scanning initiates from closed nodes.
  • The two acceleration techniques together make repeated optimal any-angle replanning practical for navigation among moving obstacles.

Where Pith is reading between the lines

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

  • The same scanning and expansion ideas could be tested on three-dimensional grids to check whether the reported speedups scale beyond two dimensions.
  • In applications that require frequent replanning, the reduced per-query time might allow the planner to incorporate updated obstacle predictions at higher rates.
  • Direct comparison of the produced paths against known optimal solutions on standardized dynamic benchmark maps would confirm the claimed factor-of-twenty improvement holds across varied obstacle densities.

Load-bearing premise

The elliptical forward expansion and field-of-view replacements preserve optimality guarantees while accelerating search in both static and dynamic obstacle settings.

What would settle it

A concrete counterexample in which Zeta*-SIPP returns a path whose length exceeds the true shortest any-angle path in a dynamic grid instance would falsify the optimality claim.

Figures

Figures reproduced from arXiv: 2607.00065 by Clark Borst, Yiyuan Zou.

Figure 1
Figure 1. Figure 1: Search trees (yellow lines) and explored nodes (blue grids or regions) of A*, Theta* and Anya [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graph vertices on grids. ments and introduces the relevant concepts and notations. Section 3 and 4 illustrate the elliptical forward expansion and field of view respectively. Section 5 describes Zeta* for static environments, while Section 6 presents Zeta*-SIPP for dynamic environments. Section 7 discusses the similarities and differences between existing approaches and the proposed algorithms. Section 8 s… view at source ↗
Figure 3
Figure 3. Figure 3: 2k -neighborhoods on grids. the grid are considered neighbors of the currently expanded node. A* with a 2∞-neighborhood is capable of finding truly optimal paths on grids (Yakovlev and Andreychuk, 2021). However, increasing k can also significantly slow down grid-based path planning, as it results in a larger branching factor for search. Although A* with a 2∞-neighborhood can achieve optimal any-angle path… view at source ↗
Figure 4
Figure 4. Figure 4: Elliptical forward expansion. The green dot is the start point and the red dot is the target point. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Octant-based symmetric shadowcasting, adapted from [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Inverted and forward scanning with elliptical forward expansion. The green dot is the start point and the red dot is the target point. The [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Naive and cost-bounded forward scanning. The blue box in the left plot is the bounding box of the elliptical search range. The blue and [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The cost of a scan range f ′ (s) (green triangle) is computed by g(n) plus the length of the yellow line to the target point t. There are four possible relative positions of the target point with respect to the scan range, denoted t1 to t4. t ′ 3 is the mirror position of t3. along with its observable and unobservable successors, can be all discarded. Furthermore, the cost functions for scan ranges and sea… view at source ↗
Figure 9
Figure 9. Figure 9: Scan range pruning in inverted and forward scanning. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Scatter plots of path length vs. runtime and scanned vertices vs. sorted elements. The path length increase is given as a ratio relative to [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Average runtime of the suboptimal algorithms under different dynamic obstacle conditions. 0 20 40 60 80 TO−AA−SIPP TO−AA−FoV−SIPP Zeta*−SIPP−i Zeta*−SIPP−f Average runtime (ms) Empty: 48 × 48 0 20 40 60 TO−AA−SIPP TO−AA−FoV−SIPP Zeta*−SIPP−i Zeta*−SIPP−f Average runtime (ms) Random: 64 × 64 0 200 400 600 TO−AA−SIPP TO−AA−FoV−SIPP Zeta*−SIPP−i Zeta*−SIPP−f Average runtime (ms) Room: 64 × 64 0 5000 10000 TO… view at source ↗
Figure 12
Figure 12. Figure 12: Average runtime of the optimal algorithms under different dynamic obstacle conditions. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Scatter plots of path time vs. runtime and scanned grids vs. sorted elements. All the metrics are normalized relative to TO-AA-SIPP. All [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
read the original abstract

Any-angle path planning extends traditional graph-based path planning by allowing movement between any pair of vertices, rather than being restricted by predefined edges. It can find straighter and shorter paths in continuous space with graphs, making it particularly suitable for navigation in open areas such as airspaces, warehouses, and oceans. Many any-angle path-planning algorithms have been proposed, but only a few can guarantee optimal solutions, especially in the presence of dynamic obstacles. To address this challenge, this article focuses on optimal any-angle path planning on grids and introduces two general techniques that accelerate computation while preserving optimality in both static and dynamic environments: 1) elliptical forward expansion, which leverages ellipse-based neighborhoods to restrict the search space, and 2) field of view, which replaces traditional line-of-sight methods to speed up visibility checks. To integrate these two techniques, inverted and forward scanning are introduced. Inverted scanning establishes visual connections from open nodes, whereas forward scanning initiates scans from closed nodes. Building on the proposed techniques, Zeta* and Zeta*-SIPP are developed for static and dynamic environments respectively. Zeta*, when combined with forward scanning, is similar to the state-of-the-art algorithm Anya and attains comparable performance. Unlike Anya, Zeta* can be readily extended to other settings, such as dynamic environments (e.g., Zeta*-SIPP). Zeta*-SIPP, with either scanning method, is more than 20 times faster than the corresponding state-of-the-art optimal planner TO-AA-SIPP. Overall, this research identifies the key requirements for achieving optimal any-angle path planning and introduces a unified approach suitable for different environments.

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 manuscript introduces Zeta* for optimal any-angle path planning in static grid environments and Zeta*-SIPP for dynamic environments with moving obstacles. It proposes two acceleration techniques—elliptical forward expansion to limit the search neighborhood and field-of-view replacement for line-of-sight visibility checks—combined via inverted scanning (from open nodes) and forward scanning (from closed nodes). The central claims are that these substitutions preserve optimality guarantees and that Zeta*-SIPP (with either scanning mode) is more than 20 times faster than the state-of-the-art optimal planner TO-AA-SIPP.

Significance. If the optimality invariants hold, the work would offer a practically significant improvement for real-time navigation in dynamic settings such as warehouses or airspaces, by delivering a unified, extensible framework that extends static any-angle methods (comparable to Anya) to dynamic cases without sacrificing guarantees. The explicit handling of safe intervals in SIPP integration is a positive aspect.

major comments (2)
  1. [Abstract] Abstract and integration paragraph: the claim that elliptical forward expansion plus field-of-view checks (under both scanning modes) preserve optimality in dynamic settings rests on an unstated invariant that a FOV scan never omits a visibility edge that exhaustive line-of-sight would have found when safe intervals change. No formal argument, proof sketch, or counter-example verification is supplied showing that shorter any-angle paths remain considered; this directly undermines the comparison of Zeta*-SIPP to TO-AA-SIPP as optimal planners.
  2. [Abstract] The manuscript asserts large speedups (more than 20x) while preserving optimality, yet provides neither the full derivation of the optimality invariant for dynamic obstacles nor the experimental tables/figures with raw timing and path-length data needed to verify the claim. Without these, the speedup result cannot be assessed as comparing equivalent optimal planners.
minor comments (2)
  1. Notation for the two scanning modes and their interaction with SIPP safe intervals should be defined explicitly before the algorithm descriptions.
  2. The relationship between Zeta* (static) and Zeta*-SIPP (dynamic) would benefit from a short table contrasting the two algorithms' data structures and termination conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback highlighting the need for clearer presentation of optimality arguments and supporting data. We address each major comment below with clarifications drawn from the manuscript and indicate where revisions can strengthen the submission without altering its core claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract and integration paragraph: the claim that elliptical forward expansion plus field-of-view checks (under both scanning modes) preserve optimality in dynamic settings rests on an unstated invariant that a FOV scan never omits a visibility edge that exhaustive line-of-sight would have found when safe intervals change. No formal argument, proof sketch, or counter-example verification is supplied showing that shorter any-angle paths remain considered; this directly undermines the comparison of Zeta*-SIPP to TO-AA-SIPP as optimal planners.

    Authors: The manuscript's sections on Zeta*-SIPP integration explicitly address how safe-interval handling combines with elliptical expansion and FOV scanning to maintain all relevant visibility edges. Inverted scanning from open nodes and forward scanning from closed nodes ensure that any visibility connection within the ellipse that would be found by exhaustive LOS is still considered, because FOV replacement is defined to replicate the same neighbor set under the current safe intervals. We acknowledge that a compact proof sketch is not highlighted in the abstract or integration paragraph and can add one in revision to make the invariant explicit, including a brief argument that no shorter any-angle path is omitted when intervals update. revision: partial

  2. Referee: [Abstract] The manuscript asserts large speedups (more than 20x) while preserving optimality, yet provides neither the full derivation of the optimality invariant for dynamic obstacles nor the experimental tables/figures with raw timing and path-length data needed to verify the claim. Without these, the speedup result cannot be assessed as comparing equivalent optimal planners.

    Authors: The full derivation of the optimality invariant for dynamic obstacles appears in the Zeta*-SIPP analysis sections, which show that the scanning modes and FOV substitution preserve the same search frontier as exhaustive methods while respecting safe intervals. The experimental results section already contains tables and figures reporting average speedups, path lengths, and success rates against TO-AA-SIPP. To aid verification we can append the underlying raw timing and path-length values in a supplementary table during revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithmic claims rest on external comparisons and stated techniques

full rationale

The paper develops Zeta* and Zeta*-SIPP by introducing elliptical forward expansion and field-of-view visibility checks, integrated via inverted/forward scanning. These are presented as extensions of standard any-angle search (e.g., similar to Anya) with explicit performance benchmarks against TO-AA-SIPP. No equations reduce claimed speedups or optimality to fitted parameters defined on the same data, no self-citation chains justify core invariants, and no renaming of known results occurs. The derivation is self-contained against external planners and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions of grid discretization and continuous-space visibility; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Grid-based discretization allows any-angle movement between vertices while preserving optimality when visibility checks are exact.
    Invoked throughout the description of any-angle planning on grids.

pith-pipeline@v0.9.1-grok · 5818 in / 1105 out tokens · 22686 ms · 2026-07-02T19:07:55.483519+00:00 · methodology

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Reference graph

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