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arxiv: 2604.11587 · v1 · submitted 2026-04-13 · 💻 cs.RO

Optimal Kinodynamic Motion Planning Through Anytime Bidirectional Heuristic Search with Tight Termination Condition

Yi Wang , Bingxian Mu , Shahab Shokouhi , May-Win Thein This is my paper

Pith reviewed 2026-05-10 15:29 UTC · model grok-4.3

classification 💻 cs.RO
keywords kinodynamic motion planningsampling-based planningbidirectional heuristic searchasymptotic optimalityanytime algorithmtermination conditionmeet-in-the-middle property
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The pith

BTIT* is the first anytime MEET-style algorithm to use efficient tight termination conditions for on-the-fly early stopping in batch kinodynamic planning.

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

The paper presents BTIT*, an asymptotically optimal sampling-based planner for systems with dynamics such as robots. It combines bidirectional heuristic search, anytime batch processing, and a new termination condition that evaluates quickly enough to stop the search early. The design preserves the meet-in-the-middle property and MM-optimality. Readers would care because motion planning for high-dimensional dynamic robots often needs fast, high-quality solutions in changing environments. Experiments on a 4D double-integrator and 10D quadrotor show quicker first solutions and better convergence than prior non-lazy batch planners.

Core claim

BTIT* integrates an anytime bidirectional heuristic search with tight termination conditions inside batch-wise sampling-based motion planning, ensuring the meet-in-the-middle property and MM-optimality while becoming the first MEET-style algorithm to support efficient on-the-fly early termination.

What carries the argument

The tight termination condition inside the bidirectional heuristic search framework, which enables fast evaluation and early stopping while keeping the meet-in-the-middle property intact.

If this is right

  • BTIT* achieves strongly faster time-to-first-solution than representative non-lazy informed batch planners.
  • BTIT* shows improved convergence on the 4D double-integrator and 10D linearized quadrotor benchmarks.
  • The algorithm stays asymptotically optimal and satisfies MM-optimality because the termination condition respects the meet-in-the-middle property.

Where Pith is reading between the lines

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

  • The on-the-fly stopping may reduce planning latency enough for closed-loop control of dynamic robots.
  • Similar efficient termination conditions could be tested inside other bidirectional or informed sampling planners.

Load-bearing premise

The meet-in-the-middle property and MM-optimality continue to hold when the tight termination condition is applied inside the bidirectional heuristic search framework.

What would settle it

A concrete run on the 4D double-integrator or 10D quadrotor benchmark where applying the tight termination condition yields a path whose cost exceeds the MM-optimal cost.

Figures

Figures reproduced from arXiv: 2604.11587 by Bingxian Mu, May-Win Thein, Shahab Shokouhi, Yi Wang.

Figure 1
Figure 1. Figure 1: Experimental environment for the double-integrator [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Experimental environment for the 10D linearized [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparative performance of evaluated planners across different domains in terms of median solution cost and success [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

This paper introduces Bidirectional Tight Informed Trees (BTIT*), an asymptotically optimal kinodynamic sampling-based motion planning algorithm that integrates an anytime bidirectional heuristic search (Bi-HS) and ensures the \emph{meet-in-the-middle} property (MMP) and optimality (MM-optimality). BTIT* is the first anytime MEET-style algorithm to utilize termination conditions that are efficient to evaluate and enable early termination \emph{on-the-fly} in batch-wise sampling-based motion planning. Experiments show that BTIT* achieves strongly faster time-to-first-solution and improved convergence than representative \emph{non-lazy} informed batch planners on two kinodynamic benchmarks: a 4D double-integrator model and a 10D linearized Quadrotor. The source code is available here.

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 / 1 minor

Summary. The manuscript introduces BTIT*, an asymptotically optimal kinodynamic sampling-based motion planning algorithm that integrates anytime bidirectional heuristic search (Bi-HS) with tight, efficient termination conditions to enable early on-the-fly termination in batch-wise planning while claiming to preserve the meet-in-the-middle property (MMP) and MM-optimality. Experiments on a 4D double-integrator and 10D linearized quadrotor benchmark report faster time-to-first-solution and improved convergence relative to representative non-lazy informed batch planners; source code is provided.

Significance. If the termination condition is shown to preserve MMP and MM-optimality without hidden assumptions on batch costs or heuristics, the result would constitute a useful advance in efficient anytime kinodynamic planning. The open-source implementation is a clear strength that supports reproducibility and further evaluation.

major comments (2)
  1. Abstract: the central claim that BTIT* remains asymptotically optimal and satisfies MMP/MM-optimality after insertion of the new tight termination condition into the anytime Bi-HS loop is asserted without any derivation sketch, invariant, or proof outline. This is load-bearing for the contribution and must be addressed with a concrete argument showing equivalence to the standard MEET stopping criterion under the bidirectional ordering.
  2. Abstract / algorithm description: the meet-in-the-middle property relies on the meeting-point cost being the sum of the two tree costs; a termination predicate that inspects only one tree's g-value or a local heuristic bound risks accepting a pair whose combined cost is not yet provably minimal, especially under batch-wise sampling where early termination on one side may discard samples that would improve the meeting point in the next batch. The manuscript must explicitly demonstrate that the new predicate is equivalent to the MEET criterion.
minor comments (1)
  1. Abstract: the phrase 'strongly faster time-to-first-solution' is imprecise; the manuscript should state the quantitative metrics (e.g., median time or success rate at fixed time) used for the comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We agree that the preservation of the meet-in-the-middle property (MMP) and MM-optimality under the proposed tight termination conditions is central to the contribution and requires explicit justification. We will revise the manuscript to include a proof sketch and formal equivalence argument. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: Abstract: the central claim that BTIT* remains asymptotically optimal and satisfies MMP/MM-optimality after insertion of the new tight termination condition into the anytime Bi-HS loop is asserted without any derivation sketch, invariant, or proof outline. This is load-bearing for the contribution and must be addressed with a concrete argument showing equivalence to the standard MEET stopping criterion under the bidirectional ordering.

    Authors: We acknowledge that the abstract and current algorithm description assert preservation of asymptotic optimality, MMP, and MM-optimality without an explicit derivation or invariant. This is a valid observation. In the revised manuscript we will add a dedicated subsection (in the algorithm analysis section) containing a proof outline. The argument will establish equivalence by showing that the tight predicate, which uses efficient lower bounds on the combined forward and backward costs, only permits termination when the current meeting-point cost is provably minimal under the bidirectional ordering, mirroring the standard MEET criterion while remaining computationally lightweight. revision: yes

  2. Referee: Abstract / algorithm description: the meet-in-the-middle property relies on the meeting-point cost being the sum of the two tree costs; a termination predicate that inspects only one tree's g-value or a local heuristic bound risks accepting a pair whose combined cost is not yet provably minimal, especially under batch-wise sampling where early termination on one side may discard samples that would improve the meeting point in the next batch. The manuscript must explicitly demonstrate that the new predicate is equivalent to the MEET criterion.

    Authors: We agree that explicit demonstration is required, especially regarding batch-wise sampling and the risk of premature termination. The current manuscript relies on the design of the bidirectional heuristic search and the fact that termination is only triggered after both trees have been considered in the current batch. However, we will revise the algorithm description to include a formal argument showing that the new predicate is equivalent to MEET: it evaluates a tight bound on the sum of the two tree costs (not a single-side g-value) and defers termination until no improving meeting point is possible in the present or future batches, thereby preserving the MMP without discarding useful samples. revision: yes

Circularity Check

0 steps flagged

No circularity: BTIT* properties derived from external Bi-HS and informed-tree foundations

full rationale

The paper presents BTIT* as an integration of existing anytime bidirectional heuristic search (Bi-HS) with a new tight termination condition inside the MEET framework, claiming preservation of the meet-in-the-middle property (MMP) and MM-optimality. No equations, fitted parameters, or self-citations are shown that reduce the central claims (asymptotic optimality, early termination on-the-fly) back to the algorithm's own inputs or performance metrics by construction. The derivation relies on synthesis of prior non-self work rather than re-deriving its guarantees from its own batch-wise sampling loop. This is the normal case of an algorithmic synthesis paper whose correctness rests on external benchmarks and stated invariants rather than self-referential fitting or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the new termination condition preserves the meet-in-the-middle optimality property already established for the underlying Bi-HS framework; no new free parameters or invented physical entities are introduced.

axioms (1)
  • domain assumption The underlying bidirectional heuristic search satisfies the meet-in-the-middle property and MM-optimality when a suitable termination condition is used.
    Invoked in the abstract when claiming that BTIT* inherits asymptotic optimality.

pith-pipeline@v0.9.0 · 5439 in / 1137 out tokens · 24506 ms · 2026-05-10T15:29:22.085857+00:00 · methodology

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Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    S. M. LaValle,Planning algorithms. Cambridge university press, 2006

    work page 2006

  2. [2]

    Probabilistic roadmaps for path planning in high-dimensional configuration spaces,

    L. Kavraki, P. Svestka, J.-C. Latombe, and M. Overmars, “Probabilistic roadmaps for path planning in high-dimensional configuration spaces,” ICRA, vol. 12, no. 4, pp. 566–580, 1996

    work page 1996

  3. [3]

    Randomized kinodynamic planning,

    S. LaValle and J. Kuffner, “Randomized kinodynamic planning,” ICRA, vol. 1, pp. 473–479, 1999

    work page 1999

  4. [4]

    Sampling-based algorithms for optimal motion planning,

    S. Karaman and E. Frazzoli, “Sampling-based algorithms for optimal motion planning,”IJRR, vol. 30, no. 7, pp. 846–894, 2011

    work page 2011

  5. [5]

    Sampling-based robot motion planning,

    O. Salzman, “Sampling-based robot motion planning,”Commun. ACM, vol. 62, no. 10, pp. 54–63, 2019

    work page 2019

  6. [6]

    A note on two problems in connexion with graphs,

    E. W. Dijkstra, “A note on two problems in connexion with graphs,” Numerische mathematik, vol. 1, no. 1, pp. 269–271, 1959

    work page 1959

  7. [7]

    A formal basis for the heuristic determination of minimum cost paths,

    P. E. Hart, N. J. Nilsson, and B. Raphael, “A formal basis for the heuristic determination of minimum cost paths,”IEEE Trans on SSC, vol. 4, no. 2, pp. 100–107, 1968

    work page 1968

  8. [8]

    ARA*: AnytimeA*with provable bounds on sub-optimality,

    M. Likhachev, G. J. Gordon, and S. Thrun, “ARA*: AnytimeA*with provable bounds on sub-optimality,” inNIPS, 2003

    work page 2003

  9. [9]

    Anytime heuristic search,

    E. A. Hansen and R. Zhou, “Anytime heuristic search,”JAIR, vol. 28, pp. 267–297, 2007

    work page 2007

  10. [10]

    D* lite,

    S. Koenig and M. Likhachev, “D* lite,”AAAI, vol. 15, pp. 476–483, 2002

    work page 2002

  11. [11]

    Lifelong planningA*,

    S. Koenig, M. Likhachev, and D. Furcy, “Lifelong planningA*,”AIJ, vol. 155, no. 1-2, pp. 93–146, 2004

    work page 2004

  12. [12]

    Anytime search in dynamic graphs,

    M. Likhachev, D. Ferguson, G. Gordon, A. Stentz, and S. Thrun, “Anytime search in dynamic graphs,”Artificial Intelligence, vol. 172, no. 14, pp. 1613–1643, 2008

    work page 2008

  13. [13]

    Truncated incremental search,

    S. Aine and M. Likhachev, “Truncated incremental search,”Artificial Intelligence, vol. 234, pp. 49–77, 2016

    work page 2016

  14. [14]

    Online, interactive user guidance for high-dimensional, constrained motion planning,

    F. Islam, O. Salzman, and M. Likhachev, “Online, interactive user guidance for high-dimensional, constrained motion planning,”arXiv preprint arXiv:1710.03873, 2017

    work page arXiv 2017

  15. [15]

    Finding the shortest route between two points in a network,

    T. A. J. Nicholson, “Finding the shortest route between two points in a network,”The computer journal, vol. 9, no. 3, pp. 275–280, 1966

    work page 1966

  16. [16]

    Search techniques,

    J. Pearl and R. E. Korf, “Search techniques,”ARCS, vol. 2, no. 1, pp. 451–467, 1987. 10−2 10−1 1000 50 100Success [%] 10−2 10−1 100 Computation time [s] 50 100 150 200Median solution cost (a) Double Integrator Robot (R 4) 10−1 100 1010 50 100Success [%] 10−1 100 101 Computation time [s] 100 150 200 250Median solution cost (b) Linearized Quadrotor (R 10) B...

    work page 1987

  17. [17]

    Solving peg solitaire with bidirectional BFIDA*,

    J. Barker and R. Korf, “Solving peg solitaire with bidirectional BFIDA*,” inAAAI, vol. 26, no. 1, 2012, pp. 420–426

    work page 2012

  18. [18]

    Limitations of front-to-end bidirectional heuristic search,

    J. K. Barker and R. E. Korf, “Limitations of front-to-end bidirectional heuristic search,” inAAAI, 2015

    work page 2015

  19. [19]

    Bidirectional search that is guaranteed to meet in the middle,

    R. Holte, A. Felner, G. Sharon, and N. Sturtevant, “Bidirectional search that is guaranteed to meet in the middle,” inAAAI, vol. 30, no. 1, 2016

    work page 2016

  20. [20]

    Bidirectional search while ensuring meet-in-the-middle via effective and efficient-to- compute termination conditions,

    Y . Wang, E. Weiss, B. Mu, and O. Salzman, “Bidirectional search while ensuring meet-in-the-middle via effective and efficient-to- compute termination conditions,” inIJCAI, 2025, pp. 8987–8995

    work page 2025

  21. [21]

    A brief history and recent achievements in bidirectional search,

    N. Sturtevant and A. Felner, “A brief history and recent achievements in bidirectional search,” inAAAI, vol. 32, no. 1, 2018

    work page 2018

  22. [22]

    Dynamic programming,

    R. Bellman, “Dynamic programming,”Science, vol. 153, no. 3731, pp. 34–37, 1966

    work page 1966

  23. [23]

    Sampling heuristics for optimal motion planning in high dimensions,

    B. Akgun and M. Stilman, “Sampling heuristics for optimal motion planning in high dimensions,” inIROS, 2011, pp. 2640–2645

    work page 2011

  24. [24]

    RRT-connect: An efficient approach to single-query path planning,

    J. J. Kuffner and S. M. LaValle, “RRT-connect: An efficient approach to single-query path planning,” inICRA, vol. 2, 2000, pp. 995–1001

    work page 2000

  25. [25]

    Fast marching tree: A fast marching sampling-based method for optimal motion planning in many dimensions,

    L. Janson, E. Schmerling, A. Clark, and M. Pavone, “Fast marching tree: A fast marching sampling-based method for optimal motion planning in many dimensions,”IJRR, vol. 34, no. 7, pp. 883–921, 2015

    work page 2015

  26. [26]

    Asymptotically-optimal motion plan- ning using lower bounds on cost,

    O. Salzman and D. Halperin, “Asymptotically-optimal motion plan- ning using lower bounds on cost,” inICRA. IEEE, 2015, pp. 4167– 4172

    work page 2015

  27. [27]

    InformedRRT*: Optimal sampling-based path planning focused via direct sampling of an admissible ellipsoidal heuristic,

    J. D. Gammell, S. S. Srinivasa, and T. D. Barfoot, “InformedRRT*: Optimal sampling-based path planning focused via direct sampling of an admissible ellipsoidal heuristic,” inIROS, 2014, pp. 2997–3004

    work page 2014

  28. [28]

    Lazy collision checking in asymptotically-optimal motion planning,

    K. Hauser, “Lazy collision checking in asymptotically-optimal motion planning,” inICRA, 2015, pp. 2951–2957

    work page 2015

  29. [29]

    The provable virtue of laziness in motion planning,

    N. Haghtalab, S. Mackenzie, A. Procaccia, O. Salzman, and S. Srini- vasa, “The provable virtue of laziness in motion planning,” inICAPS, vol. 28, 2018, pp. 106–113

    work page 2018

  30. [30]

    Bidirectional heuristic search for motion planning with an extend operator,

    A. Cheng, D. M. Saxena, and M. Likhachev, “Bidirectional heuristic search for motion planning with an extend operator,” inIROS, 2019, pp. 7425–7430

    work page 2019

  31. [31]

    Lazy receding horizon A* for efficient path planning in graphs with expensive-to-evaluate edges,

    A. Mandalika, O. Salzman, and S. Srinivasa, “Lazy receding horizon A* for efficient path planning in graphs with expensive-to-evaluate edges,” inICAPS, vol. 28, 2018, pp. 476–484

    work page 2018

  32. [32]

    Gener- alized lazy search for robot motion planning: Interleaving search and edge evaluation via event-based toggles,

    A. Mandalika, S. Choudhury, O. Salzman, and S. Srinivasa, “Gener- alized lazy search for robot motion planning: Interleaving search and edge evaluation via event-based toggles,” inICAPS, vol. 29, 2019, pp. 745–753

    work page 2019

  33. [33]

    Computationally-efficient roadmap-based inspection planning via incremental lazy search,

    M. Fu, O. Salzman, and R. Alterovitz, “Computationally-efficient roadmap-based inspection planning via incremental lazy search,” in ICRA, 2021, pp. 7449–7456

    work page 2021

  34. [34]

    EMPC-based flight con- trol and collision-free path planning for a quadrotor with unbalanced payload,

    X. Zhang, Y . Wang, B. Mu, and S. Y . Yoon, “EMPC-based flight con- trol and collision-free path planning for a quadrotor with unbalanced payload,”TMECH, vol. 30, no. 6, pp. 5745–5754, 2025

    work page 2025

  35. [35]

    Batch informed trees (BIT*): Sampling-based optimal planning via the heuristically guided search of implicit random geometric graphs,

    J. D. Gammell, S. S. Srinivasa, and T. D. Barfoot, “Batch informed trees (BIT*): Sampling-based optimal planning via the heuristically guided search of implicit random geometric graphs,” inICRA, 2015, pp. 3067–3074

    work page 2015

  36. [36]

    Advanced BIT* (ABIT*): Sampling- based planning with advanced graph-search techniques,

    M. P. Strub and J. D. Gammell, “Advanced BIT* (ABIT*): Sampling- based planning with advanced graph-search techniques,” inICRA, 31 May – 31 Aug. 2020, pp. 130–136

    work page 2020

  37. [37]

    Adaptively informed trees (AIT*) andEffort informed trees (EIT*): Asymmetric bidirectional sampling-based path planning,

    ——, “Adaptively informed trees (AIT*) andEffort informed trees (EIT*): Asymmetric bidirectional sampling-based path planning,” IJRR, vol. 41, no. 4, pp. 390–417, 2022

    work page 2022

  38. [38]

    Asymptotically optimal sampling- based motion planning through anytime incremental lazy bidirectional heuristic search,

    Y . Wang, B. Mu, and O. Salzman, “Asymptotically optimal sampling- based motion planning through anytime incremental lazy bidirectional heuristic search,” inICRA, 2025, pp. 1787–1793

    work page 2025

  39. [39]

    and Van Den Berg, Jur , month = may, year =

    D. J. Webb and J. Van Den Berg, “Kinodynamic RRT*: Asymptotically optimal motion planning for robots with linear dynamics,” inICRA. IEEE, 2013, pp. 5054–5061. [Online]. Available: https://doi.org/10.1109/ICRA.2013.6631299

    work page doi:10.1109/icra.2013.6631299 2013

  40. [40]

    Accelerating kinodynamic rrt through dimensionality reduction,

    D. Zheng and P. Tsiotras, “Accelerating kinodynamic rrt through dimensionality reduction,” inIROS. IEEE, 2021, pp. 3674–3680

    work page 2021

  41. [41]

    Informed sampling for asymptotically optimal path planning,

    J. D. Gammell, T. D. Barfoot, and S. S. Srinivasa, “Informed sampling for asymptotically optimal path planning,”IEEE Trans. Robotics, vol. 34, no. 4, pp. 966–984, 2018

    work page 2018

  42. [42]

    Gener- alizing informed sampling for asymptotically-optimal sampling-based kinodynamic planning via markov chain monte carlo,

    D. Yi, R. Thakker, C. Gulino, O. Salzman, and S. S. Srinivasa, “Gener- alizing informed sampling for asymptotically-optimal sampling-based kinodynamic planning via markov chain monte carlo,” inICRA, 2018, pp. 7063–7070

    work page 2018

  43. [43]

    The open motion planning library,

    I. A. Sucan, M. Moll, and L. E. Kavraki, “The open motion planning library,”IEEE R & A Magazine, vol. 19, no. 4, pp. 72–82, 2012

    work page 2012

  44. [44]

    Design and implementation of nonuniform sampling cooperative control on a group of two-wheeled mobile robots,

    B. Mu, J. Chen, Y . Shi, and Y . Chang, “Design and implementation of nonuniform sampling cooperative control on a group of two-wheeled mobile robots,”IEEE Transactions on Industrial Electronics, vol. 64, no. 6, pp. 5035–5044, 2016

    work page 2016