Optimal Knock-Pick Planning for Tightly Packed Tabletop Blocks With Parallel Grippers

Hao Lu; Rahul Shome

arxiv: 2605.17800 · v1 · pith:2UYKE722new · submitted 2026-05-18 · 💻 cs.RO · cs.AI

Optimal Knock-Pick Planning for Tightly Packed Tabletop Blocks With Parallel Grippers

Hao Lu , Rahul Shome This is my paper

Pith reviewed 2026-05-20 11:10 UTC · model grok-4.3

classification 💻 cs.RO cs.AI

keywords knock-pick planningtabletop manipulationparallel grippersnon-prehensile actionsmaximum-weight matchingdense packingobject rearrangementgrid-based planning

0 comments

The pith

A graphical abstraction solved by maximum-weight perfect matching computes the shortest sequence of knocks and picks needed to free tightly packed uniform blocks for parallel-gripper removal.

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

The paper examines how to rearrange densely packed tabletop blocks when a parallel gripper cannot reach an object because neighboring blocks block access. It introduces a directional knock primitive that displaces one block to create clearance and then seeks the sequence of knocks followed by picks that removes every block using the fewest total actions. The central technical step builds a graph whose vertices represent blocks and whose edges encode the minimal knocks required to resolve blocking constraints; solving a maximum-weight perfect matching on this graph yields the optimal plan in polynomial time. Experiments on synthetic grids and in IsaacSim confirm that the method scales as grid size grows. A reader would care because the reduction turns what looks like an exponential search into a fast, reliable computation for everyday dense-manipulation tasks.

Core claim

The paper claims that the optimal knock-pick plan for uniformly sized blocks on a planar grid can be obtained by first identifying the minimal set of directional knocks needed to resolve blocking relations, then reducing that identification problem to a maximum-weight perfect matching on a suitably constructed graph; the matching directly supplies both the knocks that must be performed and the order in which blocks can thereafter be picked without further interference.

What carries the argument

A graphical abstraction of knock dependencies solved via maximum-weight perfect matching, which selects the minimal set of directional knocks that enable a complete sequence of prehensile picks.

If this is right

The total number of actions required is obtained by a single polynomial-time matching computation rather than exhaustive search.
The same abstraction produces feasible plans for grids of increasing size without combinatorial blow-up.
Knock and pick actions are automatically interleaved in the order dictated by the matching.
The method supplies a concrete stepping stone for planners that mix prehensile and non-prehensile primitives in dense scenes.
Simulation results show that the generated plans succeed in a physics engine for the tested grid sizes.

Where Pith is reading between the lines

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

The same matching construction could be reused for other clearance primitives such as pushing or tilting if their dependency graphs can be written down.
Discretizing non-grid layouts into an equivalent graph might let the technique handle irregular object placements without changing the core algorithm.
Stability or friction constraints could be folded into the edge weights so the matching also avoids toppling risks.
The approach hints that many dense rearrangement problems become tractable once the minimal set of constraining interactions is isolated as a graph.

Load-bearing premise

Blocks are all the same size, sit on a flat grid, and a single directional knock creates enough clearance without creating new collisions or toppling anything else.

What would settle it

Running the computed knock sequence on a real tabletop and finding that at least one block remains unreachable or topples after the knocks are executed.

Figures

Figures reproduced from arXiv: 2605.17800 by Hao Lu, Rahul Shome.

**Figure 1.** Figure 1: A color sorting task with parallel grippers. (Right) A knock makes the subsequent picks valid. Tightly packed objects are common in logistics and automation settings [19, 27]. While object rearrangement [9] is a well-studied problem with wide applications in automation, tightly packed settings like the sorting task in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** Figure 2: Parallel gripper Pick. It is necessary to model two manipulation primitives: pick (prehensile grasp with a parallel gripper as shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Knock action (to the left). Pick feasibility. An object at a grid location v ∈ V is pickable if two adjacent locations are unoccupied in the same row or the same column. This leads to two conditions for feasibility in terms of vertex degree: (P1) degG(v) ≤ 1, or (P2) degG(v) = 2 and its two neighbors are collinear. We denote this predicate by Pick(G, v). Knock feasibility. Let D ≜ {(±1, 0),(0, ±1)}. For… view at source ↗

**Figure 4.** Figure 4: (Left) Tabletop grid arrangement of blocks. (Second) Object location graph, which greyed out ones indicating blocks that can be immediately picked (within Clean). (Third) A face is a minimal constraining set of four objects arranged in a 2 × 2 square in the object location graph. The object location graph corresponds to a set of faces. (Right) Gadgets are defined in terms of faces alongside knock candidate… view at source ↗

**Figure 5.** Figure 5: (Top) Face-graph abstraction (left) and the resulting optimal gadget cover (right). (Top-left) Each vertex represents a unit face in the cleaned instance. A selfloop at a vertex represents a square (2 × 2) gadget covering that single face; an edge between two vertices represents a horizontal (2 × 3) or vertical (3 × 2) gadget covering the corresponding adjacent face pair. Highlighted edges (including loop… view at source ↗

**Figure 6.** Figure 6: Illustration of Lemma 7 for the case σ = (−1, 1), corresponding to the topright corner. Left: the horizontal and vertical σ-outer boundaries are determined by the extremal horizontal gadget S σ h (dark green) and extremal vertical gadget S σ v (dark orange). Horizontal gadgets are shown in green, vertical gadgets in orange, and square gadgets in pink. Right: representative configurations: an unobstructed … view at source ↗

read the original abstract

Rearranging densely packed tabletop objects is challenging when parallel-gripper picks are infeasible without sufficient clearance around an object. This work studies the problem characteristics for practically motivated settings with uniformly sized blocks placed at planar tabletop grid locations. Since purely prehensile removal can become infeasible, a directional knock primitive is therefore introduced and the optimal knock-pick variant of the problem is formulated. The work proposes a series of abstractions wherein minimal constraining gadgets are covered to identify the necessary knocks. Utilizing a maximum-weight perfect matching on a graphical abstraction yields efficient polynomial-time computation of the optimal plan that minimizes the number of actions. Experiments are reported for increasing grid sizes in synthetic settings as well as in IsaacSim. The theoretical observations provide a promising stepping stone towards rigorously building efficient manipulation strategies that interleave prehensile and non-prehensile actions.

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.

Desk Editor's Note private letter to a colleague

The reduction to max-weight matching gives a fast planner for grid blocks but the optimality claim looks vulnerable to unmodeled sequential knock effects.

read the letter

The main point is that this paper takes the knock-pick planning problem for uniform blocks on a grid and reduces it to a max-weight perfect matching on a graph built from minimal constraining gadgets. That yields a polynomial-time way to minimize the total number of actions, which is useful when direct picks are blocked in dense packs. They add a directional knock primitive to create clearance and then abstract the necessary knocks into the matching setup. Experiments on growing synthetic grids plus IsaacSim runs show it handles larger instances without blowing up, which is a practical check on the approach. The formulation itself looks like a new combination for interleaving prehensile and non-prehensile moves in tabletop rearrangement. The efficiency gain is real if the abstraction works, and the sim results give some evidence it scales beyond tiny cases. The soft spot is the optimality guarantee. A static matching assumes the chosen knocks are independent enough that their total count is minimal in sequence, but a knock on one block can change clearances or blocking relations for several others. If the gadgets do not fully encode those dynamic interactions, the output plan could require extra actions once executed in order. The abstract states the matching produces optimal plans, yet without a clear derivation or counter-example checks in the text it is difficult to confirm the reduction preserves exact minimality rather than an approximation under independence. This work is aimed at robotics researchers focused on dense manipulation planning for warehouses or service tasks with parallel grippers. Readers who care about algorithmic reductions and practical sim validation would find the gadget construction and runtime numbers worth looking at. It is solid enough on the algorithmic side to deserve a serious referee who can examine the matching proof and test whether sequential dependencies are truly covered.

Referee Report

2 major / 2 minor

Summary. The manuscript studies optimal knock-pick planning for rearranging uniformly sized blocks on a planar tabletop grid using parallel grippers. When direct prehensile picks lack clearance, a directional knock primitive is introduced; the problem is then reduced via a series of abstractions that identify necessary knocks through minimal constraining gadgets. A maximum-weight perfect matching on the resulting graphical abstraction is claimed to compute the minimal-action plan in polynomial time. The approach is evaluated on synthetic grids of increasing size and in IsaacSim.

Significance. If the reduction to maximum-weight perfect matching is shown to preserve optimality, the work supplies a concrete polynomial-time algorithm for interleaving non-prehensile and prehensile actions in a restricted but practically relevant setting. The explicit use of a graphical abstraction and the reported scaling experiments constitute a clear algorithmic contribution that could serve as a stepping stone for more general cluttered-manipulation planners.

major comments (2)

[§5] §5 (Graphical Abstraction and Matching): The central claim that the maximum-weight perfect matching yields the exact minimal-action optimal plan assumes every clearance requirement and every directional-knock interaction is statically encoded as an edge weight or matching constraint. No derivation, proof sketch, or counter-example verification is supplied showing that sequential knock dependencies (one knock altering clearance for multiple subsequent blocks) are fully captured; the skeptic's concern therefore applies directly to the load-bearing optimality argument.
[§3.2] §3.2 (Problem Formulation): The weakest assumption—that a single directional knock suffices to create necessary clearance without introducing new collisions or instability—is stated for the uniform-grid case but receives no supporting analysis or edge-case enumeration. Because the matching operates on the resulting static graph, any unmodeled dynamic effect would invalidate the optimality guarantee.

minor comments (2)

[Abstract] The abstract asserts that the matching 'yields efficient polynomial-time computation of the optimal plan' without indicating where the complexity or optimality argument appears; a forward reference would help readers.
[Experiments] Table or figure captions for the IsaacSim experiments should explicitly list the grid sizes, number of trials, and quantitative metrics (e.g., action count versus baseline) rather than relying on qualitative description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. The comments highlight important points about the optimality argument and modeling assumptions that we will address by strengthening the exposition and adding supporting analysis. We respond to each major comment below.

read point-by-point responses

Referee: [§5] §5 (Graphical Abstraction and Matching): The central claim that the maximum-weight perfect matching yields the exact minimal-action optimal plan assumes every clearance requirement and every directional-knock interaction is statically encoded as an edge weight or matching constraint. No derivation, proof sketch, or counter-example verification is supplied showing that sequential knock dependencies (one knock altering clearance for multiple subsequent blocks) are fully captured; the skeptic's concern therefore applies directly to the load-bearing optimality argument.

Authors: We agree that an explicit derivation or proof sketch demonstrating how the static graphical abstraction captures sequential knock dependencies is currently absent from §5. The construction of minimal constraining gadgets is intended to encode all directional interactions required for clearance in a way that the subsequent maximum-weight perfect matching selects a non-overlapping set of knocks whose effects are sufficient for the entire plan; because each gadget represents the minimal set of blocks whose clearance must be resolved together, any downstream sequential effect is preemptively accounted for by the choice of which blocks receive knocks. Nevertheless, we will add a concise proof sketch together with a small set of counter-example verifications in the revised §5 to make this reasoning fully explicit and to directly address the concern about unmodeled sequential dependencies. revision: yes
Referee: [§3.2] §3.2 (Problem Formulation): The weakest assumption—that a single directional knock suffices to create necessary clearance without introducing new collisions or instability—is stated for the uniform-grid case but receives no supporting analysis or edge-case enumeration. Because the matching operates on the resulting static graph, any unmodeled dynamic effect would invalidate the optimality guarantee.

Authors: We acknowledge that §3.2 states the single-knock sufficiency assumption for the uniform-grid setting without an accompanying edge-case analysis. In the restricted domain of equal-sized blocks on a planar grid with parallel grippers, our IsaacSim experiments indicate that a single directional knock is sufficient to restore clearance without creating new collisions or instability for the configurations tested. To strengthen the manuscript we will expand §3.2 with a brief enumeration of potential edge cases (e.g., corner blocks, fully surrounded blocks) and a short discussion of why the assumption holds under the stated uniformity and grid constraints; if any edge case requires multiple knocks we will note the corresponding extension to the gadget library. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper reduces knock-pick planning to a maximum-weight perfect matching problem on a graph constructed from minimal constraining gadgets that encode clearance requirements. This is a direct algorithmic mapping to a standard combinatorial optimization problem whose solution is computed independently of the target plan; the optimality guarantee follows from the modeling assumptions rather than from any fitted parameter, self-definition, or self-citation chain. No equations or abstractions are shown to be equivalent to their inputs by construction, and the approach remains self-contained as a polynomial-time reduction without load-bearing reliance on prior author work that would itself presuppose the result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption of uniform grid placement and the introduction of the knock primitive; no free parameters or invented physical entities are described.

axioms (1)

domain assumption Blocks are uniformly sized and placed at planar tabletop grid locations.
Explicitly stated as the practically motivated setting in the abstract.

invented entities (1)

Directional knock primitive no independent evidence
purpose: To create clearance around an object when direct parallel-gripper pick is infeasible.
Introduced as a new action primitive to extend the feasible action space.

pith-pipeline@v0.9.0 · 5666 in / 1227 out tokens · 32021 ms · 2026-05-20T11:10:13.031118+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Utilizing a maximum-weight perfect matching on a graphical abstraction yields efficient polynomial-time computation of the optimal plan that minimizes the number of actions.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat_equivNat unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

framing the problem of minimum-knock plan computation to a special case of optimal exact cover problem

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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[1] [1]

IEEE Transactions on Robotics and Automation14(4) (August 1998)

Ben-Shahar, O., Rivlin, E.: Practical Pushing Planning for Rearrangement Tasks. IEEE Transactions on Robotics and Automation14(4) (August 1998)

work page 1998

[2] [2]

The International Journal of Robotics Research p

Chi,C.,Xu,Z.,Feng,S.,Cousineau,E.,Du,Y.,Burchfiel,B.,Tedrake,R.,Song,S.: Diffusion policy: Visuomotor policy learning via action diffusion. The International Journal of Robotics Research p. 02783649241273668 (2023)

work page 2023

[3] [3]

The International Jour- nal of Robotics Research37(10), 1134–1151 (2018) Optimal Tabletop Manipulation of Tightly Packed Blocks 17

Dantam, N.T., Kingston, Z.K., Chaudhuri, S., Kavraki, L.E.: An incremental constraint-based framework for task and motion planning. The International Jour- nal of Robotics Research37(10), 1134–1151 (2018) Optimal Tabletop Manipulation of Tightly Packed Blocks 17

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[4] [4]

Robotics: Sci- ence and Systems VII1, 65–72 (2011)

Dogar, M., Srinivasa, S.: A framework for push-grasping in clutter. Robotics: Sci- ence and Systems VII1, 65–72 (2011)

work page 2011

[5] [5]

ACM Com- puting Surveys (CSUR)18(1), 23–38 (1986)

Galil, Z.: Efficient algorithms for finding maximum matching in graphs. ACM Com- puting Surveys (CSUR)18(1), 23–38 (1986)

work page 1986

[6] [6]

In: 2019 International Conference on Robotics and Automa- tion (ICRA)

Görner, M., Haschke, R., Ritter, H., Zhang, J.: Moveit! task constructor for task- level motion planning. In: 2019 International Conference on Robotics and Automa- tion (ICRA). pp. 190–196. IEEE (2019)

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[8] [8]

In: Proceedings of the Fourteenth Annual Symposium on Computational Geometry

Halperin, D., Latombe, J.C., Wilson, R.H.: A general framework for assembly planning: The motion space approach. In: Proceedings of the Fourteenth Annual Symposium on Computational Geometry. pp. 9–18 (1998)

work page 1998

[9] [9]

International Journal of Robotics Research (2018)

Han, S.D., Stiffler, N., Krontiris, A., Bekris, K., Yu, J.: Complexity results and fast methods for optimal tabletop rearrangement with overhand grasps. International Journal of Robotics Research (2018)

work page 2018

[10] [10]

In: In- ternational Conference on Robotics and Automation (2014)

Havur, G., Ozbilgin, G., Erdem, E., Patoglu, V.: Geometric rearrangement of mul- tiple movable objects on cluttered surfaces: A hybrid reasoning approach. In: In- ternational Conference on Robotics and Automation (2014)

work page 2014

[11] [11]

IEEE Transactions on Automation Science and Engineering18(2), 398–400 (2021)

Höfer, S., Bekris, K., Handa, A., Gamboa, J.C., Mozifian, M., Golemo, F., Atkeson, C., Fox, D., Goldberg, K., Leonard, J., et al.: Sim2real in robotics and automa- tion: Applications and challenges. IEEE Transactions on Automation Science and Engineering18(2), 398–400 (2021)

work page 2021

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In: 2019 International Conference on Robotics and Automation (ICRA)

Huang, E., Jia, Z., Mason, M.T.: Large-scale multi-object rearrangement. In: 2019 International Conference on Robotics and Automation (ICRA). pp. 211–218. IEEE (2019)

work page 2019

[13] [13]

The International Journal of Robotics Research19(10), 883–894 (2000)

Huang, W.H., Mason, M.T.: Mechanics, planning, and control for tapping. The International Journal of Robotics Research19(10), 883–894 (2000)

work page 2000

[14] [14]

In: Proceedings 2000 ICRA

Kuffner, J.J., LaValle, S.M.: RRT-connect: an efficient approach to single-query path planning. In: Proceedings 2000 ICRA. Millennium Conference. IEEE Inter- national Conference on Robotics and Automation. Symposia Proceedings (Cat. No. 00CH37065). vol. 2, pp. 995–1001. IEEE (2000)

work page 2000

[15] [15]

In: Proceedings 1999 IEEE International Con- ference on Robotics and Automation (Cat

Lynch, K.: Toppling manipulation. In: Proceedings 1999 IEEE International Con- ference on Robotics and Automation (Cat. No.99CH36288C). vol. 4, pp. 2551–2557 vol.4 (1999). https://doi.org/10.1109/ROBOT.1999.773981

work page doi:10.1109/robot.1999.773981 1999

[16] [16]

Science Robotics7(66), 6074 (2022) https://doi.org/10.1126/scirobotics.abm6074

Macenski, S., Foote, T., Gerkey, B., Lalancette, C., Woodall, W.: Robot operat- ing system 2: Design, architecture, and uses in the wild. Science Robotics7(66), eabm6074 (2022). https://doi.org/10.1126/scirobotics.abm6074

work page doi:10.1126/scirobotics.abm6074 2022

[17] [17]

In: Prof

Ota, J.: Rearrangement Planning of Multiple Movable Objects. In: Prof. of the IEEE Intern. Conference on Robotics and Automation (ICRA) (2004)

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