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arxiv: 2606.22143 · v1 · pith:7F7NKS67new · submitted 2026-06-20 · 💻 cs.RO · cs.GT· cs.LG

Physics-Informed Eikonal Caging for Whole-Arm Manipulation Planning

Yan Zhang , Yiming Li , Yifei Dong , Florian T. Pokorny , Sylvain Calinon This is my paper

Pith reviewed 2026-06-26 11:39 UTC · model grok-4.3

classification 💻 cs.RO cs.GTcs.LG
keywords cagingwhole-arm manipulationphysics-informed neural networkeikonal equationescape time fieldmanipulation planningcontact model mismatch
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The pith

Reformulating caging as a minimum-time escape problem produces an escape-time field that obeys an eikonal equation and can be learned with a physics-informed neural network for manipulation planning.

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

The paper recasts the geometric idea of caging an object inside a robot arm's shape as the shortest time an object would need to escape that shape. This produces a continuous field over configurations that quantifies enclosure quality and obeys the eikonal equation from wave propagation. A physics-informed neural network then approximates the field to create a smooth, differentiable objective that planners can directly optimize. The resulting plans favor robot poses that make escape harder, allowing the use of simpler contact models while still achieving robustness to model errors in whole-arm tasks.

Core claim

Caging is reformulated as a minimum-time escape problem under the robot's enclosing geometry. The resulting escape-time field measures enclosure quality, satisfies an eikonal equation, and is approximated by a physics-informed neural network to yield a differentiable representation for embedding into continuous optimization-based manipulation planning.

What carries the argument

The escape-time field from the minimum-time escape problem, which satisfies the eikonal equation and is approximated via a physics-informed neural network to provide a differentiable enclosure metric.

If this is right

  • Whole-arm manipulation planners can optimize directly for configurations that resist object escape using the differentiable field.
  • Simplified contact models such as quasi-dynamic approximations become viable for planning without sacrificing robustness.
  • Improved resistance to disturbances and contact-model mismatch is achieved in both simulation and real-world tests compared to baselines.
  • Geometric enclosure acts as a practical robustness primitive for contact-rich whole-arm manipulation.

Where Pith is reading between the lines

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

  • The approach may generalize to other enclosure-based tasks like containment or guarding in robotics.
  • Connections to traditional path-planning methods using eikonal equations could allow hybrid planning strategies.
  • Further validation with varying object shapes and dynamic environments would test the limits of the enclosure metric.
  • Embedding the field into learning-based controllers could enable online adjustment during execution.

Load-bearing premise

That solving the minimum-time escape problem under the robot geometry accurately captures the robustness of caging to inaccuracies in contact models.

What would settle it

A real-robot experiment in which a planned manipulation using the escape-time objective fails under actual contact while a non-caging baseline succeeds would falsify the claim that the field provides the intended robustness.

Figures

Figures reproduced from arXiv: 2606.22143 by Florian T. Pokorny, Sylvain Calinon, Yan Zhang, Yifei Dong, Yiming Li.

Figure 1
Figure 1. Figure 1: Conceptual overview of whole-arm caging manipulation for robust whole-arm manipulation planning with simplified contact dynamics. Top: the robot transports a box-shaped object using distributed whole-arm contact along a manipulation trajectory planned with a simplified object model, where the box is under-approximated by a circular geometry (yellow circle). Bottom: the corresponding escape-time field over … view at source ↗
Figure 2
Figure 2. Figure 2: Illustrations of caging and partial caging with a point￾mass object (yellow star) in its configuration space R 2 . The white region indicates environment obstacles, corresponding to the collision set C col o . The escape region consists of configurations outside the boundary ∂Ωg, shown as a dashed circle. Cyan curves indicate possible collision-free paths of the object: in the caging case they remain trapp… view at source ↗
Figure 3
Figure 3. Figure 3: (a): Eikonal caging in the configuration space of a point mass (yellow star) interacting with a square-annulus obstacle (white area). The point mass lies inside the inner cavity (grey area), while the goal region is outside the red dashed boundary; No collision-free path exists from the cavity to the goal, so this region forms a cage in configuration space. Contours with values indicate the minimal escape … view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between the PINN-based eikonal caging time field and the ground-truth solution from FMM. Two representative robot configurations and point-mass positions are shown, illustrating that the learned model closely matches the true escape-time field and recovers the corresponding optimal escape paths. As shown in the previous subsection, Tˆ is a conservative lower bound of the true rigid-body escape t… view at source ↗
Figure 5
Figure 5. Figure 5: Optimal caging configurations found with CMA-ES and SQP, maximizing the learned eikonal caging metric of an object represented by three circles shown in red. Configuration optimization. We next optimize Equa￾tion (9) for an object represented by three circles, using both CMA-ES and SQP from the same initial robot configuration. As shown in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Whole-arm caging manipulation under three objective modes. Mode 1 minimizes the minimum distance between the object and the whole robot arm; Mode 2 minimizes the average distance between the object and all robot links; Mode 3 uses the learned escape-time field for whole-arm caging (our method). For each mode, we show the same initial robot configuration, the robot–object configuration at first contact, and… view at source ↗
Figure 7
Figure 7. Figure 7: Contact distribution, proximity diagnostics, and enclosure quality for the three planning modes. object–arm distance (metric used in Mode 2); and (iii) the escape-time caging metric. Results are shown in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Object–to–goal distance over time during open￾loop execution of planned trajectories under three levels of random disturbances across the three planning modes. Lower values indicate better task progress. Mode 3 shows the most concentrated and stable trajectories across disturbance levels [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Final object-to-goal distance at the terminal time step under random disturbances. Smaller values indicate better task completion. Mode 3 consistently achieves the lowest final error, while Mode 2 exhibits the largest degradation under disturbance. denote the object velocity predicted by the QP-based quasi￾dynamic model, and let v true o = f true o (q,ur) denote the velocity produced by the true contact dy… view at source ↗
Figure 10
Figure 10. Figure 10: Object trajectories under dynamics-order mismatch. Robot motions are planned using the first-order quasi-dynamic model (blue solid) and executed under second-order MuJoCo dynamics (red dashed). The goal region is shown in green. Mode 3 preserves similar overall behavior under mismatch, whereas Modes 1 and 2 exhibit larger deviations or task failure. different objectives tolerate unmodeled object motion. F… view at source ↗
Figure 11
Figure 11. Figure 11: Object trajectories under shape under-approximation. Robot motions planned for a circular object are executed on polygonal objects (octagon, hexagon, square) with increasing geometric mismatch. The circle case is the no-mismatch reference. Mode 3 remains comparatively stable as mismatch increases, while the baselines accumulate larger trajectory errors [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Illustration of shape under-approximation used during planning. The orange circle denotes the simplified object model used for optimization, while black outlines indicate the true polygonal shapes used during execution. Blue segments show maximum contact-point displacement δϕn, and green arcs show corresponding surface-normal mismatch δγd. Larger mismatch values imply greater geometric modeling error. whe… view at source ↗
Figure 13
Figure 13. Figure 13: Object trajectories under friction mismatch. Robot motions planned with nominal friction coefficient µ = 0.5 are executed using different friction values. Mode 3 exhibits smaller task-level trajectory deviations across the tested range, indicating reduced sensitivity to friction mis-specification. 7 Validation on Realistic Geometry and Real-World Systems Since the proposed method is fundamentally geometry… view at source ↗
Figure 14
Figure 14. Figure 14: Experiment setup with a Franka arm for tabletop whole-arm manipulation planning. Red dashed lines and arrows indicate the active joints. subset of joints is fixed and only joints [1, 4, 6] are actively controlled ( [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: Initial contact configurations for whole-arm caging manipulation (WACaging) and baseline contact geometries (Point, Link, WAContact). The nominal object representation used for planning (orange circle) is an under-approximation of the true box geometry (light orange). Transparent robot parts are only used for visualization and are incapable of making contact with the target object. The green marker indica… view at source ↗
Figure 15
Figure 15. Figure 15: Whole-arm caging configurations for a Franka arm and three everyday object geometries. For each object, the left panel shows the optimized caging configuration obtained with CMA-ES, with a 3D rendering of the object shown in the inset. The right panel visualizes the learned escape-time field together with the object’s 2D point-based approximation. Each object is represented by a small set of circles whose… view at source ↗
Figure 17
Figure 17. Figure 17: Robot and object configurations at keyframes with four different contact geometries [PITH_FULL_IMAGE:figures/full_fig_p017_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Robustness evaluation under random object motion disturbances. The object trajectories without any disturbances are shown as Nominal trajectories with blue lines. Red trajectories show the mean disturbed object trajectories (shown with grey thin lines) over 100 trials [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Caging metric comparison [PITH_FULL_IMAGE:figures/full_fig_p017_19.png] view at source ↗
Figure 21
Figure 21. Figure 21: Whole-arm trajectory tracking and caging perfor￾mance analysis Overall tracking and enclosure behavior. Figure 21a illustrates the resulting object trajectories and corresponding escape-time values for λ ∈ {1, 5, 10, 50, 100}. For small λ (e.g., λ = 1 or 5), the planner prioritizes enclosure and produces object paths that remain closer to the left-side arc as observed in the previous goal-reach task, desp… view at source ↗
Figure 22
Figure 22. Figure 22: Evaluation of robustness with solution generated under λ = 100. Left: Robustness to random motion disturbance with magnitude σ = 0.005m. Right: Robustness to four representative forms of contact model mismatches. Compared with the goal-reaching task, the escape-time profile differs because the robot must temporarily reduce enclosure in order to adjust its pushing configuration. As shown in Figure 21a, the… view at source ↗
Figure 23
Figure 23. Figure 23: Real-world experiment setup with increasing levels of object-shape mismatch relative to the planning model. (a,b) Initial contact configuration in simulation and on the Franka robot. During planning, the object is approximated as six cylindrical primitives corresponding to the kitchen paper rolls (white cylinders in (a)). We considered three levels of object shape mismatch (SM) relative to the simulation … view at source ↗
Figure 24
Figure 24. Figure 24: Object trajectories during sim-to-real transfer under three increasing levels of object-shape mismatch. Other execution conditions are kept unchanged across the three scenarios. The yellow dot marks the nominal initial object position used during simulation planning. For each case, the planned simulation trajectory is shown in blue, together with five real-world trials (thin dashed lines) and their mean t… view at source ↗
Figure 25
Figure 25. Figure 25: Snapshots of robot–object configurations in simulation and real-world execution under the most challenging mismatch setting (SM3: A4 paper box with an additional paper roll). The upper row shows the planning model used for whole-arm caging manipulation, while the lower row shows corresponding real-world execution. Despite multiple simultaneous contact-geometry mismatches, the transferred trajectory remain… view at source ↗
Figure 27
Figure 27. Figure 27: Object’s distance to goal at final time step. 0.5 0.0 0.5 FMM y ( m ) (a) 0.19 0.38 0.58 0.77 0.96 (b) 0.19 0.38 0.58 0.77 0.96 0.5 0.0 0.5 x (m) 0.5 0.0 0.5 PINN y ( m ) 0.19 0.38 0.58 0.77 0.5 0.0 0.5 x (m) 0.19 0.38 0.58 0.77 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Escape time (s) Point Mass Path Robot [PITH_FULL_IMAGE:figures/full_fig_p026_27.png] view at source ↗
Figure 26
Figure 26. Figure 26: Contact distribution, proximity diagnostics, and enclosure quality for the three planning modes. controller whose stiffness parameters match those used in the corresponding QP-based planning model. Each planned trajectory waypoint generated by the QP model is executed over 50 MuJoCo simulation steps, allowing the system to converge to the target configuration before advancing to the next waypoint. B Addit… view at source ↗
read the original abstract

Planning contact-rich whole-arm manipulation is challenging because interactions that involve extended robot geometry give rise to complex contact dynamics that are difficult to model accurately. This creates a need for planning principles that do not rely heavily on precise contact models. Caging offers one such geometric notion of robustness to modeling inaccuracy by restricting object escape through geometrically enclosing the object. However, existing caging formulations are difficult to incorporate into continuous optimization-based manipulation planning. We reformulate caging as a minimum-time escape problem in which the object seeks to leave an enclosing robot geometry in the shortest time. This yields a continuous escape-time field that measures the robot's enclosure quality and we show it satisfies an eikonal equation. We therefore can approximate this field using a physics-informed neural network, producing a smooth differentiable representation that can be embedded directly into manipulation planning. The resulting objective supports whole-arm manipulation planning to favor robot configurations resisting object escape. This improves the manipulation robustness to contact model mismatch, thus enabling planning with simplified contact models, including quasi-dynamic approximations and simplified object geometry. Across simulation and real-world experiments, we show improved robustness to disturbances and contact-model mismatch relative to baselines. These results suggest that geometric enclosure can serve as a practical robustness primitive for whole-arm manipulation. A supplementary video, which includes an intuitive overview of our method and experiment video results, is available on our project webpage.

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 paper reformulates caging for whole-arm manipulation as a minimum-time escape problem for an object leaving the robot's enclosing geometry. It asserts that the resulting escape-time field satisfies the eikonal equation, approximates the field via a physics-informed neural network (PINN) to obtain a smooth differentiable representation, and embeds this field as an objective in continuous manipulation planning. The approach is claimed to improve robustness to contact-model mismatch, enabling use of simplified models (e.g., quasi-dynamic), with supporting simulation and real-robot experiments.

Significance. If the escape-time-to-robustness mapping is valid, the method supplies a geometric, model-light primitive that can be directly differentiated and optimized within existing planners. This addresses a practical gap in contact-rich whole-arm tasks where accurate dynamics are unavailable. The PINN-based eikonal approximation is a clean way to obtain the required smoothness without discretizing the configuration space.

major comments (2)
  1. [Abstract] Abstract and opening paragraphs: the central claim that the min-time escape field under idealized enclosing geometry quantifies robustness to contact-model mismatch (friction, compliance, multi-contact dynamics) is asserted without an explicit argument or counter-example analysis showing why this geometric quantity dominates unmodeled effects that could permit escape even when the field value is large. This mapping is load-bearing for the practical utility of the planning objective.
  2. [Abstract / Methods] The abstract states that the escape-time field satisfies the eikonal equation and that a PINN is used to approximate it, yet supplies no derivation steps, boundary-condition enforcement details (T=0 on boundary, | abla T|=1 inside), or error analysis for the PINN training. Because the eikonal property is the justification for using the PINN inside the planner, this omission affects assessment of the method's soundness.
minor comments (1)
  1. Notation for the escape-time field T and the enclosing geometry should be introduced with a single consistent definition early in the manuscript rather than piecemeal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our contributions. We address each major comment below with specific responses and proposed revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the central claim that the min-time escape field under idealized enclosing geometry quantifies robustness to contact-model mismatch (friction, compliance, multi-contact dynamics) is asserted without an explicit argument or counter-example analysis showing why this geometric quantity dominates unmodeled effects that could permit escape even when the field value is large. This mapping is load-bearing for the practical utility of the planning objective.

    Authors: We agree that an explicit argument linking the escape-time field to robustness against contact-model mismatch would strengthen the manuscript. The underlying intuition is that a large escape time under the idealized enclosing geometry implies the object must traverse a long path in configuration space to escape, providing a temporal buffer even if unmodeled effects (e.g., reduced friction or compliance) alter the actual dynamics. In the revision we will add a dedicated paragraph in the introduction that formalizes this reasoning and includes a simple counter-example: two robot configurations with identical contact models but differing escape times, showing that the higher escape-time configuration maintains enclosure under perturbed friction coefficients where the lower one fails. This addition will make the load-bearing claim more transparent without altering the core method. revision: yes

  2. Referee: [Abstract / Methods] The abstract states that the escape-time field satisfies the eikonal equation and that a PINN is used to approximate it, yet supplies no derivation steps, boundary-condition enforcement details (T=0 on boundary, |∇T|=1 inside), or error analysis for the PINN training. Because the eikonal property is the justification for using the PINN inside the planner, this omission affects assessment of the method's soundness.

    Authors: The derivation that the minimum-time escape problem yields the eikonal equation (with T=0 on the boundary and |∇T|=1 in the free space) appears in Section III-B of the manuscript, along with the PINN loss formulation that enforces these conditions via the PDE residual. However, the abstract and early sections do not recap these steps. In the revision we will insert a concise derivation outline (one paragraph) into the introduction immediately after the problem statement, explicitly stating the boundary condition and the |∇T|=1 interior condition. We will also add a short error analysis subsection in the experiments (including L2 residual norms on a held-out grid and maximum pointwise error) to quantify PINN accuracy and support the claim that the approximation is sufficiently faithful for use inside the planner. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external PDE and experimental validation

full rationale

The paper defines caging via a min-time escape reformulation (a modeling choice), notes that the resulting field satisfies the eikonal equation by the standard Hamilton-Jacobi property of unit-speed escape (external fact), approximates it with a PINN (standard technique whose loss is independent of the planning objective), and embeds the field into an optimization cost. Robustness to contact mismatch is asserted as an empirical outcome validated in simulation and hardware experiments rather than derived by construction from any fitted parameter or self-citation. No load-bearing step reduces to a definitional equivalence or internal fit.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the mathematical fact that the minimum-time escape function satisfies the eikonal equation (standard PDE) and on the modeling assumption that geometric enclosure time is a sufficient proxy for robustness to contact mismatch. No free parameters or invented entities are mentioned.

axioms (2)
  • standard math The minimum-time escape function under enclosing robot geometry satisfies the eikonal equation.
    Invoked in the abstract when the escape-time field is introduced.
  • domain assumption Geometric enclosure quality measured by escape time provides robustness to contact-model inaccuracy.
    Stated as the motivation for using caging in the first paragraph of the abstract.

pith-pipeline@v0.9.1-grok · 5790 in / 1438 out tokens · 14660 ms · 2026-06-26T11:39:59.714658+00:00 · methodology

discussion (0)

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Reference graph

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