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arxiv: 2604.09800 · v1 · submitted 2026-04-10 · 💻 cs.RO · cs.SY· eess.SY

Kinematics of continuum planar grasping

Udit Halder , Nicolas Echeverria Zambrano , Xincheng Li This is my paper

Pith reviewed 2026-05-10 16:36 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords continuum roboticsplanar graspingkinematic modelingboundary followingshadow curveoptimal controlgrasp metricssoft robots
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The pith

Grasping with soft continuum arms reduces to a kinematic boundary-following problem in which the object boundary serves as the arm's shadow curve.

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

The paper sets out an analytical framework for planar grasping by soft continuum arms by modeling both the arm centerline and the object boundary as smooth curves. The grasping task is reformulated as a boundary-following kinematic problem where the object boundary functions as the arm's shadow curve. This reduction produces a simpler set of kinematic equations that use arm curvature as the control input. Solving the associated optimal control problem via Pontryagin's Maximum Principle identifies optimal arm shapes, from which the authors define a new class of grasp quality metrics based on the algebraic properties of the continuum grasp map. This geometric approach matters because it offers a way to analyze and plan soft grasping without relying on rigid-link approximations, potentially improving control design for compliant robots.

Core claim

The grasping problem is formulated as a kinematic boundary following problem in which the object boundary acts as the arm's shadow curve. This formulation leads to reduced kinematic equations expressed in terms of relative geometric shape variables, with the arm curvature serving as the control input. An optimal control problem is formulated to determine feasible arm shapes that achieve optimal grasping configurations, and its solution is obtained using Pontryagin's Maximum Principle. Based on the resulting optimal grasp kinematics, a class of continuum grasp quality metrics is proposed using the algebraic properties of the associated continuum grasp map.

What carries the argument

The shadow curve formulation of the object boundary, which reduces the full arm kinematics to boundary-following equations controlled solely by arm curvature.

If this is right

  • Optimal arm shapes for grasping are found by solving the reduced boundary-following optimal control problem using Pontryagin's Maximum Principle.
  • Grasp quality metrics arise directly from the algebraic properties of the continuum grasp map derived from optimal solutions.
  • Feedback control strategies for dynamic grasping follow from the kinematic model in the dynamic setting.
  • Numerical simulations confirm the method across different planar object geometries.

Where Pith is reading between the lines

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

  • The boundary-following reduction may allow real-time optimization of grasp configurations in soft robotic systems by avoiding high-dimensional state spaces.
  • Similar shadow-curve concepts could extend the framework to non-planar grasping or multi-arm coordination tasks.
  • These quality metrics might be validated against traditional force-closure measures in physical continuum robot experiments.

Load-bearing premise

The arm centerline and the object boundary can be modeled as smooth curves, and the boundary-following reduction captures all essential kinematic constraints for feasible grasping.

What would settle it

A grasping experiment with a physical continuum arm and an object having discontinuous curvature would falsify the claim if the predicted optimal shapes and contact geometries do not match the observed behavior.

Figures

Figures reproduced from arXiv: 2604.09800 by Nicolas Echeverria Zambrano, Udit Halder, Xincheng Li.

Figure 1
Figure 1. Figure 1: (a) Schematic of a continuum arm grasping a planar object with smooth boundary. The boundary of the object [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Optimal grasping configurations obtained by solving the optimal control problem (9). The problem is solved for three different [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Grasp quality maps for the metrics Q1, Q2 and Q3 for the circular, elliptical, and deformed circular objects. The quality values are normalized by the maximum quality for each object, where darker regions denote higher quality. The purple point indicates the optimal starting location, and the corresponding arm configuration is shown as a purple dashed curve. solving (16) are drawn in purple dashed lines. A… view at source ↗
Figure 4
Figure 4. Figure 4: Dynamic simulation results for a soft arm grasping a fixed [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

This paper presents an analytical framework to study the geometry arising when a soft continuum arm grasps a planar object. Both the arm centerline and the object boundary are modeled as smooth curves. The grasping problem is formulated as a kinematic boundary following problem, in which the object boundary acts as the arm's 'shadow curve'. This formulation leads to a set of reduced kinematic equations expressed in terms of relative geometric shape variables, with the arm curvature serving as the control input. An optimal control problem is formulated to determine feasible arm shapes that achieve optimal grasping configurations, and its solution is obtained using Pontryagin's Maximum Principle. Based on the resulting optimal grasp kinematics, a class of continuum grasp quality metrics is proposed using the algebraic properties of the associated continuum grasp map. Feedback control aspects in the dynamic setting are also discussed. The proposed methodology is illustrated through systematic numerical simulations.

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

3 major / 2 minor

Summary. The paper develops an analytical framework for the geometry of soft continuum planar grasping by modeling both the arm centerline and object boundary as smooth curves. It formulates grasping as a kinematic boundary-following problem in which the object boundary is the arm's 'shadow curve', yielding reduced kinematic equations in relative shape variables with arm curvature as the sole control input. An optimal control problem is solved via Pontryagin's Maximum Principle to obtain feasible grasp shapes, from which a class of algebraic continuum grasp quality metrics is derived using properties of the grasp map; dynamic feedback control is discussed and the approach is illustrated with numerical simulations.

Significance. If the shadow-curve reduction is shown to preserve all unilateral contact constraints without introducing non-physical penetrations or gaps, the work would offer a geometrically principled route to interpretable, algebraic grasp metrics for continuum arms that could complement or replace heuristic quality measures. The explicit use of PMP on the reduced system and the derivation of metrics from the grasp map are strengths that could enable systematic design and control; however, the absence of detailed validation against full kinematic constraints or physical benchmarks currently limits the immediate applicability.

major comments (3)
  1. [Kinematic reduction and shadow curve construction] The boundary-following reduction (detailed after the geometric modeling assumptions) asserts that the object boundary as shadow curve encodes all relevant kinematic constraints for feasible grasping, yet no explicit verification or inequality is provided showing that the relative shape variables prevent local interpenetration or violation of contact normals when the arm wraps only partially or encounters curvature discontinuities.
  2. [Optimal control formulation and PMP solution] While PMP is applied to the optimal control problem with curvature as input, the manuscript provides no error analysis, residual checks, or comparison of the resulting optimal shapes against the unreduced planar kinematics to confirm that solutions satisfy no-slip and no-penetration conditions throughout the contact; this is load-bearing for the claim that the derived metrics correspond to physically realizable grasps.
  3. [Numerical simulations section] The numerical simulations are described as systematic but are not shown to include quantitative validation (e.g., penetration depth metrics or contact force consistency) that would confirm the reduced equations capture the full set of unilateral constraints; without this, the proposed grasp quality metrics rest on an unverified modeling assumption.
minor comments (2)
  1. Notation for the relative shape variables and the grasp map could be introduced with a single consolidated table or diagram to improve readability across the kinematic and optimal-control sections.
  2. [Feedback control discussion] The abstract states that feedback control aspects are discussed, but the transition from the kinematic optimal solutions to the dynamic setting lacks an explicit statement of the assumed actuation model or stability guarantees.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We address each point below with clarifications on the kinematic model and indicate revisions that will be incorporated to provide the requested verifications.

read point-by-point responses

  1. Referee: [Kinematic reduction and shadow curve construction] The boundary-following reduction (detailed after the geometric modeling assumptions) asserts that the object boundary as shadow curve encodes all relevant kinematic constraints for feasible grasping, yet no explicit verification or inequality is provided showing that the relative shape variables prevent local interpenetration or violation of contact normals when the arm wraps only partially or encounters curvature discontinuities.

    Authors: We agree that an explicit verification strengthens the presentation. The shadow-curve construction defines the object boundary as the envelope of the arm centerline under the no-penetration and tangency conditions, so the relative shape variables (relative curvature and arc-length offset) are derived to enforce these by definition. For partial wrapping and curvature discontinuities we will add an appendix deriving the inequality conditions on the relative variables that guarantee non-negative separation distance and consistent normal directions. This will be included in the revised version. revision: yes

  2. Referee: [Optimal control formulation and PMP solution] While PMP is applied to the optimal control problem with curvature as input, the manuscript provides no error analysis, residual checks, or comparison of the resulting optimal shapes against the unreduced planar kinematics to confirm that solutions satisfy no-slip and no-penetration conditions throughout the contact; this is load-bearing for the claim that the derived metrics correspond to physically realizable grasps.

    Authors: The reduced system is kinematically equivalent to the full planar kinematics precisely when the boundary-following assumption holds. We will add a new subsection that integrates selected PMP-optimal shapes with the unreduced kinematic equations and reports residual norms (position and orientation errors) below 10^{-5} together with explicit checks that the minimum inter-curve distance remains non-negative and contact normals align. This will confirm that the grasp metrics are based on kinematically realizable configurations. revision: yes

  3. Referee: [Numerical simulations section] The numerical simulations are described as systematic but are not shown to include quantitative validation (e.g., penetration depth metrics or contact force consistency) that would confirm the reduced equations capture the full set of unilateral constraints; without this, the proposed grasp quality metrics rest on an unverified modeling assumption.

    Authors: The simulations are kinematic; contact forces are outside the paper's scope. We will augment the numerical section with quantitative kinematic validation: for each reported grasp we will tabulate the maximum penetration depth (minimum signed distance between arm and object curves) and the residual of the reduced kinematic equations. These metrics will be shown to remain within numerical tolerance, directly confirming that the unilateral constraints are satisfied in the reduced model. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from geometric modeling and PMP is self-contained

full rationale

The paper starts from explicit modeling assumptions (smooth curves for arm and boundary) and standard optimal-control theory (Pontryagin's Maximum Principle applied to reduced relative-shape equations with curvature as input). Grasp-quality metrics are algebraically defined from the resulting continuum grasp map after solving the OCP. No step reduces a prediction to a fitted parameter, renames a known result, or relies on a load-bearing self-citation whose content is unverified. The chain is independent of the target metrics and externally falsifiable via simulation or physical experiment.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on standard differential-geometry modeling of curves and the applicability of Pontryagin's Maximum Principle to the reduced system; no free parameters are introduced in the abstract, but the smoothness assumption is foundational.

axioms (2)
  • domain assumption Both the arm centerline and the object boundary can be represented as smooth curves
    Explicitly stated as the modeling choice that enables the boundary-following formulation.
  • domain assumption The grasping configuration can be reduced to a kinematic boundary-following problem without loss of contact constraints
    Central modeling step that produces the reduced kinematic equations.
invented entities (1)
  • shadow curve no independent evidence
    purpose: Represents the object boundary as the effective path that the arm centerline must follow during grasping
    New geometric construct introduced to simplify the contact kinematics.

pith-pipeline@v0.9.0 · 5449 in / 1216 out tokens · 25452 ms · 2026-05-10T16:36:55.524326+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Design, fabrication and control of soft robots,

    D. Rus and M. T. Tolley, “Design, fabrication and control of soft robots,”Nature, vol. 521, no. 7553, pp. 467–475, 2015

    work page 2015

  2. [2]

    Topology, dynamics, and control of a muscle- architected soft arm,

    A. Tekinalpet al., “Topology, dynamics, and control of a muscle- architected soft arm,”Proceedings of the National Academy of Sci- ences, vol. 121, no. 41, p. e2318769121, 2024

    work page 2024

  3. [3]

    Autonomous continuum grasping,

    J. Liet al., “Autonomous continuum grasping,” in2013 IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE, 2013, pp. 4569–4576

    work page 2013

  4. [4]

    A comprehensive grasp taxonomy of continuum robots,

    A. Mehrkish and F. Janabi-Sharifi, “A comprehensive grasp taxonomy of continuum robots,”Robotics and Autonomous Systems, vol. 145, p. 103860, 2021

    work page 2021

  5. [5]

    Spirobs: Logarithmic spiral-shaped robots for versatile grasping across scales,

    Z. Wanget al., “Spirobs: Logarithmic spiral-shaped robots for versatile grasping across scales,”Device, 2024

    work page 2024

  6. [6]

    R. M. Murrayet al.,A mathematical introduction to robotic manipu- lation. CRC Press, 1994

    work page 1994

  7. [7]

    Robotic grasping and contact: A review,

    A. Bicchi and V . Kumar, “Robotic grasping and contact: A review,” inProceedings of IEEE International Conference on Robotics and Automation, vol. 1. IEEE, 2000, pp. 348–353

    work page 2000

  8. [8]

    Design and kinematic modeling of constant curvature continuum robots: A review,

    R. J. Webster III and B. A. Jones, “Design and kinematic modeling of constant curvature continuum robots: A review,”The International Journal of Robotics Research, vol. 29, no. 13, pp. 1661–1683, 2010

    work page 2010

  9. [9]

    S. S. Antman,Nonlinear Problems of Elasticity. Springer, 1995

    work page 1995

  10. [10]

    Energy-shaping control of a muscular octopus arm moving in three dimensions,

    H.-S. Changet al., “Energy-shaping control of a muscular octopus arm moving in three dimensions,”Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 479, no. 2270, p. 20220593, 2023

    work page 2023

  11. [11]

    Model-based control of soft robots: A survey of the state of the art and open challenges,

    C. Della Santinaet al., “Model-based control of soft robots: A survey of the state of the art and open challenges,”IEEE Control Systems Magazine, vol. 43, no. 3, pp. 30–65, 2023

    work page 2023

  12. [12]

    Optimal control of a soft cyberoctopus arm,

    T. Wanget al., “Optimal control of a soft cyberoctopus arm,” in2021 American Control Conference (ACC). IEEE, 2021, pp. 4757–4764

    work page 2021

  13. [13]

    A sensory feedback control law for octopus arm movements,

    ——, “A sensory feedback control law for octopus arm movements,” in2022 IEEE 61st Conference on Decision and Control (CDC). IEEE, 2022, pp. 1059–1066

    work page 2022

  14. [14]

    Neural models and algorithms for sensorimotor control of an octopus arm,

    ——, “Neural models and algorithms for sensorimotor control of an octopus arm,”Biological Cybernetics, vol. 119, no. 4, p. 25, 2025

    work page 2025

  15. [15]

    A physics-informed, vision-based method to recon- struct all deformation modes in slender bodies,

    S. H. Kimet al., “A physics-informed, vision-based method to recon- struct all deformation modes in slender bodies,” in2022 International Conference on Robotics and Automation (ICRA). IEEE, 2022, pp. 4810–4817

    work page 2022

  16. [16]

    Modeling of grasping force for a soft robotic gripper with variable stiffness,

    Y . Haibinet al., “Modeling of grasping force for a soft robotic gripper with variable stiffness,”Mechanism and Machine Theory, vol. 128, pp. 254–274, 2018

    work page 2018

  17. [17]

    Cosserat-rod-based dynamic modeling of soft slender robot interacting with environment,

    L. Xunet al., “Cosserat-rod-based dynamic modeling of soft slender robot interacting with environment,”IEEE Transactions on Robotics, vol. 40, pp. 2811–2830, 2024

    work page 2024

  18. [18]

    Statics of continuum planar grasping,

    U. Halder, “Statics of continuum planar grasping,” in2025 IEEE 64th Conference on Decision and Control (CDC). IEEE, 2025, pp. 5437– 5444

    work page 2025

  19. [19]

    Boundary following using gyroscopic control,

    F. Zhanget al., “Boundary following using gyroscopic control,” in 2004 43rd IEEE Conference on Decision and Control (CDC), vol. 5. IEEE, 2004, pp. 5204–5209

    work page 2004

  20. [20]

    Liberzon,Calculus of variations and optimal control theory: a concise introduction

    D. Liberzon,Calculus of variations and optimal control theory: a concise introduction. Princeton University Press, 2011

    work page 2011

  21. [21]

    Grasp quality measures: review and performance,

    M. A. Roa and R. Su ´arez, “Grasp quality measures: review and performance,”Autonomous Robots, vol. 38, pp. 65–88, 2015

    work page 2015

  22. [22]

    Characterisation of grasp quality metrics,

    C. Rubertet al., “Characterisation of grasp quality metrics,”Journal of Intelligent & Robotic Systems, vol. 89, pp. 319–342, 2018

    work page 2018

  23. [23]

    Forward and inverse problems in the mechanics of soft filaments,

    M. Gazzolaet al., “Forward and inverse problems in the mechanics of soft filaments,”Royal Society Open Science, vol. 5, no. 6, p. 171628, 2018

    work page 2018

  24. [24]

    Gazzolalab/pyelastica,

    A. Tekinalpet al., “Gazzolalab/pyelastica,” 2024. [Online]. Available: https://doi.org/10.5281/zenodo.7658871

    work page doi:10.5281/zenodo.7658871 2024

  25. [25]

    Energy shaping control of a cyberoctopus soft arm,

    H.-S. Changet al., “Energy shaping control of a cyberoctopus soft arm,” in2020 59th IEEE Conference on Decision and Control (CDC). IEEE, 2020, pp. 3913–3920

    work page 2020

  26. [26]

    M ´emoire sur la th ´eorie des courbes `a double courbure,

    J. Bertrand, “M ´emoire sur la th ´eorie des courbes `a double courbure,” Journal de Math ´ematiques Pures et Appliqu ´ees, vol. 15, pp. 332–350, 1850

    work page

  27. [27]

    K. L. Johnson,Contact mechanics. Cambridge university press, 1987

    work page 1987

  28. [28]

    V . L. Popovet al.,Handbook of contact mechanics: exact solutions of axisymmetric contact problems. Springer Nature, 2019

    work page 2019

  29. [29]

    Steering for beacon pursuit under limited sensing,

    U. Halderet al., “Steering for beacon pursuit under limited sensing,” in 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, 2016, pp. 3848–3855

    work page 2016

  30. [30]

    Controlling a cyberoctopus soft arm with muscle- like actuation,

    H.-S. Changet al., “Controlling a cyberoctopus soft arm with muscle- like actuation,” in2021 60th IEEE Conference on Decision and Control (CDC). IEEE, 2021, pp. 1383–1390

    work page 2021

  31. [31]

    Task-oriented optimal grasping by multifin- gered robot hands,

    Z. Li and S. S. Sastry, “Task-oriented optimal grasping by multifin- gered robot hands,”IEEE Journal on Robotics and Automation, vol. 4, no. 1, pp. 32–44, 1988

    work page 1988

  32. [32]

    Optimal grasping based on non-dimensionalized performance indices,

    B.-H. Kimet al., “Optimal grasping based on non-dimensionalized performance indices,” inProceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No. 01CH37180), vol. 2. IEEE, 2001, pp. 949–956

    work page 2001

  33. [33]

    H. K. Khalil,Nonlinear Systems. Prentice Hall, 2002. APPENDIXI PROOF OFPROPOSITION4.1 Proof:Indeed, by definition, the singular values of Gare the positive square roots of the self-adjoint operator G∗G:U → U, whereG ∗ :R 3 → Uis the adjoint ofG. It is obvious that the adjoint mapG ∗ is expressed asG ∗w= G⊺(s)w, wherew∈R 3. Then, we find the eigenvaluesλof...

    work page 2002