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Manipulation with Stability Guarantees: Linear Deformable Objects with Non-negligible Physical Response Grasped at Multiple Location

Daniel Feliu-Talegon, Cosimo Della Santina

arXiv:2609.26004v1cs.RO

TL;DR

Many DLO manipulation approaches focus on quasi-static or negligible-response objects, leaving dynamic behavior under significant physical response a challenge. This paper introduces model-based closed-loop shape regulation through multiple contacts, and reports accurate dynamic manipulation in simulation and experiments, including zero steady-state error where dynamics-neglecting controllers develop error.

  • Problem

    Many deformable-object manipulation methods focus on lightweight systems with negligible mechanical response, while DLO manipulation with guaranteed stability and precision remains an open problem.

    Source: §1 · p. 1

  • Method

    The paper models DLOs as floating-base rods with localized wrench inputs and uses collocated coordinates and feedback control to regulate shape.

    Source: §1 · pp. 1–2, §4 · p. 7

  • Results

    Controllers neglecting object dynamics developed nonzero steady-state errors as stiffness and inertia increased, whereas the proposed formulation achieved zero steady-state error.

    Source: §8 · p. 18

  • Takeaways & Limitations

    A single set of identified physical parameters was used across diverse experimental scenarios, indicating potential for manipulation beyond quasi-static regimes.

    Source: §8 · p. 18

    The stiffness and damping matrices are configuration-independent under the stated Hooke-like linear elastic and viscoelastic constitutive laws.

    Source: §3.1 · p. 4

Abstract

from arXiv · show

Most research on the manipulation of deformable objects focuses on lightweight systems with negligible mechanical response, effectively restricting attention to quasi-static regimes. This assumption excludes a broad class of practically relevant objects, such as hoses, pipes, and wiring harnesses, whose dynamics cannot be ignored during manipulation. In this work, we address this limitation by introducing a closed-loop control architecture that explicitly accounts for object dynamics and recasts manipulation as a shape-regulation problem. Control is achieved by modulating forces and torques applied at multiple fixed points along the object. This approach builds on three methodological contributions: a fully dynamic model of linear deformable objects based on discrete strain parameterizations; an extension of the notion of actuation coordinates to SE(3), yielding a structured and inherently underactuated control architecture; and nonlinear feedback strategies providing explicit conditions for steady-state convergence to desired configurations. Extensive simulations on representative manipulation tasks demonstrate the performance gains enabled by the proposed modelbased formulation. We finally validate the approach experimentally through a real-time closed-loop implementation with online shape estimation, confirming its practical feasibility and effectiveness

1 INTRODUCTION

DLO manipulation remains difficult when significant stiffness, mass, or rapid motion makes physical dynamics relevant, while many approaches neglect those effects. The paper introduces model-based closed-loop shape regulation using multiple contact wrenches, a dynamic rod model, and stability-oriented control.

1 INTRODUCTION

  • Existing DLO manipulation often neglects physical response, and guaranteed stability and precision remain open problems.This is especially relevant when stiffness or mass is significant or the object moves rapidly.Source: §1 · p. 1
  • The framework treats generic three-dimensional DLOs under multiple contacts as dynamical systems regulated by applied forces and torques.Its model captures distributed inertia, elasticity, and gravity, with discretization yielding low-dimensional dynamic approximations.Source: §1 · pp. 1–2
  • An SE(3)-based extension of collocated coordinates separates contact-driven directions from internal dynamics, supporting task-consistent shape selection and stability-conditioned feedback.The framework also presents conditions for convergence to a desired equilibrium shape.Source: §1 · p. 2
  • Simulations and experiments validate a two-arm pipeline and real-time implementation for dynamic shape control beyond quasi-static regimes.The paper outlines high-level desired-shape generation and low-level wrench regulation in closed loop.Source: §1 · p. 2

2 RELATED WORKS

Prior DLO methods include physical models, geometric shape regulation, dynamic control, and learning-based strategies, but face computational, scope, or deployment constraints. The paper positions its contribution as a stable three-dimensional dynamic framework supporting multiple force and torque contacts.

2 RELATED WORKS

  • Physical DLO models range from mass–spring and elastic-rod formulations to FEM and spline-based methods, but robotics adoption is often constrained by computation [Lv et al. (2020); Chen et al. (2024)].Reduced-DoF rod models offer control-friendly ODE formulations [Della Santina and Rus (2019); Mathew et al. (2025); Webster III and Jones (2010)].Source: §2 · pp. 2–3
  • Geometry-focused regulation commonly assumes quasi-static behavior, while dynamic model-based methods often remain computationally heavy or lack real-time stability guarantees [Navarro-Alarcon et al. (2013); Lv et al. (2022); Koessler et al. (2021)].Some analytical approaches plan with static equilibrium configurations [Bretl and McCarthy (2014); Roussel et al. (2015)].Source: §2 · p. 3
  • Learning-based dynamic manipulation includes task- and object-specific demonstrations, data-intensive learned models, and local Jacobian methods restricted to small deformations [Nair et al. (2017); Yang et al. (2021); Zhu et al. (2018)].Other methods combine offline and online learning or use compact representations [Yu, Lv, Zhong, Song and Li (2022); Lan et al. (2025)].Source: §2 · p. 3
  • Prior stability-guaranteed DLO control was limited to planar endpoint regulation with one manipulator, whereas this work supports three-dimensional deformations and multiple wrench locations.The proposed framework is presented as supporting coordinated single- to multi-arm manipulation.Source: §2 · p. 3
  • The proposed framework aims for fast dynamic manipulation with provable stability and high precision when inertial effects are non-negligible.The paper frames this as a foundation for reliable real-time DLO manipulation.Source: §2 · p. 4

3 Collocation of Wrenches: extending the Actuation Coordinates in SE(3) × · · · × SE(3)

The section asks how to decouple actuated and unactuated dynamics under multiple wrench inputs, then develops coordinate transformations that enable shape regulation while accounting for forces and torques.

3 Collocation of Wrenches: extending the Actuation Coordinates in SE(3) × · · · × SE(3)

  • The section asks whether multiple wrench inputs permit a coordinate change that decouples actuated and unactuated dynamics, a result also relevant to other free-floating mechanical systems.Examples named include spaceborne platforms, aerial or underwater vehicles, and rigid bodies with multiple spatial contact wrenches.Source: §3 · p. 4

3.1 General dynamics

  • The generalized dynamics use a Lagrangian model with mass, Coriolis, damping, stiffness, gravity, and an actuation matrix mapping inputs to generalized forces.Hooke-like linear elastic and viscoelastic laws make the stiffness and damping matrices configuration-independent.Source: §3.1 · p. 4

3.2 Goal: decoupling through coordinates transformation

  • Input decoupling means finding coordinates whose input matrix is [I 0]^T, separating directly actuated coordinates from internal, unactuated motions.A system admits this representation if and only if A^T(q)q̇ is integrable in time.Source: §3.2 · p. 5

3.3.1 Integrability of pure forces

  • For a pure force, the projected generalized velocity integrates to the contact-point position, allowing an actuation coordinate to be introduced.The passage ties this result to the kinematic identity between contact-point velocity and the spatial Jacobian.Source: §3.3.1 · p. 5

3.3 Extend actuation coordinates

  • Torque directions are generally nonintegrable because their angular-velocity integral is path-dependent, unlike the integrable directions associated with pure forces.Consequently, no actuation coordinate can be associated with a pure torque in the stated sense.Source: §3.3.2 · p. 5, §3.3.1 · p. 5

3.3.2 Integrability of pure torques

  • For pure torques, the angular-velocity integral is path-dependent, so no configuration function or actuation coordinate represents it in the stated sense.A closed trajectory can have a nonzero integral, as illustrated by the rotation example.Source: §3.3.2 · p. 5, §3.3.3 · p. 5

3.3.3 Actuation coordinates by simultaneous input and configuration transformation

  • A simultaneous transformation of inputs and configuration coordinates provides a workaround when torque directions cannot be integrated, relaxing the requirement that the actuation matrix equal [I 0]^T.The construction instead seeks a full-rank block; an invertible matrix P(zi) connects nonintegrable angular velocities to integrable orientation-parameter rates.Source: §3.3.3 · pp. 5–6, §3.3 · p. 5

4 Dynamics of a DLO with non-negligible physical response

The section presents a floating-base dynamic model of a deformable linear object, represented as a Cosserat rod with its configuration and motion described along its length. Discrete strain coordinates capture deformation, with segmentation used to represent discontinuities caused by applied forces.

4 Dynamics of a DLO with non-negligible physical response

  • A floating slender body is modeled as a Cosserat rod with a free base, subject to gravity and pointwise forces and torques along its length.The model is used to represent wrenches applied by multiple manipulators.Source: §4 · p. 6
  • The rod’s cross-section configuration and motion are represented along curvilinear coordinate s by local frames, poses, twists, and a strain field.The pose and body velocity vary along the object’s backbone.Source: §4 · p. 7, Figure 3 · p. 7
  • The strain parameterization uses generalized coordinates and basis functions to represent six deformation modes, including bending, twisting, shear, and elongation.The object may be divided into elements, with the suggested segmentation matching intermediate force locations to represent strain-derivative discontinuities more accurately.Source: §4 · pp. 7–8

5 Model-based closed-loop control of DLOs

The control architecture regulates DLO shape by applying wrenches at grasp points, with the object’s dynamics informing controller design. It combines low-level feedback with a high-level task-space objective.

5 Model-based closed-loop control of DLOs

  • The general architecture uses a model-based design and analyzes stability and convergence for a broad family of controllers.Shape regulation is achieved by applying wrenches where robots grasp the DLO.Source: §5 · p. 8, §5.1 · p. 8

5.1 Low-Level Control

  • The low-level control law regulates actuated coordinates while retaining unactuated dynamics, with bounded state convergence under the stated controller and coordinate conditions.An additional condition yields a unique, globally asymptotically stable equilibrium.Source: §5.1 · p. 8
  • Partial feedback linearization guarantees exponential stability and fast actuated-coordinate convergence, but is harder to implement experimentally because it cancels collocated Coriolis and inertia effects.The convergence rate is determined by controller design.Source: §5.1 · p. 9

5.2 High-Level Control to achieve a task goal

  • The high-level controller defines a task-space output from one or more points on the DLO and optimizes control actions toward goals such as point poses or object shape.Constraints on actuated coordinates can be incorporated to prevent environmental collisions in constrained settings.Source: §5.2 · p. 9

6 Simulation results

Simulations compare controllers for DLO shape regulation and test the approach under more complex grasping conditions. Model-based controllers achieve the reported shape and pose regulation in these scenarios.

6 Simulation results

  • Simulations drive slender soft objects with different geometries toward desired shapes under varied scenarios, including different wrench application points.The evaluation emphasizes low-level shape regulation, controller robustness, stability, convergence, and model-based components.Source: §6 · p. 9, §6.1 · p. 9

6.1 Performance Comparison of the Controllers

  • PD control with model-based terms and partial feedback linearization avoid steady-state error and overshoot, unlike conventional PD and PID-salt, respectively.Conventional PD shows steady-state error; PID-salt removes it but introduces overshoot, while PFL produces a straight-line trajectory through cancellation of actuated-part dynamics.Source: §6.1 · pp. 9–10, Figure 4 · p. 10

6.2 Evaluation of the approach under more complex conditions

  • Model-based PD regulates grasp-point positions and orientations in both a clamped two-point scenario and a floating bimanual scenario.The simulations use multiple target shapes; the clamped case models the beam with two elements because a wrench is applied at an intermediate point.Source: §6.2 · p. 10, Figure 5 · p. 11

7 Experimental Evaluation

The experiments evaluate real-time shape estimation and closed-loop control on cables using two robots, motion-capture feedback, and multiple grasp configurations. They test parameter identification, control across objects and motion speeds, and task-level positioning against a model-free reference.

7.1 Implementation details

  • Two Franka Panda robots, real cables, and motion-capture markers supplied real-time shape estimates and state feedback for closed-loop control.Robot forward kinematics provided the actuated coordinates.Source: §7.1 · p. 11

7.2 Parameter identification

  • Density and Young’s modulus were identified from gravity-induced equilibrium measurements using four markers; damping was fitted from the released object’s time response.The resulting parameters for three deformable objects are reported in Table 2.Source: §7.2 · p. 12, Table 2 · p. 12

7.3 Estimation of the configuration using efficient inverse kinematics

  • A gradient-descent nonlinear least-squares estimator combines marker-position and robot-grasp pose errors and ran above 150 Hz with 6 DoF.The repeated feedback-loop optimization is described as reducing computational complexity and memory use.Source: §7.3 · pp. 12–13

7.4 Experimental results

  • Across three DLOs, endpoint control showed comparable accuracy, indicating compensation remained effective despite variation in object properties.The experiment controlled both endpoint positions while maintaining constant orientation.Source: §7.4 · pp. 13, 15
  • With one robot grasping at s = 0.6, the controller regulated an intermediate point and an endpoint, with 0.5/1 cm average/maximum position errors and 0.04/0.1 rad orientation errors.The remaining DLO segment was assumed to deform negligibly, reducing shape estimation to six generalized coordinates.Source: §7.4 · p. 15
  • For faster endpoint motions, position errors averaged 0.35 cm and peaked at 1.3 cm, while Euler-angle variation stayed below 0.08 rad.The trajectory durations ranged from 2 s to 0.5 s.Source: §7.4 · p. 15
  • Across evaluated points, mean endpoint positioning error decreased from 3.5 cm (maximum 8.8 cm) with the model-free reference to 1.5 cm (maximum 2 cm) with our method.Source: §7.4 · p. 16

8 Conclusions

The conclusion presents a model-based closed-loop framework for dynamically manipulating deformable linear objects through multiple contacts. Simulations and experiments report accurate, generalized, real-time performance, including zero steady-state error where dynamics-neglecting controllers develop error as stiffness and inertia increase.

8 Conclusions

  • The framework models deformable linear objects as floating-base rods under localized forces and torques, capturing dynamic coupling, elasticity, and gravity.Manipulation is posed as shape regulation, with Cartesian task-space objectives included when needed.Source: §8 · p. 17
  • Controllers neglecting object dynamics developed nonzero steady-state errors as stiffness and inertia increased, whereas the proposed model-based formulation achieved zero steady-state error.Source: §8 · p. 18
  • A single set of identified physical parameters was used across diverse experimental scenarios, supporting the reported robustness and generality beyond quasi-static regimes.Source: §8 · p. 18

A Collocated form in Lagrangian systems

The section describes a coordinate transformation for rewriting a Lagrangian system in collocated form. Its construction depends on an integrability condition and, in underactuated systems, a full-rank coordinate transformation.

A Collocated form in Lagrangian systems

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