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Controlling Deformable Objects with Non-negligible Dynamics: a Shape-Regulation Approach to End-Point Positioning

Sebastien Tiburzio, Tomás Coleman, Daniel Feliu-Talegon, Cosimo Della Santina

TL;DR

This work addresses end-point control of deformable linear objects with significant dynamics by developing a finite-dimensional, strain-based dynamic model (Cosserat-inspired) and casting manipulation as a shape-regulation control problem. It proves closed-loop stability for a PD-like low-level controller with gravity compensation and an elastic-dominance condition, and couples this with a model-constrained nonlinear optimization at the high level to achieve target object shapes. Extensive experiments on six heavy cables validate the modeling assumptions, parameter identification, and controller performance, demonstrating substantial improvements in end-point positioning (63.0%–78.5% error reduction versus model-free baselines) and feasible orientation control within practical angular bounds. The approach offers a principled, model-based pathway for robust, planar manipulation of DLOs with non-negligible dynamics, with potential extensions to 3D shapes, higher-order strain models, and dynamic trajectory tasks.

Abstract

Model-based manipulation of deformable objects has traditionally dealt with objects while neglecting their dynamics, thus mostly focusing on very lightweight objects at steady state. At the same time, soft robotic research has made considerable strides toward general modeling and control, despite soft robots and deformable objects being very similar from a mechanical standpoint. In this work, we leverage these recent results to develop a control-oriented, fully dynamic framework of slender deformable objects grasped at one end by a robotic manipulator. We introduce a dynamic model of this system using functional strain parameterizations and describe the manipulation challenge as a regulation control problem. This enables us to define a fully model-based control architecture, for which we can prove analytically closed-loop stability and provide sufficient conditions for steady state convergence to the desired state. The nature of this work is intended to be markedly experimental. We provide an extensive experimental validation of the proposed ideas, tasking a robot arm with controlling the distal end of six different cables, in a given planar position and orientation in space.

Controlling Deformable Objects with Non-negligible Dynamics: a Shape-Regulation Approach to End-Point Positioning

TL;DR

This work addresses end-point control of deformable linear objects with significant dynamics by developing a finite-dimensional, strain-based dynamic model (Cosserat-inspired) and casting manipulation as a shape-regulation control problem. It proves closed-loop stability for a PD-like low-level controller with gravity compensation and an elastic-dominance condition, and couples this with a model-constrained nonlinear optimization at the high level to achieve target object shapes. Extensive experiments on six heavy cables validate the modeling assumptions, parameter identification, and controller performance, demonstrating substantial improvements in end-point positioning (63.0%–78.5% error reduction versus model-free baselines) and feasible orientation control within practical angular bounds. The approach offers a principled, model-based pathway for robust, planar manipulation of DLOs with non-negligible dynamics, with potential extensions to 3D shapes, higher-order strain models, and dynamic trajectory tasks.

Abstract

Model-based manipulation of deformable objects has traditionally dealt with objects while neglecting their dynamics, thus mostly focusing on very lightweight objects at steady state. At the same time, soft robotic research has made considerable strides toward general modeling and control, despite soft robots and deformable objects being very similar from a mechanical standpoint. In this work, we leverage these recent results to develop a control-oriented, fully dynamic framework of slender deformable objects grasped at one end by a robotic manipulator. We introduce a dynamic model of this system using functional strain parameterizations and describe the manipulation challenge as a regulation control problem. This enables us to define a fully model-based control architecture, for which we can prove analytically closed-loop stability and provide sufficient conditions for steady state convergence to the desired state. The nature of this work is intended to be markedly experimental. We provide an extensive experimental validation of the proposed ideas, tasking a robot arm with controlling the distal end of six different cables, in a given planar position and orientation in space.
Paper Structure (29 sections, 3 theorems, 25 equations, 18 figures, 2 tables, 1 algorithm)

This paper contains 29 sections, 3 theorems, 25 equations, 18 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Strain-Based Dynamic Modeling of DLOs
  3. Additional assumptions
  4. Object's Backbone Kinematics
  5. Object's Full Kinematics Under Cosserat Assumption
  6. Object's Dynamics
  7. Complete Dynamical Model
  8. Manipulation as Closed-loop Control
  9. Task Goal
  10. Low-level Control
  11. Stability of the zero dynamics
  12. Control strategy
  13. High-level Control
  14. Experimental Validation of the Model
  15. Experimental Setup
  16. ...and 14 more sections

Key Result

Proposition 1

The state $(\Theta^*, 0) \in \mathbb{R}^{2n}$ is an asymptotically stable equilibrium of (eq:zd) if an open neighborhood $\mathcal{N}(\Theta)$$\subseteq \mathbb{R}^{2}$ of $\Theta^*$ exists such that $\forall \Theta \in \mathcal{N}(\Theta^*)/\{\Theta^*\}$ where $U_{G_{\Theta}}$ and $U_{K}$ represent the potential energy associated with gravity and elasticity of the DLO, respectively.

Figures (18)

  • Figure 1: Pictorial representation of the task that we investigate in this work: the manipulation of slender deformable objects via a generic manipulator holding them at one of their ends by its end effector. We superimpose the important reference frames.
  • Figure 2: A block diagram outline of the control strategy. The input is a task goal $\zeta^*$ which can be achieved by regulation of the object's shape. We use a nonlinear model-constrained optimization to map this to a Cartesian object base pose $(x^*, y^*, \phi^*)$, providing a control input to the coupled manipulator-object system.
  • Figure 3: The test objects used in the experiments, OB1-OB6 from left to right. These are high-voltage electric cables used in the context of the electric car industry. The cables end with a plug connector.
  • Figure 4: The experimental setup used for validation of the proposed model and control architecture. Relevant components are highlighted. Note that for simplicity and reproducibility, we are connecting the object directly to the robot's end effector instead of grasping it via a gripper.
  • Figure 5: Distributions of computation times for the inverse kinematics algorithm \ref{['alg:num_IK']}, for various order of approximation. The assumed DoFs for the object are equal to the polynomial order plus one. The time taken to converge to a solution was calculated for 1000 values randomly selected throughout the $\theta$-space.
  • ...and 13 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Proof 1
  • Proposition 2
  • Proof 2
  • Proposition 3
  • Proof 3
  • Remark 1