A Tutorial on Shape Servoing of Deformable Objects-Part I: Fundamentals

Ignacio Cuiral-Zueco; Alessio Caporali; Miguel Aranda; Gianluca Palli; Gonzalo López-Nicolás; Youcef Mezouar

cs.ROOct 29, 2026

A Tutorial on Shape Servoing of Deformable Objects-Part I: Fundamentals

Authors: Ignacio Cuiral-Zueco, Alessio Caporali, Miguel Aranda, Gianluca Palli, Gonzalo López-Nicolás, Youcef Mezouar

Organizations: Instituto de Investigación en Ingeniería de Aragón (I3A), University of Zaragoza, Spain · DEI, University of Bologna, Italy · Institut Pascal, Clermont Auvergne INP, France

Abstract

In robotics, shape control (or shape servoing) refers to the automatic manipulation of a deformable object to achieve a desired shape. It involves a feedback loop that exploits the relation between robot motion and changes in the measured shape. This relation, known as the interaction Jacobian or, in this context, the deformation Jacobian, depends on the object's mechanical properties, geometry, and boundary conditions (e.g., where the object is grasped). Because these factors are often difficult to model accurately, the deformation Jacobian can be estimated locally from observed shape changes produced by commanded robot motions, without requiring a fine-tuned mechanical model. This article is the first of a two-part tutorial on data-driven Jacobian-based shape control of deformable objects. It introduces the central elements of the approach: the formulation of the shape-control problem, shape representation, estimation and online update of the deformation Jacobian, and a standard control law from the literature. It then assembles these elements into a complete shape-servoing loop, derives a local stability condition for this loop, and illustrates why some target shapes cannot be reached. Part II covers more advanced topics. Throughout both parts, a simulated planar example in which two robot arms manipulate an elastic rod observed by a camera illustrates the concepts.

Figures & tables

Fig. 1: Illustration of a shape-servoing loop and of the setup (plant) used as the running example in this tutorial. Left: shape-servoing loop, with the measured shape obtained from the camera image and the shape error converted into robot motion using a local deformation model represented by the deformation Jacobian. The Jacobian is estimated from the applied robot motions and the corresponding measured shape changes. Right: running-example setup, in which two robot arms deform an elastic rod

\mathbf{x}

(blue), observed by a stationary camera, toward a target shape

\mathbf{x}^{*}

(dashed). For simplicity, in the running example, the grippers and the rod move in a plane parallel to the image plane of the camera, although this tutorial’s formulation generalizes to three-dimensional actuation. The arrows at the grippers indicate the six in-plane actuation degrees of freedom

[{\color[rgb]{1,0,0}x_{L}},{\color[rgb]{0,1,0}y_{L}},{\color[rgb]{0,0,1}\theta_{L}},{\color[rgb]{1,0,0}x_{R}},{\color[rgb]{0,1,0}y_{R}},{\color[rgb]{0,0,1}\theta_{R}}]^{\top}

. Bottom: schematic definition of shape servoing, with an illustration of the object shape approaching the target over time

t

Section	Main content
Section II	Shape representation and shape error with respect to the target, the deformation Jacobian relating robot motion to shape change, and the control law mapping shape error to robot motion.
Section III	Estimation of the initial Jacobian from exploratory motions, online adaptation from closed-loop data, and model-based alternatives.
Section IV	Integration of the components into a complete feedback loop, characterization of reachable target shapes, and conditions for local decrease of the shape error.

TABLE I: Contents of Part I

Fig. 2: Discrete-time shape-servoing loop of the running example, with the sections and equations defining each block. Perception extracts the image coordinates

\mathbf{s}_{k}

of the tracked points from the camera measurement

\mathbf{y}_{k}

, and the controller computes the command

\mathbf{u}_{k}

from the shape error

\mathbf{e}_{k}

. The operators

\Delta t\,z^{-1}

and

1-z^{-1}

, where

z^{-1}

denotes a one-iteration delay, compute the latest robot-state increment

\Delta\mathbf{q}_{k-1}

and feature change

\Delta\mathbf{s}_{k}

. These are used by the online update ( 7 ) to correct the Jacobian estimate, initialized from the offline estimate ( 6 )

\hat{\mathbf{J}}_{0}

(Fig. 3 ). Right: current features

\mathbf{s}_{k}

(yellow), target features

\mathbf{s}^{*}

, and shape error

\mathbf{e}_{k}

in the image; the error arrows are also shown on the rod in the scene.

Fig. 3: Offline Jacobian estimation in the running example. Left: a small robot perturbation $\Delta\mathbf{q}_{c}$ produces a measured feature change $\Delta\mathbf{s}_{c}$ . Center: the $C=24$ action–response pairs are stacked as the columns of $\Delta\mathbf{Q}$ and $\Delta\mathbf{S}$ . Right: least-squares regression gives the initial Jacobian estimate $\hat{\mathbf{J}}_{0}$ .

Fig. 4: Targets with different closed-loop outcomes in the running example. Top: rod configurations from

k=0

(light grey) to

k=90

(blue, with yellow feature points) for two target shapes. Left: a target recorded at another gripper configuration, which is reached from the tested initial state. Right: a modified target obtained by displacing three feature points, for which the controller settles with a nonzero residual. Dashed curves with grey rings indicate the target shapes, and red arrows denote the residual feature error. Bottom: normalized error histories for the two cases.

Fig. 5: Closed-loop behavior for the arch target of Fig. 1 . Top: rod configurations (blue, with yellow feature points) and dashed target at the indicated iterations

k

, selected at approximately equal intervals of travel in feature space; lighter curves show two preceding iterations. Bottom, from left to right: normalized error with online Jacobian updates and with

\hat{\mathbf{J}}_{0}

held fixed (

\alpha=0.05

); commanded robot velocities (

\alpha=0.05

;

y_{L}

and

y_{R}

coincide by symmetry); and normalized error for three control gains (solid), each with its nominal decay

(1-\alpha)^{k}

(dotted, same shade).

Case	Sufficient local condition	Stability and convergence
$n\leq m$ , maximum rank	$\operatorname{rank}(\mathbf{J})=n$ and $\operatorname{Sym}(\mathbf{J}\hat{\mathbf{J}}^{+})\succ 0$	$\mathbf{e}=\mathbf{0}$ is locally exponentially stable and $\mathbf{e}\to\mathbf{0}$ . For $n=m$ , full row rank corresponds to the nonsingular square case.
$n>m$ , maximum rank	$\operatorname{rank}(\mathbf{J})=m$ and $\operatorname{Sym}(\hat{\mathbf{J}}^{+}\mathbf{J})\succ 0$	The controllable closed-loop modes converge exponentially, and $\mathbf{e}$ approaches a residual in $\operatorname{Null}(\hat{\mathbf{J}}^{\top})$ . For exact Jacobian knowledge ( $\hat{\mathbf{J}}=\mathbf{J}$ ), this becomes the local least-squares residual in $\operatorname{Null}(\mathbf{J}^{\top})$ and is zero when $\mathbf{e}\in\operatorname{Range}(\mathbf{J})$ .
Rank deficient	$\operatorname{rank}(\mathbf{J})<\min(n,m)$ , with $\operatorname{Sym}(\mathbf{J}\hat{\mathbf{J}}^{+})$ positive definite when restricted to $\operatorname{Range}(\mathbf{J})$ for $n\leq m$ , or $\operatorname{Sym}(\hat{\mathbf{J}}^{+}\mathbf{J})$ positive definite when restricted to $\operatorname{Range}(\mathbf{J}^{\top})$ for $n>m$	Local exponential convergence is guaranteed only on the remaining controllable subspace. Error components outside $\operatorname{Range}(\mathbf{J})$ cannot be reduced locally; rank loss in $\hat{\mathbf{J}}$ may remove additional corrective directions.

TABLE II: Local Stability and Convergence with an Estimated Deformation Jacobian

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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

Department of Cognitive Robotics, Delft University of Technology, 2628 CN Delft, The Netherlands · Institute of Robotics and Mechatronics, German Aerospace Center (DLR), 82234 Oberpfaffenhofen, Germany

Deformable Object Manipulation Feedback Control Dynamic Robotic Manipulation

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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

Abstract

Deformable Object Manipulation

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