3D-Force Estimation for Dual-arm Manipulation of Deformable Linear Objects via Factor Graphs

Annika Delucchi; Vincenzo Di Paola; Bruno Marco Quadrelli; Matteo Zoppi

cs.ROOct 26, 2026

3D-Force Estimation for Dual-arm Manipulation of Deformable Linear Objects via Factor Graphs

Authors: Annika Delucchi, Vincenzo Di Paola, Bruno Marco Quadrelli, Matteo Zoppi

Organizations: University of Genova, Department of Mechanical, Energy, Management and Transportation Engineering, Via all’opera Pia 15A, Genova, Italy. · Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, United States.

Abstract

Estimating forces acting on Deformable Linear Objects (DLOs) is challenging due to their underactuated nature. However, reliable contact estimation is crucial to precisely control them and enable safe manipulation tasks. Differently from the various approaches presented in the literature, this work adopts a probabilistic formulation where the DLO state and sensor data are related through conditional distributions. In particular, factor graphs are used to determine the magnitude, direction, and location of a 3D force acting on a DLO, while simultaneously reconstructing its shape. Since DLOs are not actuated and are typically thin, like cables and wires, strain gauges and actuation readings are usually impractical. Therefore, to overcome this issue, this work is tested using one type of sensing information only, thus developing a generic technique suited for a wider variety of tasks. Realistic tests are performed by introducing noise into the measurements while the DLO follows a prescribed path in the presence of external forces that are accurately estimated.

Figures & tables

Panel of Figure 1 — Fig. 1 : Left: two manipulators hold the ends of a DLO to change its shape. Right (zoomed view): the DLO is modelled as a generic rod undergoing large displacements and small deformations. Each cross-section, at arc length $s$ , is associated with a reference frame $\mathcal{F}_{s}$ whose $x$ -axis is tangent to the centreline. The cross-section position $p(s)$ and orientation $R(s)$ are expressed with respect to the world frame $\mathcal{F}_{0}$ .

Fig. 2 : Graphical representation of the factor graph used to solve the force and shape estimation problem in superimposition over a gray rod. Squares indicate factors nodes, while variable nodes are circles. Red squares represent prior factors, blue ones equilibrium, cyan ones kinematics and magenta ones are sensors measurements. Yellow nodes encode the rod state, accounting for pose and internal wrench, while the green one represents force.

Panel of Figure 3 — Fig. 3 : The force $f_{ext}$ applied at a location $s$ along the rod and expressed in the local frame $\mathcal{F}_{s}$ is rotated in the world frame $\mathcal{F}_{0}$ to extract its direction. The force direction is indicated in terms of two angles $\alpha$ and $\varepsilon$ , representing the azimuth angle and the elevation, respectively.

Panel of Figure 4 — Fig. 4 : Intermediate configurations of the pull trajectory with a force acting at $s=0.5$ and markers measurement noise of $\sigma=1\times 10^{-4}$ m. Left: 3D view at steps $t_{1}$ – $t_{10}$ . Right: top view at steps $t_{2},t_{4},t_{6},t_{8},t_{10}$ . Bottom: estimated curvature components $\kappa_{x}$ (torsion), $\kappa_{y},\kappa_{z}$ (bendings). Through the graph-based algorithm, node poses at discrete points along the rod and the external force are estimated and shown in red at each step.

Panel of Figure 5 — Fig. 5 : Intermediate configurations of the push trajectory with a force acting at $s=0.6$ and markers measurement noise of $\sigma=1\times 10^{-4}$ m. Left: 3D view at steps $t_{1}$ – $t_{10}$ . Right: top view at steps $t_{2},t_{4},t_{6},t_{8},t_{10}$ . Bottom: estimated curvature components $\kappa_{x}$ (torsion), $\kappa_{y},\kappa_{z}$ (bendings). Through the graph-based algorithm, node poses at discrete points along the rod and the external force are estimated and shown in red at each step.

	True Force [N]			Est. Force [N]			True Loc.	Est. Loc.	True Dir. [ ∘ ]		Est. Dir. [ ∘ ]
Step	$f_{x}$	$f_{y}$	$f_{z}$	$\hat{f}_{x}$	$\hat{f}_{y}$	$\hat{f}_{z}$	$s$	$\hat{s}$	$\alpha$	$\varepsilon$	$\hat{\alpha}$	$\hat{\varepsilon}$
1	0.104	0.000	0.263	0.111	-0.003	0.270	0.500	0.473	-0.167	68.380	-1.680	67.717
2	0.118	0.023	0.287	0.128	0.026	0.304	0.500	0.480	10.917	67.298	11.485	66.744
3	0.129	0.052	0.311	0.158	0.055	0.337	0.500	0.474	21.976	65.803	19.274	63.625
4	0.147	0.090	0.336	0.155	0.098	0.365	0.500	0.476	31.547	62.875	32.191	63.343
5	0.156	0.134	0.362	0.161	0.137	0.395	0.500	0.472	40.806	60.401	40.340	61.816

TABLE I : Pull Trajectory – Comparison Between True and Estimated Values

	True Force [N]			Est. Force [N]			True Loc.	Est. Loc.	True Dir. [ ∘ ]		Est. Dir. [ ∘ ]
Step	$f_{x}$	$f_{y}$	$f_{z}$	$\hat{f}_{x}$	$\hat{f}_{y}$	$\hat{f}_{z}$	$s$	$\hat{s}$	$\alpha$	$\varepsilon$	$\hat{\alpha}$	$\hat{\varepsilon}$
1	-0.127	-0.065	-0.248	-0.133	-0.060	-0.262	0.600	0.577	-152.905	-60.056	-155.728	-60.896
2	-0.188	-0.062	-0.272	-0.211	-0.054	-0.281	0.600	0.575	-161.875	-53.960	-165.676	-52.223
3	-0.255	-0.057	-0.285	-0.285	-0.041	-0.306	0.600	0.575	-167.338	-47.435	-171.736	-46.717
4	-0.326	-0.053	-0.287	-0.337	-0.037	-0.297	0.600	0.576	-170.811	-41.055	-173.804	-41.192
5	-0.399	-0.046	-0.278	-0.408	-0.031	-0.301	0.600	0.577	-173.358	-34.726	-175.712	-36.350

TABLE II : Push Trajectory – Comparison Between True and Estimated Values

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Multi-Contact Force Estimation for Continuum Robots via Gaussian-Parameterized Factor Graphs

Aditya Prakash, Panagiotis Tsiotras

School of Aerospace Engineering Georgia Institute of Technology Atlanta, GA, USA

Contact Force Estimation Robot State Estimation Continuum Robot Shape Estimation

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Continuum robots offer key advantages in navigating unstructured environments, but their safe operation requires accurate estimation of the external contact forces acting anywhere along the robot body. Estimating these forces at unknown locations is an ill-conditioned problem, particularly for multiple contacts. We propose a unified shape and force estimation framework formulated on a factor graph. By incorporating a Gaussian mixture force parameterization into a discretized probabilistic Cosserat rod model, we reduce the dimensionality of the unknown external forces and mitigate the ill-conditioning of node-wise force estimation. The framework fuses strain, tendon tension, and pose measurements to simultaneously estimate the robot's shape and external forces while accounting for modeling and sensor uncertainties. Numerical simulations demonstrate that the proposed method outperforms existing methods in terms of force location and magnitude estimation for both single and multi-contact scenarios. We further present a progressive variant that introduces basis functions on demand to estimate contact forces sequentially during a simulated confined-navigation task.

Abstract

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