Performance and Experimental Analysis of Strain-based Models for Continuum Robots

Annika Delucchi; Vincenzo Di Paola; Andreas Müller; and Matteo Zoppi

cs.ROFeb 26, 2026

Performance and Experimental Analysis of Strain-based Models for Continuum Robots

Authors: Annika Delucchi, Vincenzo Di Paola, Andreas Müller, and Matteo Zoppi

Organizations: Department of Mechanical, Energy, Management and Transportation Engineering (DIME), University of Genova, Italy · Institute of Robotics, Johannes Kepler University (JKU) Linz, Austria

Abstract

Although strain-based models have been widely adopted in robotics, no comparison beyond the uniform bending test is commonly recognized to assess their performance. In addition, the increasing effort in prototyping continuum robots highlights the need to assess the applicability of these models and the necessity of comprehensive performance evaluation. To address this gap, this work investigates the shape reconstruction abilities of a third-order strain interpolation method, examining its ability to capture both individual and combined deformation effects. These results are compared and discussed against the Geometric-Variable Strain approach. Subsequently, simulation results are experimentally verified by reshaping a slender rod while recording the resulting configurations using cameras. The rod configuration is imposed using a manipulator displacing one of its tips and extracted through reflective markers, without the aid of any other external sensor -- i.e. strain gauges or wrench sensors placed along the rod. The experiments demonstrate good agreement between the model predictions and observed shapes, with average error of 0.58% of the rod length and average computational time of 0.32s per configuration, outperforming existing models.

Figures & tables

$L$	Rod length
$s$	Curvilinear abscissa
$\tau$	Normalised curvilinear abscissa
$\mathbf{R}(\tau)$	Rotation matrix
$\mathbf{r}(\tau)$	Position vector
$\mathbf{H}(\tau)$	Transformation matrix
$\mathbf{X}(\tau)$	Canonical coordinates

TABLE I: Notation

Fig. 1: Rod description and parameters: a generic rod undergoing large deformations, where at each $\tau$ corresponds a reference frame $\mathcal{F}_{\tau}$ whose $x$ -axis is aligned with the rod centreline.

MODEL	DEFORMATION
Cosserat	$\boldsymbol{\chi}=\begin{pmatrix}\boldsymbol{\kappa}_{1}&\boldsymbol{\kappa}_{2}&\boldsymbol{\kappa}_{3}&\boldsymbol{\rho}_{1}&\boldsymbol{\rho}_{2}&\boldsymbol{\rho}_{3}\end{pmatrix}^{\text{T}}$
Kirchhoff	$\boldsymbol{\chi}=\begin{pmatrix}\boldsymbol{\kappa}_{1}&\boldsymbol{\kappa}_{2}&\boldsymbol{\kappa}_{3}&\boldsymbol{\rho}_{1}&0&0\end{pmatrix}^{\text{T}}$
Inextensible Kirchhoff	$\boldsymbol{\chi}=\begin{pmatrix}\boldsymbol{\kappa}_{1}&\boldsymbol{\kappa}_{2}&\boldsymbol{\kappa}_{3}&1&0&0\end{pmatrix}^{\text{T}}$
Torsion Free Kirchhoff	$\boldsymbol{\chi}=\begin{pmatrix}0&\boldsymbol{\kappa}_{2}&\boldsymbol{\kappa}_{3}&\boldsymbol{\rho}_{1}&0&0\end{pmatrix}^{\text{T}}$

TABLE II: Deformation Models

Fig. 4: Two bending test: rod configurations (a) and evaluation metrics plots (b). The GVS model is solved with thirds degree monomial basis.

Fig. 5: Bending-torsion test: rod configurations for the exact model (a-left), GVS model (a-middle) and interpolated model (a-left), together with evaluation metrics (b). The GVS model is solved with third degree Legendre polynomial basis.

TEST CASE	MAX POSITION ERR [%]		MEAN POSITION ERR [%]		COMPUTATION
	Interp.	GVS	Interp.	GVS	Time [s]	Iter.
Bending ( $\kappa_{2}$ )	1.64	1.03	0.40	0.35	0.10 / 0.38	4 / 15
Bending ( $\kappa_{2},\kappa_{3}$ )	0.43	0.35	0.12	0.08	0.45 / 0.99	28 / 42
Torsion ( $\kappa_{1}$ , $\theta\leq\pi$ )	$<0.01$	$<0.01$	$<0.01$	$<0.01$	0.14 / 0.25	5 / 9
Bending-torsion ( $\kappa_{1},\kappa_{2},\kappa_{3}$ )	3.01	2.30	0.75	0.62	2.82 / 4.35	24 / 41

TABLE III: Simulation results Summary. Errors are expressed as percentage of the total rod length

L

PARAMETER	VALUE
Rod Length	$0.955$ m
Rod Diameter	$1.5\times 10^{-3}$ m
Number of Markers	$7$
Distance Between Markers	$0.119$ m
Young’s Modulus † † footnotemark: $E$	$80$ GPa
Shear Modulus † † footnotemark: $G$	$30$ GPa

TABLE IV: Nitinol Rod parameters

PARAMETER	VALUE
Rod Length	$0.885$ m
Rod Diameter	$2\times 10^{-3}$ m
Number of Markers	$6$
Distance Between Markers	$0.126$ m
Young’s Modulus 3 3 footnotemark: 3 $E$	$36.5$ GPa
Shear Modulus † † footnotemark: $G$	$3$ GPa

TABLE V: Fibreglass Rod parameters

Fig. 6: Experimental set-up: a nitinol rod is actuated by a KUKA robotic manipulator, while the 3D positions of reflective markers are captured by a multi-camera motion tracking system (VICON). Both clamps are 3D-printed in PLA and are designed with an ad-hoc geometry that matches the rod diameter, ensuring precise alignment and repeatable boundary conditions.

Fig. 9: Two-bending experiment: configurations (a) and performances (b) in terms of iteration number, execution time and total configuration error. To be notice the symmetry of the results on figures (c) and (d) due to the imposed trajectory. The same happened above in Figure 4. The GVS model is solved with second degree monomial basis.

Fig. 10: Snapshot of a trajectory inducing bending and torsional effects simultaneously.

TEST CASE	MAX POSITION ERR [%]		MEAN POSITION ERR [%]		COMPUTATIONAL TIME [s]
	Interp.	GVS	Interp.	GVS	Interp.	GVS
Fibreglass Bending ( $\kappa_{2},\kappa_{3}$ )	1.38	1.42	0.86	0.86	0.34	0.42
Nitinol Bending ( $\kappa_{2}$ , $\kappa_{3}$ )	1.39	1.38	0.58	0.59	0.32	0.48
Nitinol Bending-torsion ( $\kappa_{1},\kappa_{2},\kappa_{3}$ )	3.62	3.03	1.86	1.32	0.64	1.14

TABLE VI: Experiments Results Summary. Errors are expressed as percentage of the total rod length

L

Fig. 12: Fibreglass experiments: configurations total error (a) and markers mean error (b). The GVS model solutions are found with second degree monomial basis.

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Reconstructing the shape of continuum manipulators from sparse, noisy sensor data is a challenging task, owing to the infinite-dimensional nature of such systems. Existing approaches broadly trade off between parametric methods that yield compact state representations but lack probabilistic structure, and Cosserat rod inference on factor graphs, which provides principled uncertainty quantification at the cost of a state dimension that grows with the spatial discretization. This letter combines the strength of both paradigms by estimating the coefficients of a low-dimensional Geometric Variable Strain (GVS) parameterization within a factor graph framework. A novel kinematic factor, derived from the Magnus expansion of the strain field, encodes the closed-form rod geometry as a prior constraint linking the GVS strain coefficients to the backbone pose variables. The resulting formulation yields a compact state vector directly amenable to model-based control, while retaining the modularity, probabilistic treatment and computational efficiency of factor graph inference. The proposed method is evaluated in simulation on a 0.4 m long tendon-driven continuum robot under three measurement configurations, achieving mean position errors below 2 mm for all three scenarios and demonstrating a sixfold reduction in orientation error compared to a Gaussian process regression baseline when only position measurements are available.

Abstract

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