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
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
τ
Normalised curvilinear abscissa
R(τ)
Rotation matrix
r(τ)
Position vector
H(τ)
Transformation matrix
X(τ)
Canonical coordinates
TABLE I: Notation
Fig. 1: Rod description and parameters: a generic rod undergoing large deformations, where at each τ corresponds a reference frame Fτ whose x -axis is aligned with the rod centreline.
MODEL
DEFORMATION
Cosserat
χ=(κ1κ2κ3ρ1ρ2ρ3)T
Kirchhoff
χ=(κ1κ2κ3ρ100)T
Inextensible Kirchhoff
χ=(κ1κ2κ3100)T
Torsion Free Kirchhoff
χ=(0κ2κ3ρ100)T
TABLE II: Deformation Models
Figure 4
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 ( κ2 )
1.64
1.03
0.40
0.35
0.10 / 0.38
4 / 15
Bending ( κ2,κ3 )
0.43
0.35
0.12
0.08
0.45 / 0.99
28 / 42
Torsion ( κ1 , θ≤π )
<0.01
<0.01
<0.01
<0.01
0.14 / 0.25
5 / 9
Bending-torsion ( κ1,κ2,κ3 )
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
1.5×10−3 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
2×10−3 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.
Figure 11
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 ( κ2,κ3 )
1.38
1.42
0.86
0.86
0.34
0.42
Nitinol Bending ( κ2 , κ3 )
1.39
1.38
0.58
0.59
0.32
0.48
Nitinol Bending-torsion ( κ1,κ2,κ3 )
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.
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.
Lorenzo Ticozzi, Patricio A. Vela, Panagiotis Tsiotras
School of Aerospace Engineering · School of Electrical and Computer Engineering
In this paper, we present a fully discrete approach for the accurate and numerically efficient dynamical modeling and state estimation of continuum robots. The model is based on geometrically exact beams in a minimal, strain-based formulation and derived in the framework of Lie group variational integrators, allowing to preserve important geometric properties that we exploit to achieve high accuracy and numerical efficiency. We then propose a disturbance observer based on an extended Kalman filter formulation that reliably estimates system states as well as model uncertainties and external disturbances. Experiments on a real system validate the accuracy and efficiency of the proposed model and observer.
Maximilian Herrmann, Leander Pfeiffer, Paul Kotyczka
Technical University of Munich, TUM School of Engineering and Design, Munich Institute of Robotics and Machine Intelligence (MIRMI), Chair of Automatic Control.
Forward kinetostatics of spatial tendon-driven continuum robots typically requires a nonlinear equilibrium solve for each actuation input. This paper develops a force-to-Cartesian-configuration model with a closed-form solution in quadratures for spatial multi-segment robots under tendon actuation. The Cartesian backbone centerline and accumulated material twist serve as generalized coordinates, from which the strain measures and tendon geometry are derived. Variational equilibrium yields explicit axial and bending relations and establishes zero equilibrium material twist within the proposed model for admissible longitudinal non-helical routing. The solution is propagated segment by segment without an iterative equilibrium solve, while retaining axial deformation, spatially varying axial and bending stiffnesses and tendon-routing diameter, and segment-dependent tendon participation. Numerical comparisons with a full-strain geometric variable-strain model (GVS) yield maximum length-normalized tip-position discrepancies of 8.91 x 10^-6 and 1.01 x 10^-5 for the single- and three-segment robots, respectively. Mean evaluation times of 1.52 μs and 2.94 μs, with corresponding speedups of approximately 1864x and 3348x over the baseline, demonstrate the computational advantage of the explicit force-to-configuration mapping in the reported benchmark.