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Cosserat Modeling of Trimmed Helicoid Soft Arms with a Separated-Section Constitutive Law
Zhihang Qin, Linxin Hou, Zeyu Zhong, Yuchen Sun, Wenci Xin, Yueheng Zhang, Ji Qi, Jie Wang, Muhammad Sunny Nazeer, Yu Jun Tan, Federico Renda, Cecilia Laschi
TL;DR
Trimmed-helicoid arms have separated load-bearing helix domains that can deform relative to one another, challenging common-cross-section stiffness models. The paper maps local helix responses to a backbone and accounts for sparse-fusion compliance in a dynamic Cosserat model. The resulting stiffness is strongly anisotropic, and the model reports pooled position errors below 8% across 103 measured configurations.
Problem
Trimmed-helicoid arms contain separated load-bearing helix domains joined at sparse fused crossings, so common-motion sectional models do not represent their relative deformation.
Source: §I · p. 1Method
The paper pulls each helix domain’s local constitutive response back to the backbone, adds sparse-fusion mechanics, and embeds the resulting sectional law in a dynamic GVS Cosserat model.
Source: §I · pp. 1–3Results
Across 103 measured configurations, pooled normalized position errors are 7.7%, 6.7%, and 7.8%; bending and extension stiffness fall to roughly one tenth of rigid-transport values while torsion remains closer.
Source: C. Full-Arm Prediction and… · p. 6, §IV · p. 8Takeaways & Limitations
At approximately 0.3 s per simulated state on one CPU core, the model resolves distributed curvature, internal wrench, tendon loading, and lattice compaction.
Source: §IV · p. 8
Abstract
from arXiv · showhide
Cosserat rod models for soft robots usually construct sectional stiffness by summing material properties over a common cross-section. This assumption becomes inaccurate for trimmed helicoid arms, where load-bearing helix domains are separated and connected only through sparse fused crossings. This paper formulates a separated-section constitutive law that evaluates each helix domain in its local frame and pulls its constitutive response back to the backbone, yielding an effective backbone stiffness. Sparse-fusion mechanics captures the additional compliance caused by relative motion between neighboring domains and determines channel-wise reduction profiles $η_c(s/L)$ for bending, torsion, and extension. The resulting effective sectional stiffness is strongly anisotropic: bending and extension are reduced by about one order of magnitude, whereas torsion remains close to the effective backbone stiffness. The resulting sectional law is embedded in a geometrically exact dynamic Cosserat model with GVS discretization and routed-tendon actuation. Across 103 measured configurations, the three datasets give pooled normalized position errors of \SI{7.7}{\percent}, \SI{6.7}{\percent}, and \SI{7.8}{\percent}, while each full-arm solve requires approximately \SI{0.3}{s} on one CPU core (Intel Xeon, Cascade Lake, \SI{2.8}{GHz}), enabling rapid model-based planning, state and load estimation, and morphology--control co-design for architected soft robots.
I. INTRODUCTION
Trimmed-helicoid arms have separated load-bearing helix domains that can move relative to one another, unlike sections governed by a common-motion assumption. The paper pulls each domain’s constitutive response back to a common backbone and adds sparse-fusion mechanics before integrating the resulting law into a dynamic Cosserat model.
I. INTRODUCTION
- Sparse fused crossings permit relative deformation between neighboring helix domains, a topology not represented by prior common-motion sectional treatments.Existing trimmed-helicoid stiffness models estimate module-level stiffness but do not provide a separated-section constitutive treatment integrated into a Cosserat rod framework.Source: §I · p. 1
- The formulation evaluates each helix domain in its local frame, pulls its response back to the backbone, and uses sparse-fusion mechanics to determine channel-wise stiffness reductions.The resulting sectional law is embedded in a dynamic GVS Cosserat model with routed-tendon actuation.Source: §I · pp. 1–3
- The separated-section law gives stiffness ratios ηc of 0.10 for extension, 0.11 for bending, and 0.98 for torsion; under 9 N single-tendon tension, end rotations are 71° versus 8° for the summed-section model.Source: Figure 2 · p. 2
- The work integrates the sectional law into a dynamic GVS Cosserat model and reports validation over 103 measured configurations against a conventional summed-section baseline.The contribution also includes analysis of distributed mechanical response and computational performance.Source: §I · p. 3
II. MODELING FRAMEWORK
The modeling framework obtains a sectional stiffness from separated helix domains and uses it in a dynamic GVS Cosserat model of the tendon-driven arm. In the mapping step, local helix responses are transported to the backbone under a common-motion assumption to define the effective backbone stiffness.
II. MODELING FRAMEWORK
- The complete tendon-driven arm is modeled by deriving sectional stiffness from separated helix domains, then integrating it into a dynamic GVS Cosserat model.The prototype comprises three tapered sections, M1–M3, separated by rigid platforms, with three tendons actuating each section.Source: §II · p. 3
1) Helix-to-backbone sectional mapping:
- Each helix domain is evaluated on its own normal section, whose sectional properties are obtained from the corresponding CAD sections.The listed properties include Young’s and shear moduli, area, principal moments of area, and torsional constant.Source: 1) Helix-to-backbone sectional mapping · p. 3
- Each helix domain has a local reference frame aligned with its helix tangent, and its local strain response is mapped to the backbone strain.The backbone strain is defined as deviation from the printed reference twist; cos βk converts backbone-based strain to helix arc-length measure.Source: 1) Helix-to-backbone sectional mapping · p. 3
- KV is the effective sectional stiffness obtained by pulling helix-domain responses back to the backbone under common sectional motion, rather than directly summing material properties on a backbone-normal cut.The text states that KV contains no identified parameter.Source: 1) Helix-to-backbone sectional mapping · p. 3
2) Sparse-fusion correction and effective sectional stiffness:
Sparse fused crossings allow neighboring helix domains to deform relative to one another, adding compliance beyond the effective backbone stiffness. The model accounts for this by relaxing internal domain motion and deriving channel-specific effective stiffnesses.
2) Sparse-fusion correction and effective sectional stiffness:
- Relative deformation between neighboring helix domains is permitted between sparse fused crossings, adding compliance to the free-span structure.The fusion contribution is included alongside helix-domain stiffness in the internal tangent stiffness.Source: 2) Sparse-fusion correction and… · pp. 3–4
- A single κw corrects nominal fusion stiffness and is calibrated from the tendon-free hanging configuration, then fixed across channels and subsequent simulations.Source: 2) Sparse-fusion correction and… · p. 4
- Internal sectional degrees of freedom are determined by minimizing elastic energy subject to static equilibrium, using a second-order approximation under small local material strains.The reference configuration is stress-free, with zero elastic energy.Source: 2) Sparse-fusion correction and… · p. 4
- Channel-wise condensed stiffnesses are related to the directly calculated effective backbone stiffness; measured load–strain pairs determine the reduction.The passage defines Rc as equal to the corresponding external force or moment resultant under static equilibrium.Source: 2) Sparse-fusion correction and… · p. 4
- Bending and extension are reduced by approximately one order of magnitude, while torsion remains close to the effective backbone stiffness.This comparison uses the same calibrated κw for all three channels; the local material response is assumed to remain in the small-strain regime.Source: 2) Sparse-fusion correction and… · p. 4
B. Dynamic Cosserat Model and Numerical Method
The section pose is represented as an element of SE(3), comprising its orientation and backbone position.
B. Dynamic Cosserat Model and Numerical Method
- The deformable section pose g(s, t) is represented by orientation R in SO(3) and backbone position p in R^3.Source: 1) GVS dynamics · p. 4
1) GVS dynamics:
The GVS dynamics represent strain and body velocity using a reduced basis, then project momentum balance into generalized coordinates. The model incorporates effective sectional stiffness, viscous response, and external loads.
1) GVS dynamics:
- GVS represents the strain field as a reference strain plus a basis Φ(s) weighted by generalized coordinates q(t), with body velocity represented through J and q̇.Source: 1) GVS dynamics · p. 4
- The internal wrench combines elastic response K(s)(ξ − ξ*) with a viscous term Υ(s)ξ̇; the effective sectional stiffnesses form K(s).Source: 1) GVS dynamics · p. 4
- Linear and angular momentum balance are combined in body coordinates and Galerkin-projected onto the GVS basis to obtain generalized dynamics.The generalized loads include gravity, tendon, and other external contributions.Source: 1) GVS dynamics · pp. 4–5
- The pooled normalized position errors reported for Datasets 0, A, and B are 7.7 %, 6.7 %, and 7.8 %, respectively.Fig. 3 also reports station-wise position and rotation errors at C1, C2, and the tip.Source: Figure 3 · p. 5
- The modeled system includes rigid platforms whose Jacobians and lumped inertias contribute to the reduced operators.The configuration-dependent mass matrix supplies the GVS velocity-dependent inertial matrix.Source: 1) GVS dynamics · p. 5
2) Routed-tendon actuation:
Tendon actuation is represented through routed paths and their local tangents, with channel friction determining tension along each tendon and a pull-only constraint limiting actuation.
2) Routed-tendon actuation:
- Each tendon follows a material offset ρi(s), with its spatial path and material-frame tangent used to describe the routed geometry.Source: 2) Routed-tendon actuation · p. 5
- Channel friction determines the cumulative turning angle and tendon tension field, with entry tension Ti0 and friction coefficient μc specifying the relation.Source: 2) Routed-tendon actuation · p. 5
- The pull-only tendon constraint uses a tension bound Tmax and projects tensions onto the interval [0, Tmax].Source: 2) Routed-tendon actuation · p. 5
3) Geometric saturation:
The geometric saturation rule monitors clearances between non-fused helix pairs and constrains section strains when a geometry-based compaction threshold is reached.
3) Geometric saturation:
- The minimum clearance is measured between sampled helix centerlines, adjusted by the effective helix radius, over monitored non-fused pairs.The threshold is prescribed from lattice geometry rather than fitted to validation data.Source: 3) Geometric saturation · p. 5
- At the threshold, that section’s generalized strain coordinates are held fixed while the remaining coordinates continue to evolve.This geometric saturation constraint approximates section locking without an additional contact or jamming constitutive model.Source: 3) Geometric saturation · p. 5
4) Discretization and numerical propagation:
The discretization uses generalized strain coordinates for the modules and transitions, resolves spatial variation by split-interval quadrature, and propagates pose and geometric Jacobian together.
4) Discretization and numerical propagation:
- Each module uses shifted Legendre strain bases of orders (6, 6, 6, 2, 2, 2), giving 30 generalized coordinates per module.The orders correspond to (κ1, κ2, κ3, ν1, ν2, ν3).Source: 4) Discretization and numerical… · p. 5
- Gauss–Legendre quadrature intervals are split at module boundaries, rigid connectors, and stiffness-profile breakpoints.The splits resolve spatial variation in K(s) and routed tendon geometry.Source: 4) Discretization and numerical… · p. 5
- Pose and geometric Jacobian are propagated together using fourth-order Zanna–Magnus integration.The interval propagation uses the Magnus twist increment and basis increment, with the differential of the exponential map.Source: 4) Discretization and numerical… · p. 6
III. MODEL ANALYSIS AND EXPERIMENTAL VALIDATION
The experimental platform is a three-module trimmed-helicoid arm actuated by routed tendons, with platform and tip motion measured optically.
III. MODEL ANALYSIS AND EXPERIMENTAL VALIDATION
- Three tapered trimmed-helicoid modules printed from TPU 95A are joined by rigid PLA connectors and mounted vertically on a bolted support.Each module has three tendons driven by servo motors.Source: A. Experimental Setup · p. 6
- An OptiTrack system measures position and orientation at the two inter-module platforms and distal platform using passive markers.The deformable modules are enclosed by a thin flexible sleeve reported to have negligible influence on motion.Source: A. Experimental Setup · p. 6
B. Validation Protocol and Parameter Identification
The validation compares the effective sectional law with summed properties using measured tendon inputs, then reports errors across three datasets and representative states. The results favor the effective law, while the ablation attributes the difference to the sectional closure.
B. Validation Protocol and Parameter Identification
- The validation covers 103 configurations spanning distal-tendon deep bending, combined deformation, and axial contraction, with 309 platform-pose samples.Simulations reproduce the tendon ramp durations and holds, and measured tendon-length changes drive both forward models.Source: B. Validation Protocol and Parameter… · p. 6
C. Full-Arm Prediction and Sectional-Stiffness Ablation
- The effective law reproduces deep coiling, combined multi-module deformation, and axial contraction, whereas summed properties underpredict all three deformation modes.The comparison uses identical experimentally measured tendon inputs.Source: C. Full-Arm Prediction and… · p. 6
- 1.5–10.3% versus 25.3–91.5% are the normalized position-error ranges for the effective law and summed properties across three stations in representative cases.Dataset A’s summed-properties errors exceed 83% at all three plates; Table I reports the representative-state comparison.Source: C. Full-Arm Prediction and… · p. 6, Table I · p. 6
- 7.7%, 6.7%, and 7.8% are the pooled position errors for Datasets 0, A, and B across the complete actuation sequences.The errors are reported for the effective sectional law model.Source: C. Full-Arm Prediction and… · p. 6
- Figure 4 maps actuation level and spatial response, including strain, internal wrench, tendon line loads, helix gaps, and post-processed frictional transmission.Its caption specifies three-tendon retractions over ten actuation levels and marks a 0.10 mm compaction threshold.Source: Figure 4 · p. 7
- The ablation uses the same measured tendon inputs and otherwise identical formulations, isolating the sectional closure as the source of the difference.The conventional summed-property assumption cannot recover the measured compliance.Source: C. Full-Arm Prediction and… · p. 7
D. Distributed Mechanical Response
The model resolves distributed deformation and loading in trimmed-helicoid arms, while its reduced-order solver maintains convergence and low computational cost. Its separated-domain constitutive law captures anisotropic stiffness and reproduces measured arm configurations.
D. Distributed Mechanical Response
- Increasing tendon actuation redistributes curvature, strain energy, and internal wrench across the three modules, revealing the mechanical state underlying full-arm deformation.This distribution is shown in Fig. 4(a–e).Source: D. Distributed Mechanical Response · p. 7
- Tendon line loads increase where tendon direction changes rapidly, especially near highly curved regions and module transitions, as the minimum gap between non-fused domains decreases toward its prescribed threshold.These patterns link large bending with progressive lattice compaction.Source: D. Distributed Mechanical Response · p. 7
- Deformed tendon paths determine accumulated wrap angle and frictional tension transmission; Fig. 4(j–l) illustrates the effect for µc = 0.15 and its sensitivity over µc ∈ [0.10, 0.20].Source: D. Distributed Mechanical Response · p. 7
E. Numerical Convergence and Computational Performance
- Relative to degree ten, tip discrepancy falls from approximately 2.4 mm/1.2° at degree one to 0.03 mm/0.016° at degree eight, placing the degree-six basis in the converged regime.Degree six is the basis used throughout the simulations.Source: Figure 5 · p. 7, E. Numerical Convergence and… · p. 8
- For representative deep bending, the projected augmented-Lagrangian procedure reduces the inner residual by approximately ten orders of magnitude and tendon-length violation below 1 × 10^-8 m within three outer updates.The commanded tendon lengths therefore remain accurately enforced during large deformation.Source: E. Numerical Convergence and… · p. 8
IV. CONCLUSION
- The separated-domain law evaluates each helix on its own normal section, transports its response to the backbone frame, and models compliance from relative domain motion at sparse fused crossings.The resulting bending, torsional, and axial stiffnesses close a dynamic GVS Cosserat model with routed-tendon actuation.Source: §IV · p. 8
- Across 103 measured configurations, pooled normalized position errors remain below 8%, and each simulated state requires approximately 0.3 s on one CPU core.Under the same measured tendon inputs, the effective sectional law reproduces deep bending, coupled deformation, and axial contraction, while the summed-property model systematically underpredicts deformation.Source: §IV · p. 8
- Bending and extension decrease to roughly one tenth of rigid-transport values, whereas torsion remains much closer to the rigid-transport reference.A single calibrated fusion correction produces this strongly anisotropic sectional prediction.Source: §IV · p. 8