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Cosserat Modeling of Trimmed Helicoid Soft Arms with a Separated-Section Constitutive Law

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 · N/A · 2026

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Summary

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.

Abstract Summary

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 $\eta_c(s/L)$ 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.

Key Points

  • 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 $\eta_c(s/L)$ 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.

Modeling Framework

Figure 1 (b–d) summarizes the prototype and the kinematic and dynamic descriptions used in the model. The arm is represented as a serial assembly of three tapered arm sections, denoted M1, M2, and M3, separated by rigid platforms. Each arm section is actuated by three tendons, while the distal tendons are routed through guide bores in the proximal sections.

The backbone geometry, helix layout, tendon routes, mass distribution, and material properties are obtained from the CAD model and the fabricated structure.

Modeling Framework — Cosserat Modeling of Trimmed Helicoid Soft Arms with a Separated-Section Constitutive Law
Fig. 3: Experimental validation over three deformation modes. (a–c) Deep bending, combined deformation, and axial contraction, with predictions from the effective sectional law and summed properties compared against experiment. (d,e) Position and rotation errors at C1, C2, and the tip. (f) Pooled normalized position errors of $7.7\text{\,}\mathrm{\%}$, $6.7\text{\,}\mathrm{\%}$, and $7.8\text{\,}\mathrm{\%}$ for Datasets 0, A, and B.

Model Analysis and Experimental Validation

The experimental platform consists of three tapered trimmed helicoid modules printed from TPU 95A using a Bambu Lab H2D printer, with rigid PLA connectors and mounting components. The arm is mounted vertically on a stainless-steel frame through a rigid bolted support.

Each module is actuated by three tendons driven by ROBOTIS DYNAMIXEL XM430-W350-R servo motors through a ROBOTIS U2D2 interface, with tendon commands sent from Python using the pySerial library.

Model Analysis and Experimental Validation — Cosserat Modeling of Trimmed Helicoid Soft Arms with a Separated-Section Constitutive Law
Fig. 4: Distributed response during three-tendon actuation. Tendons $\mathrm{M1}_{0^{\circ}}$, $\mathrm{M2}_{180^{\circ}}$, and $\mathrm{M3}_{120^{\circ}}$ retract by $u\times(140,185,270)\,\mathrm{mm}$, respectively, at ten levels spanning $u=0.12$–$1.00$. (a) Predicted configurations at $u=0.32$, $0.61$, and $1.00$. (b) Backbone evolution: hue indicates elastic energy per unit length $w(s)$ and lightness indicates $u$. (c–e) Rotational strain, internal moment, and internal force magnitudes. (f–h) Tendon line-load magnitudes $\/\mathrm{d}f/\mathrm{d}s\/$ along the three routes. Darker curves indicate increasing $u$ in (b–h,j,k); spatial plots shade module spans and omit interface segments. (i) Minimum non-fused helix gap in each module versus $u$; the dashed line marks the $0.10\text{\,}\mathrm{m}\mathrm{m}$ compaction threshold. (j–l) Frictional transmission post-processed from frictionless equilibria: (j) cumulative wrap angle $\Theta(s)$ of $\mathrm{M3}_{120^{\circ}}$; (k) its tension $T=T_{0}e^{-\mu_{c}\Theta(s)}$ at $\mu_{c}=0.15$, with entry tension $T_{0}$; (l) $T/T_{0}$ along all three routes at $u=1$, with solid curves for $\mu_{c}=0.15$ and bands for $\mu_{c}\in[0.10,0.20]$.

Sources and Demonstrations

Figures are reproduced from Qin et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.

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