Cosserat rod theory

Overview

A Cosserat rod describes a slender body by a centreline r(X) and a rotating cross-section R(X), with X the rest-length coordinate 2112.03645 §III.B. Force and moment balances along the rod give its static and dynamic equations. Internal force and moment follow from a linear Hookean law, with stiffness diagonals built from the axial, shear, area and inertia terms of the section 2112.03645 §III.B. The same model can be written on SE(3), using a homogeneous pose g(X) and a twist ξ 2112.03645 §III.B. Kirchhoff and non-linear Euler–Bernoulli beams are presented as special cases of the Cosserat model 2112.03645 §III.B.

Classic Cosserat models keep each cross-section rigid. They therefore cannot represent in-plane change of the section under lateral traction, or a hollow tube 2412.03046 §1.

The shared core is the set of balance laws and the constitutive law. Variants differ in how those equations are solved, in how actuators enter them, and in whether the cross-section can deform.

Variants

Boundary-value solvers

These discretise the boundary-value problem directly. The review lists four options: finite differences, shooting, a continuous Newton–Euler scheme and collocation 2112.03645 §V.A. The review does not rank them against each other.

Energetic Ritz reductions

These reduce the configuration to a set of Lagrange ordinary differential equations 2112.03645 §V.B. The review names three examples:

• the geometrically-exact finite element method (GE-FEM), based on Simo's formulation 2112.03645 §V.B. Card: Soft Robots Modeling;
• absolute and relative modal Ritz reductions 2112.03645 §V.B;
• the geometric variable-strain approach (GVS), which describes the rod by strain modes. Its reduced dynamics combine a mass term, a velocity-dependent term, and internal and external force terms 2112.03645 §V.B. The review recommends GVS over the constant-curvature models it covers 2112.03645 §X. Its validation against GE-FEM is cited to Boyer et al. and is not reproduced in the review 2112.03645 §V.B.

Cross-section inflation

This variant relaxes the rigid-section assumption. It adds an in-plane inflation ratio ρ(s) at each arclength, so the section can grow or shrink under lateral traction 2412.03046 §2.1. The elastic law is Saint-Venant Kirchhoff with Kelvin-Voigt damping. It is made quadratic in the strain variables by keeping ρ close to one, and it is used for a nearly incompressible material 2412.03046 §2.2. The inflation adds a second order ODE in ρ, which needs two boundary conditions 2412.03046 §2.3. The authors reduce the system with GVS, using Legendre bases for the strains and Hermite splines for ρ 2412.03046 §4. Card: Real-time Dynamics of Soft Manipulators with Cross-section Inflation.

Actuation by tendons

Tendons can enter the model in two ways. Either they act as external wrenches, summed over the tendons with each tension and routing vector, or they are part of an active constitutive law made of the passive law plus the same tendon terms 2112.03645 §III.B. The inflation variant uses the same tendon wrench, with force T t and moment T d × t for a route with unit tangent t 2412.03046 §3.2.

For transverse actuation, the inflation variant applies an inward pressure on the section boundary. For a circular section and a uniform pressure, this gives a lateral traction of 2πz²f₀, with z the reference radius. The formula does not hold for other pressure distributions or section shapes 2412.03046 §3.1.

When to use which

The review judges these options qualitatively. It gives no measured accuracy or run-time figures for them, so the rows below cite the review's statements, not benchmark results. The inflation row cites the authors' own reported results, which have not been checked independently.

The main drawbacks the review gives for the Cosserat family are that it is hard to implement and that absolute variables are hard to reduce [arxiv:2112.03645 tab. 1]. The inflation variant adds its own limits. It assumes an axisymmetric cross-section, and it does not cover anisotropic sections or out-of-plane deformation 2412.03046 §2.1 2412.03046 §6. Its octopus results use planar bending only 2412.03046 §5.

Implementation sketch

The review gives no algorithm. The sketch below is our paraphrase of the common core. The strain definitions are not restated on this card, so check 2112.03645 §III.B before implementing them.

# Cosserat rod core, from the balance and Hookean laws [arxiv:2112.03645 §III.B]
# Input:  rest-length coordinate X, stiffness diagonals H_l and H_a, loads
# Output: centreline r(X) and cross-section rotation R(X)

def rod_residual(r, R, loads):
    # strains from the centreline and the rotation (definitions: see §III.B)
    eps_l, eps_a = strains(r, R)
    # internal force and moment from the linear Hookean law
    n = R @ H_l @ eps_l
    m = R @ H_a @ eps_a
    # residuals of the force and moment balances, static or dynamic
    return force_balance(n, loads), moment_balance(m, r, n, loads)

# Solve either with a boundary-value solver [arxiv:2112.03645 §V.A],
# or project onto a Ritz basis and solve the reduced ODEs [arxiv:2112.03645 §V.B]

The inflation variant adds one scalar ρ(s) per section, with its own ODE and boundary conditions. The sketch above does not include them 2412.03046 §2.3.

Contradictions

No direct contradiction between the two cards. The review does not discuss cross-section inflation, so it neither supports nor disputes the extension. The review recommends GVS over the constant-curvature models 2112.03645 §X. The extension keeps GVS as its reduced solver, and the card reports no head-to-head comparison between the two.

Open points and source discrepancies, each side cited:

• The review does not state the strain definitions used with the Hookean law, and it does not define the boundary conditions of its Euler–Bernoulli example 2112.03645 §III.B 2112.03645 §V.C.
• The extension paper uses a Poisson ratio of 0.4999 2412.03046 tab. 2. Its released configs use 0.499, which changes the Lamé constant λ by about a factor of ten. The card's paper-vs-code section lists this and other differences 2412.03046 tab. 2.
• The extension paper does not state the boundary condition of ρ used in its octopus run 2412.03046 §5.

Papers

• Soft Robots Modeling: a Structured Overview: a review that sorts soft-robot models into four families and gives the Cosserat balance laws, their SE(3) form and the numerical methods used to solve them.
• Real-time Dynamics of Soft Manipulators with Cross-section Inflation: extends Cosserat rod dynamics with an in-plane cross-section inflation ratio, reduces it with GVS, and reports octopus-arm reaching and stiffness-tuning results.