Poincaré’s Equations for Cosserat Media: Application to Shells

Abstract

In 1901 Henri Poincaré discovered a new set of equations for mechanics. These equations are a generalization of Lagrange's equations for a system whose conguration space is a Lie group which is not necessarily commutative. Since then, this result has been extensively rened through the Lagrangian reduction theory. In the present contribution, we apply an extended version of these equations to continuous Cosserat media, i.e. media in which the usual point particles are replaced by small rigid bodies, called micro-structures. In particular, we will see how the Shell balance equations used in nonlinear structural dynamics, can be easily deduced from this extension of the Poincaré's result. In future, these results will be used as foundations for the study of squid locomotion, which is an emerging topic relevant to soft robotics. AcknowledgmentThe nal publication is available at: Transformation at time t from material to geometric space

A Sliding-rod Variable-strain Model for Concentric Tube Robots

“…where we have used the strain equation 12, the identity J T θ2 ad * ξ θ 2 F i2 = 0 and the boundary conditions (17), (18) at L. Note that, since the generalized coordinates q are independent, the unknown constraint forces F λj cancel out as expected.…”

“…The static equilibrium of a Cosserat rod in a concentric tube setting can be derived from the general equilibrium equation (see [17] for a derivation) with the addition of the constraints wrenches due to the concentricity constraint. Thus, for a tube j we obtain:…”

First-Order Dynamic Modeling and Control of Soft Robots

“…Note that for the non-redundant cases, the configuration-space (q) can be interchanged with the task-space variable (x), as done in this paper. The continuous models of the position, velocity and acceleration of a soft body can be derived from the Cosserat rod theory, which gives (Boyer and Renda, 2016):…”

Screw-Based Modeling of Soft Manipulators With Tendon and Fluidic Actuation

“…The equation of motion of a continuous Cosserat arm has been derived in [20] (together with shell and 3D body) from the extension to continuum media of the Poincaré equations of mechanics by taking a Lagrangian density T(η) − U(ξ ), being T and U the densities of kinetic and elastic energy of the Cosserat beam per unit of material length. For the purpose of this paper, only the steady-state equilibrium with respect to the (micro-)body frame is reported here:…”

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