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Geometrically Exact Hard Magneto-Elastic Cosserat Shells: Static Formulation for Shape Morphing

Mohammadjavad Javadi, Robin Chhabra

Abstract

Cosserat rod theory is the popular approach to modeling ferromagnetic soft robots as 1-Dimensional (1D) slender structures in most applications, such as biomedical. However, recent soft robots designed for locomotion and manipulation often exhibit a large width-to-length ratio that categorizes them as 2D shells. For analysis and shape-morphing control purposes, we develop an efficient coordinate-free static model of hard-magnetic shells found in soft magnetic grippers and walking soft robots. The approach is based on a novel formulation of Cosserat shell theory on the Special Euclidean group ($\mathbf{SE}(3)$). The shell is assumed to be a 2D manifold of material points with six degrees of freedom (position & rotation) suitable for capturing the behavior of a uniformly distributed array of spheroidal hard magnetic particles embedded in the rheological elastomer. The shell's configuration manifold is the space of all smooth embeddings $\mathbb{R}^2\rightarrow\mathbf{SE}(3)$. According to a novel definition of local deformation gradient based on the Lie group structure of $\mathbf{SE}(3)$, we derive the strong and weak forms of equilibrium equations, following the principle of virtual work. We extract the linearized version of the weak form for numerical implementations. The resulting finite element approach can avoid well-known challenges such as singularity and locking phenomenon in modeling shell structures. The proposed model is analytically and experimentally validated through a series of test cases that demonstrate its superior efficacy, particularly when the shell undergoes severe rotations and displacements.

Geometrically Exact Hard Magneto-Elastic Cosserat Shells: Static Formulation for Shape Morphing

Abstract

Cosserat rod theory is the popular approach to modeling ferromagnetic soft robots as 1-Dimensional (1D) slender structures in most applications, such as biomedical. However, recent soft robots designed for locomotion and manipulation often exhibit a large width-to-length ratio that categorizes them as 2D shells. For analysis and shape-morphing control purposes, we develop an efficient coordinate-free static model of hard-magnetic shells found in soft magnetic grippers and walking soft robots. The approach is based on a novel formulation of Cosserat shell theory on the Special Euclidean group (). The shell is assumed to be a 2D manifold of material points with six degrees of freedom (position & rotation) suitable for capturing the behavior of a uniformly distributed array of spheroidal hard magnetic particles embedded in the rheological elastomer. The shell's configuration manifold is the space of all smooth embeddings . According to a novel definition of local deformation gradient based on the Lie group structure of , we derive the strong and weak forms of equilibrium equations, following the principle of virtual work. We extract the linearized version of the weak form for numerical implementations. The resulting finite element approach can avoid well-known challenges such as singularity and locking phenomenon in modeling shell structures. The proposed model is analytically and experimentally validated through a series of test cases that demonstrate its superior efficacy, particularly when the shell undergoes severe rotations and displacements.

Paper Structure

This paper contains 29 sections, 2 theorems, 83 equations, 17 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The Lie Group Structure of $\mathbf{SE}(3)$
  5. Cosserat Shell Media
  6. Kinematics
  7. Cosserat Shell Balance Equations
  8. Internal Energy
  9. Magnetized Cosserat Shell Body
  10. Linearization of the Weak Form of Balance Equations
  11. Linearization of Internal Energy
  12. Linearization of External Magnetic Energy
  13. Finite Element Formulation
  14. Cosserat Shell Discretization
  15. Out-of-Balance Forces Computation and The Element Tangent Stiffness Matrices
  16. ...and 14 more sections

Key Result

Lemma 1

A Cosserat shell with Lagrangian $L$ satisfies the variational principle in Eq. E18 if and only if $\ell_0$ in Eq. E17 satisfies the variational principle: for variations of type ($\alpha=1,2$):

Figures (17)

  • Figure 1: Shell structures for shape morphing soft robots:(A) Spherical dielectric elastomer shell in the untethered soft robot cao2018untet. (B) Cylindrical shell structure in the modeling of a caterpillar-inspired robot goldberg2019planar. (C) Deformable shell structure in a multifunctional morphing drone hwang2022shape. (D) Soft flexible surface for robotic applications habibi2020lumped. (E) The JWST sunshield in its deployed position. arenberg2008design.
  • Figure 2: Cosserat shell configuration spaces (Note: The term $\mathbf{g_0^{-1}}$ in the definition of $\boldsymbol{\chi}$ is the inverse of function $\mathbf{g_0}\colon\mathcal{A}\rightarrow\mathbf{SE}(3)$ and not the group inverse)
  • Figure 3: Commutative diagram
  • Figure 4: Load-deflection of the cantilever plate
  • Figure 5: Roll-up of the clamped plate for the end rotation $2\pi$ with 150 elements
  • ...and 12 more figures

Theorems & Definitions (11)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 1 more