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.

翻译：Cosserat杆理论是当前将铁磁软体机器人建模为一维细长结构的常用方法，广泛应用于生物医学等领域。然而，近期为运动与操控设计的软体机器人常具有较大的宽长比，应归类为二维壳结构。为满足分析与形状变形控制需求，本文针对软磁夹持器与行走软机器人中的硬磁壳，建立了一种高效的无坐标静态模型。该方法基于特殊欧几里得群（$\mathbf{SE}(3)$）上的新型Cosserat壳理论框架。该壳被定义为具有六个自由度（位置与旋转）的材料点二维流形，适用于描述嵌入流变弹性体中均匀分布的球状硬磁颗粒阵列的力学行为。壳的构型流形为所有光滑嵌入映射$\mathbb{R}^2\rightarrow\mathbf{SE}(3)$构成的空间。基于$\mathbf{SE}(3)$的李群结构，我们提出了局部变形梯度的新定义，并依据虚功原理推导了平衡方程的强形式与弱形式。进一步提取弱形式的线性化版本以用于数值实现。所得有限元方法能够有效避免壳结构建模中常见的奇异性与闭锁现象。通过系列测试案例，从解析与实验两方面验证了所提模型的有效性，尤其证明了其在壳结构经历大旋转与大位移时的优越性能。