Soft magneto-elastic robots in locomotion: A geometrically exact dynamic cosserat shell formulation

Soft robotic systems with embedded magnetic actuation are increasingly being designed with broad, compliant geometries that defy traditional slender-body assumptions. To address the modeling limitations posed by these shell-like architectures, we propose a geometrically exact, nonlinear dynamic formulation for magneto-active soft Cosserat shells. Our model captures full six-degree-of-freedom motion at every material point, encompassing both large rotations and deformations, by leveraging the Special Euclidean group SE (3) as the configuration space. This coordinate-free representation is particularly suited for analyzing 2-dimensional soft robotic structures embedded with arrays of hard magnetic particles, such as those found in soft robotic grippers and walking robots. To integrate dynamics over time, we develop a structure-preserving extension of the Newmark method, incorporating the Baker-Campbell-Hausdorff formula for efficient configuration updates on SE (3). The resulting linearized weak form of the equilibrium equations is solved via a robust Newton-Raphson scheme. Our framework eliminates common numerical artifacts such as shear locking and coordinate singularities without resorting to heuristic corrections. Through a series of experimental validations and simulated benchmarks, we demonstrate the model’s ability to accurately capture the complex shape-morphing behavior of magneto-elastic shells under dynamic excitation. The results confirm both the physical fidelity and computational efficiency of the approach, highlighting its potential for enabling predictive design and control of next-generation soft robotic grippers, walking robots, and morphing surfaces.