Three-dimensional modeling of hard-magnetic soft continuum robots with composite magnetoactive elastomers under nonuniform magnetic fields

This study presents a novel theoretical and experimental investigation through the development of a comprehensive three-dimensional analytical framework for hard-magnetic soft continuum robots (HMSCRs) actuated by nonuniform magnetic fields, explicitly incorporating the magnetic field gradient generated by a permanent magnet through both magnetic torque and body force, while also accounting for axial strain and gravity. The permanent magnet’s five degrees of freedom, including three translational and two rotational motions, are embedded in the formulation to capture realistic field–structure coupling for arbitrary poses. The geometrically nonlinear behavior of the HMSCR, involving coupled stretching, twisting, and nonplanar bending, is represented using Euler angles. To address Euler singularities, an adaptive switching mechanism is designed to automatically switch between the ZYX and YZX Euler sequences, effectively mitigating gimbal lock. The model is derived from the principle of minimum potential energy and solved using the Galerkin method with a dogleg optimization algorithm. A deep neural network surrogate, trained on finite element magnetic field data and fine-tuned with experimental measurements, enables rapid prediction of nonuniform magnetic fields. A novel experimental setup is developed, featuring a precision-molded HMSCR actuated by a six-degree-of-freedom robotic arm that positions and orients the magnet within a calibrated workspace. The proposed model is validated through benchmark studies, including comparative analyses with quaternion-based formulations and new experiments, all demonstrating excellent agreement between the developed model and experimental and numerical results. Moreover, numerical analyses, including bifurcation analysis, are conducted to assess the three-dimensional nonlinear response of the HMSCR under realistic nonuniform magnetic fields.