Modeling magnetic soft continuum robot in nonuniform magnetic fields via energy minimization

The emerging magnetic soft continuum robots (MSCRs) – a type of slender and soft rods with embedded hard-magnetic particles – hold great promise in endovascular intervention via remote magnetic actuation. Although numerous advantages of using permanent magnets have been demonstrated for manipulating MSCRs (e.g., simple systems with high actuation force, large operating workspace), the magneto-mechanical behavior of MSCRs in nonuniform magnetic fields, particularly those generated by permanent magnets, remains largely unexplored. In this work, a systematic study of MSCRs in the nonuniform field is presented, which includes theoretical modeling using hard-magnetic elastica theory, numerical analyses by energy minimization method, finite element simulations using ABAQUS user element (UEL), and experimental validation. Without solving governing differential equations, the large deflection of MSCRs is efficiently obtained via the minimization of the total potential energy using sequential quadratic programming (SQP). This efficient modeling method offers insights into the control of MSCRs using nonuniform magnetic fields. Two practical strategies are provided for precisely controlling MSCRs by manipulating a cubic magnet through the adjustment of the actuation distance, rotation angle, and spin angle, laying the foundation for applications of magnetically-assisted endovascular intervention.