Hierarchical Control for Real-Time 3D Manipulation of Magnetic Bead Using a Single Permanent Magnet

Permanent magnet (PM) actuated microrobotics offers significant advantages for minimally invasive medicine, but faces three critical challenges: nonlinear magnetic force relationships, directional control asymmetry between horizontal and vertical motion, and imaging-capturing frequency mismatch. This letter presents a hierarchical control framework enabling precise 3D manipulation of magnetic beads using a single PM. Our approach addresses these challenges through: (1) a decoupled XY-Z control architecture that independently handles horizontal damped dynamics and vertical unstable equilibrium, (2) feedback linearization that transforms nonlinear magnetic forces into linearly controllable systems, and (3) predictive feedforward compensation that bridges the 120 ms imaging interval with 10 ms actuation requirements. Experimental validation demonstrates sub-millimeter positioning accuracy (maximum oscillations: 0.210 mm in the Z-direction, 0.013 mm in the XY-plane) across diverse environments, including free liquid space, narrow maze channels, and tubular structures. The system successfully navigates complex 3D trajectories while maintaining contactless control, achieving 60%-80% cost reduction compared with electromagnetic alternatives. Applications include targeted drug delivery, minimally invasive diagnostics, and precise tissue manipulation in constrained anatomical environments.