Closed-Loop Magnetic Control of Medical Soft Continuum Robots for Deflection

TL;DR

This work develops a differential kinematic model for magnetic soft continuum robots actuated by a single rotatable permanent magnet in non-uniform magnetic fields. It introduces a quasi-static control framework that integrates a linear extended state observer and a tracking differentiator to achieve closed-loop distal deflection with Jacobian-based control, and it proves existence/uniqueness of the Jacobian while examining singularities. The approach is validated through simulations and hardware experiments, showing that the quasi-static controller outperforms a PD controller, reduces chattering near singularities, and attains sub-millimeter distal-end accuracy in path-following and positional tasks. The results demonstrate the potential for precise, compact, magnetically actuated intravascular navigation, with extensions to end-position control and considerations for future angiographic image guidance.

Abstract

Magnetic soft continuum robots (MSCRs) have emerged as powerful devices in endovascular interventions owing to their hyperelastic fibre matrix and enhanced magnetic manipulability. Effective closed-loop control of tethered magnetic devices contributes to the achievement of autonomous vascular robotic surgery. In this article, we employ a magnetic actuation system equipped with a single rotatable permanent magnet to achieve closed-loop deflection control of the MSCR. To this end, we establish a differential kinematic model of MSCRs exposed to non-uniform magnetic fields. The relationship between the existence and uniqueness of Jacobian and the geometric position between robots is deduced. The control direction induced by Jacobian is demonstrated to be crucial in simulations. Then, the corresponding quasi-static control (QSC) framework integrates a linear extended state observer to estimate model uncertainties. Finally, the effectiveness of the proposed QSC framework is validated through comparative trajectory tracking experiments with the PD controller under external disturbances. Further extensions are made for the Jacobian to path-following control at the distal end position. The proposed control framework prevents the actuator from reaching the joint limit and achieves fast and low error-tracking performance without overshooting.

Figures (8)

• Figure 1: Small rotatable magnetic equipment. (A) Cylindrical N52-grade NdFeB magnet loaded in a rotatable hammer-like mold. (B) (left) Axis-symmetric magnetic field distribution around the magnet with remanence $B_{re}=1.44$$\mathrm{T}$. (right) Comparison of the analytical solutions of the magnetic flux density $\|\mathbf{b}\|$ (noted in $B$) along the principle axis at distance $d$ with the finite element method (FEM) and experiments. (C) Comparison of the spatial gradient magnetic field along the principle axis of the magnet in the distance $d$ between analytical solutions and FEM results ($\hat{\mathbf{x}}_{\mathcal{A}}$-$\hat{\mathbf{y}}_{\mathcal{A}}$ plane).
• Figure 2: Magnetic soft continuum robot (MSCR). (A) The material composition of the axially magnetized MSCR of length $L$ and diameter $D$. (B) A kinematic schematic of the MSCR in configuration space of $(s,\theta,\varphi)$.
• Figure 3: A feasible scheme for deflecting the MSCR in a specific plane. The geometry of the $\varphi$-plane is characterized by the normal $\hat{\mathbf{n}}$ and the unit vector of intersection line $\hat{\mathbf{o}}$ in frame $\{\mathcal{G}\}$. The cylindrical magnet with desired configuration $\mathrm{T}_{\mathcal{GA}}^d$ is located in the $\varphi$-plane and the rotation axis $\hat{\mathbf{z}}_{\mathcal{A}}$ coincides with $\hat{\mathbf{n}}$.
• Figure 4: Heat map of (A) the rotation angle $\theta_L$ and (B) the Jacobian $J_\psi$ with the RME origin located at the distal end of the MSCR, varying in height $(H)$ from $0.18$$\mathrm{m}$ to $0.22$$\mathrm{m}$. (C) Comparison of the analytical Jacobian ($\bar{J}_\psi$) with the experimental result and numerical Jacobian when the RME positioned at $H=0.18 \mathrm{m}$ (see Movie S1). (D) Comparison of the experimental, numerical, and analytical Jacobian singularities.
• Figure 5: A photo shot of the experiments with instrumentation. The distal rotation angle of the MSCR is measured by conic section fitting.

Theorems & Definitions (11)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3: On the C-space Representation
• Theorem 2
• Remark 4
• Theorem 3
• Remark 5
• proof
• proof