Efficient Optimization of a Permanent Magnet Array for a Stable 2D Trap

TL;DR

The paper tackles the problem of stably confining a millirobot in a 2D plane using a permanent-magnet array positioned on one side at distances relevant to biomedical contexts. It introduces a GPU-accelerated, first-order optimization framework based on the Adam algorithm to design magnet angles that produce a target in-plane force field, without resorting to neural networks or second-order Hessians. The approach achieves stable 2D trapping at a distance of 89 mm with a two-magnet setup and demonstrates rapid optimization for up to 100 magnets, along with experimental validation showing precise millirobot trajectory control. This work provides a scalable, efficient pathway for designing permanent-magnet configurations to generate desired force-vector fields for wireless millirobot manipulation, with potential extensions to more complex fields and trajectories.

Abstract

Untethered magnetic manipulation of biomedical millirobots has a high potential for minimally invasive surgical applications. However, it is still challenging to exert high actuation forces on the small robots over a large distance. Permanent magnets offer stronger magnetic torques and forces than electromagnetic coils, however, feedback control is more difficult. As proven by Earnshaw's theorem, it is not possible to achieve a stable magnetic trap in 3D by static permanent magnets. Here, we report a stable 2D magnetic force trap by an array of permanent magnets to control a millirobot. The trap is located in an open space with a tunable distance to the magnet array in the range of 20 - 120mm, which is relevant to human anatomical scales. The design is achieved by a novel GPU-accelerated optimization algorithm that uses mean squared error (MSE) and Adam optimizer to efficiently compute the optimal angles for any number of magnets in the array. The algorithm is verified using numerical simulation and physical experiments with an array of two magnets. A millirobot is successfully trapped and controlled to follow a complex trajectory. The algorithm demonstrates high scalability by optimizing the angles for 100 magnets in under three seconds. Moreover, the optimization workflow can be adapted to optimize a permanent magnet array to achieve the desired force vector fields.

Figures (6)

• Figure 1: Schematic of the stable 2D trap generated in XY-plane, induced by the magnetic force vector field of permanent magnets positioned on the Z-axis. The magnet array located at a far distance and only on one side of the trapping position, making it suitable for biomedical applications.
• Figure 2: Optimization workflow. A) The input parameters, such as the magnets' edge length, the magnetization, and the desired distance of the magnetic trap are defined. B) The target force field is generated based on the given trap position, and the optimization algorithm iteratively identifies the optimal angles of the magnets in the array to minimize the mean squared error (MSE) between the resulting and target force fields. C) The physical setup is built based on the optimization results to trap and control a magnetic object in the XY-plane.
• Figure 3: Qualitative comparison between the developed algorithm and numerical simulation. A) The magnetic flux density vectors in the middle vertical YZ-plane (X = 0). B) The flux density vectors in the middle horizontal XY-plane (Z = 0). The arrows are normalized to represent B-field directions (mainly in the negative Z-direction). The red cross labels the trap position with the surrounding optimization area. C) The simulated $B_z$ component along the center line (X = 0 and Z = 0) compared to the magnetometer measurements.
• Figure 4: Magnetic force field analysis. The force vectors in the $20\text{ mm}\times 20\text{ mm}$ optimized region, displayed in A) the YZ-plane X = 0, and B) the XY-plane Z = 0. The arrows represent $\mathbf{F}_{{output}_{ij}}$ after optimization.
• Figure 5: Properties of the magnetic trapping system. A) Angle - trapping distance configurations. The lines represent the $\alpha$ angles of two magnets as a function of the resulting trapping position. Two solutions can be found to realize one desired trap position. B) Visualization of Solution 1: As the magnetic moments rotate in the direction of the circular arrows, the trap position moves further away. The arrows represent magnetic moment directions. C) Trapping force strength variation. The average force magnitude within a 10 mm radius region is plotted over an increasing trap distance, for different numbers of magnets with an edge length of 50.8 mm. D) Aspect ratio of the trap area as a function of the trap distance, for different numbers of magnets with edge length of 50.8 mm.