Simulation of controlled motion of a spherical magnetically active object in an elastic channel

Purpose of reseach. Mathematical modeling of the dynamics of controlled motion of a spherical magnetically active object in a curved channel by means of an external magnetic field created by a movable permanent magnet

Tasks. Development of a system of differential equations describing the controlled motion of a magnetically active object in a curved elastic channel. Development of algorithms for localization of a magnetically active object inside a channel, calculation of the normal and magnitude of deformation during contact interaction. Setting up computational experiments in order to determine the nature of the movement of a magnetically active object in a curved channel and obtain the maximum values of the system parameters that ensure the controllability of the microrobot due to the movement of a permanent magnet.

Methods. When modeling the motion of a magnetically active microrobot inside a biologically inspired curved channel, a system of differential equations and equations for an external inhomogeneous magnetic field is used. The model takes into account magnetic forces, environmental resistance forces, inertia forces, and gravity. Numerical integration methods are used to solve the equations of system dynamics. In the framework of this study, the model was implemented using the MATLAB.

Results. The paper presents a mathematical model of the motion of a controlled magnetically active spherical object in a curved elastic channel simulating a blood vessel. The developed model takes into account hydrodynamic resistance, interaction with deformable channel walls and external magnetic influence. The numerical experiments performed demonstrate the possibility of predicting the trajectory of an object and reveal the limiting values of the system parameters at which controllability by a magnetically active microrobot is maintained.

Conclusions. The movement of the particle along the sinusoidal channel is effectively ensured by the action of a permanent magnet. The resulting normal reaction of the canal wall does not exceed the values allowed for vascular structures, amounting to 5 mN at the peak and about 1 mN with prolonged exposure. Taking into account key physical and geometric parameters, such as the channel shape, magnetically active object properties, viscosity of the medium, friction forces and ponderomotor action, ensures the versatility of the model. The proposed methodology can be used to optimize magnetic navigation algorithms for endovascular embolization, targeted drug delivery, and other promising medical techniques.

1. Nguyen K.T., Go G., Jin Z., et al. A magnetically guided self‐rolled microrobot for targeted drug delivery, real‐time X‐Ray imaging, and microrobot retrieval. Advanced Healthcare Materials. 2021; 10(6): 2001681. https://doi.org/10.1002/adhm.202001681

2. Ullrich F., Bergeles C., Pokki J., et al. Mobility experiments with microrobots for minimally invasive intraocular surgery. Investigative Ophthalmology & Visual Science. 2013; 54(4): 2853–2863. https://doi.org/10.1167/iovs.12-11176

3. Ergeneman O., Bergeles C., Kummer M.P., et al. Wireless intraocular microrobots: Opportunities and challenges. Surgical Robotics: Systems Applications and Visions. 2010; 271-311. https://doi.org/10.1007/978-1-4419-1126-1_13

4. Arcese L., Fruchard M., Ferreira A. Adaptive controller and observer for a magnetic microrobot. IEEE Transactions on Robotics. 2013; 29(4): 1060-1067. https://doi.org/10.1109/TRO.2013.2257581

5. Xie M., Zhang W., Fan C., et al. Bioinspired soft microrobots with precise magnetocollective control for microvascular thrombolysis. Advanced Materials. 2020; 32(26): 2000366. https://doi.org/10.1002/adma.202000366

6. Pries A.R., Secomb T.W., Gaehtgens P. Biophysical aspects of blood flow in the microvasculature. Cardiovascular Research. 1996; 32(4): 654-667. https://doi.org/10.1016/ S0008-6363(96)00065-X

7. Lomax H., Pulliam T.H., Zingg D.W. Fundamentals of computational fluid dynamics. Applied Mechanics Reviews. 2002; 55(4): B61-B67. https://doi.org/10.1115/1.1483340

8. Yang Z., Zhang L. Magnetic actuation systems for miniature robots: A review. Advanced Intelligent Systems. 2020; 2(9): 2000082. https://doi.org/10.1002/aisy.202000082

9. Malchikov A.V., Jatsun S.F., Ryapolov P.A., et al. An experimental stand for studying the motion of magnetically active objects under the influence of the magnetic field of a movable magnet. Robotics and Technical Cybernetics. 2024; 12(4): 270-279. https://doi.org/10.31776/RTCJ.12401

10. Malchikov A., Jatsun S., Ryapolov P. Contactless Motion Control System for Magnetoactive Micro-Objects. 2025 International Russian Automation Conference (RusAuto-Con). 2025; 631-636. https://doi.org/10.1109/RusAutoCon65989.2025.11177342

11. Malchikov A.V., Yatsun S.F., Ryapolov P.A., et al. Studying the controlled motion of a magnetically active object in a viscous medium. Bulletin of the Russian Academy of Sciences: Physics. 2025; 89(7): 1111-1117. https://doi.org/10.1134/S106287382571178X

12. Kim Y., Zhao X. Magnetic soft materials and robots. Chemical Reviews. 2022; 122(5): 5317-5364. https://doi.org/10.1021/acs.chemrev.1c00481

13. Wu Z., Li L., Yang Y., et al. Medical micro/nanorobots in complex media. Chemical Society Reviews. 2020; 49(22): 8088-8112. https://doi.org/10.1039/D0CS00309C Известия Юго-Западного государственного университета 92-110

14. Aziz A., Pena-Francesch A., Jung I., et al. Medical imaging of microrobots: Toward in vivo applications. ACS Nano. 2020; 14(9): 10865-10893. https://doi.org/10.1021/acsnano.0c05530

15. Hu C., Pané S., Nelson B.J. Soft micro-and nanorobotics. Annual Review of Control, Robotics, and Autonomous Systems. 2018; 1(1): 53-75. https://doi.org/10.1146/annurevcontrol-060117-104947

16. Dong X., Sitti M. Controlling two-dimensional collective formation and cooperative behavior of magnetic microrobot swarms. The International Journal of Robotics Research. 2020; 39(5): 617-638. https://doi.org/10.1177/02783649209031

17. Ebrahimi N., Bi C., Cappelleri D.J., et al. Magnetic actuation methods in bio/soft robotics. Advanced Functional Materials. 2021; 31(11): 2005137. https://doi.org/10.1002/adfm.202005137

18. Yang L., Jiang J., Gao X., et al. Autonomous environment-adaptive microrobot swarm navigation enabled by deep learning-based real-time distribution planning. Nature Machine Intelligence. 2022; 4(5): 480-493. https://doi.org/10.1038/s42256-022-00482-8

19. Behrens M.R., Ruder W.C. Smart magnetic microrobots learn to swim with deep reinforcement learning. Advanced Intelligent Systems. 2022; 4(10): 2200023. https://doi.org/10.1002/aisy.202200023

20. Xu J., Wu T., Zhang Y. Soft microrobots in microfluidic applications. Biomedical Materials & Devices. 2023; 1(2): 1028-1034. https://doi.org/10.1007/s44174-023-00071-2