In Memory of Alexey V. Borisov. On His 60th Birthday

Spherical Robot with Electromagnetic Drive: Modeling and Control

D. V. Balandin^ab, A. M. Tuzikov^ab, R. S. Biryukov^b, A. V. Polovinkin<sup>b

a</sup> Sirius University of Science and Technology, Olympiiskiy prosp. 1, Sirius Federal Territory, 354340 Russia
^b Lobachevsky State University of Nizhny Novgorod, prosp. Gagarina 23, Nizhny Novgorod, 603022 Russia

Abstract: Spherical robots are a type of robots that have a spherical shape, allowing them to move by rolling. The drive for this category of robots can be implemented in various ways. This paper proposes the use of a new electromagnetic drive for this type of robot, resembling a spherical motor in structure. It includes permanent magnets and electromagnets, and the robot’s motion is achieved through electromagnetic interaction between them. The main goal of this work is to substantiate the fundamental possibility of controlling the motion of a spherical robot using an electromagnetic drive. To this end, first, a mathematical model of the quasi-stationary electromagnetic interaction of magnets placed on different moving shells was constructed. Second, based on this model, the dependence of the torque acting on the inner spherical shell of the robot on the angular displacement between the permanent magnet and the electromagnet was derived; using this dependence, the parameters of an effective configuration of the spherical robot with electromagnetic drive were calculated, along with the energy-optimal distribution of currents in the electromagnets. Third, using Kirchhoff’s equations, as well as results from previous works on optimal control of the motion of a spherical robot with a mechanical drive, equations for currents and voltages were derived that enable the implementation of optimal motion of the spherical robot on an uneven surface without slipping. Finally, the motions of the spherical robot on flat and bell-shaped surfaces were considered, and the dependences of currents and voltages in the electromagnets necessary for implementing the specified motions were obtained, containing information on the dynamic and frequency ranges of the control signals supplied to the electromagnets.
To track the trajectory of the spherical robot’s motion, a regulator based on linearization of the robot’s dynamics equations near the reference trajectory was used. To suppress disturbances and minimize deviations, a regulator based on the generalized $H_2$-norm was applied.

Keywords: spherical robot, optimal control, electromagnetism, mathematical model

MSC: 70E60

Received: 31.07.2025
Accepted: 30.10.2025

Language: English

DOI: 10.20537/nd251102