Dmitry Balandin

This paper extends the modeling and control of a spherical robot with electromagnetic drive (SRED) from planar to fully three-dimensional motion on uneven surfaces. The robot consists of two concentric spherical shells: an outer shell carrying permanent magnets (PMs) and an inner shell with solenoid-type electromagnets (EMs); rolling is produced by the controlled electromagnetic interaction between the two magnet arrays. To enable omnidirectional rolling, the magnet placement is generalized to vertex-transitive polyhedra with cubic or icosahedral symmetry, which preserves the scalarity of the inertia tensors and allows the existing optimal control framework to be applied without modification. Within this class, $M=60$ PMs on a rhombicosidodecahedron and $S=24$ EMs on a truncated octahedron are selected; numerical evaluation over $10^{4}$ relative orientations confirms that a nonzero torque can be generated about an arbitrary axis. The drive model is reformulated as a three-dimensional torque map matrix, and coil currents and terminal voltages are recovered from a reference torque via the generalized-$H_{2}$ optimal controller, a Tikhonov-regularized pseudoinverse, and Kirchhoff's circuit equations with back-EMF terms. Simulations of straight-line rolling, lateral-offset correction, and three-dimensional path tracking on a Gaussian bell-shaped surface confirm convergence of the tracking error to zero, with currents and voltages within the range of standard low-voltage motor drivers.