Fig. 5

Shape-dependent behaviour of magnetic rotor in a rotating magnetic field. a Magnetostatic torque map in coordinates of the rotor magnet geometry generating points for the initial design (b), an optimal design (c) of the magnets and a design with good assembly behaviour but low \(\overrightarrow {\tau _z}\). b Free energy landscapes (free energy surfaces (FES)) of an initial design with characteristic hills and energy minima and potential relaxation pathways of the rotor aligning the structure’s z-axis along the field. Green arrows show the pathways the structures have to follow during rotation of the magnet. Red arrows show misaligning relaxation pathways. c, d FES of best designs with characteristic energy minima and potential relaxation pathways of the rotor aligning the structure’s y axis always along the field. For d, \(\overrightarrow {\tau _y}\) is 30% smaller, which has a direct impact on the maximum rolling length. Green lines on the FES represent rotation pathways for the tube driven by an external motorized magnet. White arrow shows another possible stable position, but it is inaccessible as the initial orientation of the magnetic field is 90°. e Misassembly with non-optimized shape of magnetic stripes due to torque that aligns the tube along the field. f Magnetic Origami of capacitor array with optimal shape of magnetic stripes where \(\overrightarrow {\tau _y}\) aligns the structure perpendicular to the field. e, f Scale bars are 500 μm