The dynamics of a spherical robot rolling on a rough support due to the movement of internal masses is studied. The possibility of paradoxical situations of two types when the equations of motion are incompatible or ambiguous is shown. The singularity of the first type is due to the unrealistic assumption of an arbitrarily large friction force during rolling. This situation does not arise in practice because the body will start sliding earlier. As for the ambiguity paradox, it has been found in numerical experiments for values of the sliding friction coefficient of the order of unity. This creates a situation of bistability when two types of movement are realistic: rolling and jumping. These general results are applied to the analysis of robot-ball dynamics with constant or variable eccentricity and moment of inertia.
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