A mobile Mathieu oscillator model for vibrational locomotion of a bristlebot

Terrestrial locomotion that is produced by creating and exploiting frictional anisotropy is common amongst animals such as snakes, gastropods, limbless lizards. In this paper we present a model of a bristle bot that locomotes by generating frictional anisotropy due to the oscillatory motion of an internal mass and show that this is equivalent to a stick-slip Mathieu oscillator. Such vibrational robots have been available as toys and theoretical curiosities and have seen some applications such as the well known kilobot and in pipe line inspection, but much remains unknown about this type of terrestrial locomotion. In this paper, motivated by a toy model of a bristle bot made from a toothbrush, we derive a theoretical model for its dynamics and show that its dynamics can be classified into four modes of motion : purely stick (no locomotion), slip, stick-slip and hopping. In the stick mode, the dynamics of the system are those of a nonlinear Mathieu oscillator and large amplitude resonance oscillations lead to the slip mode of motion. The mode of motion depends on the amplitude and frequency of the periodic forcing. We compute a phase diagram that captures this behavior, that is reminiscent of the tongues of instability seen in a Mathieu oscillator. The broader result that emerges in this paper is that mobile limbless continuum or soft robots can exploit high frequency parametric oscillations to generate fast and efficient terrestrial motion.