  Bristle-bots are legged robots that can be easily made out of a toothbrush
head and a small vibrating engine. Despite their simple appearance, the
mechanism enabling them to propel themselves by exploiting friction with the
substrate is far from trivial. Numerical experiments on a model bristle-bot
have been able to reproduce such a mechanism revealing, in addition, the
ability to switch direction of motion by varying the vibration frequency. This
paper provides a detailed account of these phenomena through a fully analytical
treatment of the model. The equations of motion are solved through an expansion
in terms of a properly chosen small parameter. The convergence of the expansion
is rigorously proven. In addition, the analysis delivers formulas for the
average velocity of the robot and for the frequency at which the direction
switch takes place. A quantitative description of the mechanism for the
friction modulation underlying the motility of the bristle-bot is also
provided.
