Toothpicks and toy football players: frictional jamming causes locomotion of vertically-shaken assymetrical objects

In a vertically shaken vertical round container of loosely packed nominally-vertical toothpicks, the toothpicks gradually progress horizontally. This vibration-driven locomotion is also observed in vintage toy football players on a buzzing support and in modern "Bristlebots" which are toothbrush heads with phone buzzers mounted on top. In all these cases the objects, or their lower parts, have a slant. With experiments, simulations and simple approximate theories we identify the two key features of horizontal locomotion induced by vertical-shaking: (1) the normal-force dependence of Coulomb friction and (2) some, but not too much, rotational freedom of the slanted legs. The small rotational freedom causes the necessary elastic or inertial anisotropy of the slanted objects. On the other hand, the slight restriction on rotation keeps the orientation near this anisotropic state. A key factor is jamming, which tends to occur if the slant of the principal axes of the contact mass or stiffness matrix is less than the friction angle <i>ϕ</i>_f ≡ arctan(<i>μ</i>), where <i>μ</i> is the friction coefficient.