Novel Periodic and Turning Motions in the Simplest Passive Walking Model

The ability to move along curved paths is an essential feature for biped walkers to move around obstacles. This study is aimed at extending passive walking concept for curved walking and turning to generate more natural and effective motion. Hence three-dimensional (3D) motion of a rimless spoked-wheel, as the simplest walking model, about a general vertical fixed coordinate system has been derived. Then, two kinds of a stable passive turning, i.e. limited and circular continuous have been considered and discussed. The first kind is actually transferring from a 2D periodic motion to another, and can be implemented on a straight slope surface. While, it was shown that the second kind is just related to novel 3D periodic motions and can be recognized on a special surface profile namely “helical slope” introduced here. The latter are interpreted as 3D fixed points of a Poincare return map again. So, their stability was evaluated numerically by a Jacobian analysis and demonstrated through several simulations. Results show asymptotical stability of such motions and their considerable basin of attraction with respect to initial states. In addition, the characteristic of passive turning is shown to be strictly connected with the value of the initial perturbed condition, for instance, to the initial inclination of the wheel. It is then surprising to note that the stability of a 3D passive periodic motion (turning) is higher than 2D one (straight walking) which is actually a special case just with an infinite radius of turn.