Path-following analysis of the dynamical response of a piecewise-linear capsule system

The dynamical response of a piecewise-linear capsule system is studied by means of path-following techniques in this paper. As the capsule model belongs to the class of piecewise-smooth dynamical systems involving impact and friction, a special care is taken in order to divide the trajectory of the system into a smooth vector field in each disjoint subregion. Specifically we study a two-sided drifting system focusing on directional control and energy consumption. We aim to address two practical problems which are maximizing the rate of progression and directional control of the system by following a typical period-1 trajectory. The one-parameter analysis shows that two types of bifurcations, grazing bifurcation and boundary-intersection crossing bifurcation are found, and the maximal rate of progression is achieved when the capsule performs the oscillations without sticking phases. In our two-parameter study, the control parameters for which the rate of progression is maximal are identified using fixed value of power consumption, and the curves which divide the motion of the capsule between forward and backward progression are obtained.