Periodic gait classification and control of a biped model with telescopic legs and pulse thrust

In this paper, an impulsive hybrid nonlinear dynamics for a walking bipedal model with telescopic legs is introduced. The biped robot is controlled by a pulse thrust, which takes the form of a first-order polynomial related to the angular velocity of the support leg and serves as the energy source for walking on horizontal ground. By considering continuous and sudden changes in movement during walking, an explicit map is constructed to investigate the walking dynamics. The conditions for the existence and stability of period-1 gait are obtained, the flip bifurcation of period-1 gait is investigated, and then the period-2 gait is discussed. By designing a predetermined trajectory criterion, period-1 gaits are divided into 2 categories, and period-2 gaits into 3 categories. By reducing the leg extension ratio, the occurrence the flip bifurcation is delayed and the gait can be controlled to a stable period-1 gait.