An Explicit Analytical Expression of the Poincaré Map for Analyzing Passive Dynamic Walking of the Compass-Gait Biped Model

It is known nowadays that the planar compass-gait biped robot can walk steadily and indefinitely down an inclined surface without any actuation and with a passive dynamic walking. Such biped robot is a two-degree-of-freedom impulsive mechanical system known to possess cyclic motions, called as limit cycle walking. Moreover, the bipedal walking is described by an impulsive hybrid nonlinear dynamics, which is complex enough to be handled theoretically as well as numerically. Thus, the Poincaré map method has been commonly used in analysis of bipedal locomotion. However, it is difficult to represent the Poincaré map with an explicit form. In the present paper, we develop, for the first time, an explicit mathematical expression of the Poincaré map for the passive compass-gait model. Our design methodology is based mainly on the linearization of the impulsive dynamics around some desired hybrid limit cycle. As a result, we develop an explicit analytical expression of a linear hybrid Poincaré map, which is a discrete system constrained by an equality. Using a certain approximation, we show that such constrained map is transformed into a classical nonlinear Poincaré map. A comparison between these two maps is presented in the end of this work to show the efficiency and the validity of the developed Poincaré map.