Design of an explicit expression of the Poincaré map for the passive dynamic walking of the compass-gait biped model.

• We study the passive dynamic walking of the compass-gait biped model. • A design of an explicit analytical expression of the Poincaré map having the classical form is achieved. • A linearization of the impulsive hybrid nonlinear dynamics around a desired period-1 limit cycle is realized. • We develop a simplified expression of the Poincaré map with a reduced dimension and its Jacobian matrix. • An extensive portfolio of numerical studies, simulations and comparisons is presented showing the validity of the Poincaré map. This paper presents a design method of an explicit analytical classical expression of the Poincaré map for the passive dynamic walking (PDW) of the planar compass-gait biped model. Our methodology is based chiefly on a time-piecewise linearization of the impulsive hybrid nonlinear dynamics of the PDW around a desired one-periodic hybrid limit cycle. We start by linearizing the continuous dynamics of the swing phase and also the discrete dynamics of the impact phase. Thus, we obtain a simplified impulsive hybrid linear dynamics. By means of the first-order Taylor approximation, we design an explicit expression of the Poincaré map. Moreover, we provide a simplified expression of the Poincaré map with a reduced dimension. The expression of the Jacobian matrix of this reduced Poincaré map is also developed. Finally, some numerical results and graphical simulations are presented in order to compare between the impulsive hybrid nonlinear dynamics and the developed Poincaré map. These results show the similarity between the two models and then the efficiency and the validity of the designed Poincaré map in the analysis of the PDW of the compass-gait biped robot. [ABSTRACT FROM AUTHOR]